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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Rational iterated function system for positive/monotonic shape preservation

AKB Chand1*, N Vijender1 and RP Agarwal23

Author Affiliations

1 Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India

2 Department of Mathematics, Texas A&M University - Kingsville, 700 University Blvd, Kingsville, TX, 78363-8202, USA

3 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

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Advances in Difference Equations 2014, 2014:30  doi:10.1186/1687-1847-2014-30

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2014/1/30


Received:2 July 2013
Accepted:16 December 2013
Published:27 January 2014

© 2014 Chand et al.; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we consider the (inverse) problem of determining the iterated function system (IFS) which produces a shaped fractal interpolant. We develop a new type of rational IFS by using functions of the form <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M1">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M2">View MathML</a> are cubics and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M3">View MathML</a> are preassigned quadratics having 3-shape parameters. The fixed point of the developed rational cubic IFS is in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4">View MathML</a>, but its derivative varies from a piecewise differentiable function to a continuous nowhere differentiable function. An upper bound of the uniform error between the fixed point of a rational IFS and an original function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M5">View MathML</a> is deduced for the convergence results. The automatic generations of the scaling factors and shape parameters in the rational IFS are formulated so that its fixed point preserves the positive/monotonic features of prescribed data. The presence of scaling factors provides additional freedom to the shape of the fractal interpolant over its classical counterpart in the modeling of discrete data.

1 Introduction

Setting a novel platform for the approximation of natural objects such as trees, clouds, feathers, leaves, flowers, landscapes, glaciers, galaxies, and torrents of water, Mandelbrot [1] introduced the term fractal in the literature. Since fractals capture the non-linear structures of various objects effectively, the fractal geometry has been successfully used in different problems in applied sciences and engineering [1-4]. The iterated function system (IFS) was introduced by Hutchinson [5] for the construction of various types of fractal sets, and popularized by Barnsley [6]. An IFS is a dynamical system consisting of a finite collection of continuous maps. Based on the IFS theory, Barnsley [7] constructed a class of functions that are known as FIFs. The graph of a FIF is the fixed point of an IFS. Also a FIF is the fixed point of the Read-Bajraktarević operator on a suitable function space. Common features between a FIF and a piecewise polynomial interpolation are that they are geometrical in nature, and they can be computed rapidly, but the main difference is the fractal character, i.e., a FIF satisfies a functional relation related to the self-similarity on smaller scales. In the direction of smooth fractal curves, Barnsley and Harington [8] initiated the construction of a restricted class of differentiable FIF or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M6">View MathML</a>-FIF that interpolates the prescribed data if the values of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M7">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M8">View MathML</a>, at the initial end point of the interval are given, where Φ is the original function. This method is based on the recursive nature of an algorithm, and specifying the boundary conditions similar to the classical splines was found to be quite difficult to handle in this construction. The fractal splines with general boundary conditions have been studied recently [9-13] by restricting their IFSs parameters suitably.

The motivation of this work is the research on different types of splines by several authors; see, for instance, Schmidt and Heß [14], Fritsch and Carlson [15], Schumaker [16], and Brodlie and Butt [17], and references therein. The uniqueness of spline representation for a given data set turns out to be a disadvantage for shape modification problems. The use of rational functions with the shape parameters was introduced by Späth [18] to preserve different geometric properties attached to a given set of data. Rational interpolants are often used in data visualization problems due to their excellent asymptotic properties, capability to model complicated smooth structures, better interpolation properties, and excellent extrapolating powers. Gregory and Delbourgo [19] introduced the rational cubic spline with one family of shape parameters, and this work inspired a large amount of research in shape-preserving rational spline interpolations, see [20,21] and references therein.

In this paper, we introduce the rational cubic IFS with 3-shape parameters in each subinterval of the interpolation domain such that its fixed point generalizes the corresponding classical rational cubic spline functions [20]. The developed rational cubic spline FIF is bounded, and is unique by fixed point theory for a given set of scaling factors and shape parameters. Because of the recursive nature of FIF, the necessary conditions for monotonicity on the derivative values at knots alone may not ensure the monotonicity of a rational cubic fractal interpolant for a given monotonic data. Based on the appropriate condition on the rational IFS parameters: (i) the scaling factors that depend only on given data, and (ii) the shape parameters that depend on both the interpolation data and scaling factors, we construct the shape-preserving rational cubic FIFs for a prescribed positive and/or monotonic data. By varying the scaling factors (within the shape-preserving interval) and shape parameters (according to the conditions derived in our theory), we can make the fixed point of a rational cubic IFS more pleasant and suitable for aesthetic requirements in a modeling problem. The proposed method is suitable for the shape-preserving interpolation problems where a data set originates from an unknown function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M9">View MathML</a> and its derivative <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M10">View MathML</a> is a continuous nowhere differentiable function, for instance, the motion of single inverted pendulum in non-linear control theory [22].

Comparison of the proposed rational cubic FIF over some existing schemes:

• When all the scaling factors are zero, the proposed rational cubic FIF reduces to the classical rational cubic interpolant [20], see Remark 1, Section 3.

• To generate shape-preserving interpolants, our construction does not need additional knots in contrast to methods due to Schumaker [16] and Brodlie and Butt [17], which require additional knots for the shape-preserving interpolants.

• The classical interpolants [15,23] are suitable only for monotonicity interpolation whereas the proposed rational cubic FIF is suitable for both monotonicity and positivity interpolation. Moreover, the rational quadratic interpolant [23] is a special case of our rational cubic FIF for the particular choice of the scaling factors and shape parameters, see Remark 2, Section 3.

• For given monotonic data, the monotonic curve generated by the rational quadratic interpolant [23] is unique for fixed shape parameters, whereas for the same monotonic data an infinite number of monotonic curves will be obtained using our rational FIF by suitable modifications in the associated scaling factors. Thus, when the shape parameters are incapable to change the shape of an interpolant in given intervals, then the scaling factors can be used to alter the shape of the interpolant in our method.

• Where monotonicity is concerned, our construction does not need an additional condition on derivatives at knots except for the necessary conditions. But the construction of Fritsch and Carlson [15] needs some restrictions on derivatives at knots apart from the necessary conditions for the same problem.

• The derivatives of the shape-preserving interpolants [15-17,23] are piecewise smooth, whereas the derivative of our rational cubic FIF may be piecewise smooth to a non-differentiable function according to the choice of the scaling factors. Owing to this special feature, the proposed method is preferable over the classical shape-preserving interpolants when the approximation is taken for data originating with an unknown function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M9">View MathML</a> having a shape with fractality in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M10">View MathML</a>.

This paper is organized as follows. In Section 2, the general constructions of fractal interpolants and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M13">View MathML</a>-rational cubic FIFs based on IFSs are summarized. Section 3 is devoted to the construction of a suitable rational IFS so that its fixed point is the desired interpolant that can be used for shape preservation. Then we deduce an upper bound of the uniform error bound between the original function and the rational cubic FIF. The fixed point of this rational IFS does not follow any shape constraints. The restrictions on the rational IFS parameters are deduced for a positivity shape in Section 4, and the results are illustrated with suitably chosen examples. In Section 5, the monotonicity problem is considered through the developed rational cubic IFS.

2 IFS for fractal functions

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M14">View MathML</a> be a partition of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M15">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M16">View MathML</a> be the value of original function at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M17">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M18">View MathML</a>. Denote <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M19">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M20">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M21">View MathML</a>, and let D be a compact sub-set of ℝ such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M22">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M23">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M24">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M25">View MathML</a>, be the contractive homeomorphisms such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M26">View MathML</a>

(1)

It is easy to verify that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M27">View MathML</a> is a just touching hyperbolic IFS whose unique fixed point is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M28">View MathML</a>

(2)

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M29">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M30">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31">View MathML</a>, be the continuous real-valued functions on C such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M32">View MathML</a>

(3)

and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M33">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M34">View MathML</a>, are the suitable continuous functions. Now define the functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M35">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M36">View MathML</a>, as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M37">View MathML</a> for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M38">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M39">View MathML</a> is called an IFS related to a given interpolation data <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M40">View MathML</a>. According to [7], the IFS ℐ has a unique fixed point G which is the graph of a continuous function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M41">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M42">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M43">View MathML</a>. The function ϕ is called a FIF generated by the IFS ℐ, and it takes the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M44">View MathML</a>

The existence of a spline FIF based on a polynomial IFS is given in [8]. We have extended this result to the rational IFS with 3-shape parameters in the following.

Theorem 1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M45">View MathML</a>be a given data set, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M46">View MathML</a>are the slope at<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M17">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M48">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M49">View MathML</a>) are thekth derivative values at<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M17">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M51">View MathML</a>. Consider the rational IFS<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M52">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M53">View MathML</a>satisfies equation (1), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M54">View MathML</a>is a suitable compact sub-set of ℝ. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M55">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M56">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M57">View MathML</a>is a polynomial containing<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M58">View MathML</a>arbitrary constants, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M59">View MathML</a>is a non-vanishing quadratic polynomial with 3-shape parameters in each subinterval defined onI, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M60">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M61">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M62">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M63">View MathML</a>represents the<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M64">View MathML</a>derivative of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M65">View MathML</a>with respect tox. With the setting<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M66">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M67">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M68">View MathML</a>, if

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M69">View MathML</a>

(4)

then the fixed point of the rational IFS<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M70">View MathML</a>is the graph of the<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M13">View MathML</a>-rational FIF.

Proof Suppose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M72">View MathML</a>. Now <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M73">View MathML</a> is a complete metric space, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M74">View MathML</a> is the metric on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M75">View MathML</a> induced by the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M13">View MathML</a>-norm on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M77">View MathML</a>. Define the Read-Bajraktarević operator U on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M75">View MathML</a> as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M79">View MathML</a>

(5)

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M80">View MathML</a>, the conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M81">View MathML</a> and (4) imply that U is a contractive operator on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M82">View MathML</a>. The fixed point ψ of U is a fractal function that satisfies the functional equation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M83">View MathML</a>

(6)

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M84">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M85">View MathML</a> satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M86">View MathML</a>

(7)

Using equation (4) in equation (7), we get the following system of equations for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31">View MathML</a>:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M88">View MathML</a>

(8)

When all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M58">View MathML</a> arbitrary constants in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M65">View MathML</a> are determined from equation (8), then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M91">View MathML</a> exists. By using similar arguments as in [7], it can be shown that IFS <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M70">View MathML</a> has a unique fixed point, and that it is the graph of the rational FIF <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M93">View MathML</a>. □

3 Rational cubic IFS

The construction of the desired rational cubic IFS is given in Section 3.1 such that its fixed point is used for shape preservation in the sequel. The error analysis of the fixed point of rational cubic IFS with an original function is studied in Section 3.2 for convergence results.

3.1 Construction

In the proposed rational cubic IFS, we assume <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M94">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31">View MathML</a>) are the rational functions with 3-shape parameters, whose denominators are preassigned quadratics. Based on Theorem 1, with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M96">View MathML</a>, consider the following fixed point equation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M97">View MathML</a>

(9)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M81">View MathML</a>, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M61">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M100">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M101">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M102">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M103">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M104">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M105">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M106">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M107">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M108">View MathML</a> are arbitrary constants, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111">View MathML</a> are the shape parameters such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M112">View MathML</a>. From this condition, it is easy to see that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M113">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M114">View MathML</a>. To make the fixed point ψ a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4">View MathML</a>-interpolant, the following Hermite interpolatory conditions are imposed:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M116">View MathML</a>

After evaluation of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M105">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M106">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M107">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M108">View MathML</a> using the above Hermite interpolatory conditions, we get the desired rational cubic FIF:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M121">View MathML</a>

(10)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M122">View MathML</a>

Now it is easy to see that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4">View MathML</a>-rational cubic FIF ψ is the fixed point of the following rational cubic IFS:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M124">View MathML</a>

(11)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M125">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M126">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M127">View MathML</a> are evaluated by using equation (1),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M128">View MathML</a>

The fixed point ψ of the above rational cubic IFS is unique for every fixed set of scaling factors and shape parameters. Thus by taking different sets of scaling and shape parameters, we can generate an infinite number of fixed points for the above rational cubic IFS. In most applications, the derivatives <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M46">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M130">View MathML</a>) are not given, and hence they must be calculated either from the given data or by using numerical approximation methods [24].

Remark 1 If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M131">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31">View MathML</a>, then the rational cubic FIF (10) coincides with the corresponding classical rational cubic interpolation function S as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M133">View MathML</a>

described in the literature [20] with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M134">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M135">View MathML</a>.

Remark 2 Substituting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M136">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M137">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M138">View MathML</a> in equation (10), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M139">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M140">View MathML</a>

After some rigorous calculations, we have found that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M141">View MathML</a>

(12)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M142">View MathML</a>

Now from equation (12), we conclude that for the above choice of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111">View MathML</a>, our rational cubic FIF ψ reduces to a monotonicity preserving rational quadratic FIF [25] constructed by our group. Also it is easy to verify that, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M136">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M148">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M61">View MathML</a>, then the rational cubic FIF reduces to the rational quadratic function as in [23].

3.2 Error analysis of fixed point of rational cubic IFS

Theorem 2LetψandS, respectively, be the fixed point of rational cubic IFS (11) and the classical rational cubic function with respect to the data<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M150">View MathML</a>obtained from the original function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M151">View MathML</a>. Denote<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M152">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M153">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M154">View MathML</a>. Let the shape parameters satisfy<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M155">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M156">View MathML</a>

(13)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M157">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M158">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M159">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M160">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M161">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M162">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M163">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M164">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M165">View MathML</a>.

Proof Since the coefficients of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M166">View MathML</a> in equation (9) depend on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167">View MathML</a>, we can write <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M168">View MathML</a>. From equation (5), the Read-Bajraktarević operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M169">View MathML</a> (cf. Section 2 with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M170">View MathML</a>) is re-written as for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M171">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M172">View MathML</a>

(14)

Let ξ and e be the non-zero and zero vectors in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M173">View MathML</a> respectively. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M175">View MathML</a> is the only function of x for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31">View MathML</a>, then the classical rational cubic interpolant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M177">View MathML</a> is the fixed point of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M178">View MathML</a>. Let us assume that ψ is a fixed point of a rational cubic IFS (11) associated with a non-zero scale vector ξ. Consequently, ψ is the fixed point of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M179">View MathML</a>. From equation (14), it is easy to verify that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M179">View MathML</a> is a contractive operator for a fixed scaling vector ξ:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M181">View MathML</a>

(15)

Now,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M182">View MathML</a>

(16)

Using the mean value theorem for functions of several variables, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M183">View MathML</a> such that for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M185">View MathML</a>

(17)

Using equation (17) in equation (16), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M186">View MathML</a>

(18)

Now we wish to calculate the bounds of each term in the right-hand side of equation (18). From Remark 1, it is easy to see that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M187">View MathML</a>

(19)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M188">View MathML</a>

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M189">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M34">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M191">View MathML</a>

(20)

Now using equation (20) in equation (19), we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M192">View MathML</a>.

Since the above inequality is true for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M34">View MathML</a>, we get the following estimation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M194">View MathML</a>

(21)

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M195">View MathML</a> is independent of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167">View MathML</a>, it easy to see that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M197">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M198">View MathML</a>

By using similar arguments as used in the estimation of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M199">View MathML</a>, we have found that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M200">View MathML</a>

(22)

By using equations (21) and (22) in equation (18), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M201">View MathML</a>

Since the above inequality is true for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M34">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M203">View MathML</a>

(23)

Combining equations (15) and (23) with the inequality

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M204">View MathML</a>

we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M205">View MathML</a>

(24)

From equation (24), it is evident that for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M207">View MathML</a>, the fixed point of rational cubic IFS (11) coincides with the corresponding classical rational cubic interpolant.

Since the original function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M151">View MathML</a>, it is known that [20]

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M209">View MathML</a>

(25)

Therefore, using equations (24)-(25) together with the inequality <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M210">View MathML</a>, we get the bound for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M211">View MathML</a>, and it completes the proof of theorem. □

Corollary 1 (Convergence results)

Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M212">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M213">View MathML</a>are bounded as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M214">View MathML</a>. Then we have the following results:

(i) Since<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M215">View MathML</a>, we conclude from equation (13) that the fixed point of rational cubic IFS equation (11) converges uniformly to the original function Φ as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M216">View MathML</a>.

(ii) Again from the error estimation (13), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M217">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M218">View MathML</a>) convergence can be obtained if the derivative values are available such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M219">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M218">View MathML</a>), and the scaling factors are chosen as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M221">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M218">View MathML</a>) for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M223">View MathML</a>.

4 Positivity preserving rational cubic FIF

The <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4">View MathML</a>-rational cubic fractal interpolation function developed in Section 3 has deficiencies as far as the positivity preserving issue is concerned. Because of the recursive nature of FIFs, we assume all the scaling factors are non-negative so that it is easy to derive the sufficient conditions for a positive fixed point of the rational cubic IFS (11). It requires one to assign appropriate restrictions on the scaling factors <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167">View MathML</a> and shape parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M227">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111">View MathML</a>, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31">View MathML</a>, so that the positivity feature of a given set of positive data is preserved in the fixed point of the rational cubic IFS (11). In Section 4.1, the suitable restrictions are developed on the scaling factors and shape parameters for a positivity preserving <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4">View MathML</a>-rational cubic spline FIF. The importance of suitable restrictions on the rational IFS parameters is illustrated in Section 4.2.

4.1 Restrictions on IFS parameters for positivity

Theorem 3Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M231">View MathML</a>be a given positive data. If

(i) the scaling factors<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M233">View MathML</a>, are selected as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M234">View MathML</a>

(26)

(ii) with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M235">View MathML</a>, the shape parameters<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M233">View MathML</a>, are chosen as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M239">View MathML</a>

(27)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M240">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M241">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M242">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M243">View MathML</a>, then for fixed<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M247">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M248">View MathML</a>), the unique fixed pointψof the rational IFS (11) is positive.

Proof From equation (10), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M249">View MathML</a>

It is easy to verify that using equation (2), if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M250">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M25">View MathML</a>, the sufficient conditions for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M252">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M253">View MathML</a> are <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M254">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M114">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M25">View MathML</a>. If we assume <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M257">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M258">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M235">View MathML</a>, then it is easy to see that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M260">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M261">View MathML</a>. Thus the initial conditions on the scaling factor and shape parameters are <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M250">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M257">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M258">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M265">View MathML</a>, respectively, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31">View MathML</a>. Including the initial conditions on the shape parameters, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M267">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M268">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M25">View MathML</a>. Thus our problem reduces to finding conditions on the scaling factors and shape parameters for which <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M270">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M261">View MathML</a>. From equation (10), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M272">View MathML</a> is re-written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M273">View MathML</a>

(28)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M274">View MathML</a>

By substituting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M275">View MathML</a> in equation (28), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M270">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M261">View MathML</a> is equivalent to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M278">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M279">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M280">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M281">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M282">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M283">View MathML</a>.

From [14], we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M284">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M285">View MathML</a> if and only if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M286">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M287">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M288">View MathML</a>, then we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M289">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M290">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M291">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M292">View MathML</a>. Now <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M289">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M290">View MathML</a> if and only if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M295">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M296">View MathML</a>, respectively. Hence, the restriction on the scaling factor <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167">View MathML</a> is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M298">View MathML</a>

(29)

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M299">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M300">View MathML</a> is true from equation (29), and in this case <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M257">View MathML</a> can be chosen arbitrarily. Otherwise, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M302">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M303">View MathML</a>. Similarly <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M304">View MathML</a> is true when (i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M305">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M258">View MathML</a> arbitrary (ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M307">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M308">View MathML</a>. Another set of restrictions on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111">View MathML</a> can be derived if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M313">View MathML</a>. But we have not considered it here due to the complexity involved in the calculations. The above discussions yield equation (27).

Therefore, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M314">View MathML</a> whenever equations (29) and (27) are true. Now it is easy to see that the fixed point of the rational cubic IFS (11) is positive if the scaling factors and shape parameters involved in the IFS (11) satisfy equations (26) and (27), respectively. □

4.2 Examples and discussion

In order to demonstrate the positive interpolation using our rational cubic IFS, consider the positive data set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M315">View MathML</a>, which is taken from a function (see Figure 1(a)) with an irregular derivative function as described in Figure 2(a). Such type of data arises in the motion of a single inverted pendulum in the field of non-linear control theory [22]. The position of the cart can be taken as smooth and positive in a short time interval with an irregular cart velocity. To approximate such data, we have employed the rational cubic IFS (11). The derivative values at the knots are approximated by the arithmetic mean method [24] as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M316">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M317">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M318">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M319">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M320">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M321">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M322">View MathML</a>. The scaling factors are constrained as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M323">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M324">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M325">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M326">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M327">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M328">View MathML</a> by equation (26) with a choice of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M329">View MathML</a>.

thumbnailFigure 1. Illustration of positive rational fractal interpolants with shape parameters.

thumbnailFigure 2. Derivatives of positive rational fractal interpolants and classical interpolant.

The IFS parameters of the original function are given in Table 1, and aesthetic modifications are illustrated by varying the scaling factors and shape parameters. In order to explain the sensitiveness of a rational cubic FIF with respect to the scaling factors, we have taken a fixed set of shape parameters in the construction of Figures 1(b)-(c), see Table 1. By comparing Figure 1(b) with Figure 1(a), we observe that the fractal curve pertaining to the first subinterval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M330">View MathML</a> converges to a convex shape as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M331">View MathML</a>, and changes in other subintervals are negligible. By comparing the shapes of Figure 1(a) and Figure 1(c), we notice perceptible variations in the second subinterval, and variations in other subintervals are negligible. By comparing Figures 1(d)-(e) with Figure 1(a) and Figure 1(c), respectively, we can observe the sensitivity of the positive FIF with respect to its shape parameters. Finally, we have constructed the classical rational cubic interpolant in Figure 1(f) with the zero scaling vector. From the above discussion, we conclude that the effects due to the scaling factors <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M332">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M333">View MathML</a>, and shape parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M334">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M335">View MathML</a> are very local in nature for the given positive data set.

Table 1. Rational IFS parameters for positive fractal interpolants

From equations (4) and (10), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M394">View MathML</a> interpolates the data <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M395">View MathML</a>. In this example, the interpolation data for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M394">View MathML</a> is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M397">View MathML</a>. The derivative functions of rational cubic FIFs in Figures 1(a)-(e) are constructed in Figures 2(a)-(e), respectively, and they are typically irregular fractal functions close to a continuous function, but at least they differ from a piecewise differentiable function. We have calculated the uniform errors between this original function Φ in Figure 1(a) and the rational cubic FIFs in Figures 1(b)-(f) (see Table 2). Also we have calculated the uniform errors between their derivatives (see Table 2). The effects of the scaling factors <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M332">View MathML</a> are very prominent in the first subinterval of Figure 1(b), but they also render major effects in its derivative (see Figure 2(b) and Table 2). The effects of the scaling factor <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M333">View MathML</a> are prominent in the second and third subintervals of Figure 2(c) in comparison with Figure 2(a). Hence the scaling factor <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M333">View MathML</a> is moderately local in nature in the derivative of rational cubic FIF. The shape parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M334">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M335">View MathML</a> produce similar effects (see the corresponding figures and Table 2). The rational fractal functions in Figures 2(a)-(e) are irregular in nature over the interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M403">View MathML</a>, whereas the derivative of a classical interpolant is piecewise differentiable in the interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M403">View MathML</a> (see Figure 2(f)). Comparing the uniform distances in Table 2, if the original function is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4">View MathML</a>-smooth and positive but its derivative is very irregular, then our rational cubic IFS is an ideal tool for approximating such a function instead of the classical rational cubic interpolant whose derivative is a piecewise smooth function.

Table 2. Uniform errors betweenΦand rational fractal interpolants, and their derivatives

5 Monotonicity preserving rational cubic FIF

The fixed point ψ of a rational cubic IFS may not preserve the monotonic feature of a given set of monotonic data. For an automatic generation of rational IFS parameters, we restrict them in Section 5.1, and the results are implemented in Section 5.2 through suitable examples.

5.1 Restrictions on IFS parameters for monotonicity

Theorem 4Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M231">View MathML</a>be a given monotonic data. Let the derivative values satisfy the necessary conditions for monotonicity, namely

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M407">View MathML</a>

(30)

If (i) the scaling factors<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M233">View MathML</a>, are chosen as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M410">View MathML</a>

(31)

(ii) the shape parameters<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M233">View MathML</a>, are selected as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M415">View MathML</a>

(32)

then for a fixedξ, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M247">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M419">View MathML</a>), the unique fixed pointψof the rational cubic IFS (11) is monotonic in nature.

Proof Differentiating equation (10) with respect to x, after some mathematical manipulations, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M420">View MathML</a>

(33)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M421">View MathML</a>

Due to the recursive nature of rational fractal function (33), the necessary conditions (30) are not sufficient to ensure the monotonicity of fixed point ψ of the rational cubic IFS (11). We impose additional restrictions on the scaling factors <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M422">View MathML</a>, and shape parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M34">View MathML</a>, so that these conditions together with the necessary conditions (30) yield the monotonic feature of the fixed point ψ of the IFS (11).

Case I: Monotonically increasing data

Suppose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M427">View MathML</a> is a given monotonically increasing data set. Due to the recursive nature of IFS and equation (2), it is assumed that all the scaling factors <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31">View MathML</a>, are non-negative for a monotonic fixed point of rational cubic IFS (11). For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M430">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M431">View MathML</a>, which is monotone on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M432">View MathML</a> (choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M433">View MathML</a>). Otherwise for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M434">View MathML</a>, the sufficient conditions for the monotonicity of the fixed point of rational cubic IFS (11) are <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M435">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M436">View MathML</a>. From equation (33),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M437">View MathML</a>

(34)

Similarly,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M438">View MathML</a>

(35)

From equation (33), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M439">View MathML</a> is re-written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M440">View MathML</a>

Without loss of generality, assume that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M441">View MathML</a>

(36)

i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M442">View MathML</a>. We search for sufficient conditions that make <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M443">View MathML</a>. For this purpose, we make each term in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M439">View MathML</a> non-negative. The selection of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M167">View MathML</a> with respect to equations (34) and (36) gives <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M446">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M447">View MathML</a>, respectively. Now it remains to make <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M448">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M449">View MathML</a>. In these two inequalities, the product of the shape parameters is involved. Therefore these inequalities are true if we restrict the shape parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M34">View MathML</a>, respectively, as in equation (32).

Justification for equation (32)

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M454">View MathML</a> be negative, then from equations (32) and (34)-(36), we can conclude that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M455">View MathML</a> is negative. Therefore, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M448">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M449">View MathML</a>. Similarly, it can be shown that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M458">View MathML</a> being positive gives similar results.

The above discussion led to the following procedure to make <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M443">View MathML</a>: first choose the scaling factors with respect to equations (34)-(36), then select the shape parameters according to equation (32).

Again from equation (33), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M460">View MathML</a> is re-arranged as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M461">View MathML</a>. Similarly, it is easy to verify that equations (32) and (34)-(36) are sufficient for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M462">View MathML</a>. For simplicity, denote <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M463">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M464">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M465">View MathML</a>. From equation (33), and with the above notations, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M466">View MathML</a> is re-written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M467">View MathML</a>

Substituting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111">View MathML</a> (see equation (32)) in the above expression, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M469">View MathML</a>

From the final expression of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M470">View MathML</a>, it is easy to verify that equations (32) and (34)-(36) are sufficient for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M471">View MathML</a>. Hence we have proved that the fixed point ψ of the rational cubic IFS (11) is monotonically increasing over <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M472">View MathML</a>, if the scaling factors and shape parameters are chosen according to equation (31) and equation (32), respectively. In the case of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M430">View MathML</a>, the fixed point of the rational cubic IFS (11) is a constant throughout that subinterval with the value <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M16">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146">View MathML</a>.

Case II: Monotonically decreasing data

Suppose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M476">View MathML</a> is a given monotonically decreasing data set. It is easy to see that the sufficient conditions for monotonicity of equation (10) on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M432">View MathML</a> are <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M478">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M479">View MathML</a>. As explained in Case I, it is easy to verify that selections of the scaling factors and shape parameters according to equation (31) and (32), respectively, are sufficient for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M478">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M481">View MathML</a>.

Therefore from the arguments in Case I and Case II, we conclude that if the scaling factors and shape parameters are chosen according to (31) and (32), respectively, then the fixed point ψ of the rational cubic IFS (11) is monotone for given monotonic data. □

Remark 3 Convergence results in Corollary 1 are valid for the shape-preserving rational cubic FIFs.

5.2 Examples and discussion

We construct the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4">View MathML</a>-rational cubic fractal interpolation functions (RCFIFs) for the standard increasing Akima data [26]<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M483">View MathML</a>. The <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M4">View MathML</a>-rational cubic FIFs are generated iteratively (Figures 3(a)-(i)) as the fixed points of rational cubic IFS (11). Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M485">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M486">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M488">View MathML</a>, and there is no need to choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M109">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M110">View MathML</a>, consequently there is no need to calculate <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M111">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M486">View MathML</a> using equation (32). The derivatives values <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M46">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M494">View MathML</a>) are approximated by the arithmetic mean method [24] as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M495">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M496">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M497">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M498">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M499">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M500">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M501">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M502">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M503">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M504">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M505">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M506">View MathML</a> in equation (31). The scaling factors are restricted as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M507">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M508">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M509">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M510">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M511">View MathML</a> to capture the monotonicity of the Akima data. A standard rational cubic FIF Φ in Figure 3(a) is generated with a suitable choice of the scaling factors (see Table 3). By comparing Figure 3(b) with Figure 3(a), we observe that the graph of the rational cubic FIF in the subinterval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M512">View MathML</a> converges to a convex shape as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M513">View MathML</a>, and the changes in other intervals are negligible. By analyzing the shapes of Figure 3(a) and Figure 3(c), we observe that visually pleasing effects are produced in the ninth subinterval. By analyzing Figure 3(d) with respect to Figure 3(a), we have found excellent variations in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M514">View MathML</a>. The individual effects of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M515">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M516">View MathML</a>, respectively, from Figure 3(b) and Figure 3(c) are reflected in Figure 3(d), and thereby we conclude that all the non-zero scaling factors are very much local in nature for this Akima data. Next to visualize the effects of change in the shape parameters, we construct the monotonically increasing rational cubic FIFs in Figures 3(e)-(h). By comparing the shapes of Figure 3(e) and Figure 3(a), we notice perceptible variations in the ninth subinterval, and variations in other subintervals are negligible. Again analyzing Figure 3(a), Figure 3(c), and Figure 3(e), we observe that to get appropriate deviations in the rational cubic FIF in the ninth subinterval, one has to vary the scaling parameter <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M516">View MathML</a>, the shape parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M518">View MathML</a> and/or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M519">View MathML</a> suitably.

thumbnailFigure 3. Monotonicity preserving rational cubic FIFs and their derivatives.

Table 3. Rational IFS parameters for monotonic fractal interpolants

Next a monotonic rational FIF in Figure 3(f) is constructed as per the data in Table 3 and the variations in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M512">View MathML</a> of Figure 3(f) with respect to Figure 3(a) are more evident than those of Figure 3(f) with respect to Figure 3(b). So, we can say that the scaling factor <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M515">View MathML</a> is dominant over the shape parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M629">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M630">View MathML</a> in this case at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M512">View MathML</a>. Hence, for major changes at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M512">View MathML</a>, one has to modify <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M515">View MathML</a>, and for minor changes (or fine tuning), one has to alter <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M629">View MathML</a> and/or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M630">View MathML</a>. This observation is useful for aesthetic requirements in various engineering design problems. By analyzing Figures 3(g)-(h) with respect to Figure 3(c), we have noticed that the graphs of rational cubic FIFs in the ninth subinterval in Figures 3(g)-(h) are concave and convex, respectively. Finally, we construct the classical rational cubic interpolant in Figure 3(i) with respect to the shape parameters of Figure 3(g), and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146">View MathML</a> for all i. Since the shape parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M518">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M519">View MathML</a> are the same in Figure 3(g) and Figure 3(i), there is some visual similarity between these two curves in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M512">View MathML</a>, whereas the same effects are missing in Figure 3(h) due to a variation in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M519">View MathML</a>. Also, one gets a classical rational cubic interpolant which is similar to Figure 3(h) in our fractal scheme, whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M146">View MathML</a> for all i, and the shape parameters are chosen according to Figure 3(h). The presence of scaling factors in a monotonic rational FIF with shape parameters gives an additional advantage in the choice of interpolant over the classical interpolants with shape parameters. Our construction gives an extra freedom for aesthetic modifications in local shape over the classical rational cubic interpolants to an user. For a qualitative study of the derivatives of monotonic fractal interpolants, the readers are invited to check the effects of rational IFS parameters in Figures 3(a)-(g). The uniform errors between monotonic fractal interpolants and their derivatives are given in Table 4 to show the importance of our rational cubic IFS (11).

Table 4. Uniform errors betweenΦand rational fractal interpolants, and their derivatives

From the examples in Sections 4-5, it is observed that proper interactive adjustments of the scaling factors and shape parameters give us a wide variety of positivity and/or monotonicity preserving fixed points of our rational cubic IFS (11) that can be used in various scientific and engineering problems for aesthetic modifications. In order to get an optimal choice of the fixed point of our rational cubic IFS (11), one can employ a genetic algorithm interactively until the desired accuracy is obtained with the original function.

6 Conclusion

A new type of rational cubic IFS with 3-shape parameters is introduced in this work such that its fixed point can be used for shaped data. The developed FIF in this paper includes the corresponding classical rational cubic interpolant as a special case. An upper bound of uniform error between the rational cubic FIF ψ and an original function Φ in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M642">View MathML</a> is estimated, and consequently we have found that ψ converges uniformly to Φ as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M216">View MathML</a>. When the accurate derivatives of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M644">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M645">View MathML</a> are available, and the scaling factors are chosen as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M646">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M31">View MathML</a>, it is possible to get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M648">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M218">View MathML</a>) convergence for the rational cubic FIF. Automatic data dependent restrictions are derived on the scaling factors and shape parameters of rational cubic IFS so that its fixed point preserves the positivity or monotonicity features of a given set of data. The effects of a change in the scaling factors and shape parameters on the local control of the shape of rational cubic FIF are demonstrated through various examples. Our rational cubic FIFs are more flexible and more suitable for shape related problems in computer graphics, CAD/CAM, CAGD, medical imaging, finance, and engineering applications, and apply equally well to data with or without derivatives. In particular, the proposed method will be an ideal tool in shape-preserving interpolation problems where the data set originates from a positive and/or monotonic function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M9">View MathML</a>, but its derivative <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/30/mathml/M10">View MathML</a> is a continuous and nowhere differentiable function.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

Acknowledgements

AKBC is thankful to the Department of Science and Technology, Govt. of India for the SERC DST Project No. SR/S4/MS: 694/10. The authors are grateful to the anonymous referees for the valuable comments and suggestions, which improved the presentation of the paper.

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