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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

An AQCQ-functional equation in matrix Banach spaces

Choonkil Park1, Jung Rye Lee2 and Dong Yun Shin3*

Author Affiliations

1 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea

2 Department of Mathematics, Daejin University, Pocheon, Kyeonggi, 487-711, Korea

3 Department of Mathematics, University of Seoul, Seoul, 130-743, Korea

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Advances in Difference Equations 2013, 2013:146  doi:10.1186/1687-1847-2013-146

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/146


Received:2 April 2013
Accepted:8 May 2013
Published:23 May 2013

© 2013 Park 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

Using the fixed point method, we prove the Hyers-Ulam stability of an additive-quadratic-cubic-quartic functional equation in matrix normed spaces.

MSC: 47L25, 47H10, 39B82, 46L07, 39B52.

Keywords:
operator space; fixed point; Hyers-Ulam stability; additive-quadratic-cubic-quartic functional equation

1 Introduction and preliminaries

The abstract characterization given for linear spaces of bounded Hilbert space operators in terms of matricially normed spaces[1] implies that quotients, mapping spaces and various tensor products of operator spaces may again be regarded as operator spaces. Owing in part to this result, the theory of operator spaces is having an increasingly significant effect on operator algebra theory (see [2]).

The proof given in [1] appealed to the theory of ordered operator spaces [3]. Effros and Ruan [4] showed that one can give a purely metric proof of this important theorem by using the technique of Pisier [5] and Haagerup [6] (as modified in [7]).

The stability problem of functional equations originated from a question of Ulam [8] concerning the stability of group homomorphisms.

The functional equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M1">View MathML</a>

is called the Cauchy additive functional equation. In particular, every solution of the Cauchy additive functional equation is said to be an additive mapping. Hyers [9] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [10] for additive mappings and by TM Rassias [11] for linear mappings by considering an unbounded Cauchy difference. A generalization of the TM Rassias theorem was obtained by Găvruta [12] by replacing the unbounded Cauchy difference by a general control function in the spirit of TM Rassias’ approach.

In 1990, TM Rassias [13] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M2">View MathML</a>. In 1991, Gajda [14], following the same approach as in TM Rassias [11], gave an affirmative solution to this question for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M3">View MathML</a>. It was shown by Gajda [14], as well as by TM Rassias and Šemrl [15], that one cannot prove a TM Rassias’ type theorem when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M4">View MathML</a> (cf. the books of Czerwik [16], Hyers et al.[17]).

In 1982, JM Rassias [18] followed the innovative approach of the TM Rassias’ theorem [11] in which he replaced the factor <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M5">View MathML</a> by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M6">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M7">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M8">View MathML</a>.

The functional equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M9">View MathML</a>

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [19] for mappings <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a>, where X is a normed space and Y is a Banach space. Cholewa [20] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [21] proved the Hyers-Ulam stability of the quadratic functional equation.

In [22], Jun and Kim considered the following cubic functional equation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M11">View MathML</a>

(1.1)

It is easy to show that the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M12">View MathML</a> satisfies functional equation (1.1), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping.

In [23], Lee et al. considered the following quartic functional equation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M13">View MathML</a>

(1.2)

It is easy to show that the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M14">View MathML</a> satisfies functional equation (1.2), which is called a quartic functional equation, and every solution of the quartic functional equation is said to be a quartic mapping. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [24-33]).

We will use the following notations:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M15">View MathML</a>;

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M16">View MathML</a> is that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M17">View MathML</a>-component is 1 and the other components are zero;

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M18">View MathML</a> is that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M17">View MathML</a>-component is x and the other components are zero;

For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M20">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M21">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M22">View MathML</a>

Note that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M23">View MathML</a> is a matrix normed space if and only if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M24">View MathML</a> is a normed space for each positive integer n and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M25">View MathML</a> holds for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M26">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M27">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M28">View MathML</a>, and that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M29">View MathML</a> is a matrix Banach space if and only if X is a Banach space and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M30">View MathML</a> is a matrix normed space.

Let E, F be vector spaces. For a given mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M31">View MathML</a> and a given positive integer n, define <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M32">View MathML</a> by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M33">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M34">View MathML</a>.

Let X be a set. A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M35">View MathML</a> is called a generalized metric on X if d satisfies

(1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M36">View MathML</a> if and only if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M37">View MathML</a>;

(2) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M38">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M39">View MathML</a>;

(3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M40">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M41">View MathML</a>.

We recall a fundamental result in fixed point theory.

Theorem 1.1[34,35]

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M42">View MathML</a>be a complete generalized metric space and let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M43">View MathML</a>be a strictly contractive mapping with Lipschitz constant<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44">View MathML</a>. Then, for each given element<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M45">View MathML</a>, either

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M46">View MathML</a>

for all nonnegative integersnor there exists a positive integer<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M47">View MathML</a>such that

(1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M48">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M49">View MathML</a>;

(2) the sequence<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M50">View MathML</a>converges to a fixed point<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M51">View MathML</a>ofJ;

(3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M51">View MathML</a>is the unique fixed point ofJin the set<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M53">View MathML</a>;

(4) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M54">View MathML</a>for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M55">View MathML</a>.

In 1996, Isac and Rassias [36] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [37-43]).

In this paper, we prove the Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M56">View MathML</a>

(1.3)

in matrix normed spaces by using the fixed point method.

One can easily show that an odd mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a> satisfies (1.3) if and only if the odd mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a> is an additive-cubic mapping, i.e.,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M59">View MathML</a>

It was shown in [[44], Lemma 2.2] that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M60">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M61">View MathML</a> are cubic and additive, respectively, and that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M62">View MathML</a>.

One can easily show that an even mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a> satisfies (1.3) if and only if the even mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a> is a quadratic-quartic mapping, i.e.,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M65">View MathML</a>

It was shown in [[45], Lemma 2.1] that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M66">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M67">View MathML</a> are quartic and quadratic, respectively, and that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M68">View MathML</a>.

Throughout this paper, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M69">View MathML</a> be a matrix normed space and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M70">View MathML</a> be a matrix Banach space.

2 Hyers-Ulam stability of AQCQ-functional equation (1.3) in matrix normed spaces: odd mapping case

In this section, we prove the Hyers-Ulam stability of AQCQ-functional equation (1.3) in matrix normed spaces for an odd mapping case.

Lemma 2.1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M23">View MathML</a>be a matrix normed space. Then:

(1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M72">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M45">View MathML</a>.

(2) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M74">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M75">View MathML</a>.

(3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M76">View MathML</a>if and only if<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M77">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M78">View MathML</a>.

Proof (1) Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M79">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M80">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M81">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M82">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M83">View MathML</a>. So <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M84">View MathML</a>.

(2) Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M85">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M86">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M87">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M88">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M89">View MathML</a>

(3) By

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M90">View MathML</a>

we get the result. □

For a mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M91">View MathML</a>, define <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M92">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M93">View MathML</a> by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M94">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95">View MathML</a> and all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M96">View MathML</a>.

Theorem 2.2Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97">View MathML</a>be a function such that there exists an<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44">View MathML</a>with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M99">View MathML</a>

(2.1)

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a>be an odd mapping satisfying

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M102">View MathML</a>

(2.2)

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M96">View MathML</a>. Then there exists a unique additive mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M104">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M105">View MathML</a>

(2.3)

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M107">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M108">View MathML</a> except for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M109">View MathML</a> in (2.2).

Putting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M110">View MathML</a> in (2.2), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M111">View MathML</a>

(2.4)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M112">View MathML</a>.

Replacing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M113">View MathML</a> by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M114">View MathML</a> in (2.2), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M115">View MathML</a>

(2.5)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M112">View MathML</a>.

By (2.4) and (2.5),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M117">View MathML</a>

(2.6)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M112">View MathML</a>. Replacing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M119">View MathML</a> by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M113">View MathML</a> and letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M121">View MathML</a> in (2.6), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M122">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M124">View MathML</a>

(2.7)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123">View MathML</a>.

Consider the set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M126">View MathML</a>

and introduce the generalized metric on S:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M127">View MathML</a>

where, as usual, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M128">View MathML</a>. It is easy to show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129">View MathML</a> is complete (see [46,47]).

Now we consider the linear mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M130">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M131">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132">View MathML</a>.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M133">View MathML</a> be given such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M134">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M135">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132">View MathML</a>. Hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M137">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M138">View MathML</a>. So <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M134">View MathML</a> implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M140">View MathML</a>. This means that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M141">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M142">View MathML</a>.

It follows from (2.7) that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M143">View MathML</a>.

By Theorem 1.1, there exists a mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M104">View MathML</a> satisfying the following:

(1) A is a fixed point of J, i.e.,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M145">View MathML</a>

(2.8)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132">View MathML</a>. The mapping A is a unique fixed point of J in the set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M147">View MathML</a>

This implies that A is a unique mapping satisfying (2.8) such that there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M148">View MathML</a> satisfying

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M149">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132">View MathML</a>;

(2) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M151">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M152">View MathML</a>. This implies the equality

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M153">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132">View MathML</a>;

(3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M155">View MathML</a>, which implies the inequality

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M156">View MathML</a>

So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M157">View MathML</a>

(2.9)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132">View MathML</a>.

It follows from (2.1) and (2.2) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M159">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M160">View MathML</a>. Hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M161">View MathML</a> for all a, b. So <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M162">View MathML</a> is additive.

By Lemma 2.1 and (2.9),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M163">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M164">View MathML</a>. Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M104">View MathML</a> is a unique additive mapping satisfying (2.3), as desired. □

Corollary 2.3Letr, θbe positive real numbers with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M166">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167">View MathML</a>be an odd mapping such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M168">View MathML</a>

(2.10)

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M96">View MathML</a>. Then there exists a unique additive mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M104">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M171">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof The proof follows from Theorem 2.2 by taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95">View MathML</a>. Then we can choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M175">View MathML</a> and we get the desired result. □

Theorem 2.4Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97">View MathML</a>be a function such that there exists an<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44">View MathML</a>with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M178">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a>be an odd mapping satisfying (2.2). Then there exists a unique additive mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M104">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M182">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129">View MathML</a> be the generalized metric space defined in the proof of Theorem 2.2.

Now we consider the linear mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M130">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M186">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M132">View MathML</a>.

It follows from (2.7) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M188">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123">View MathML</a>. Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M190">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M191">View MathML</a>

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 2.5Letr, θbe positive real numbers with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M192">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167">View MathML</a>be an odd mapping satisfying (2.10). Then there exists a unique additive mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M104">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M195">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof The proof follows from Theorem 2.4 by taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95">View MathML</a>. Then we can choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M199">View MathML</a> and we get the desired result. □

Theorem 2.6Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97">View MathML</a>be a function such that there exists an<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44">View MathML</a>with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M202">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a>be an odd mapping satisfying (2.2). Then there exists a unique cubic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M205">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M206">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129">View MathML</a> be the generalized metric space defined in the proof of Theorem 2.2.

Replacing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M119">View MathML</a> by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M113">View MathML</a> and letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M211">View MathML</a> in (2.6), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M212">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M214">View MathML</a>

(2.11)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123">View MathML</a>. Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M216">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M217">View MathML</a>

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 2.7Letr, θbe positive real numbers with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M218">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167">View MathML</a>be an odd mapping satisfying (2.10). Then there exists a unique cubic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M205">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M221">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof The proof follows from Theorem 2.6 by taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95">View MathML</a>. Then we can choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M225">View MathML</a> and we get the desired result. □

Theorem 2.8Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97">View MathML</a>be a function such that there exists an<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44">View MathML</a>with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M228">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a>be an odd mapping satisfying (2.2). Then there exists a unique cubic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M231">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M232">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129">View MathML</a> be the generalized metric space defined in the proof of Theorem 2.2.

It follows from (2.11) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M235">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123">View MathML</a>. Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M237">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M238">View MathML</a>

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 2.9Letr, θbe positive real numbers with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M239">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167">View MathML</a>be an odd mapping satisfying (2.10). Then there exists a unique cubic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M205">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M242">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof The proof follows from Theorem 2.8 by taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95">View MathML</a>. Then we can choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M246">View MathML</a> and we get the desired result. □

3 Hyers-Ulam stability of AQCQ-functional equation (1.3) in matrix normed spaces: even mapping case

In this section, we prove the Hyers-Ulam stability of AQCQ-functional equation (1.3) in matrix normed spaces for an even mapping case.

Theorem 3.1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97">View MathML</a>be a function such that there exists an<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44">View MathML</a>with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M249">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a>be an even mapping satisfying<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M252">View MathML</a>and (2.2). Then there exists a unique quadratic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M253">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M254">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129">View MathML</a> be the generalized metric space defined in the proof of Theorem 2.2.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M107">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M108">View MathML</a> except for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M109">View MathML</a> in (2.2).

Putting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M110">View MathML</a> in (2.2), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M261">View MathML</a>

(3.1)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M112">View MathML</a>.

Replacing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M113">View MathML</a> by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M114">View MathML</a> in (2.2), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M265">View MathML</a>

(3.2)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M112">View MathML</a>.

By (3.1) and (3.2),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M267">View MathML</a>

(3.3)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M112">View MathML</a>. Replacing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M119">View MathML</a> by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M113">View MathML</a> and letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M271">View MathML</a> in (3.3), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M272">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M274">View MathML</a>

(3.4)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123">View MathML</a>. Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M276">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M277">View MathML</a>

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 3.2Letr, θbe positive real numbers with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M278">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167">View MathML</a>be an even mapping satisfying (2.10). Then there exists a unique quadratic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M253">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M281">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof The proof follows from Theorem 3.1 by taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95">View MathML</a>. Then we can choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M285">View MathML</a> and we get the desired result. □

Theorem 3.3Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97">View MathML</a>be a function such that there exists an<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44">View MathML</a>with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M288">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a>be an even mapping satisfying<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M252">View MathML</a>and (2.2). Then there exists a unique quadratic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M253">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M293">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129">View MathML</a> be the generalized metric space defined in the proof of Theorem 2.2.

It follows from (3.4) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M296">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123">View MathML</a>. Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M298">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M299">View MathML</a>

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 3.4Letr, θbe positive real numbers with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M300">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167">View MathML</a>be an even mapping satisfying (2.10). Then there exists a unique quadratic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M253">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M303">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof The proof follows from Theorem 3.3 by taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95">View MathML</a>. Then we can choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M307">View MathML</a> and we get the desired result. □

Theorem 3.5Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97">View MathML</a>be a function such that there exists an<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44">View MathML</a>with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M310">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a>be an even mapping satisfying<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M252">View MathML</a>and (2.2). Then there exists a unique quartic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M314">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M315">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129">View MathML</a> be the generalized metric space defined in the proof of Theorem 2.2.

Replacing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M119">View MathML</a> by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M113">View MathML</a> and letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M320">View MathML</a> in (3.3), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M321">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M323">View MathML</a>

(3.5)

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123">View MathML</a>. Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M325">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M326">View MathML</a>

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 3.6Letr, θbe positive real numbers with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M327">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167">View MathML</a>be an even mapping satisfying (2.10). Then there exists a unique quartic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M314">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M330">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof The proof follows from Theorem 3.5 by taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95">View MathML</a>. Then we can choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M334">View MathML</a> and we get the desired result. □

Theorem 3.7Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M97">View MathML</a>be a function such that there exists an<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M44">View MathML</a>with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M337">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M100">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M10">View MathML</a>be an even mapping satisfying<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M252">View MathML</a>and (2.2). Then there exists a unique quartic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M314">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M342">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M129">View MathML</a> be the generalized metric space defined in the proof of Theorem 2.2.

It follows from (3.5) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M345">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M123">View MathML</a>. Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M347">View MathML</a>. So

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M348">View MathML</a>

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 3.8Letr, θbe positive real numbers with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M349">View MathML</a>. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M167">View MathML</a>be an even mapping satisfying (2.10). Then there exists a unique quartic mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M314">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M352">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M106">View MathML</a>.

Proof The proof follows from Theorem 3.7 by taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M173">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M95">View MathML</a>. Then we can choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/146/mathml/M356">View MathML</a> and we get the desired result. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Acknowledgements

CP was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299), and DYS was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

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