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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 Article

A nonlocal multi-point multi-term fractional boundary value problem with Riemann-Liouville type integral boundary conditions involving two indices

Ahmed Alsaedi1, Sotiris K Ntouyas2, Ravi P Agarwal13 and Bashir Ahmad1*

Author Affiliations

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

2 Department of Mathematics, University of Ioannina, Ioannina, 451 10, Greece

3 Department of Mathematics, Texas A&M University, Kingsville, 78363-8202, USA

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Advances in Difference Equations 2013, 2013:369  doi:10.1186/1687-1847-2013-369


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/369


Received:6 November 2013
Accepted:1 December 2013
Published:13 December 2013

© 2013 Alsaedi et al.; licensee Springer.

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

Abstract

In this paper, we study the existence of solutions for fractional differential equations of arbitrary order with multi-point multi-term Riemann-Liouville type integral boundary conditions involving two indices. The Riemann-Liouville type integral boundary conditions considered in the problem address a more general situation in contrast to the case of a single index. Our results are based on standard fixed point theorems. Some illustrative examples are also presented.

MSC: 26A33, 34A08.

Keywords:
fractional differential equations; nonlocal boundary conditions; fixed point theorems

1 Introduction

In the last few decades, the subject of fractional differential equations has become a hot topic for the researchers due to its intensive development and applications in the field of physics, mechanics, chemistry, engineering, etc. For a reader interested in the systematic development of the topic, we refer the books [1-6]. A fractional-order differential operator distinguishes itself from the integer-order differential operator in the sense that it is nonlocal in nature, that is, the future state of a dynamical system or process involving fractional derivative depends on its current state as well its past states. In other words, differential equations of arbitrary order describe memory and hereditary properties of various materials and processes. As a matter of fact, this characteristic of fractional calculus makes the fractional-order models more realistic and practical than the classical integer-order models. There has been a great surge in developing the theoretical aspects such as periodicity, asymptotic behavior and numerical methods for fractional equations. For some recent work on the topic, see [7-25] and the references therein. In particular, the authors studied nonlinear fractional differential equations and inclusions of arbitrary order with multi-strip boundary conditions in [21], while a boundary value of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multi-strip boundary conditions was investigated in [22]. Sudsutad and Tariboon [26] obtained some existence results for an integro-differential equation of fractional order <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M1">View MathML</a> with m-point multi-term fractional-order integral boundary conditions.

In this paper, we study a boundary value problem of fractional differential equations of arbitrary order <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M2">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M3">View MathML</a> with m-point multi-term Riemann-Liouville type integral boundary conditions involving two indices given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M4">View MathML</a>

(1.1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M5">View MathML</a> denotes the Caputo fractional derivative of order q, f is a given continuous function, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M6">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M7">View MathML</a> is the Riemann-Liouville fractional integral of order <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M8">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M9">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M10">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M11">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M12">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M13">View MathML</a> is such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M14">View MathML</a>

Here we emphasize that Riemann-Liouville type integral boundary conditions involving two indices give rise to a more general situation in contrast to the case of a single index [22]. Furthermore, the present work dealing with an arbitrary-order problem generalizes the results for the problem of order <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M1">View MathML</a> obtained in [26]. Several examples are considered to show the worth of the results established in this paper.

We develop some existence results for problem (1.1) by using standard techniques of fixed point theory. The paper is organized as follows. In Section 2 we recall some preliminary facts that we need in the sequel, and Section 3 contains the main results. Section 4 provides some examples for the illustration of the main results.

2 Preliminaries from fractional calculus

Let us recall some basic definitions of fractional calculus [2-4].

Definition 2.1 For an at least n-times continuously differentiable function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M16">View MathML</a>, the Caputo derivative of fractional order q is defined as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M17">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M18">View MathML</a> denotes the integer part of the real number q.

Definition 2.2 The Riemann-Liouville fractional integral of order q is defined as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M19">View MathML</a>

provided the integral exists.

Lemma 2.3For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M20">View MathML</a>, the fractional boundary value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M21">View MathML</a>

(2.1)

has a unique solution given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M22">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M23">View MathML</a>

(2.3)

Proof The general solution of fractional differential equations in (2.1) can be written as [4]

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M24">View MathML</a>

(2.4)

Using the given boundary conditions, it is found that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M25">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M26">View MathML</a>, …, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M27">View MathML</a>. Applying the Riemann-Liouville integral operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M7">View MathML</a> on (2.4), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M29">View MathML</a>

Using the concept of beta function, we find that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M30">View MathML</a>

Now using the condition

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M31">View MathML</a>

we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M32">View MathML</a>

which yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M33">View MathML</a>

where δ is given by (2.3). Substituting the values of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M34">View MathML</a> in (2.4), we obtain (2.2). This completes the proof. □

3 Main results

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M35">View MathML</a> denote the Banach space of all continuous functions defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M36">View MathML</a> endowed with a topology of uniform convergence with the norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M37">View MathML</a>.

To prove the existence results for problem (1.1), we need the following known results.

Theorem 3.1 (Leray-Schauder alternative [[27], p.4])

LetXbe a Banach space. Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M38">View MathML</a>is a completely continuous operator and the set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M39">View MathML</a>

is bounded. ThenThas a fixed point inX.

Theorem 3.2[28]

LetXbe a Banach space. Assume that Ω is an open bounded subset of Xwith<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M40">View MathML</a>, and let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M41">View MathML</a>be a completely continuous operator such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M42">View MathML</a>

ThenThas a fixed point in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M43">View MathML</a>.

By Lemma 2.3, we define an operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M44">View MathML</a> as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M45">View MathML</a>

(3.1)

Observe that problem (1.1) has a solution if and only if the associated fixed point problem <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M46">View MathML</a> has a fixed point.

For the sake of convenience, we set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M47">View MathML</a>

(3.2)

Theorem 3.3Assume that there exists a positive constant<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M48">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M49">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M50">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M51">View MathML</a>. Then problem (1.1) has at least one solution.

Proof First of all, we show that the operator is completely continuous. Note that the operator is continuous in view of the continuity of f. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M54">View MathML</a> be a bounded set. By the assumption that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M55">View MathML</a>, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M56">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M57">View MathML</a>

which implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M58">View MathML</a>. Further, we find that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M59">View MathML</a>

Hence, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M60">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M61">View MathML</a>

This implies that is equicontinuous on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M63">View MathML</a>. Thus, by the Arzela-Ascoli theorem, the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M64">View MathML</a> is completely continuous.

Next, we consider the set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M65">View MathML</a>

and show that the set V is bounded. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M66">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M67">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M68">View MathML</a>. For any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M50">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M70">View MathML</a>

Thus, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M71">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M50">View MathML</a>. So, the set V is bounded. Thus, by the conclusion of Theorem 3.1, the operator has at least one fixed point, which implies that (1.1) has at least one solution. □

Theorem 3.4Let there exist a small positive numberτsuch that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M74">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M75">View MathML</a>, with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M76">View MathML</a>, whereϑis given by (3.2). Then problem (1.1) has at least one solution.

Proof Let us define <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M77">View MathML</a> and take <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M78">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M79">View MathML</a>, that is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M80">View MathML</a>. As before, it can be shown that is completely continuous and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M82">View MathML</a>

which, in view of the given condition <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M83">View MathML</a>, gives <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M84">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M85">View MathML</a>. Therefore, by Theorem 3.2, the operator has at least one fixed point, which in turn implies that problem (1.1) has at least one solution. □

Our next result is based on Leray-Schauder nonlinear alternative.

Lemma 3.5 (Nonlinear alternative for single-valued maps [[27], p.135])

LetEbe a Banach space, Cbe a closed, convex subset ofE, Ube an open subset ofCand<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M87">View MathML</a>. Suppose that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M88">View MathML</a>is a continuous, compact (that is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M89">View MathML</a>is a relatively compact subset ofC) map. Then either

(i) Fhas a fixed point in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M90">View MathML</a>, or

(ii) there are<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M91">View MathML</a> (the boundary ofUinC) and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M92">View MathML</a>with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M93">View MathML</a>.

Theorem 3.6Assume that

(A1) there exist a function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M94">View MathML</a>and a nondecreasing function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M95">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M96">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M97">View MathML</a>;

(A2) there exists a constant<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M98">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M99">View MathML</a>

Then boundary value problem (1.1) has at least one solution on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M63">View MathML</a>.

Proof Consider the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M101">View MathML</a> defined by (3.1). We show that maps bounded sets into bounded sets in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M103">View MathML</a>. For a positive number r, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M104">View MathML</a> be a bounded set in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M103">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M106">View MathML</a>

Next we show that Fmaps bounded sets into equicontinuous sets of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M107">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M108">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M109">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M110">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M111">View MathML</a> is a bounded set of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M107">View MathML</a>. Then we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M113">View MathML</a>

Obviously the right-hand side of the above inequality tends to zero independently of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M110">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M115">View MathML</a>. As <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M116">View MathML</a> satisfies the above assumptions, therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.

Let x be a solution. Then, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M50">View MathML</a>, and following the similar computations as before, we find that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M119">View MathML</a>

In consequence, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M120">View MathML</a>

Thus, by (A2), there exists M such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M121">View MathML</a>. Let us set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M122">View MathML</a>

Note that the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M123">View MathML</a> is continuous and completely continuous. From the choice of V, there is no <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M124">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M125">View MathML</a> for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M126">View MathML</a>. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.5), we deduce that has a fixed point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M128">View MathML</a> which is a solution of problem (1.1). This completes the proof. □

Finally we prove an existence and uniqueness result by means of Banach’s contraction mapping principle.

Theorem 3.7Suppose that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M129">View MathML</a>is a continuous function and satisfies the following assumption:

(A3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M130">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M131">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M132">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M133">View MathML</a>.

Then boundary value problem (1.1) has a unique solution provided

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M134">View MathML</a>

(3.3)

whereϑis given by (3.2).

Proof With <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M135">View MathML</a>, we define <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M136">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M137">View MathML</a> and ϑ is given by (2.3). Then we show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M138">View MathML</a>. For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M110">View MathML</a>, by means of the inequality <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M140">View MathML</a>, it can easily be shown that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M141">View MathML</a>

Now, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M142">View MathML</a> and for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M143">View MathML</a>, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M144">View MathML</a>

Note that ϑ depends only on the parameters involved in the problem. As <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M145">View MathML</a>, therefore is a contraction. Hence, by Banach’s contraction mapping principle, problem (1.1) has a unique solution on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M147">View MathML</a>. □

4 Examples

In this section, we present some examples for the illustration of the results established in Section 3 by choosing the nonlinear function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M148">View MathML</a> appropriately. Let us consider the following nonlocal boundary value problem:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M149">View MathML</a>

(4.1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M150">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M151">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M152">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M153">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M154">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M155">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M156">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M157">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M158">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M159">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M160">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M161">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M162">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M163">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M164">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M165">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M166">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M167">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M168">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M169">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M170">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M171">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M172">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M173">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M174">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M175">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M176">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M177">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M178">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M179">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M180">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M181">View MathML</a>. Using the given data, we find that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M182">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M183">View MathML</a>

(a) As a first example, let us take

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M184">View MathML</a>

(4.2)

Observe that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M185">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M186">View MathML</a>. Thus the hypothesis of Theorem 3.3 is satisfied. Hence, by the conclusion of Theorem 3.3, problem (4.1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M187">View MathML</a> given by (4.2) has at least one solution.

(b) Let us consider

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M188">View MathML</a>

(4.3)

For sufficiently small x (ignoring <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M189">View MathML</a> and higher powers of x), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M190">View MathML</a>

Choosing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M191">View MathML</a>, all the assumptions of Theorem 3.4 hold. Therefore, the conclusion of Theorem 3.4 implies that problem (4.1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M187">View MathML</a> given by (4.3) has at least one solution.

(c) Consider

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M193">View MathML</a>

(4.4)

with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M194">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M195">View MathML</a>. Using <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M196">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M197">View MathML</a>, we find by condition (A2) that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M198">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M199">View MathML</a>. Thus all the assumptions of Theorem 3.6 are satisfied. Hence, it follows by Theorem 3.6 that problem (4.1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M187">View MathML</a> defined by (4.4) has at least one solution.

(d) For the illustration of the existence-uniqueness result, we choose

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M201">View MathML</a>

(4.5)

Clearly, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M202">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M203">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M204">View MathML</a>. Therefore all the conditions of Theorem 3.7 hold, and consequently there exists a unique solution for problem (4.1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/369/mathml/M187">View MathML</a> given by (4.5).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, AA, SKN, RPA and BA, contributed to each part of this work equally and read and approved the final version of the manuscript.

Acknowledgements

This research was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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