This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Analysis of Abel-type nonlinear integral equations with weakly singular kernels

JinRong Wang12, Chun Zhu2 and Michal Fečkan34*

Author Affiliations

1 School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou, 550018, P.R. China

2 Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, P.R. China

3 Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynská dolina, Bratislava, 842 48, Slovakia

4 Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, Bratislava, 814 73, Slovakia

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Boundary Value Problems 2014, 2014:20  doi:10.1186/1687-2770-2014-20


Dedicated to Professor Ivan Kiguradze


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/20


Received:14 October 2013
Accepted:19 December 2013
Published:17 January 2014

© 2014 Wang 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 investigate Abel-type nonlinear integral equations with weakly singular kernels. Existence and uniqueness of nontrivial solution are presented in an order interval of a cone by using fixed point methods. As a byproduct of our method, we improve a gap in the proof of Theorem 5 in Buckwar (Nonlinear Anal. TMA 63:88-96, 2005). As an extension, solutions in closed form of some Erdélyi-Kober-type fractional integral equations are given. Finally theoretical results with three illustrative examples are presented.

MSC: 26A33, 45E10, 45G05.

Keywords:
Abel-type nonlinear integral equations; weakly singular kernels; existence; numerical solutions

1 Introduction

Abel-type integral equations are associated with a wide range of physical problems such as heat transfer [1], nonlinear diffusion [2], propagation of nonlinear waves [3], and they can also be applied in the theory of neutron transport and in traffic theory. In the past 70 years, many researchers investigated the existence and uniqueness of nontrivial solutions for a large number of Abel-type integral equations by using various analysis methods (see [4-16] and references therein).

Fractional calculus provides a powerful tool for the description of hereditary properties of various materials and memory processes. In particular, integral equations involving fractional integral operators (which can be regarded as an extension of Abel integral equations) appear naturally in the fields of biophysics, viscoelasticity, electrical circuits, and etc. There are some remarkable monographs that provide the main theoretical tools for the qualitative analysis of fractional order differential equations, and at the same time, show the interconnection as well as the contrast between integer order differential models and fractional order differential models [17-24].

It is remarkable that many researchers pay attention to the study of the existence and attractiveness of solutions for fractional integral equations by using functional analysis methods such as the contraction principle, the Schauder fixed point theorem and a Darboux-type fixed point theorem involving a measure of noncompactness (see [25-33] and references therein).

A completely different approach is given in Buckwar [13] to discussing the existence and uniqueness of nontrivial solutions for Abel-type nonlinear integral equation with power-law nonlinearity on an order interval as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M1">View MathML</a>

(1)

Many analysis techniques are used to construct the suitable order interval (see Lemma 2, [13]) and the spaces with suitable weighted norms.

Motivated by [6,11,13,33], we extend to study the following Abel-type nonlinear integral equation with weakly singular kernels:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M2">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M3">View MathML</a> are given functions for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M4">View MathML</a>, h is increasing, g is nondecreasing such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M5">View MathML</a>

(3)

for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M11">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M12">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M13">View MathML</a> is non-negative and it has either the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M14">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M15">View MathML</a> for some function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M17">View MathML</a> specified later. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M18">View MathML</a> is the Gamma function. Of course, we suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M19">View MathML</a>

(4)

It is obvious that equation (1) or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M20">View MathML</a>

(5)

are special cases of equation (2), which of course all have trivial solutions.

Thus, the main purpose of this paper is to prove the existence and uniqueness of nontrivial solutions for equation (2). The key difficult comes from the weakly singular kernels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M21">View MathML</a> and nonlinear terms in equation (2). Although we are motivated by [13], we have to introduce novel techniques and results to overcome the difficult from the weakly singular kernels and nonlinear terms h and g. For example, the first important step is how to construct a suitable order interval to help us to apply the fixed point theorem in such an order interval. More details of the novel techniques and results will be found in the proof. As a byproduct of our method, we improve a gap in the proof of [[13], Theorem 5]. So even for equation (1) (or (5)) we get a new result.

As an extension, we find general solutions in closed form of some Erdélyi-Kober-type fractional integral equations (the special case of equation (5) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M22">View MathML</a>):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M23">View MathML</a>

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M25">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M26">View MathML</a> and the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M27">View MathML</a> denotes the Erdélyi-Kober-type fractional integrals [19] of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M28">View MathML</a>, which is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M29">View MathML</a>

The plan of this paper is as follows. In Section 2, some notation and preparation results are given. Existence and uniqueness results of a nontrivial solution of equation (2) in an order interval are given in Section 3. In Section 4, we find general solutions in closed form of some Erdélyi-Kober-type fractional integral equations, and finally theoretical results with three illustrate examples are presented in Section 5.

2 Preliminary

Let ℳ be the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M30">View MathML</a> with the supremum-norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M31">View MathML</a>. Clearly, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M32">View MathML</a> is a closed subspace of Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M33">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M32">View MathML</a> is a Banach space.

Let q be a continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M35">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M36">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M37">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M38">View MathML</a> be the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M39">View MathML</a>

with the weighted norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M40">View MathML</a>

(7)

Remark 2.1 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M41">View MathML</a>, then the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M38">View MathML</a> is the same as the set ℳ, but with an equivalent norm, and the constants can be determined in the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M43">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M45">View MathML</a>. Note that the similar inequality (5) of [13] is incorrect.

Consider the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M46">View MathML</a> in ℳ. The so-called partial ordering induced by the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M47">View MathML</a> is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M48">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M49">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M50">View MathML</a>. In general [34,35], a set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M51">View MathML</a> is called an order interval where E is an ordered Banach space. We know that every order interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M52">View MathML</a> is closed. Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M53">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M54">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M55">View MathML</a>, then every order interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M52">View MathML</a> is bounded.

We introduce some conditions on the functions K, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M58">View MathML</a> as follows:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M59">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M60">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M61">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M62">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M64">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M65">View MathML</a>.

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M66">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M67">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M63">View MathML</a>.

• For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M70">View MathML</a>, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M71">View MathML</a>

(8)

• For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M15">View MathML</a>, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M73">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M74">View MathML</a>

(9)

Similarly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M73">View MathML</a>.

We note [34] an important estimate on the function K, which will be used in the sequel.

Lemma 2.2The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M77">View MathML</a>has the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M78">View MathML</a>

(10)

Proof We only check the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M14">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M70">View MathML</a>, since the other cases are trivial.

Integrating n times step-by-step all sides of the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M81">View MathML</a>

from 0 to t and using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M66">View MathML</a> we immediately derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M83">View MathML</a>

Replacing t by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M84">View MathML</a>, we obtain the desired result. □

To end this section, we collect the following basic facts, which will be used several times in the next section.

Lemma 2.3Letλ, γ, μ, andνbe constants such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M87">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M88">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M89">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M90">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M91">View MathML</a>

is the well-known Beta function.

Proof The first result have been reported in [36] or [[37], Formula 3.251]. We only verify the second inequality. In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M93">View MathML</a>, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M94">View MathML</a>

The proof is completed. □

3 Existence and uniqueness of nontrivial solution in an order interval

In this section, we will use the fixed point method to prove the existence and uniqueness of nontrivial solution for equation (2) in an order interval.

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M50">View MathML</a>, we introduce the following functions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M96">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M97">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M98">View MathML</a> are defined in equation (8) or (9).

Remark 3.1 Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M99">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M100">View MathML</a>. Next, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M101">View MathML</a>.

The following result is clear.

Lemma 3.2If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M102">View MathML</a>

(11)

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M103">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M50">View MathML</a>. Consequently, the order interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M105">View MathML</a>is well defined.

Remark 3.3 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M107">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M108">View MathML</a> then equation (11) reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M109">View MathML</a>

which is satisfied, since equation (3) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M111">View MathML</a> (see equation (4)) and clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M112">View MathML</a>. This case occurs for instance when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M114">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M117">View MathML</a>.

From now on, we suppose that all above assumptions hold: equations (3), (4), (i)-(iii), and (11).

Lemma 3.4Any solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M118">View MathML</a>of equation (2), with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M119">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M63">View MathML</a>, satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M121">View MathML</a>.

Proof Step 1: We prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M122">View MathML</a> for a solution x of equation (2).

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M123">View MathML</a>. Then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M124">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M125">View MathML</a>

(12)

Next we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M126">View MathML</a>

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M127">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M128">View MathML</a>

hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M129">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M130">View MathML</a>

consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M131">View MathML</a>

(13)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M132">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M133">View MathML</a>, estimate (13) is an improvement of equation (12).

Step 2: We prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M134">View MathML</a>. Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M135">View MathML</a> and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M136">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M137">View MathML</a>. Then like above, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M138">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M139">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M140">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M141">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M142">View MathML</a>

Consequently, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M143">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M135">View MathML</a> is arbitrarily, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M145">View MathML</a>

Hence we can complete the proof. □

To solve equation (2), we introduce an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M146">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M147">View MathML</a>

(14)

Lemma 3.5The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M148">View MathML</a>maps the order interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M149">View MathML</a>into itself.

Proof To achieve our aim, we only need to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M150">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M151">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M152">View MathML</a>

(15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M153">View MathML</a>

(16)

First we show equation (15):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M154">View MathML</a>

Secondly we derive equation (16):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M155">View MathML</a>

Since obviously, the operator S is strictly increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M149">View MathML</a> and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M121">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M50">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M160">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M161">View MathML</a>

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M148">View MathML</a> is well defined and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M163">View MathML</a>. The proof is completed. □

From the Arzela-Ascoli theorem and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M164">View MathML</a> is nondecreasing, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M148">View MathML</a> is compact, so the Schauder fixed point theorem implies the following existence result [35,38,39].

Theorem 3.6Equation (2) has a solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M149">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M167">View MathML</a>

are fixed points of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M148">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M169">View MathML</a>

Now we are ready to state the following uniqueness result. But first we note that the above considerations can be repeated for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M170">View MathML</a>, so we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M175">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M176">View MathML</a> as continuous functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M177">View MathML</a>. Note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M171">View MathML</a> is nonincreasing, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M172">View MathML</a> is nondecreasing, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M172">View MathML</a> can be continuously extended to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M182">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M183">View MathML</a>. We still keep the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M186">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M187">View MathML</a>.

Theorem 3.7If there are constantsψ, χand continuous functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M188">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M189">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M35">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M191">View MathML</a>

(17)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M194">View MathML</a>then equation (2) has a unique solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M149">View MathML</a>provided we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M196">View MathML</a>

(18)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M197">View MathML</a>

(19)

where we set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M198">View MathML</a>.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M199">View MathML</a> we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M201">View MathML</a>. Clearly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M202">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M203">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M205">View MathML</a> specified below, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M206">View MathML</a>. Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M63">View MathML</a>, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M208">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M209">View MathML</a>

consequently, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M210">View MathML</a>

(20)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M211">View MathML</a>

Since (note equation (18))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M212">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M213">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M214">View MathML</a>, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M215">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M204">View MathML</a> sufficiently large uniformly for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M205">View MathML</a>. So we take and fix such a ϖ. Next, by equation (19) there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M218">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M219">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M220">View MathML</a>. Furthermore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M221">View MathML</a>, we have (note equation (18))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M222">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M205">View MathML</a> sufficiently large, so we fix such <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M205">View MathML</a>. Consequently we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M225">View MathML</a>

Summarizing we see that there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M205">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M228">View MathML</a>

This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M164">View MathML</a> is a contraction with respect to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M230">View MathML</a> with a constant L. By the contraction mapping principle, one can obtain the result immediately. □

Remark 3.8 Consider equation (5). Of course, we can suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M231">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M234">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M235">View MathML</a>. Moreover, Remark 3.3 can be applied to get an existence result. If in addition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M236">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M237">View MathML</a>, and it is not difficult to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M239">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M240">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M241">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M242">View MathML</a>, so equation (18) holds. Next, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M243">View MathML</a>

Hence condition (19) is satisfied and then we get a uniqueness result by Theorem 3.7. Note there is gap in the proof of [[13], Theorem 5]. So here we give its correct proof.

4 General solutions of Erdélyi-Kober-type integral equations

This section is devoted to a derivation of explicit solutions of some Erdélyi-Kober-type integral equations. In order to establish this, we introduce the following useful result.

Lemma 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M244">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M245">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M246">View MathML</a>

Proof Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M247">View MathML</a>. By using Lemma 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M248">View MathML</a>

This completes the proof. □

Now we are ready to present our main result of this section.

Theorem 4.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M252">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M253">View MathML</a>. Then equation (6) is solvable and its solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M254">View MathML</a>can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M255">View MathML</a>

(21)

where the constantCsatisfies the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M256">View MathML</a>

(22)

Proof With the help of Lemma 4.1, substituting equation (21) into (6), we find that C satisfies equation (22) which completes the proof. □

5 Illustrative examples

In this section, we pay our attention to show three numerical performance results.

Example 5.1 We consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M257">View MathML</a>

(23)

First, Theorem 4.2 gives the exact solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M258">View MathML</a> of equation (23). Next, by changing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M259">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M260">View MathML</a>

(24)

Of course, we get a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M261">View MathML</a>. In equation (5) for (24), we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M263">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M268">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M269">View MathML</a>. After some computation, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M270">View MathML</a>

Obviously, all the assumptions in Theorem 3.7 are satisfied. Numerical result is given in Figure 1.

thumbnailFigure 1. Solution of equation (23) and the boundariesFandGfor Example 5.1 coincide with the unique solution.

Example 5.2 In equation (5), we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M271">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M272">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M273">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M268">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M269">View MathML</a>. Now, we turn to consider the following homogeneous Abel-type integral equation with weakly singular kernels and power-law nonlinearity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M278">View MathML</a>

(25)

After some computation, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M279">View MathML</a>

Obviously, all the assumptions in Theorem 3.7 are satisfied. Then, the problem (5.2) has a unique solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M149">View MathML</a>. Numerical results are given in Figure 2.

thumbnailFigure 2. Solution (black line) of equation (25) and the boundariesF(red line) andG(blue line) for Example 5.2.

Example 5.3 In equation (2), we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M271">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M272">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M273">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M286">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M289">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M290">View MathML</a>. Now, we turn to considering the following homogeneous Abel-type integral equation with weakly singular kernels and polynomial law nonlinearity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M291">View MathML</a>

(26)

It is clear that now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M292">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M293">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M294">View MathML</a>, so equation (4) holds. After some computation, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M295">View MathML</a>

Since now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M296">View MathML</a>, obviously, all the assumptions in Theorem 3.6 are satisfied. Then, the problem (5.3) has a solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/20/mathml/M149">View MathML</a>. Numerical results are given in Figure 3.

thumbnailFigure 3. Solution (black line) of equation (26) and the boundariesF(red line) andG(blue line) for Example 5.3.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented here was carried out in collaboration between the authors. The authors contributed to every part of this study equally and read and approved the final version of the manuscript.

Acknowledgements

The first and second authors acknowledge the support by National Natural Science Foundation of China (11201091) and Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169), Key Support Subject (Applied Mathematics), Key Project on the Reforms of Teaching Contents and Course System of Guizhou Normal College and Doctor Project of Guizhou Normal College (13BS010). The third author acknowledges the support by Grants VEGA-MS 1/0071/14, VEGA-SAV 2/0029/13 and APVV-0134-10.

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