Open Access Research

Existence of solutions for nonlocal p-Laplacian thermistor problems on time scales

Wenjing Song1 and Wenjie Gao2*

Author affiliations

1 Institute of Applied Mathematics, Jilin University of Finance and Economics, Changchun, 130117, P.R. China

2 Institute of Mathematics, Jilin University, Changchun, 130012, P.R. China

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Citation and License

Boundary Value Problems 2013, 2013:1  doi:10.1186/1687-2770-2013-1


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


Received:1 September 2012
Accepted:18 December 2012
Published:4 January 2013

© 2013 Song and Gao; 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, a nonlocal initial value problem to a p-Laplacian equation on time scales is studied. The existence of solutions for such a problem is obtained by using the topological degree method.

Keywords:
existence; p-Laplacian; time scales; topological degree

1 Introduction

In this paper, we are concerned with the existence of solutions of the following nonlocal p-Laplacian dynamic equation on a time scale <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M1">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M2">View MathML</a>

(1.1)

with integral initial value

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M3">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M4">View MathML</a> is the p-Laplace operator defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M7">View MathML</a> with q the Hölder conjugate of p, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M11">View MathML</a> is continuous (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M12">View MathML</a> denotes positive real numbers), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M13">View MathML</a> is left dense continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M14">View MathML</a> and A is a real constant.

This model arises in ohmic heating phenomena, which occur in shear bands of metals which are deformed at high strain rates [1,2], in the theory of gravitational equilibrium of polytropic stars [3], in the investigation of the fully turbulent behavior of real flows, using invariant measures for the Euler equation [4], in modeling aggregation of cells via interaction with a chemical substance (chemotaxis) [5]. For the one-dimensional case, problems with the nonlocal initial condition appear in the investigation of diffusion phenomena for a small amount of gas in a transparent tube [6,7]; nonlocal initial value problems in higher dimension are important from the point of view of their practical applications to modeling and investigating of pollution processes in rivers and seas, which are caused by sew-age [8].

The study of dynamic equations on time scales has led to some important applications [9-11], and an amount of literature has been devoted to the study the existence of solutions of second-order nonlinear boundary value problems (e.g., see [12-18]).

Motivated by the above works, in this paper, we study the existence of solutions to Problem (1.1), (1.2). Compared with the works mentioned above, this article has the following new features: firstly, the main technique used in this paper is the topological degree method; secondly, Problem (1.1), (1.2) involves the integral initial condition.

The paper is organized as follows. We introduce some necessary definitions and lemmas in the rest of this section. In Section 2, we provide some necessary preliminaries, and in Section 3, the main results are stated and proved.

Definition 1.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M16">View MathML</a>, define the forward jump operator σ and the backward jump operator ρ, respectively,

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

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M18">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M19">View MathML</a>, t is said to be right scattered, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M20">View MathML</a>, r is said to be left scattered. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M21">View MathML</a>, t is said to be right dense, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M22">View MathML</a>, r is said to be left dense. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M1">View MathML</a> has a right scattered minimum m, define ; otherwise, set . If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M1">View MathML</a> has a left scattered maximum M, define ; otherwise, set .

Definition 1.2 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M29">View MathML</a> and , we define the delta derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M32">View MathML</a>, to be the number (when it exists) with the property that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M33">View MathML</a>, there is a neighborhood U of t such that

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

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M35">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M29">View MathML</a> and , we define the nabla derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M39">View MathML</a>, to be the number (when it exists) with the property that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M33">View MathML</a>, there is a neighborhood V of t such that

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

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

Definition 1.3 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M43">View MathML</a>, then we define the delta integral by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M44">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M45">View MathML</a>, then we define the nabla integral by

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

Throughout this paper, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M1">View MathML</a> is a nonempty closed subset of ℝ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M49">View MathML</a>.

Lemma 1.1 (Alternative theorem)

Suppose thatXis a Banach space andAis a completely continuous operator fromXto X. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M50">View MathML</a>, only one of the following statements holds:

(i) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M51">View MathML</a>, there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M52">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M53">View MathML</a>

(ii) There exists an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M55">View MathML</a>, such that

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

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M57">View MathML</a> be a Banach space equipped with the maximum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M58">View MathML</a>.

Consider the following problem:

(2.1)

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M62">View MathML</a>.

Integrating Eq. (2.1) from 0 to t, one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M63">View MathML</a>

Using the initial condition (2.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M64">View MathML</a>

Integrating the above equality from 0 to t again, we obtain

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

(2.3)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M66">View MathML</a>.

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M67">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M68">View MathML</a>

then (2.3) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M69">View MathML</a>

(2.4)

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M31">View MathML</a> is a solution to (2.1), (2.2) if and only if it is a solution to (2.4).

Lemma 2.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M71">View MathML</a>is a Fredholm operator.

Proof To prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M71">View MathML</a> is a Fredholm operator, we need only to show that K is completely continuous.

It is easy to see from the definition of K that K is a bounded linear operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M73">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M73">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M75">View MathML</a>. So, K is a completely continuous operator. This completes the proof. □

Lemma 2.2Problem (2.1), (2.2) admits a unique solution.

Proof Since Problem (2.1), (2.2) is equivalent to Problem (2.4), we need only to show that Problem (2.4) has a unique solution.

Using Lemma 2.1 and the alternative theorem, it is sufficient to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M76">View MathML</a>

(2.5)

has a trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M77">View MathML</a> only.

On the contrary, suppose (2.5) has a nontrivial solution μ, then μ is a constant, and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M78">View MathML</a>

The definition of K and the above equality yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M79">View MathML</a>

which is a contradiction to the assumptions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M81">View MathML</a>.

Thus, we complete the proof. □

3 Main results

Throughout this section, we assume that the following conditions hold.

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M82">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M11">View MathML</a> is continuous;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M13">View MathML</a> is left dense continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M85">View MathML</a>;

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M88">View MathML</a>, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M89">View MathML</a>;

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M92">View MathML</a>, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M93">View MathML</a>.

From Lemma 2.2 we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M94">View MathML</a> is a solution to Problem (1.1), (1.2) if and only if it is a solution to the following integral equation:

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

(3.1)

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M96">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M97">View MathML</a>

then (3.1) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M98">View MathML</a>

In order to prove the existence of solutions to (3.1), we need the following lemmas.

Lemma 3.1Fis completely continuous.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M99">View MathML</a> be an arbitrary positive real number and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M100">View MathML</a>. Then we have for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M101">View MathML</a>,

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

This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M103">View MathML</a> is uniformly bounded.

Moreover, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M104">View MathML</a>, we have

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

Thus, it is easy to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M103">View MathML</a> is equicontinuous. This together with the Ascoli-Arzelà theorem guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M103">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M73">View MathML</a>.

Therefore, F is completely continuous. The proof of Lemma 3.1 is completed. □

Theorem 3.1Assume that conditions (H1)-(H5) hold. Then Problem (1.1), (1.2) has at least one solution.

Proof Lemma 2.1 and Lemma 3.1 imply that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M109">View MathML</a> is completely continuous. It suffices for us to prove that the equation

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

(3.2)

has at least one solution.

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

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

and it is clear that H is completely continuous.

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

To apply the Leray-Schauder degree to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M115">View MathML</a>, we need only to show that there exists a ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M116">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M117">View MathML</a>, whose radius R will be fixed later, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M118">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M89">View MathML</a>, choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M120">View MathML</a>, then for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M121">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M122">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M123">View MathML</a>. By direct calculation, we have

(3.3)

From (H4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M125">View MathML</a>

(3.4)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M93">View MathML</a>, choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M127">View MathML</a>, then for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M121">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M122">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M123">View MathML</a>. From (H5), we have

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

(3.5)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M132">View MathML</a>, choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M133">View MathML</a>, then for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M121">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M122">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M123">View MathML</a>. By direct calculation, we have

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

(3.6)

This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M138">View MathML</a> and hence we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M118">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M140">View MathML</a>, we know that (3.2) admits a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M141">View MathML</a>, which implies that (1.1), (1.2) also admits a solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/1/mathml/M116">View MathML</a>. □

Competing interests

All authors declare that they have no competing interests.

Authors’ contributions

WS dfafted this paper and WG checked and corrected the manuscript.

Acknowledgements

This work was supported by NSFC (11271154) and by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education and by the 985 program of Jilin University, and the first author is also supported by the Youth Studies Program of Jilin University of Finance and Economics (XJ2012006).

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