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Solvability for a coupled system of fractional differential equations with impulses at resonance

Xiaozhi Zhang, Chuanxi Zhu* and Zhaoqi Wu

Author Affiliations

Department of Mathematics, Nanchang University, Nanchang, 330031, P.R. China

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Boundary Value Problems 2013, 2013:80  doi:10.1186/1687-2770-2013-80


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/80


Received:9 November 2012
Accepted:18 March 2013
Published:8 April 2013

© 2013 Zhang 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, some Banach spaces are introduced. Based on these spaces and the coincidence degree theory, a 2m-point boundary value problem for a coupled system of impulsive fractional differential equations at resonance is considered, and the new criterion on existence is obtained. Finally, an example is also given to illustrate the availability of our main results.

MSC: 34A08, 34B10, 34B37.

Keywords:
coupled system; impulsive fractional differential equations; at resonance; coincidence degree

1 Introduction

Recently, Wang et al.[1] presented a counterexample to show an error formula of solutions to the traditional boundary value problem for impulsive differential equations with fractional derivative in [2-5]. Meanwhile, they introduced the correct formula of solutions for an impulsive Cauchy problem with the Caputo fractional derivative. Shortly afterwards, many works on the better formula of solutions to the Cauchy problem for impulsive fractional differential equations have been reported by Li et al.[6], Wang et al.[7], Fečkan [8], etc.

Fractional differential equations have been paid much attention to in recent years due to their wide applications such as nonlinear oscillations of earthquakes, Nutting’s law, charge transport in amorphous semiconductors, fluid dynamic traffic model, non-Markovian diffusion process with memory etc.[9-11]. For more details, see the monographs of Hilfer [12], Miller and Ross [13], Podlubny [14], Lakshmikantham et al.[15], Samko et al.[16], and the papers of [2,17-19] and the references therein.

In recent years, many researchers paid much attention to the coupled system of fractional differential equations due to its applications in different fields [20-25]. Zhang et al.[25] investigated a three-point boundary value problem at resonance for a coupled system of nonlinear fractional differential equations given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M6">View MathML</a> is the standard Riemann-Liouville fractional derivative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M7">View MathML</a> are continuous. And Wang et al.[23] considered a 2m-point boundary value problem (BVP) at resonance for a coupled system as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M8">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M2">View MathML</a>. With the help of the coincidence degree theory, many existence results have been given in the above literatures. It is worth mentioning that the orders of derivative in the nonlinear function on the right-hand of equal signs are all fixed in the above works, but the opposite case is more difficult and complicated, then this work attempts to deal exactly with this case. What is more, this case of arbitrary order derivative included in the nonlinear functions is very important in many aspects [20,22].

There are significant developments in the theory of impulses especially in the area of impulsive differential equations with fixed moments, which provided a natural description of observed evolution processes, regarding as important tools for better understanding several real word phenomena in applied sciences [1,7,26-29]. In addition, motivated by the better formula of solutions cited by the work of Zhou et al.[1,7,8], the aim of this work is to discuss a boundary value problem for a coupled system of impulsive fractional differential equation. Exactly, this paper deals with the 2m-point boundary value problem of the following coupled system of impulsive fractional differential equations at resonance:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M10">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M17">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M18">View MathML</a> satisfy Carathéodory conditions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M19">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M21">View MathML</a>, here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M25">View MathML</a> denote the right and left limits of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M26">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M27">View MathML</a>, respectively, and the fractional derivative is understood in the Riemann-Liouville sense. k, m, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M31">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M32">View MathML</a>) are fixed constant satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M33">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M34">View MathML</a>.

The coupled system (1.1) happens to be at resonance in the sense that the associated linear homogeneous coupled system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M35">View MathML</a>

has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M38">View MathML</a> as a nontrivial solution. To solve this interesting and important problem and to overcome the difficulties caused by the impulses, we will construct some Banach spaces, then we shall obtain the new solvability results for the coupled system (1.1) with the help of a coincidence degree continuation theorem. The main contributions of this work are Lemma 2.1 and Lemma 3.1 in Section 3 since the calculations are disposed well.

The plan of this work is organized as follows. Section 2 contains some necessary notations, definitions and lemmas that will be used in the sequel. In Section 3, we establish a theorem on the existence of solutions for the coupled system (1.1) based on the coincidence degree theory due to Mawhin [30,31].

2 Background materials and preliminaries

For the convenience of the readers, we recall some notations and an abstract existence theorem [30,31].

Let Y, Z be real Banach spaces, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M39">View MathML</a> be a Fredholm map of index zero and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M41">View MathML</a> be continuous projectors such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M45">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M46">View MathML</a> is invertible. We denote the inverse of the map by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M47">View MathML</a>. If Ω is an open bounded subset of Y such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M48">View MathML</a>, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M49">View MathML</a> will be called L-compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M50">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M51">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M52">View MathML</a> is compact.

The main tool we used is Theorem 2.4 of [30].

Theorem 2.1LetLbe a Fredholm operator of index zero, and letNbeL-compact on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M50">View MathML</a>. Assume that the following conditions are satisfied:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M54">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M55">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M56">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M57">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M58">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M59">View MathML</a>is a projection as above with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M60">View MathML</a>.

Then the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M61">View MathML</a>has at least one solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M62">View MathML</a>.

Now, we present some basic knowledge and definitions about fractional calculus theory, which can be found in the recent works [13,16,32].

Definition 2.1 The fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M63">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M64">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M65">View MathML</a>

provided the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M66">View MathML</a>.

Definition 2.2 The fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M63">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M64">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M69">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M70">View MathML</a>, provided the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M66">View MathML</a>.

Remark 2.1 It can be directly verified that the Riemann-Liouville fractional integration and fractional differentiation operators of the power functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M72">View MathML</a> yield power functions of the same form. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M74">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M75">View MathML</a>

(2.1)

Proposition 2.1[17]

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M76">View MathML</a>with a fractional derivative of order<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M63">View MathML</a>that belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M78">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M79">View MathML</a>

(2.2)

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M81">View MathML</a>, whereNis the smallest integer grater than or equal toα.

Proposition 2.2[32]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M83">View MathML</a>, then the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M84">View MathML</a>

is satisfied for a continuous functiony.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M86">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M87">View MathML</a>, the fractional derivatives<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M88">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M89">View MathML</a>exist, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M90">View MathML</a>

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M63">View MathML</a>, then the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M92">View MathML</a>

is satisfied for a continuous functiony.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M93">View MathML</a>, then the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M94">View MathML</a>

holds for a continuous functiony.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M95">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M96">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M97">View MathML</a>

with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M98">View MathML</a>. Denote

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M101">View MathML</a> with the norm

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M103">View MathML</a> is a Banach space with the norm defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M104">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M105">View MathML</a> equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M106">View MathML</a>

thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M107">View MathML</a> is a Banach space with the norm defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M108">View MathML</a>.

Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M111">View MathML</a>, where

with

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M114">View MathML</a> be defined as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M115">View MathML</a>, where

Then the coupled system of boundary value problem (1.1) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M117">View MathML</a>

For the sake of simplicity, we define the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M118">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M119">View MathML</a> as follows:

(2.3)

(2.4)

By the same way, we define the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M122">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M123">View MathML</a> as follows:

(2.5)

(2.6)

In what follows, we present the following lemmas which will be used to prove our main results.

Lemma 2.1If the following condition is satisfied:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M127">View MathML</a>, where

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M129">View MathML</a>is a Fredholm operator of index zero. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M130">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M131">View MathML</a>

(2.7)

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M132">View MathML</a>, here

(2.8)

(2.9)

Proof It is clear that (2.7) holds. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M135">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M136">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M138">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M140">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M130">View MathML</a>. Similarly, it is not difficult to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M132">View MathML</a>. Next, we will show that (2.8) and (2.9) hold.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M144">View MathML</a>, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M145">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M146">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M147">View MathML</a>

(2.10)

and

(2.11)

(2.12)

Proposition 2.1 together with (2.10)-(2.12) gives that

(2.13)

(2.14)

Substituting the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M152">View MathML</a> into (2.13), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M153">View MathML</a>

(2.15)

and substituting the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M154">View MathML</a> into (2.13), one has

(2.16)

By the same way, if we substitute the condition (2.12) into (2.14), then we can obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M156">View MathML</a>

(2.17)

and

(2.18)

Conversely, if (2.15)-(2.18) hold, set

It is easy to check that the above u, v satisfy equation (2.10)-(2.12). Thus, (2.8) and (2.9) hold.

Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M160">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M162">View MathML</a>, here

In what follows, we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M164">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M165">View MathML</a> are linear projectors. By some direct computations, we have

As a result,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M167">View MathML</a>

Similarly, we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M168">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M169">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M170">View MathML</a>. It means that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M171">View MathML</a> is a projector.

Now, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M172">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M173">View MathML</a>. On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M174">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M175">View MathML</a> implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M176">View MathML</a>

The condition (H1) guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M178">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M179">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M172">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M181">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M182">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M184">View MathML</a>, it means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M185">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M172">View MathML</a> gives that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M187">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M188">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M189">View MathML</a>, L is a Fredholm map of index zero. □

Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M190">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M191">View MathML</a>, here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M193">View MathML</a> are defined as follows:

Moreover, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M195">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M196">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M197">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M198">View MathML</a> is defined as follows:

Lemma 2.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M200">View MathML</a>is an open bounded subset with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M201">View MathML</a>, thenNisL-compact on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M50">View MathML</a>.

Proof Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M203">View MathML</a>. By a direct computation, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M204">View MathML</a>

Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M205">View MathML</a>. This gives that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M206">View MathML</a>, that is to say, the operator P is a linear projector. It is easy to check from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M207">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M208">View MathML</a>. Moreover, we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M209">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M210">View MathML</a>.

In what follows, we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M47">View MathML</a> defined above is the inverse of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M212">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M179">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M215">View MathML</a>, which gives that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M216">View MathML</a>

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M217">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M218">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M219">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M220">View MathML</a>, then

(2.19)

(2.20)

By some calculations, (2.19) and (2.20) imply that

It means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M224">View MathML</a>. Analogously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M225">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M226">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M47">View MathML</a> is the inverse of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M212">View MathML</a>.

Finally, we show that N is L-compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M50">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M231">View MathML</a>, where

Then we can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M233">View MathML</a>

where

So, we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M235">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M236">View MathML</a> is uniformly bounded.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M237">View MathML</a>, we have

(2.21)

(2.22)

The equicontinuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M241">View MathML</a> together with (2.21) and (2.22) gives that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M242">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M243">View MathML</a>, which yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M236">View MathML</a> is equicontinuous. By the Ascoli-Arzela theorem, we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M236">View MathML</a> is compact. By the same way, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M246">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M247">View MathML</a> is compact. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M248">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M249">View MathML</a>, then QN is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M250">View MathML</a> is compact. This means that N is L-compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M50">View MathML</a>. □

3 Main results

In this section, we present the existence results of the coupled system (1.1). To do this, we need the following hypotheses.

(H2) There exist functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M198">View MathML</a>, such that

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M255">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M256">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M198">View MathML</a>) satisfy

here

(H3) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M260">View MathML</a>, there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M261">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M262">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M263">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M198">View MathML</a>) such that

(1) if either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M265">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M266">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M267">View MathML</a>, then either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M268">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M269">View MathML</a>;

(2) if either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M270">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M271">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M272">View MathML</a>, then either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M273">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M274">View MathML</a>.

(H4) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M135">View MathML</a>, there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M276">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M198">View MathML</a>) such that if either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M278">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M279">View MathML</a>, either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M280">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M281">View MathML</a>, then either (1) or (2) holds, where

(1)

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M283">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M284">View MathML</a> are positive constants;

(2)

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M286">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M287">View MathML</a> are negative constants.

Lemma 3.1Suppose that (H2)-(H3) hold. Then the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M288">View MathML</a>

is bounded inY.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M289">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M290">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M260">View MathML</a>, we have

(3.1)

(3.2)

(3.3)

(3.4)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M296">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M297">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M298">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M299">View MathML</a>. Then we can see, from the condition (H3), that there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M300">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M302">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M303">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M304">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M305">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M306">View MathML</a>. So, we can see from (3.1) and (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M307">View MathML</a>

(3.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M308">View MathML</a>

(3.6)

Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M309">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M310">View MathML</a>, we have

(3.7)

(3.8)

(3.9)

Similarly, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M310">View MathML</a>, we have that

(3.10)

(3.11)

(3.12)

Substitute (3.11) and (3.12) into (3.8), then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M318">View MathML</a>

(3.13)

It means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M319">View MathML</a>

similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M320">View MathML</a>

Substituting the above two into (3.9) and (3.12), we can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M321">View MathML</a>

(3.14)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M322">View MathML</a>

(3.15)

From the condition (H2), (3.14) and (3.15) give that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M323">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M324">View MathML</a> are bounded, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M325">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M326">View MathML</a> are also bounded. Thus, by the definition of the norm on Y, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M327">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M328">View MathML</a> are bounded. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M329">View MathML</a> is bounded in Y. □

Lemma 3.2Suppose that the condition (H3) holds. Then the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M330">View MathML</a>

is bounded inY.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M135">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M332">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M333">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M334">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M335">View MathML</a>, so we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M336">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M337">View MathML</a>

From (H3), there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M338">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M339">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M340">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M341">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M342">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M343">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M344">View MathML</a>

which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M345">View MathML</a>. And for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M346">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M347">View MathML</a>

which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M348">View MathML</a>. So, we can see that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M309">View MathML</a>,

The above two arguments imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M351">View MathML</a> is bounded. In the same way, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M352">View MathML</a> is bounded. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M353">View MathML</a> is bounded in Y. □

Lemma 3.3The set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M354">View MathML</a>

is bounded inY, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M355">View MathML</a>is the linear isomorphism given by

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M357">View MathML</a>

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M135">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M359">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M360">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M361">View MathML</a> implies that

(3.16)

(3.17)

(3.18)

(3.19)

From (3.16) and (3.17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M366">View MathML</a>

the condition (H4) gives that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M367">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M368">View MathML</a>

which is a contradiction. As a result, there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M369">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M370">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M371">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M372">View MathML</a>. Similarly, from (3.18)-(3.19) and the second part of (1) or (2) of (H4), there exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M373">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M374">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M375">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M376">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M377">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M378">View MathML</a> are bounded, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M379">View MathML</a> is bounded in Y. □

Theorem 3.1Suppose that (H1)-(H4) hold. Then the problem (1.1) has at least one solution inY.

Proof Let Ω be a bounded open set of Y such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M380">View MathML</a>. It follows from Lemma 2.2 that N is L-compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M50">View MathML</a>. By means of above Lemmas 3.1-3.3, one obtains that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M382">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M383">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M384">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M385">View MathML</a>.

Then we need only to prove

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M386">View MathML</a>.

Take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M387">View MathML</a>

According to Lemma 3.3, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M388">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M389">View MathML</a>. Thus, the homotopy invariance property of degree theory gives that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M390">View MathML</a>

Then, by Theorem 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M391">View MathML</a> has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M392">View MathML</a>, i.e., the problem (1.1) has at least one solution in Y, which completes the proof. □

4 An example

Example 4.1

Consider the following boundary value problem for coupled systems of impulsive fractional differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M393">View MathML</a>

(4.1)

where

and

Due to the coupled problem (1.1), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M396">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M397">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M398">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M399">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M400">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M401">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M402">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M403">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M404">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M405">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M406">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M407">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M408">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M409">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M410">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M411">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M412">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M413">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M414">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M415">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M416">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M417">View MathML</a>. By direct calculation, we obtain that

It is easy to see that

where

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M421">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M422">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M423">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M424">View MathML</a>. And

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M426">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M427">View MathML</a>. Thus, the condition (H2) holds.

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M428">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M146">View MathML</a>, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M430">View MathML</a> holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M431">View MathML</a>. Thus either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M432">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M433">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M431">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M432">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M431">View MathML</a>, then

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M438">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M431">View MathML</a>, then

Similarly, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M441">View MathML</a> holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M442">View MathML</a>. Thus either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M443">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M444">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M442">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M432">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M442">View MathML</a>, then

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M433">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M442">View MathML</a>, then

So, from the above arguments, the first part of the condition (H3) is true for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M428">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M431">View MathML</a>.

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M454">View MathML</a>, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M455">View MathML</a> holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M456">View MathML</a>. Then either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M457">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M458">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M459">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M457">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M459">View MathML</a>, then

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M458">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M459">View MathML</a>, then

By the same way, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M466">View MathML</a>, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M467">View MathML</a> holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M468">View MathML</a>. Then either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M469">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M470">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M468">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M469">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M468">View MathML</a>, then

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M469">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M468">View MathML</a>, then

So, from the above arguments, the second part of the condition (H3) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M459">View MathML</a>.

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M480">View MathML</a>, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M481">View MathML</a>, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M482">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M38">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M484">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M485">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M486">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M487">View MathML</a>. And <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M488">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M442">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M490">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M431">View MathML</a>. Then we can see, from the above arguments, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M492">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M493">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M494">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M495">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M496">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M38">View MathML</a>. Thus,

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M499">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M500">View MathML</a>. So, the condition (H4) holds. Hence, from Theorem 3.1, the coupled problem (4.1) has at least one solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/80/mathml/M501">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

CZ conceived the main idea of the study, XZ carried out the main parts of the draft. CZ gave many valuable suggestions and corrected the main theorems in the discussion. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the editor and referee for their valuable comments and remarks which lead to a great improvement of the article. This research is supported by the National Natural Science Foundation of China (11071108), the Provincial Natural Science Foundation of Jiangxi, China (2010GZS0147, 20114BAB201007).

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