SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Existence of periodic solutions for a class of p-Laplacian equations

Xiaojun Chang12* and Yu Qiao3

Author Affiliations

1 School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, P.R. China

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

3 College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, P.R. China

For all author emails, please log on.

Boundary Value Problems 2013, 2013:96  doi:10.1186/1687-2770-2013-96


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/96


Received:26 September 2012
Accepted:5 April 2013
Published:19 April 2013

© 2013 Chang and Qiao; 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

This paper is devoted to the existence of periodic solutions for the one-dimensional p-Laplacian equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M3">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M4">View MathML</a>. By using some asymptotic interaction of the ratios <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M6">View MathML</a> with the Fučík spectrum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M7">View MathML</a> related to periodic boundary condition, we establish a new existence theorem of periodic solutions for the one-dimensional p-Laplacian equation.

Keywords:
periodic solutions; p-Laplacian; Fučík spectrum; Leray-Schauder degree; Borsuk theorem

1 Introduction and main results

In this paper, we are concerned with the existence of solutions for the following periodic boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M8">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M3">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M4">View MathML</a>. A solution u of problem (1.1) means that u is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M13">View MathML</a> is absolutely continuous such that (1.1) is satisfied for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>.

Existence and multiplicity of solutions of the periodic problems driven by the p-Laplacian have been obtained in the literature by many people (see [1-5]). Many solvability conditions for problem (1.1) were established by using the asymptotic interaction at infinity of the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M15">View MathML</a> with the Fučík spectrum for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M16">View MathML</a> under periodic boundary condition (see e.g., [2,4,6-9]). In [6], Del Pino, Manásevich and Murúa firstly defined the Fučík spectrum for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M16">View MathML</a> under periodic boundary value condition as the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18">View MathML</a> consisting of all the pairs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M19">View MathML</a> such that the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M20">View MathML</a>

admits at least one nontrivial 2π-periodic solution (see [10] for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M21">View MathML</a>). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M22">View MathML</a>

By [6], it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M23">View MathML</a>

Then they applied the Sturm’s comparison theorem and Leray-Schauder degree theory to prove that problem (1.1) is solvable if the following relations hold:

uniformly for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M26">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M27">View MathML</a>

Clearly, in this case, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M28">View MathML</a>, which is usually called that the nonlinearity f is nonresonant with respect to the Fučík spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18">View MathML</a>. In [11], Anane and Dakkak obtained a similar result by using the property of nodal set for eigenfunctions. If f is resonant with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18">View MathML</a>, i.e., there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M31">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M33">View MathML</a> uniformly for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>, together with the Landesman-Lazer type condition, Jiang [9] obtained the existence of solutions of (1.1) by applying the variational methods and symplectic transformations. In these works, either f is resonant or nonresonant with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18">View MathML</a>, the solvability of problem (1.1) was assured by assuming that the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M5">View MathML</a> stays at infinity in the pointwise sense asymptotically between two consecutive curves of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M38">View MathML</a>

we can see that the conditions on the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M39">View MathML</a> are more general than that on the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M40">View MathML</a>. Recently, Liu and Li [2] studied the nondissipative p-Laplacian equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M41">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M42">View MathML</a> is a constant. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M43">View MathML</a>. They proved that (1.2) is solvable under the following assumptions:

(1) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M44">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M45">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M46">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M48">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M49">View MathML</a>.

Here, the potential function G is nonresonant with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18">View MathML</a> and the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M51">View MathML</a> is not required to stay at infinity in the pointwise sense asymptotically between two consecutive branches of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18">View MathML</a> and it may even cross at infinity multiple Fučík spectrum curves.

In this paper, we want to obtain the solvability of problem (1.1) by using the asymptotic interaction at infinity of both the ratios <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M54">View MathML</a> with the Fučík spectrum for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M16">View MathML</a> under periodic boundary condition. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M56">View MathML</a>. The goal is to obtain the existence of solutions of (1.1) by requiring neither the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M15">View MathML</a> stays at infinity in the pointwise sense asymptotically between two consecutive branches of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18">View MathML</a> nor the limits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M59">View MathML</a> exist. We shall prove that problem (1.1) admits a solution under the assumptions that the nonlinearity f has at most <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M60">View MathML</a>-linear growth at infinity and the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M61">View MathML</a> has a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M62">View MathML</a> limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M63">View MathML</a>, while the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M54">View MathML</a> stays at infinity in the pointwise sense asymptotically between two consecutive branches of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M18">View MathML</a>. Our result will complement the results in the literature on the solvability of problem (1.1) involving the Fučík spectrum.

For related works on resonant problems involving the Fučík spectrum, we also refer the interested readers to see [12-19] and the references therein.

Our main result for problem (1.1) now reads as follows.

Theorem 1.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M66">View MathML</a>and the following conditions hold:

(i) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M67">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M68">View MathML</a>

(1.3)

(ii) There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M69">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M70">View MathML</a>

(1.4)

(iii) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M71">View MathML</a>such that

(1.5)

(1.6)

hold uniformly for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M75">View MathML</a>

(1.7)

Then problem (1.1) admits a solution.

Remark If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M76">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M77">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M79">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M71">View MathML</a> satisfy (1.7), e is continuous on ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M81">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M83">View MathML</a>. By Theorem 1.1, it follows that problem (1.1) admits a solution. It is easily seen that the result of [16] cannot be applied to this case. Note that one can also obtain the solvability of (1.1) in this case by the result of [6], while in Theorem 1.1 we do not require the pointwise limit at infinity of the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M5">View MathML</a> as in [6].

For convenience, we introduce some notations and definitions. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M85">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M86">View MathML</a>) denotes the usual Sobolev space with inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M87">View MathML</a> and norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M88">View MathML</a>, respectively. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M89">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M90">View MathML</a>) denotes the space of m-times continuous differential real functions with norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M91">View MathML</a>

2 Proof of the main result

Denote by deg the Leray-Schauder degree. To prove Theorem 1.1, we need the following results.

Lemma 2.1[20]

Let Ω be a bounded open region in a real Banach spaceX. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M92">View MathML</a>is completely continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M93">View MathML</a>. Then the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M94">View MathML</a>has a solution in Ω if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M95">View MathML</a>.

Lemma 2.2 (Borsuk Theorem [20])

Assume thatXis a real Banach space. Let Ω be a symmetric bounded open region with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M96">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M92">View MathML</a>is completely continuous and odd with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M98">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M99">View MathML</a>is odd.

Proof of Theorem 1.1 Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M100">View MathML</a>. Consider the following homotopy problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M101">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M102">View MathML</a>.

By (1.3) and the regularity arguments, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M103">View MathML</a>, and furthermore there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M104">View MathML</a> such that, if u is a solution of problem (2.1), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M105">View MathML</a>

(2.2)

In what follows, we shall prove that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M106">View MathML</a> independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M102">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M108">View MathML</a> for all possible solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M109">View MathML</a> of (2.1). Assume by contradiction that there exist a sequence of number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M110">View MathML</a> and corresponding solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M111">View MathML</a> of (2.1) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M112">View MathML</a>

(2.3)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M113">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M114">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M115">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M116">View MathML</a>

By (1.3), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M117">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M118">View MathML</a>

Then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M119">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M120">View MathML</a>

(2.4)

In addition, using (1.3) and the regularity arguments, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M121">View MathML</a> such that, for each n, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M122">View MathML</a>, and thus there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M123">View MathML</a> such that, passing to a subsequence if possible,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M124">View MathML</a>

(2.5)

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M125">View MathML</a>. In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M110">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M127">View MathML</a> such that, passing to a subsequence if possible,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M128">View MathML</a>

(2.6)

Note that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M129">View MathML</a>, problem (2.1) has only the trivial solution, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M130">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M132">View MathML</a>. It is easily seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M133">View MathML</a> is a nontrivial solution of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M134">View MathML</a>

(2.7)

We now distinguish three cases:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M133">View MathML</a> changes sign in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M136">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M138">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M140">View MathML</a>.

In the following, it will be shown that each case leads to a contradiction.

Case (i). Let

Then, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M142">View MathML</a>, we get

In addition, as shown in [11], we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M144">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M145">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M146">View MathML</a>

By (1.4) and (2.4), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M147">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M133">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M149">View MathML</a>

(2.8)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M151">View MathML</a>.

Now we prove that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M153">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M154">View MathML</a>

(2.9)

In fact, if not, we assume, by contradiction, that there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M111">View MathML</a>, we still denote it as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M111">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M158">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M159">View MathML</a>

Combing with (2.5), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M125">View MathML</a> and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M133">View MathML</a> changes sign, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M162">View MathML</a>

A contradiction. Hence, (2.9) holds.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M163">View MathML</a>, define

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M165">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M167">View MathML</a>. Then by (2.9) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M169">View MathML</a>. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M170">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M172">View MathML</a> is the nearest point satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M173">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M174">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M175">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M176">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M177">View MathML</a>

(2.10)

By (2.5), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M179">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M180">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M182">View MathML</a>. Hence, together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M183">View MathML</a> and (1.4), there exist subsequences of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M185">View MathML</a>, we still denote them as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M185">View MathML</a>, such that, for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M189">View MathML</a>

Using (1.3), for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M191">View MathML</a> is uniformly bounded with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M192">View MathML</a>, we obtain by the Lebesgue dominated convergence theorem that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M193">View MathML</a>

Thus,

(2.11)

By (1.4) and (2.2), we get

In view of (2.11), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M196">View MathML</a>

(2.12)

holds uniformly for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>. Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M198">View MathML</a>

(2.13)

holds uniformly for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>.

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M200">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M201">View MathML</a> satisfying (2.12)-(2.13), denoting

we obtain by (1.5)-(1.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M203">View MathML</a>

(2.14)

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M183">View MathML</a>, we have

(2.15)

(2.16)

We claim that there exists subinterval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M207">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M208">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M209">View MathML</a>

(2.17)

or subinterval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M210">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M211">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M212">View MathML</a>

(2.18)

Indeed, if not, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M214">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>. Together with the choosing of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M217">View MathML</a> and (2.14), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M218">View MathML</a>

Then by (1.7), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M219">View MathML</a>. A contradiction. Combining (2.12)-(2.13) with (2.15)-(2.18), we obtain a contradiction.

Case (ii). In this case, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M220">View MathML</a>

Using similar arguments as in Case (i), by (1.4) and (2.4) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M138">View MathML</a>. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M224">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>. We can see that there exists subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M200">View MathML</a>, which is still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M200">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M228">View MathML</a>

(2.19)

holds uniformly for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>. On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M200">View MathML</a> satisfying (2.19), denoting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M231">View MathML</a>

we obtain by (1.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M232">View MathML</a>

(2.20)

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M183">View MathML</a>, we have

(2.21)

We shall show that there exists subinterval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M235">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M236">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M237">View MathML</a>

(2.22)

In fact, if not, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M238">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M239">View MathML</a>. By the choosing of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M216">View MathML</a> and (2.20), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M241">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M133">View MathML</a> is a nontrivial solution of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M244">View MathML</a>

(2.23)

Taking 1 as test function in problem (2.23), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M245">View MathML</a>

(2.24)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M246">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M248">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M14">View MathML</a>, which is contrary to that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M125">View MathML</a>. Hence, (2.22) holds. Clearly, (2.21)-(2.22) contradict (2.19).

Case (iii). In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M169">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M200">View MathML</a> is uniformly bounded. Similar arguments as in Case (ii) imply a contradiction.

In a word, (2.3) cannot hold, and hence by (2.2) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M106">View MathML</a> independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M102">View MathML</a> such that, if u is a solution of problem (2.1), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M255">View MathML</a>

(2.25)

Note that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M256">View MathML</a>, the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M257">View MathML</a>

(2.26)

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M258">View MathML</a>. Clearly, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M259">View MathML</a> seen as an operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M260">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M261">View MathML</a> is completely continuous. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M262">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M263">View MathML</a>. Then solving problem (1.1) is equivalent to finding solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M261">View MathML</a> of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M265">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M266">View MathML</a>. Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M267">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M268">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M269">View MathML</a>. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M270">View MathML</a> is well defined for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M271">View MathML</a>. Owing to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M272">View MathML</a>, there is a continuous curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M273">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M274">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M275">View MathML</a>, whose image is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M276">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M277">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M279">View MathML</a>. From the invariance property of Leray-Schauder degree under compact homotopies, it follows that the degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M280">View MathML</a> is constant for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M275">View MathML</a>. Obviously, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M282">View MathML</a> is odd. By the Borsuk’s theorem, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M283">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M271">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M285">View MathML</a>

Consider the following homotopy:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M286">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M287">View MathML</a>. By (2.25), we can see that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M288">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M289">View MathML</a>

From the invariance property of Leray-Schauder degree, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/96/mathml/M290">View MathML</a>

Hence, problem (1.1) has a solution. The proof is complete. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

The first author sincerely thanks Professor Yong Li and Doctor Yixian Gao for their many useful suggestions and the both authors thank Professor Zhi-Qiang Wang for many helpful discussions and his invitation to Chern Institute of Mathematics. The first author is partially supported by the NSFC Grant (11101178), NSFJP Grant (201215184) and FSIIP of Jilin University (201103203). The second author is partially sup- ported by NSFC Grant (11226123).

References

  1. Aizicovici, S, Papageorgiou, NS, Staicu, V: Nonlinear resonant periodic problems with concave terms. J. Math. Anal. Appl.. 375, 342–364 (2011). Publisher Full Text OpenURL

  2. Liu, W, Li, Y: Existence of periodic solutions for p-Laplacian equation under the frame of Fučík spectrum. Acta Math. Sin. Engl. Ser.. 27, 545–554 (2011). Publisher Full Text OpenURL

  3. Manásevich, R, Mawhin, J: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Differ. Equ.. 145, 367–393 (1998). Publisher Full Text OpenURL

  4. Reichel, W, Walter, W: Sturm-Liouville type problems for the p-Laplacian under asymptotic nonresonance conditions. J. Differ. Equ.. 156, 50–70 (1999). Publisher Full Text OpenURL

  5. Yang, X, Kim, Y, Lo, K: Periodic solutions for a generalized p-Laplacian equation. Appl. Math. Lett.. 25, 586–589 (2012). Publisher Full Text OpenURL

  6. Del Pino, M, Manásevich, R, Murúa, A: Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE. Nonlinear Anal.. 18, 79–92 (1992). Publisher Full Text OpenURL

  7. Fabry, C, Fayyad, D: Periodic solutions of second order differential equations with a p-Laplacian and asymmetric nonlinearities. Rend. Ist. Mat. Univ. Trieste. 24, 207–227 (1992)

  8. Fabry, C, Manásevich, R: Equations with a p-Laplacian and an asymmetric nonlinear term. Discrete Contin. Dyn. Syst.. 7, 545–557 (2001)

  9. Jiang, M: A Landesman-Lazer type theorem for periodic solutions of the resonant asymmetric p-Laplacian equation. Acta Math. Sin.. 21, 1219–1228 (2005). Publisher Full Text OpenURL

  10. Drábek, P: Solvability and Bifurcations of Nonlinear Equations (1992)

  11. Anane, A, Dakkak, A: Nonexistence of nontrivial solutions for an asymmetric problem with weights. Proyecciones. 19, 43–52 (2000)

  12. Boccardo, L, Drábek, P, Giachetti, D, Kućera, M: Generalization of Fredholm alternative for nonlinear differential operators. Nonlinear Anal.. 10, 1083–1103 (1986). Publisher Full Text OpenURL

  13. Drábek, P, Invernizzi, S: On the periodic boundary value problem for forced Duffing equation with jumping nonlinearity. Nonlinear Anal.. 10, 643–650 (1986). Publisher Full Text OpenURL

  14. Fonda, A: On the existence of periodic solutions for scalar second order differential equations when only the asymptotic behaviour of the potential is known. Proc. Am. Math. Soc.. 119, 439–445 (1993). Publisher Full Text OpenURL

  15. Habets, P, Omari, P, Zanolin, F: Nonresonance conditions on the potential with respect to the Fučík spectrum for the periodic boundary value problem. Rocky Mt. J. Math.. 25, 1305–1340 (1995). Publisher Full Text OpenURL

  16. Liu, W, Li, Y: Existence of 2π-periodic solutions for the non-dissipative Duffing equation under asymptotic behaviors of potential function. Z. Angew. Math. Phys.. 57, 1–11 (2006)

  17. Omari, P, Zanolin, F: Nonresonance conditions on the potential for a second-order periodic boundary value problem. Proc. Am. Math. Soc.. 117, 125–135 (1993). Publisher Full Text OpenURL

  18. Zhang, M: Nonresonance conditions for asymptotically positively homogeneous differential systems: the Fučík spectrum and its generalization. J. Differ. Equ.. 145, 332–366 (1998). Publisher Full Text OpenURL

  19. Zhang, M: The rotation number approach to the periodic Fučík spectrum. J. Differ. Equ.. 185, 74–96 (2002). Publisher Full Text OpenURL

  20. Deimling, K: Nonlinear Functional Analysis, Springer, New York (1985)