This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

An order-type existence theorem and applications to periodic problems

Jifeng Chu1* and Feng Wang12

Author Affiliations

1 Department of Mathematics, Hohai University, Nanjing, 210098, China

2 School of Mathematics and Physics, Changzhou University, Changzhou, 213164, China

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Boundary Value Problems 2013, 2013:37  doi:10.1186/1687-2770-2013-37


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/37


Received:8 November 2012
Accepted:5 February 2013
Published:21 February 2013

© 2013 Chu and Wang; 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

Based on the fixed point index and partial order method, one new order-type existence theorem concerning cone expansion and compression is established. As applications, we present sufficient existence conditions for the first- and second-order periodic problems.

MSC: 34B15.

Keywords:
fixed point index; order-type existence theorem; cone expansion and compression; positive solutions; periodic boundary value problems

1 Introduction and preliminaries

Let X, Y be real Banach spaces. Consider a linear mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M1">View MathML</a> and a nonlinear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M2">View MathML</a>. Here we assume that L is a Fredholm operator of index zero, that is, ImL is closed and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M3">View MathML</a>. Then the solvability of the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M4">View MathML</a>

has been studied by many researchers in the literature; see [1-8] and the references therein. In [1], Cremins established a fixed point index for A-proper semilinear operators defined on cones which includes and improves the results in [5,8,9]. Using the fixed point index and the concept of a quasi-normal cone introduced in [10], Cremins established a norm-type existence theorem concerning cone expansion and compression in [11], which generalizes some corresponding results contained in [12].

In this paper, we will use the properties of the fixed point index in [1] and partial order to present a new order-type existence theorem concerning cone expansion and compression which extends the corresponding results in [12]. We recall that a partial order in X induced by a cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M5">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M6">View MathML</a>

As applications, we study the first- and second-order periodic boundary problems and obtain new existence results. During the last few decades, periodic boundary value problems have been studied by many researchers in the literature; see, for example, [13-19] and the references therein. Our new results improve those contained in [13,18].

Next we recall some notations and results which will be needed in this paper. Let X and Y be Banach spaces, D be a linear subspace of X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M8">View MathML</a> be the sequences of oriented finite dimensional subspaces such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M9">View MathML</a> in Y for every y and dist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M10">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M11">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M13">View MathML</a> are sequences of continuous linear projections. The projection scheme <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M14">View MathML</a> is then said to be admissible for maps from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M15">View MathML</a> to Y. A map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M16">View MathML</a> is called approximation-proper (abbreviated A-proper) at a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M17">View MathML</a> with respect to an admissible scheme Γ if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M18">View MathML</a> is continuous for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M19">View MathML</a> and whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M20">View MathML</a> is bounded with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M21">View MathML</a>, then there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M22">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M24">View MathML</a>. T is simply called A-proper if it is A-proper at all points of Y. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M1">View MathML</a> is a Fredholm operator of index zero if ImL is closed and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M26">View MathML</a>. As a consequence of this property, X and Y may be expressed as direct sums; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M28">View MathML</a> with continuous linear projections <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M30">View MathML</a>. The restriction of L to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M31">View MathML</a>, denoted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M32">View MathML</a>, is a bijection onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M33">View MathML</a> with continuous inverse <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M34">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M36">View MathML</a> have the same finite dimension, there exists a continuous bijection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M37">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M38">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M39">View MathML</a> is a linear bijection with bounded inverse. Let K be a cone in a Banach space X. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M40">View MathML</a> is a cone in Y. In [20], Petryshyn has shown that an admissible scheme <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M41">View MathML</a> can be constructed such that L is A-proper with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M41">View MathML</a>. The following properties of the fixed point index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M43">View MathML</a> and two lemmas can be found in [1].

Proposition 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M44">View MathML</a>be open and bounded and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M45">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M47">View MathML</a>mapsKtoK, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M48">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M49">View MathML</a>.

(P1) (Existence property) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M50">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M51">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M52">View MathML</a>.

(P2) (Normality) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M53">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M54">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M55">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M56">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M57">View MathML</a>.

(P3) (Additivity) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M48">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M59">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M60">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M61">View MathML</a>are disjoint relatively open subsets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M62">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M63">View MathML</a>

with equality if either of indices on the right is a singleton.

(P4) (Homotopy invariance) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M64">View MathML</a>is an A-proper homotopy on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M62">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M67">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M68">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M70">View MathML</a>is independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M72">View MathML</a>.

Lemma 1.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M73">View MathML</a>is Fredholm of index zero, Ω is an open bounded set and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M75">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M76">View MathML</a>be A-proper for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66">View MathML</a>. Assume thatNis bounded and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M47">View MathML</a>mapsKtoK. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M79">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M49">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M81">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M82">View MathML</a>

Lemma 1.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M73">View MathML</a>is Fredholm of index zero, Ω is an open bounded set and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M74">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M76">View MathML</a>be A-proper for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66">View MathML</a>. Assume thatNis bounded and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M47">View MathML</a>mapsKtoK. If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M88">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M89">View MathML</a>

for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M90">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M91">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M92">View MathML</a>

2 An abstract result

We will establish an abstract existence theorem concerning cone expansion and compression of order type, which reads as follows.

Theorem 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M73">View MathML</a>is Fredholm of index zero, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M76">View MathML</a>be A-proper for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M95">View MathML</a>. Assume thatNis bounded and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M47">View MathML</a>mapsKtoK. Suppose further that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M60">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M61">View MathML</a>are two bounded open sets inXsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M100">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M101">View MathML</a>. If one of the following two conditions is satisfied:

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M102">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M103">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M104">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M105">View MathML</a>;

(C2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M104">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M107">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M102">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M105">View MathML</a>.

Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M110">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M52">View MathML</a>.

Proof We assume that (C1) is satisfied. First we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M112">View MathML</a>

(2.1)

In fact, otherwise, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M114">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M115">View MathML</a>

then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M116">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M117">View MathML</a>

which contradicts condition (C1). From (2.1) and Lemma 1.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M118">View MathML</a>

(2.2)

Choosing an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M88">View MathML</a>, next we prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M120">View MathML</a>

(2.3)

In fact, otherwise, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M122">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M123">View MathML</a>

then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M124">View MathML</a>

in which the partial order is induced by the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M125">View MathML</a> in Y. So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M126">View MathML</a>

which is a contradiction to condition (C1). Hence (2.3) holds, and then by Lemma 1.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M127">View MathML</a>

(2.4)

It follows therefore from (2.2), (2.4) and the additivity property (P3) of Proposition 1.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M128">View MathML</a>

(2.5)

Since the index is nonzero, the existence property (P1) of Proposition 1.1 implies that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M110">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M52">View MathML</a>.

Similarly, when (C2) is satisfied, instead of (2.2), (2.4) and (2.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M131">View MathML</a>

and therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M132">View MathML</a>

Also, we can assert that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M133">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M52">View MathML</a>. □

3 Applications

3.1 First-order periodic boundary value problems

We consider the following first-order periodic boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M135">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M136">View MathML</a> is continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M137">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M138">View MathML</a>.

Consider the Banach spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M139">View MathML</a> endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M140">View MathML</a>. Define the cone K in X by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M141">View MathML</a>

Let L be the linear operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M142">View MathML</a> to Y with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M143">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M144">View MathML</a>

Let us define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M2">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M146">View MathML</a>

Then (3.1) is equivalent to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M147">View MathML</a>

It is obvious that L is a Fredholm operator of index zero with

Next we define the projections <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M150">View MathML</a> by

and the isomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M152">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M153">View MathML</a>. Note that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M154">View MathML</a>, the inverse operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M155">View MathML</a>

of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M156">View MathML</a>

is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M157">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M158">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M159">View MathML</a>

We can verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M160">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M161">View MathML</a>

To state the existence result, we introduce two conditions:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M162">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M163">View MathML</a>,

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M164">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M165">View MathML</a>.

Theorem 3.1Assume that there exist two positive numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M166">View MathML</a>such that (H1), (H2) and

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M167">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M168">View MathML</a>

hold. Then (3.1) has at least one positive periodic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M169">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M170">View MathML</a>.

Proof First, we note that L, as defined, is Fredholm of index zero, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M171">View MathML</a> is compact by the Arzela-Ascoli theorem and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M76">View MathML</a> is A-proper for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66">View MathML</a> by [[20], Lemma 2(a)].

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M174">View MathML</a>, then by condition (H3),

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M176">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M177">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M61">View MathML</a> are bounded open sets and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M180">View MathML</a>

We now show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M181">View MathML</a>

(3.2)

In fact, if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M182">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M183">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M184">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M185">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M186">View MathML</a>. Clearly, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M187">View MathML</a> attains a maximum on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M188">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M189">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M190">View MathML</a>. As a consequence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M191">View MathML</a>

which is a contradiction to (H1). Therefore (3.2) holds.

On the other hand, we claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M192">View MathML</a>

(3.3)

In fact, if not, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M193">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M194">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M193">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M196">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M197">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M163">View MathML</a>. By condition (H2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M199">View MathML</a>

which is a contradiction. As a result, (3.3) is verified.

It follows from (3.2), (3.3) and Theorem 2.1 that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M200">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M201">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M170">View MathML</a>. □

Remark 3.1 In [18], the following condition is required instead of (H2):

(H) there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M203">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M205">View MathML</a>, and continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M207">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M208">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M211">View MathML</a> is nonincreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M212">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M213">View MathML</a>

Obviously, our condition (H2) is much weaker and less strict compared with (H). Moreover, (H2) is easier to check than (H). So, our result generalizes and improves [[18], Theorem 5].

Remark 3.2 From the proof of Theorem 3.1, we can see that condition (H2) can be replaced by one of the following two relatively weaker conditions:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M214">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M215">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M216">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M217">View MathML</a> is positive for almost everywhere on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M218">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M219">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M220">View MathML</a>.

Remark 3.3 Finally in this section, we note that conditions (H1) and (H2) can be replaced by the following asymptotic conditions:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M221">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M222">View MathML</a> uniformly for t;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M223">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M224">View MathML</a> uniformly for t.

Example 3.1 Let the nonlinearity in (3.1) be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M225">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M227">View MathML</a> are positive 1-periodic functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M228">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M229">View MathML</a> is a positive parameter. Then (3.1) has at least one positive 1-periodic solution for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M230">View MathML</a>, here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M231">View MathML</a> is some positive constant.

Proof We will apply Theorem 3.1 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M232">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M228">View MathML</a>, it is easy to see that (H3) holds. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M234">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M235">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M226">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M237">View MathML</a>

One may easily see that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M238">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M239">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M240">View MathML</a>

Then, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M241">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M242">View MathML</a>

which implies that (H1) holds.

On the other hand, we have

which implies that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M223">View MathML</a>) holds. Now we have the desired result. □

3.2 Second-order periodic boundary value problems

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M136">View MathML</a> be continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M137">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M138">View MathML</a>. We will discuss the existence of positive solutions of the second-order periodic boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M248">View MathML</a>

(3.4)

Since some parts of the proof are in the same line as that of Theorem 3.1, we will outline the proof with the emphasis on the difference.

Let X, Y be Banach spaces and the cone K be as in Section 3.1. In this case, we may define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M249">View MathML</a>

and let the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M73">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M251">View MathML</a>

Then L is Fredholm of index zero,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M252">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M253">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M254">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M255">View MathML</a>

Thus it is clear that (3.4) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M256">View MathML</a>

We use the same projections P, Q as in Section 3.1 and define the isomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M152">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M258">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M259">View MathML</a>. It is easy to verify that the inverse operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M260">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M261">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M262">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M263">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M264">View MathML</a>

We can verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M265">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M266">View MathML</a>

Theorem 3.2Assume that there exist two positive numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M166">View MathML</a>such that (H1), (H2) and

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M268">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M269">View MathML</a>

hold. Then (3.4) has at least one positive periodic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M169">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M170">View MathML</a>.

Proof It is again easy to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M76">View MathML</a> is A-proper for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M66">View MathML</a> by [[20], Lemma 2(a)].

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M174">View MathML</a>, then by condition (H4),

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M176">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M277">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M278">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M279">View MathML</a> are bounded and open sets and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M280">View MathML</a>

Next, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M281">View MathML</a>

(3.5)

On the contrary, suppose that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M282">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M283">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M284">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M285">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M286">View MathML</a>. Using the boundary conditions, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M287">View MathML</a>. In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M289">View MathML</a>. This gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M290">View MathML</a>

which is a contradiction to condition (H1). Therefore (3.5) holds.

Finally, similar to the proof of (3.3), it follows from condition (H2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M291">View MathML</a>

Consequently all conditions of Theorem 2.1 are satisfied. Therefore, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M292">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M201">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M169">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/37/mathml/M170">View MathML</a> and the assertion follows. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors read and approved the final manuscript.

Acknowledgements

JC was supported by the National Natural Science Foundation of China (Grant No. 11171090, No. 11271333 and No. 11271078), the Program for New Century Excellent Talents in University (Grant No. NCET-10-0325), China Postdoctoral Science Foundation funded project (Grant No. 2012T50431). FW was supported by the National Natural Science Foundation of China (Grant No. 10971179) and the Natural Science Foundation of Changzhou University (Grant No. JS201008).

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