Existence of Solutions to a Nonlocal Boundary Value Problem with Nonlinear Growth

Boundary Value Problems20102011:416416

DOI: 10.1155/2011/416416

Received: 17 July 2010

Accepted: 17 October 2010

Published: 24 October 2010

Abstract

This paper deals with the existence of solutions for the following differential equation: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq2_HTML.gif , subject to the boundary conditions: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq4_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq7_HTML.gif is a continuous function, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq8_HTML.gif is a nondecreasing function with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq9_HTML.gif . Under the resonance condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq10_HTML.gif , some existence results are given for the boundary value problems. Our method is based upon the coincidence degree theory of Mawhin. We also give an example to illustrate our results.

1. Introduction

In this paper, we consider the following second-order differential equation:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ1_HTML.gif
(1.1)
subject to the boundary conditions:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ2_HTML.gif
(1.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq11_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq12_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq13_HTML.gif is a continuous function, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq14_HTML.gif is a nondecreasing function with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq15_HTML.gif . In boundary conditions (1.2), the integral is meant in the Riemann-Stieltjes sense.

We say that BVP (1.1), (1.2) is a problem at resonance, if the linear equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ3_HTML.gif
(1.3)

with the boundary condition (1.2) has nontrivial solutions. Otherwise, we call them a problem at nonresonance.

Nonlocal boundary value problems were first considered by Bicadze and Samarskiĭ [1] and later by Il'pin and Moiseev [2, 3]. In a recent paper [4], Karakostas and Tsamatos studied the following nonlocal boundary value problem:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ4_HTML.gif
(1.4)
Under the condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq16_HTML.gif (i.e., nonresonance case), they used Krasnosel'skii's fixed point theorem to show that the operator equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq17_HTML.gif has at least one fixed point, where operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq18_HTML.gif is defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ5_HTML.gif
(1.5)

However, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq19_HTML.gif (i.e., resonance case), then the method in [4] is not valid.

As special case of nonlocal boundary value problems, multipoint boundary value problems at resonance case have been studied by some authors [511].

The purpose of this paper is to study the existence of solutions for nonlocal BVP (1.1), (1.2) at resonance case (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq20_HTML.gif ) and establish some existence results under nonlinear growth restriction of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq21_HTML.gif . Our method is based upon the coincidence degree theory of Mawhin [12].

2. Main Results

We first recall some notation, and an abstract existence result.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq22_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq23_HTML.gif be real Banach spaces, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq24_HTML.gif be a linear operator which is Fredholm map of index zero (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq25_HTML.gif , the image of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq26_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq27_HTML.gif , the kernel of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq28_HTML.gif are finite dimensional with the same dimension as the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq29_HTML.gif ), and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq31_HTML.gif be continuous projectors such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq32_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq34_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq37_HTML.gif . It follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq38_HTML.gif is invertible; we denote the inverse by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq39_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq40_HTML.gif be an open bounded, subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq41_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq42_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq43_HTML.gif , the map http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq44_HTML.gif is said to be http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq45_HTML.gif -compact on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq46_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq47_HTML.gif is bounded, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq48_HTML.gif is compact. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq49_HTML.gif be a linear isomorphism.

The theorem we use in the following is Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq50_HTML.gif of [12].

Theorem 2.1.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq51_HTML.gif be a Fredholm operator of index zero, and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq52_HTML.gif be http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq53_HTML.gif -compact on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq54_HTML.gif . Assume that the following conditions are satisfied:

(i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq55_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq56_HTML.gif ,

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq57_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq58_HTML.gif ,

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq59_HTML.gif ,

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq60_HTML.gif is a projection with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq61_HTML.gif . Then the equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq62_HTML.gif has at least one solution in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq63_HTML.gif .

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq64_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq65_HTML.gif , we use the norms http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq66_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq67_HTML.gif and denote the norm in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq68_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq69_HTML.gif . We will use the Sobolev space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq70_HTML.gif which may be defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ6_HTML.gif
(2.1)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq71_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq72_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq73_HTML.gif is a linear operator defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ7_HTML.gif
(2.2)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ8_HTML.gif
(2.3)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq74_HTML.gif be defined as
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ9_HTML.gif
(2.4)

Then BVP (1.1), (1.2) is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq75_HTML.gif .

We will establish existence theorems for BVP (1.1), (1.2) in the following two cases:

case (i): http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq76_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq77_HTML.gif ;

case (ii): http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq78_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq79_HTML.gif .

Theorem 2.2.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq80_HTML.gif be a continuous function and assume that

(H1) there exist functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq81_HTML.gif and constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq82_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq83_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq84_HTML.gif , it holds that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ10_HTML.gif
(2.5)
(H2) there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq85_HTML.gif , such that for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq86_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq87_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq88_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ11_HTML.gif
(2.6)
(H3) there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq89_HTML.gif , such that either
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ12_HTML.gif
(2.7)
or else
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ13_HTML.gif
(2.8)
Then BVP (1.1), (1.2) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq90_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq91_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq92_HTML.gif has at least one solution in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq93_HTML.gif provided that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ14_HTML.gif
(2.9)

Theorem 2.3.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq94_HTML.gif be a continuous function. Assume that assumption (H1) of Theorem 2.2 is satisfied, and

(H4) there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq95_HTML.gif , such that for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq96_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq97_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq98_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ15_HTML.gif
(2.10)
(H5) there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq99_HTML.gif , such that either
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ16_HTML.gif
(2.11)

or else

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ17_HTML.gif
(2.12)
Then BVP (1.1), (1.2) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq100_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq101_HTML.gif has at least one solution in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq102_HTML.gif provided that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ18_HTML.gif
(2.13)

3. Proof of Theorems 2.2 and 2.3

We first prove Theorem 2.2 via the following Lemmas.

Lemma 3.1.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq103_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq104_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq105_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq106_HTML.gif is a Fredholm operator of index zero. Furthermore, the linear continuous projector operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq107_HTML.gif can be defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ19_HTML.gif
(3.1)
and the linear operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq108_HTML.gif can be written by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ20_HTML.gif
(3.2)
Furthermore,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ21_HTML.gif
(3.3)

Proof.

It is clear that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ22_HTML.gif
(3.4)
Obviously, the problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ23_HTML.gif
(3.5)
has a solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq109_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq110_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq111_HTML.gif , if and only if
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ24_HTML.gif
(3.6)
which implies that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ25_HTML.gif
(3.7)
In fact, if (3.5) has solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq112_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq113_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq114_HTML.gif , then from (3.5) we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ26_HTML.gif
(3.8)
According to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq115_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq116_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ27_HTML.gif
(3.9)
then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ28_HTML.gif
(3.10)
On the other hand, if (3.6) holds, setting
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ29_HTML.gif
(3.11)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq117_HTML.gif is an arbitrary constant, then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq118_HTML.gif is a solution of (3.5), and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq119_HTML.gif , and from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq120_HTML.gif and (3.6), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ30_HTML.gif
(3.12)

Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq121_HTML.gif . Hence (3.7) is valid.

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq122_HTML.gif , define
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ31_HTML.gif
(3.13)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq123_HTML.gif , and we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ32_HTML.gif
(3.14)
then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq124_HTML.gif , thus http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq125_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq126_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq127_HTML.gif , also http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq128_HTML.gif . So we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq129_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ33_HTML.gif
(3.15)

Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq130_HTML.gif is a Fredholm operator of index zero.

We define a projector http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq131_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq132_HTML.gif . Then we show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq133_HTML.gif defined in (3.2) is a generalized inverse of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq134_HTML.gif .

In fact, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq135_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ34_HTML.gif
(3.16)
and, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq136_HTML.gif , we know
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ35_HTML.gif
(3.17)
In view of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq137_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq138_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq139_HTML.gif , thus
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ36_HTML.gif
(3.18)
This shows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq140_HTML.gif . Also we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ37_HTML.gif
(3.19)

then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq141_HTML.gif . The proof of Lemma 3.1 is finished.

Lemma 3.2.

Under conditions (2.5) and (2.9), there are nonnegative functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq142_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ38_HTML.gif
(3.20)

Proof.

Without loss of generality, we suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq143_HTML.gif . Take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq144_HTML.gif , then there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq145_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ39_HTML.gif
(3.21)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ40_HTML.gif
(3.22)
Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq146_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ41_HTML.gif
(3.23)
Then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ42_HTML.gif
(3.24)
and from (2.5) and (3.21), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ43_HTML.gif
(3.25)
Hence we can take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq147_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq148_HTML.gif , 0, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq149_HTML.gif to replace http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq150_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq151_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq152_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq153_HTML.gif , respectively, in (2.5), and for the convenience omit the bar above http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq154_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq155_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq156_HTML.gif , that is,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ44_HTML.gif
(3.26)

Lemma 3.3.

If assumptions (H1), (H2) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq157_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq158_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq159_HTML.gif hold, then the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq160_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq161_HTML.gif is a bounded subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq162_HTML.gif .

Proof.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq163_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq164_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq165_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq166_HTML.gif , so that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ45_HTML.gif
(3.27)
thus by assumption (H2), there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq167_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq168_HTML.gif . In view of
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ46_HTML.gif
(3.28)
then, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ47_HTML.gif
(3.29)
Again for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq169_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq170_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq171_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq172_HTML.gif thus from Lemma 3.1, we know
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ48_HTML.gif
(3.30)
From (3.29) and (3.30), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ49_HTML.gif
(3.31)
If (2.5) holds, from (3.31), and (3.26), we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ50_HTML.gif
(3.32)
Thus, from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq173_HTML.gif and (3.32), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ51_HTML.gif
(3.33)
From http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq174_HTML.gif , (3.32), and (3.33), one has
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ52_HTML.gif
(3.34)
that is,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ53_HTML.gif
(3.35)
From (3.35) and (3.33), there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq175_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ54_HTML.gif
(3.36)
Thus
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ55_HTML.gif
(3.37)
Again from (2.5), (3.35), and (3.36), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ56_HTML.gif
(3.38)

Then we show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq176_HTML.gif is bounded.

Lemma 3.4.

If assumption (H2) holds, then the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq177_HTML.gif is bounded.

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq178_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq179_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq180_HTML.gif ; therefore,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ57_HTML.gif
(3.39)

From assumption (H2), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq181_HTML.gif , so http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq182_HTML.gif , clearly http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq183_HTML.gif is bounded.

Lemma 3.5.

If the first part of condition (H3) of Theorem 2.2 holds, then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ58_HTML.gif
(3.40)
for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq184_HTML.gif . Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ59_HTML.gif
(3.41)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq185_HTML.gif is the linear isomorphism given by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq186_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq187_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq188_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq189_HTML.gif is bounded.

Proof.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq190_HTML.gif , then we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ60_HTML.gif
(3.42)
or equivalently
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ61_HTML.gif
(3.43)
If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq191_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq192_HTML.gif . Otherwise, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq193_HTML.gif , in view of (3.40), one has
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ62_HTML.gif
(3.44)

which contradicts http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq194_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq195_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq196_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq197_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq198_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq199_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq200_HTML.gif and we obtain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq201_HTML.gif ; therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq202_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq203_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq204_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq205_HTML.gif is bounded.

The proof of Theorem 2.2 is now an easy consequence of the above lemmas and Theorem 2.1.

Proof of Theorem 2.2.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq206_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq207_HTML.gif . By the Ascoli-Arzela theorem, it can be shown that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq208_HTML.gif is compact; thus http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq209_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq210_HTML.gif -compact on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq211_HTML.gif . Then by the above Lemmas, we have the following.

(i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq212_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq213_HTML.gif .

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq214_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq215_HTML.gif .

(iii)Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq216_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq217_HTML.gif as in Lemma 3.5. We know http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq218_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq219_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq220_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq221_HTML.gif . Thus, by the homotopy property of degree, we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ63_HTML.gif
(3.45)
According to definition of degree on a space which is isomorphic to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq222_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq223_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ64_HTML.gif
(3.46)
We have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ65_HTML.gif
(3.47)
and then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ66_HTML.gif
(3.48)

Then by Theorem 2.1, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq224_HTML.gif has at least one solution in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq225_HTML.gif , so that the BVP (1.1), (1.2) has at least one solution in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq226_HTML.gif . The proof is completed.

Remark 3.6.

If the second part of condition (H3) of Theorem 2.2 holds, that is,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ67_HTML.gif
(3.49)
for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq227_HTML.gif , then in Lemma 3.5, we take
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ68_HTML.gif
(3.50)
and exactly as there, we can prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq228_HTML.gif is bounded. Then in the proof of Theorem 2.2, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ69_HTML.gif
(3.51)

since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq229_HTML.gif . The remainder of the proof is the same.

By using the same method as in the proof of Theorem 2.2 and Lemmas 3.1–3.5, we can show Lemma 3.7 and Theorem 2.3.

Lemma 3.7.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq230_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq231_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq232_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq233_HTML.gif is a Fredholm operator of index zero. Furthermore, the linear continuous projector operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq234_HTML.gif can be defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ70_HTML.gif
(3.52)
and the linear operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq235_HTML.gif can be written by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ71_HTML.gif
(3.53)
Furthermore,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ72_HTML.gif
(3.54)
Notice that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ73_HTML.gif
(3.55)

Proof of Theorem 2.3.

Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ74_HTML.gif
(3.56)
Then, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq236_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq237_HTML.gif ; thus http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq238_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq239_HTML.gif ; hence
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ75_HTML.gif
(3.57)
thus, from assumption (H4), there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq240_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq241_HTML.gif and in view of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq242_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ76_HTML.gif
(3.58)
From http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq243_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq244_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq245_HTML.gif . Thus, from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq246_HTML.gif , one has
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ77_HTML.gif
(3.59)
We let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq247_HTML.gif ; hence from (3.58) and (3.59), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ78_HTML.gif
(3.60)

thus, by using the same method as in the proof of Lemmas 3.2 and 3.3, we can prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq248_HTML.gif is bounded too. Similar to the other proof of Lemmas 3.4–3.7 and Theorem 2.2, we can verify Theorem 2.3.

Finally, we give two examples to demonstrate our results.

Example 3.8.

Consider the following boundary value problem:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ79_HTML.gif
(3.61)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq249_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ80_HTML.gif
(3.62)
and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq250_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq251_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq252_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq253_HTML.gif , then we can choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq254_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq255_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq256_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq257_HTML.gif ; thus
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ81_HTML.gif
(3.63)
Since
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ82_HTML.gif
(3.64)

and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq258_HTML.gif has the same sign as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq259_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq260_HTML.gif , we may choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq261_HTML.gif , and then the conditions (H1)–(H3) of Theorem 2.2 are satisfied. Theorem 2.2 implies that BVP (3.61) has at least one solution, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq262_HTML.gif .

Example 3.9.

Consider the following boundary value problem:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ83_HTML.gif
(3.65)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq263_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ84_HTML.gif
(3.66)
and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq264_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq265_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq266_HTML.gif , then we can choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq267_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq268_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq269_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq270_HTML.gif ; thus
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ85_HTML.gif
(3.67)
Similar to Example 3.8, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ86_HTML.gif
(3.68)

and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq271_HTML.gif has the same sign as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq272_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq273_HTML.gif , we may choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq274_HTML.gif , and then all conditions of Theorem 2.3 are satisfied. Theorem 2.3 implies that BVP (3.65) has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq275_HTML.gif .

Declarations

Acknowledgment

This work was sponsored by the National Natural Science Foundation of China (11071205), the NSF of Jiangsu Province Education Department, NFS of Xuzhou Normal University.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Xuzhou Normal University

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Copyright

© Xiaojie Lin. 2011

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