Open Access

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: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq2_HTML.gif , subject to the boundary conditions: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq4_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq7_HTML.gif is a continuous function, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq8_HTML.gif is a nondecreasing function with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq9_HTML.gif . Under the resonance condition https://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:
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq13_HTML.gif is a continuous function, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq14_HTML.gif is a nondecreasing function with https://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
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ4_HTML.gif
(1.4)
Under the condition https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq18_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ5_HTML.gif
(1.5)

However, if https://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., https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq22_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq23_HTML.gif be real Banach spaces, let https://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., https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq25_HTML.gif , the image of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq26_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq27_HTML.gif , the kernel of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq29_HTML.gif ), and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq30_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq31_HTML.gif be continuous projectors such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq32_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq34_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq37_HTML.gif . It follows that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq39_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq40_HTML.gif be an open bounded, subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq41_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq42_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq43_HTML.gif , the map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq44_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq45_HTML.gif -compact on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq46_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq47_HTML.gif is bounded, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq48_HTML.gif is compact. Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq50_HTML.gif of [12].

Theorem 2.1.

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq52_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq53_HTML.gif -compact on https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq55_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq56_HTML.gif ,

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

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

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

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq64_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq65_HTML.gif , we use the norms https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq66_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq67_HTML.gif and denote the norm in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq68_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq69_HTML.gif . We will use the Sobolev space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq70_HTML.gif which may be defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ6_HTML.gif
(2.1)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq71_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq72_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq73_HTML.gif is a linear operator defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ7_HTML.gif
(2.2)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ8_HTML.gif
(2.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq74_HTML.gif be defined as
https://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 https://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): https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq76_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq77_HTML.gif ;

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

Theorem 2.2.

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq81_HTML.gif and constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq82_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq83_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq84_HTML.gif , it holds that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq85_HTML.gif , such that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq86_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq87_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq88_HTML.gif , then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq89_HTML.gif , such that either
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ12_HTML.gif
(2.7)
or else
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq90_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq91_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq92_HTML.gif has at least one solution in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq93_HTML.gif provided that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ14_HTML.gif
(2.9)

Theorem 2.3.

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq95_HTML.gif , such that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq96_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq97_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq98_HTML.gif , then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq99_HTML.gif , such that either
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ16_HTML.gif
(2.11)

or else

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq100_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq101_HTML.gif has at least one solution in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq102_HTML.gif provided that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq103_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq104_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq105_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq107_HTML.gif can be defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ19_HTML.gif
(3.1)
and the linear operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq108_HTML.gif can be written by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ20_HTML.gif
(3.2)
Furthermore,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ22_HTML.gif
(3.4)
Obviously, the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ23_HTML.gif
(3.5)
has a solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq109_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq110_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq111_HTML.gif , if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ24_HTML.gif
(3.6)
which implies that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq112_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq113_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq114_HTML.gif , then from (3.5) we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ26_HTML.gif
(3.8)
According to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq115_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq116_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ27_HTML.gif
(3.9)
then
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ29_HTML.gif
(3.11)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq117_HTML.gif is an arbitrary constant, then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq119_HTML.gif , and from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq120_HTML.gif and (3.6), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ30_HTML.gif
(3.12)

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

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

Thus, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq131_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq132_HTML.gif . Then we show that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq134_HTML.gif .

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

then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq142_HTML.gif satisfying
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq143_HTML.gif . Take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq144_HTML.gif , then there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq145_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ39_HTML.gif
(3.21)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ40_HTML.gif
(3.22)
Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq146_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ41_HTML.gif
(3.23)
Then
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ43_HTML.gif
(3.25)
Hence we can take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq147_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq148_HTML.gif , 0, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq149_HTML.gif to replace https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq150_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq151_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq152_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq154_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq155_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq156_HTML.gif , that is,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq157_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq158_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq159_HTML.gif hold, then the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq160_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq161_HTML.gif is a bounded subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq162_HTML.gif .

Proof.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq163_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq164_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq165_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq166_HTML.gif , so that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq167_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq168_HTML.gif . In view of
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ46_HTML.gif
(3.28)
then, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ47_HTML.gif
(3.29)
Again for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq169_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq170_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq171_HTML.gif , https://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
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ50_HTML.gif
(3.32)
Thus, from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq173_HTML.gif and (3.32), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ51_HTML.gif
(3.33)
From https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ52_HTML.gif
(3.34)
that is,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq175_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ54_HTML.gif
(3.36)
Thus
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ56_HTML.gif
(3.38)

Then we show that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq177_HTML.gif is bounded.

Proof.

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

From assumption (H2), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq181_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq182_HTML.gif , clearly https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ58_HTML.gif
(3.40)
for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq184_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ59_HTML.gif
(3.41)

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

Proof.

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

which contradicts https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq194_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq195_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq196_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq197_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq198_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq199_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq200_HTML.gif and we obtain https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq201_HTML.gif ; therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq202_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq203_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq204_HTML.gif https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq206_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq208_HTML.gif is compact; thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq209_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq210_HTML.gif -compact on https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq212_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq213_HTML.gif .

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

(iii)Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq216_HTML.gif , with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq218_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq219_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq220_HTML.gif https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq222_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq223_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ64_HTML.gif
(3.46)
We have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ65_HTML.gif
(3.47)
and then
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq224_HTML.gif has at least one solution in https://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 https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ67_HTML.gif
(3.49)
for all https://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
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ69_HTML.gif
(3.51)

since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq230_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq231_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq232_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq234_HTML.gif can be defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ70_HTML.gif
(3.52)
and the linear operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq235_HTML.gif can be written by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ71_HTML.gif
(3.53)
Furthermore,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ72_HTML.gif
(3.54)
Notice that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ74_HTML.gif
(3.56)
Then, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq236_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq237_HTML.gif ; thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq238_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq239_HTML.gif ; hence
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq240_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq241_HTML.gif and in view of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq242_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ76_HTML.gif
(3.58)
From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq243_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq244_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq245_HTML.gif . Thus, from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq246_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ77_HTML.gif
(3.59)
We let https://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
https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ79_HTML.gif
(3.61)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq249_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ80_HTML.gif
(3.62)
and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq250_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq251_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq252_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq253_HTML.gif , then we can choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq254_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq255_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq256_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq257_HTML.gif ; thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ81_HTML.gif
(3.63)
Since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ82_HTML.gif
(3.64)

and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq258_HTML.gif has the same sign as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq259_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq260_HTML.gif , we may choose https://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, https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ83_HTML.gif
(3.65)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq263_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ84_HTML.gif
(3.66)
and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq264_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq265_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq266_HTML.gif , then we can choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq267_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq268_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq269_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq270_HTML.gif ; thus
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_Equ86_HTML.gif
(3.68)

and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq271_HTML.gif has the same sign as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq272_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F416416/MediaObjects/13661_2010_Article_38_IEq273_HTML.gif , we may choose https://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 https://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

References

  1. Bicadze AV, Samarskiĭ AA: Some elementary generalizations of linear elliptic boundary value problems. Doklady Akademii Nauk SSSR 1969, 185: 739-740.MathSciNet
  2. Il'pin VA, Moiseev EI: Nonlocal boundary value problems of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential Equations 1987, 23(7):803-810.
  3. Il'cprimein VA, Moiseev EI: Nonlocal boundary value problems of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential Equations 1987, 23(8):979-987.
  4. Karakostas GL, Tsamatos PCh: Sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem. Applied Mathematics Letters 2002, 15(4):401-407. 10.1016/S0893-9659(01)00149-5View ArticleMathSciNetMATH
  5. Du Z, Lin X, Ge W: On a third-order multi-point boundary value problem at resonance. Journal of Mathematical Analysis and Applications 2005, 302(1):217-229. 10.1016/j.jmaa.2004.08.012View ArticleMathSciNetMATH
  6. Du Z, Lin X, Ge W: Some higher-order multi-point boundary value problem at resonance. Journal of Computational and Applied Mathematics 2005, 177(1):55-65. 10.1016/j.cam.2004.08.003View ArticleMathSciNetMATH
  7. Feng W, Webb JRL: Solvability of three point boundary value problems at resonance. Nonlinear Analysis 1997, 30(6):3227-3238. 10.1016/S0362-546X(96)00118-6View ArticleMathSciNetMATH
  8. Liu B: Solvability of multi-point boundary value problem at resonance. II. Applied Mathematics and Computation 2003, 136(2-3):353-377. 10.1016/S0096-3003(02)00050-4View ArticleMathSciNetMATH
  9. Gupta CP: A second order m -point boundary value problem at resonance. Nonlinear Analysis 1995, 24(10):1483-1489. 10.1016/0362-546X(94)00204-UView ArticleMathSciNetMATH
  10. Zhang X, Feng M, Ge W: Existence result of second-order differential equations with integral boundary conditions at resonance. Journal of Mathematical Analysis and Applications 2009, 353(1):311-319. 10.1016/j.jmaa.2008.11.082View ArticleMathSciNetMATH
  11. Du B, Hu X: A new continuation theorem for the existence of solutions to p -Laplacian BVP at resonance. Applied Mathematics and Computation 2009, 208(1):172-176. 10.1016/j.amc.2008.11.041View ArticleMathSciNetMATH
  12. Mawhin J: opological degree and boundary value problems for nonlinear differential equations. In Topological Methods for Ordinary Differential Equations, Lecture Notes in Mathematics. Volume 1537. Edited by: Fitzpertrick PM, Martelli M, Mawhin J, Nussbaum R. Springer, New York, NY, USA; 1991.

Copyright

© Xiaojie Lin. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.