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Boundary Value Problems

, 2011:416416 | Cite as

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

Open Access
Research Article
Part of the following topical collections:
  1. Nonlocal Boundary Value Problems

Abstract

This paper deals with the existence of solutions for the following differential equation: Open image in new window , Open image in new window , subject to the boundary conditions: Open image in new window , Open image in new window , where Open image in new window , Open image in new window , Open image in new window is a continuous function, Open image in new window is a nondecreasing function with Open image in new window . Under the resonance condition Open image in new window , 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.

Keywords

Linear Operator Existence Result Fixed Point Theorem Fredholm Operator Nonlocal Boundary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

In this paper, we consider the following second-order differential equation:
subject to the boundary conditions:

where Open image in new window , Open image in new window , Open image in new window is a continuous function, Open image in new window is a nondecreasing function with Open image in new window . 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

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:
Under the condition Open image in new window (i.e., nonresonance case), they used Krasnosel'skii's fixed point theorem to show that the operator equation Open image in new window has at least one fixed point, where operator Open image in new window is defined by

However, if Open image in new window (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 [5, 6, 7, 8, 9, 10, 11].

The purpose of this paper is to study the existence of solutions for nonlocal BVP (1.1), (1.2) at resonance case (i.e., Open image in new window ) and establish some existence results under nonlinear growth restriction of Open image in new window . 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 Open image in new window , Open image in new window be real Banach spaces, let Open image in new window be a linear operator which is Fredholm map of index zero (i.e., Open image in new window , the image of Open image in new window , Open image in new window , the kernel of Open image in new window are finite dimensional with the same dimension as the Open image in new window ), and let Open image in new window , Open image in new window be continuous projectors such that Open image in new window = Open image in new window , Open image in new window = Open image in new window and Open image in new window , Open image in new window . It follows that Open image in new window is invertible; we denote the inverse by Open image in new window . Let Open image in new window be an open bounded, subset of Open image in new window such that Open image in new window Open image in new window , the map Open image in new window is said to be Open image in new window -compact on Open image in new window if Open image in new window is bounded, and Open image in new window is compact. Let Open image in new window be a linear isomorphism.

The theorem we use in the following is Theorem Open image in new window of [12].

Theorem 2.1.

Let Open image in new window be a Fredholm operator of index zero, and let Open image in new window be Open image in new window -compact on Open image in new window . Assume that the following conditions are satisfied:

(i) Open image in new window for every Open image in new window ,

(ii) Open image in new window for every Open image in new window ,

(iii) Open image in new window ,

where Open image in new window is a projection with Open image in new window . Then the equation Open image in new window has at least one solution in Open image in new window .

Then BVP (1.1), (1.2) is Open image in new window .

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

case (i): Open image in new window , Open image in new window ;

case (ii): Open image in new window , Open image in new window .

Theorem 2.2.

Let Open image in new window be a continuous function and assume that

(H1) there exist functions Open image in new window and constant Open image in new window such that for all Open image in new window , Open image in new window , it holds that
(H3) there exists a constant Open image in new window , such that either
Then BVP (1.1), (1.2) with Open image in new window , Open image in new window , and Open image in new window has at least one solution in Open image in new window provided that

Theorem 2.3.

Let Open image in new window be a continuous function. Assume that assumption (H1) of Theorem 2.2 is satisfied, and

(H5) there exists a constant Open image in new window , such that either

or else

Then BVP (1.1), (1.2) with Open image in new window , and Open image in new window has at least one solution in Open image in new window provided that

3. Proof of Theorems 2.2 and 2.3

We first prove Theorem 2.2 via the following Lemmas.

Lemma 3.1.

If Open image in new window , Open image in new window , and Open image in new window , then Open image in new window is a Fredholm operator of index zero. Furthermore, the linear continuous projector operator Open image in new window can be defined by
and the linear operator Open image in new window can be written by
Furthermore,

Proof.

It is clear that
Obviously, the problem
which implies that
In fact, if (3.5) has solution Open image in new window satisfying Open image in new window , Open image in new window , then from (3.5) we have
On the other hand, if (3.6) holds, setting
where Open image in new window is an arbitrary constant, then Open image in new window is a solution of (3.5), and Open image in new window , and from Open image in new window and (3.6), we have

Then Open image in new window . Hence (3.7) is valid.

Thus, Open image in new window is a Fredholm operator of index zero.

We define a projector Open image in new window by Open image in new window . Then we show that Open image in new window defined in (3.2) is a generalized inverse of Open image in new window .

This shows that Open image in new window . Also we have

then Open image in new window . The proof of Lemma 3.1 is finished.

Lemma 3.2.

Under conditions (2.5) and (2.9), there are nonnegative functions Open image in new window satisfying

Proof.

Without loss of generality, we suppose that Open image in new window . Take Open image in new window , then there exists Open image in new window such that
and from (2.5) and (3.21), we have

Lemma 3.3.

If assumptions (H1), (H2) and Open image in new window , Open image in new window , and Open image in new window hold, then the set Open image in new window for some Open image in new window is a bounded subset of Open image in new window .

Proof.

thus by assumption (H2), there exists Open image in new window , such that Open image in new window . In view of
then, we have
From (3.29) and (3.30), we have
If (2.5) holds, from (3.31), and (3.26), we obtain
Thus, from Open image in new window and (3.32), we have
From Open image in new window , (3.32), and (3.33), one has
that is,
From (3.35) and (3.33), there exists Open image in new window , such that
Again from (2.5), (3.35), and (3.36), we have

Then we show that Open image in new window is bounded.

Lemma 3.4.

If assumption (H2) holds, then the set Open image in new window is bounded.

Proof.

From assumption (H2), Open image in new window , so Open image in new window , clearly Open image in new window is bounded.

Lemma 3.5.

If the first part of condition (H3) of Theorem 2.2 holds, then

where Open image in new window is the linear isomorphism given by Open image in new window , for all Open image in new window , Open image in new window . Then Open image in new window is bounded.

Proof.

Suppose that Open image in new window , then we obtain
or equivalently
If Open image in new window , then Open image in new window . Otherwise, if Open image in new window , in view of (3.40), one has

which contradicts Open image in new window . Then Open image in new window = Open image in new window Open image in new window Open image in new window Open image in new window Open image in new window and we obtain Open image in new window ; therefore, Open image in new window Open image in new window Open image in new window Open image in new window 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 Open image in new window such that Open image in new window . By the Ascoli-Arzela theorem, it can be shown that Open image in new window is compact; thus Open image in new window is Open image in new window -compact on Open image in new window . Then by the above Lemmas, we have the following.

(i) Open image in new window for every Open image in new window .

(ii) Open image in new window for every Open image in new window .

According to definition of degree on a space which is isomorphic to Open image in new window , Open image in new window , and
and then

Then by Theorem 2.1, Open image in new window has at least one solution in Open image in new window , so that the BVP (1.1), (1.2) has at least one solution in Open image in new window . The proof is completed.

Remark 3.6.

If the second part of condition (H3) of Theorem 2.2 holds, that is,
for all Open image in new window , then in Lemma 3.5, we take
and exactly as there, we can prove that Open image in new window is bounded. Then in the proof of Theorem 2.2, we have

since Open image in new window . 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 Open image in new window , Open image in new window , and Open image in new window , then Open image in new window is a Fredholm operator of index zero. Furthermore, the linear continuous projector operator Open image in new window can be defined by
and the linear operator Open image in new window can be written by
Furthermore,
Notice that

Proof of Theorem 2.3.

thus, from assumption (H4), there exists Open image in new window , such that Open image in new window and in view of Open image in new window , we obtain
We let Open image in new window ; hence from (3.58) and (3.59), we have

thus, by using the same method as in the proof of Lemmas 3.2 and 3.3, we can prove that Open image in new window 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:

and Open image in new window has the same sign as Open image in new window when Open image in new window , we may choose Open image in new window , 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, Open image in new window .

Example 3.9.

Consider the following boundary value problem:
Similar to Example 3.8, we have

and Open image in new window has the same sign as Open image in new window when Open image in new window , we may choose Open image in new window , and then all conditions of Theorem 2.3 are satisfied. Theorem 2.3 implies that BVP (3.65) has at least one solution Open image in new window .

Notes

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.

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Copyright information

© 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.

Authors and Affiliations

  1. 1.School of Mathematical SciencesXuzhou Normal UniversityXuzhou, JiangsuChina

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