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

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## 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## 1. Introduction

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.

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

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

Theorem 2.3.

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

or else

## 3. Proof of Theorems 2.2 and 2.3

We first prove Theorem 2.2 via the following Lemmas.

Lemma 3.1.

Proof.

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 .

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

Lemma 3.2.

Proof.

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.

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.

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.

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 .

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.

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.

Proof of Theorem 2.3.

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.

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.

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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