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

- Xiaojie Lin
^{1}Email author

**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: , , subject to the boundary conditions: , , where , , is a continuous function, is a nondecreasing function with . Under the resonance condition , 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

## 1. Introduction

where , , is a continuous function, is a nondecreasing function with . 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 (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 [5–11].

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

The theorem we use in the following is Theorem of [12].

Theorem 2.1.

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

where is a projection with . Then the equation has at least one solution in .

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

Theorem 2.2.

Let be a continuous function and assume that

Theorem 2.3.

Let 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.

Thus, is a Fredholm operator of index zero.

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

then . The proof of Lemma 3.1 is finished.

Lemma 3.2.

Proof.

Lemma 3.3.

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

Proof.

Lemma 3.4.

If assumption (H2) holds, then the set is bounded.

Proof.

From assumption (H2), , so , clearly is bounded.

Lemma 3.5.

where is the linear isomorphism given by , for all , . Then is bounded.

Proof.

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

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

Remark 3.6.

since . 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 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 has the same sign as when , we may choose , 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, .

Example 3.9.

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

## 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

## References

- Bicadze AV, Samarskiĭ AA: Some elementary generalizations of linear elliptic boundary value problems.
*Doklady Akademii Nauk SSSR*1969, 185: 739-740.MathSciNetGoogle Scholar - 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.Google Scholar - 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.Google Scholar - 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 ArticleMathSciNetMATHGoogle Scholar - 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 ArticleMathSciNetMATHGoogle Scholar - 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 ArticleMathSciNetMATHGoogle Scholar - 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 ArticleMathSciNetMATHGoogle Scholar - 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 ArticleMathSciNetMATHGoogle Scholar - 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 ArticleMathSciNetMATHGoogle Scholar - 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 ArticleMathSciNetMATHGoogle Scholar - 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 ArticleMathSciNetMATHGoogle Scholar - 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.Google Scholar

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