## Boundary Value Problems

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# Solvability for p-Laplacian boundary value problem at resonance on the half-line

Boundary Value Problems20132013:207

DOI: 10.1186/1687-2770-2013-207

Accepted: 23 August 2013

Published: 11 September 2013

## Abstract

The existence of solutions for p-Laplacian boundary value problem at resonance on the half-line is investigated. Our analysis relies on constructing the suitable Banach space, defining appropriate operators and using the extension of Mawhin’s continuation theorem. An example is given to illustrate our main result.

MSC:70K30, 34B10, 34B15.

### Keywords

p-Laplacian resonance half-line multi-point boundary value problem continuation theorem

## 1 Introduction

A boundary value problem is said to be a resonance one if the corresponding homogeneous boundary value problem has a non-trivial solution. Resonance problems can be expressed as an abstract equation , where L is a noninvertible operator. When L is linear, Mawhin’s continuation theorem [1] is an effective tool in finding solutions for these problems, see [210] and references cited therein. But it does not work when L is nonlinear, for instance, p-Laplacian operator. In order to solve this problem, Ge and Ren [11] proved a continuation theorem for the abstract equation when L is a noninvertible nonlinear operator and used it to study the existence of solutions for the boundary value problems with a p-Laplacian:

where , , . is nonlinear when .

As far as the boundary value problems on unbounded domain are concerned, there are many excellent results, see [1215] and references cited therein.

To the best of our knowledge, there are few papers that study the p-Laplacian boundary value problem at resonance on the half-line. In this paper, we investigate the existence of solutions for the boundary value problem
(1.1)

where , , .

In order to obtain our main results, we always suppose that the following conditions hold.

(H1) , , .

(H2) is continuous, , and for any , there exists a nonnegative function such that

## 2 Preliminaries

For convenience, we introduce some notations and a theorem. For more details, see [11].

Definition 2.1 [11]

Let X and Y be two Banach spaces with the norms , , respectively. A continuous operator is said to be quasi-linear if
1. (i)

is a closed subset of Y,

2. (ii)

is linearly homeomorphic to , , where domM denote the domain of the operator M.

Let and be the complement space of in X, then . On the other hand, suppose that is a subspace of Y, and that is the complement of in Y, i.e., . Let and be two projectors and an open and bounded set with the origin .

Definition 2.2 [11]

Suppose that , is a continuous operator. Denote by N. Let . is said to be M-compact in if there exist a vector subspace of Y satisfying and an operator being continuous and compact such that for ,
1. (a)

,

2. (b)

,

3. (c)

is the zero operator and ,

4. (d)

.

Theorem 2.1 [11]

Let X and Y be two Banach spaces with the norms , , respectively, and an open and bounded nonempty set. Suppose that

is a quasi-linear operator and , M-compact. In addition, if the following conditions hold:

(C1) , , ,

(C2) ,

then the abstract equation has at least one solution in , where , is a homeomorphism with .

## 3 Main result

Let with norm , where . with norm . Then and are Banach spaces.

Define operators and as follows:
where

Then the boundary value problem (1.1) is equivalent to .

Obviously,

It is clear that KerM is linearly homeomorphic to , and is closed. So, M is a quasi-linear operator.

Define , as

where , . We can easily obtain that , are projectors. Set , .

Define an operator :

where , . By (H1) and (H2), we get that is continuous.

Lemma 3.1 [15]

is compact if and are both equicontinuous on any compact intervals of and equiconvergent at infinity.

Lemma 3.2 is compact.

Proof Let be nonempty and bounded. There exists a constant such that , . It follows from (H2) that there exists a nonnegative function such that
For any , , , , we have
Since are equicontinuous on , we get that are equicontinuous on .
Let
Then
(3.1)
For , , , we have

It follows from the absolute continuity of integral that are equicontinuous on . Since is uniformly continuous on , by (3.1), we can obtain that are equicontinuous on .

For , since
and is uniformly continuous on , for any , there exists a constant such that if , then
(3.2)
Since
(3.3)
there exists a constant such that if , then
(3.4)
For , by (3.2), (3.3) and (3.4), we have
and

By Lemma 3.1, we get that is compact. The proof is completed. □

In the spaces X and Y, the origin . In the following sections, we denote the origin by 0.

Lemma 3.3 Let be nonempty, open and bounded. Then is M-compact in .

Proof By (H2), we know that is continuous. Obviously, . For , since is a zero operator, we get . For , . So, we have . It is clear that
and , . means that and , thus,
For , we have

These, together with Lemma 3.2, mean that is M-compact in . The proof is completed. □

In order to obtain our main results, we need the following additional conditions.

(H3) There exist nonnegative functions , , with and such that
(H4) There exists a constant such that if , then one of the following inequalities holds:
Lemma 3.4 Assume that (H3) and (H4) hold. The set

is bounded in X.

Proof If , then , i.e., . By (H4), there exists such that . It follows from that
Considering (H3), we have
(3.5)
Since , we get
Thus,
(3.6)
By (3.5), (3.6) and (H3), we get
So,

This, together with (3.6), means that is bounded. The proof is completed. □

Lemma 3.5 Assume that (H4) holds. The set

is bounded in X.

Proof means that , and , i.e.,

By (H4), we get that . So, is bounded. The proof is completed. □

Theorem 3.1 Suppose that (H1)-(H4) hold. Then problem (1.1) has at least one solution.

Proof Let , where . It follows from the definition of and that , , and , .

Define a homeomorphism as . If for , take the homotopy
For , we have . Then
Obviously, , . For , , if , we have
A contradiction with , . If , , take

and the contradiction follows analogously. So, we obtain , , .

By the homotopy of degree, we get that

By Theorem 2.1, we can get that has at least one solution in . The proof is completed. □

## 4 Example

Let us consider the following boundary value problem at resonance
(4.1)

where , , .

Corresponding to problem (1.1), we have , .

Take , , , . By simple calculation, we can get that conditions (H1)-(H4) hold. By Theorem 3.1, we obtain that problem (4.1) has at least one solution.

## Declarations

### Acknowledgements

This work is supported by the National Science Foundation of China (11171088) and the Natural Science Foundation of Hebei Province (A2013208108). The author is grateful to anonymous referees for their constructive comments and suggestions, which led to the improvement of the original manuscript.

## Authors’ Affiliations

(1)
College of Sciences, Hebei University of Science and Technology

## References

1. Mawhin J NSFCBMS Regional Conference Series in Mathematics. In Topological Degree Methods in Nonlinear Boundary Value Problems. Am. Math. Soc., Providence; 1979.View Article
2. Mawhin J: Resonance problems for some non-autonomous ordinary differential equations. Lecture Notes in Mathematics 2065. In Stability and Bifurcation Theory for Non-Autonomous Differential Equatons. Edited by: Johnson R, Pera MP. Springer, Berlin; 2013:103-184.View Article
3. Feng W, Webb JRL: Solvability of m -point boundary value problems with nonlinear growth. J. Math. Anal. Appl. 1997, 212: 467-480. 10.1006/jmaa.1997.5520
4. Ma R: Existence results of a m -point boundary value problem at resonance. J. Math. Anal. Appl. 2004, 294: 147-157. 10.1016/j.jmaa.2004.02.005
5. Zhang X, Feng M, Ge W: Existence result of second-order differential equations with integral boundary conditions at resonance. J. Math. Anal. Appl. 2009, 353: 311-319. 10.1016/j.jmaa.2008.11.082
6. Du Z, Lin X, Ge W: Some higher-order multi-point boundary value problem at resonance. J. Comput. Appl. Math. 2005, 177: 55-65. 10.1016/j.cam.2004.08.003
7. Kosmatov N: A boundary value problem of fractional order at resonance. Electron. J. Differ. Equ. 2010, 135: 1-10.
8. Jiang W: The existence of solutions to boundary value problems of fractional differential equations at resonance. Nonlinear Anal. TMA 2011, 74: 1987-1994. 10.1016/j.na.2010.11.005
9. Jiang W: Solvability for a coupled system of fractional differential equations at resonance. Nonlinear Anal., Real World Appl. 2012, 13: 2285-2292. 10.1016/j.nonrwa.2012.01.023
10. Jiang W:Solvability of conjugate boundary-value problems at resonance. Electron. J. Differ. Equ. 2012, 114: 1-10.MathSciNet
11. Ge W, Ren J: An extension of Mawhin’s continuation theorem and its application to boundary value problems with a p -Laplacian. Nonlinear Anal. 2004, 58: 477-488. 10.1016/j.na.2004.01.007
12. Liu Y, Li D, Fang M: Solvability for second-order m -point boundary value problems at resonance on the half-line. Electron. J. Differ. Equ. 2009., 2009: Article ID 13
13. Liu Y: Boundary value problem for second order differential equations on unbounded domain. Acta Anal. Funct. Appl. 2002, 4(3):211-216. (in Chinese)
14. Agarwal RP, O’Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht; 2001.
15. Kosmatov N: Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal. 2008, 68: 2158-2171. 10.1016/j.na.2007.01.038