Abstract and Applied Analysis

Volume 2012 (2012), Article ID 215617, 10 pages

doi:10.1155/2012/215617

Research Article

Existence and Multiplicity Results for Nonlinear Differential Equations Depending on a Parameter in Semipositone Case

Hailong Zhu¹ and Shengjun Li²

¹School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China

²College of Information Sciences and Technology, Hainan University, Haikou 570228, China

Received 27 September 2012; Accepted 13 October 2012

Academic Editor: Jifeng Chu

Copyright © 2012 Hailong Zhu and Shengjun Li. 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.

Abstract.

The existence and multiplicity of solutions for second-order differential equations with a parameter are discussed in this paper. We are mainly concerned with the semipositone case. The analysis relies on the nonlinear alternative principle of Leray-Schauder and Krasnosel'skii's fixed point theorem in cones.

1. Introduction

In this paper, we consider the problem of existence, multiplicity, and nonexistence of positive solutions for the following boundary value problem (BVP):

where , is a positive parameter, , and , are real-valued measurable functions defined on and satisfy the following condition:

Here, the symbol denotes the set of functions satisfying the Carathédory conditions on ; that is,

(i) is Lebesgue integrable for each fixed , and (ii) is continuous for a.e. .

Due to a wide range of applications in physics and engineering, second-order boundary value problems have been extensively investigated by numerous researchers in recent years. For a small sample of such work, we refer the reader to [1–18] and the references therein. When , , of , in [11, 18], by using Krasnosel'skii's fixed point theorem, the existence and multiplicity of positive solutions are established to the periodic boundary value problem:

where .

In [8], Graef et al. consider the second-order periodic boundary value problem:

where is continuous and is continuous and for . Under different combinations of superlinearity and sublinearity of the function , various existence, multiplicity, and nonexistence results for positive solutions are derived in terms of different value of via Krasnosel'skii's fixed point theorem.

Hao et al. [9] use the Global continuation theorem, fixed point index theory, and approximate method to study the following periodic boundary value problems:

where and .

In [10], by using the fixed point index theory, He et al. study the existence and multiplicity of positive solutions to BVP . Motivated by the above works, we establish the results of existence, multiplicity, and nonexistence of positive solutions for BVP via Leray-Schauder alternative principle and Krasnosel'skii's fixed point in the semipositone case, that is, for some . Notice that we do not need for any and , which is an essential condition of [9, 10].

The main result of the present paper is summarized as follows.

Theorem 1.1. Assume that

Then, there exist such that has no positive solution for and at least two positive solutions for .

Remark 1.2. The main result above is a generalization of [9, Theorem 1.2] and [10, Theorem 1.2] and some other known results, in which and must be zero, besides is positive.

The remaining part of the paper is organized as follows. Some preliminary results will be given in Section 2. In Section 3, existence results are obtained using a nonlinear alternative of Leray-Schauder and fixed point theorem in cones when is large enough; the proof of Theorem 1.1 is also given.

2. Preliminaries and Lemmas

In this section, we present some preliminary results which will be needed in subsequent sections. Denote by and the solutions of the corresponding homogeneous equation:

under the initial conditions

Lemma 2.1 (see [2, Theorem 2.4], [10, Lemma 2.1]). Assume that holds and . Then for the solution of the BVP

the formula

holds, where

and .

Lemma 2.2 (see [2, Theorem 2.5], [10, Lemma 2.2]). Under condition , the Green's function of the BVP (2.3) is positive, that is, for .

Remark 2.3. We denote

Thus, and . In this paper, we use to denote the unique periodic solution of (2.3) with , that is, . Obviously, .

Remark 2.4. If , then the Green's function of the boundary value problem (2.3) has the form

It is obvious that for , and a direct calculation shows that

In the obtention of the second periodic solution of , we need the following well-known fixed point theorem of compression and expansion of cones [19].

Lemma 2.5 (see Krasnosel'skii [19]). Let be a Banach space and a cone. Assume that , are open subsets of with , , and let

be a continuous and compact operator such that either

(i), and , , or (ii), and , .

Then has a fixed point in .

In the applications below, we take with the supremum norm and define

where .

One may readily verify that is a cone in . Finally, we define an operator by

for and , where is continuous and is the Green function defined above.

Lemma 2.6 (see [12, Lemmas 2.2, 2.3], [13, Lemma 2.4]). is well defined and maps into . Moreover, is continuous and completely continuous.

3. Proof of Theorem 1.1

In this section we establish the existence, multiplicity, and nonexistence of positive solutions to the periodic boundary problem . The first existence result is based on the following nonlinear alternative of Leray-Schauder, which can be found in [15].

Lemma 3.1. Assume is a relatively compact subset of a convex set in a normed space . Let be a compact map with . Then one of the following two conclusions holds:

(I) has at least one fixed point in .(II)There exist and such that .

Since we are mainly interested in the semipositone case, without loss of generality, we may assume that satisfies the following.

(F1) There is a constant such that for all and let . Besides, we introduce the following assumption on .(F2) there exists a continuous, nonnegative function on such that

that is,

and is nondecreasing in .

Theorem 3.2. Suppose satisfies (F1) and (F2). Suppose further that

(F3) there exists such that

where and are as in Section 2.

Then has at least one positive periodic solution with .

Proof. The existence is proved using the Leray-Schauder alternative principle. Consider the following equation:

where . Problem (3.4) is equivalent to the following fixed point problem in :

where denotes the operator defined by (2.11), with replaced by .

We claim that any fixed point of (3.5) for any must satisfy .

Then we have from condition (F2), for all ,

Therefore,

This is a contradiction to the condition (F3). From this claim, the nonlinear alternative of Leray-Schauder guarantees that (3.5) (with ) has a fixed point, denoted by , that is,

Using Lemma 2.5 and condition (F3), for all , we have

that is,

Let

It is easy to see that is a solution of which satisfies . Thus, the proof of Theorem 3.2 is completed.

Theorem 3.3. Suppose that conditions (F1)–(F3) hold. In addition, it is assumed that the following two conditions are satisfied.

(F4) There exists a continuous, nonnegative function on such that

that is,

and is nondecreasing in .

(F5) There exists a positive number such that

Then, besides the periodic solution constructed in Theorem 3.2, has another positive periodic solution with .

Proof. As in the proof of Theorem 3.2, we only need to show that (3.8) has a periodic solution with with and .

Let and the cone in in Section 2. Let and be balls in . The operator is defined by (2.11), with replaced by . Note that any satisfies , thus is well defined.

First we have for . In fact, if , then . Now the estimate can be obtained almost following the same ideas in proving (3.7). We omit the details here.

Next we show that for . To see this, let , then and ; it follows from conditions (F4) and (F5) that, for ,

Now Lemma 2.5 guarantees that has a fixed point , thus .

Finally, will be the another desired positive periodic solution of . We omit the details because they are much similar to that in the proof of Theorem 3.2.

Now we are in a position to present the proof of Theorem 1.1.

Proof of Theorem 1.1. Consider be an eigenfunction satisfying

corresponding to the principal eigenvalue . Let be a positive solution of . Multiplying (3.16) by and by , and subtracting we obtain

Since and , there exist positive numbers , , , and such that and

with . Let the positive number be defined by

Then

Thus, there exists a , for satisfying . (3.17) cannot hold, and hence has no positive solution for .

Note that the sublinearity of near , we can construct a suitable in (F2) which satisfies . This means that there exists satisfying (3.3) with being large enough. There also exists satisfying (3.14). Let . Thus, with the help of Theorems 3.2 and 3.3, has at least two positive solution for . This completes the proof of the theorem.

Example 3.4. Let the nonlinearity in be

with , is a continuous function for all and is a real coefficient polynomial function which has zero constant term. Then Theorem 1.1 is valid.

Proof. In this case, with the function , it is easy to verify

Then the conclusion follows from Theorem 1.1 that there exists such that has no positive solution for and at least two positive solutions for .

Acknowledgments

H. Zhu was supported by the NSF of the Educational Bureau of Anhui province (nos. KJ2012B002, KJ2012B004, and 1208085QA11). S. Li was supported by the NSF of China (no. 11161017).

References

R. P. Agarwal, D. O'Regan, and B. Yan, “Multiple positive solutions of singular Dirichlet second-order boundary-value problems with derivative dependence,” Journal of Dynamical and Control Systems, vol. 15, no. 1, pp. 1–26, 2009.

F. M. Atici and G. S. Guseinov, “On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions,” Journal of Computational and Applied Mathematics, vol. 132, no. 2, pp. 341–356, 2001.

A. Cabada and J. Cid, “Existence and multiplicity of solutions for a periodic Hill's equation with parametric dependence and singularities,” Abstract and Applied Analysis, vol. 2011, Article ID 545264, 19 pages, 2011.

A. Cabada and J. J. Nieto, “Extremal solutions of second order nonlinear periodic boundary value problems,” Applied Mathematics and Computation, vol. 40, no. 2, pp. 135–145, 1990.

J. Chu, P. J. Torres, and M. Zhang, “Periodic solutions of second order non-autonomous singular dynamical systems,” Journal of Differential Equations, vol. 239, no. 1, pp. 196–212, 2007.

J. Chu and M. Li, “Positive periodic solutions of Hill's equations with singular nonlinear perturbations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 1, pp. 276–286, 2008.

A. Feichtinger, I. Rachůnková, S. Staněk, and E. Weinmüller, “Periodic BVPs in ODEs with time singularities,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 2058–2070, 2011.

J. R. Graef, L. Kong, and H. Wang, “Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem,” Journal of Differential Equations, vol. 245, no. 5, pp. 1185–1197, 2008.

X. Hao, L. Liu, and Y. Wu, “Existence and multiplicity results for nonlinear periodic boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 9-10, pp. 3635–3642, 2010.

T. He, F. Yang, C. Chen, and S. Peng, “Existence and multiplicity of positive solutions for nonlinear boundary value problems with a parameter,” Computers & Mathematics with Applications, vol. 61, no. 11, pp. 3355–3363, 2011.

D. Jiang, “On the existence of positive solutions to second order periodic BVPs,” Acta Mathematica Sinica, vol. 18, pp. 31–35, 1998.

D. Jiang, J. Chu, D. O'Regan, and R. P. Agarwal, “Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces,” Journal of Mathematical Analysis and Applications, vol. 286, no. 2, pp. 563–576, 2003.

D. Jiang, J. Chu, and M. Zhang, “Multiplicity of positive periodic solutions to superlinear repulsive singular equations,” Journal of Differential Equations, vol. 211, no. 2, pp. 282–302, 2005.

R. Ma, J. Xu, and X. Han, “Global bifurcation of positive solutions of a second-order periodic boundary value problem with indefinite weight,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 10, pp. 3379–3385, 2011.

D. O'Regan, Existence Theory for Nonlinear Ordinary Differential Equations, Kluwer Academic Publishers, Dodrecht, The Netherlands, 1997.

I. Rachůnková, “Existence of two positive solution of a singular nonlinear periodic boundary value problem,” Journal of Computational and Applied Mathematics, vol. 113, no. 1-2, pp. 24–34, 2000.

P. J. Torres, “Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem,” Journal of Differential Equations, vol. 190, no. 2, pp. 643–662, 2003.

Z. Zhang and J. Wang, “On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations,” Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 99–107, 2003.

M. Krasnosel'skii, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, The Netherlands, 1964.