Journal of Applied MathematicsVolume 2012 (2012), Article ID 160891, 23 pagesdoi:10.1155/2012/160891

Research Article

Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term

Yuefeng Han,¹ Xinguang Zhang,² Lishan Liu,3,4 and Yonghong Wu⁴

¹College of International Economics and Trade, Jilin University of Finance and Economics, Jilin, Changchun 130117, China²School of Mathematical and Informational Sciences, Yantai University, Shandong, Yantai 264005, China³School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China⁴Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845 , Australia

Received 13 July 2011; Accepted 3 November 2011

Academic Editor: Andrew Pickering

Copyright © 2012 Yuefeng Han et al. 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.

By employing a well-known fixed point theorem, we establish the existence of multiple positive solutions for the following fourth-order singular differential equation , with and , where denotes the linear operator , and . This equation is viewed as a perturbation of the fourth-order Sturm-Liouville problem, where the perturbed term only satisfies the global Carathéodory conditions, which implies that the perturbed effect of on is quite large so that the nonlinearity can tend to negative infinity at some singular points.

1. Introduction

In this paper, we consider the existence of multiple positive solutions for the following fourth-order singular Sturm-Liouville boundary value problem involving a perturbed term where and , and denotes the linear operator and and . The perturbed term, , satisfies global Carathéodory’s conditions.

Equation (1.1) arises from many branches of applied mathematics and physics; for details, see [1–16]. It mainly describes the deformation of an elastic beam for ; for example, under the Lidstone boundary condition, problem (1.1) is used to model such phenomena as the deflection of an elastic beam simply supported at the endpoints; see [1, 3, 5, 7–11]. Also, if the boundary condition of (1.1) is a Focal boundary condition, then it describes the deflection of an elastic beam having both end-points fixed, or having one end simply supported and the other end clamped with sliding clamps. In addition, the derivative in is the bending moment term which represents the bending effect, see [1, 3, 5, 7–11, 13, 14, 16]. A brief discussion of the physical interpretation under some boundary conditions associated with the linear beam equation can be found in Zill and Cullen [17].

Recently, for the case where the nonlinearity does not contain the bending moment term , Ma and Wang [1] studied the existence of positive solutions for (1.1) subject to boundary conditions and if is superlinear or sublinear. In the case where contains the bending moment term and under the particular boundary conditions, the authors of papers [9, 12] studied the existence of positive solutions for (1.1) when satisfies the following growth condition: where is enough small. But most of the above works were done on base of the assumptions that the nonlinearity is nonnegative and has no any singularity. In recent years, one found that the fourth-order changing-sign nonlinear problems also occur to the classical model for the elastic beam fixed at both ends, especially in the medium span or large span bridge constructions, this implies that it is necessary and quite natural to study fourth-order changing-sign boundary value problems.

In this paper, we focus on the particularly difficult and interesting situation, when (1.1) is singularly perturbed, so that the nonlinearity is allowed to change sign, even may tend to negative infinity. This problem has essential difference from those unperturbed problems of [1–16]. We quote in the sequel some papers from the relevant bibliography devoted to this subject. In [18], Loud considered the existence of -periodic solutions for a first-order perturbed system of ordinary differential equations by employing the so-called bifurcation function Moreover, the author of [18] also considered the case when is not a simple zero of , and the existence of -periodic solutions of the above problem is associated with the existence of the roots of a certain quadratic equation. Recently, by using the exponential dichotomies and contraction mapping principle, Xia et al. [19] established some sufficient conditions of the existence and uniqueness of almost periodic solution for a forced perturbed system with piecewise constant argument. The other works, such as Khanmamedov [20], Wu and Gan [21], Makarenkov and Nistri [22], Liu and Yang [23], Clavero et al. [24], and Cui and Geng [25], are rich sources for application of perturbed problems.

Our main tool used for the analysis here is known as Guo-Krasnoselskii’s fixed point theorem, for the convenience of the reader, we now state it as follows.

Lemma 1.1 (see, [26]). Let be a real Banach space, a cone. Assume are two bounded open subsets of with , and let be a completely continuous operator such that either , and , or, and .Then, has a fixed point in ).

2. Preliminaries and Lemmas

The following definition introduces global Carathéodory’s conditions imposed on a map.

Definition 2.1. A map is said to satisfy global Crathéodory’s conditions if the following conditions hold: (i)for each , the mapping is Lebesgue measurable; (ii)for a.e. , the mapping is continuous on ;(iii)there exists a such that, for a.e. and , we have

The following lemmas play an important role in proving our main results.

Lemma 2.2 (see, [27]). Let and be the solutions of the linear problems respectively. Then, (i) is strictly increasing on and on ; (ii) is strictly decreasing on and on .

Set by Liouville’s formula, one can easily show constant .

As [27], we define Green’s function for the BVP: by then we have the following lemma.

Lemma 2.3. For any , we have where

Proof. It follows from the monotonicity of and that the right-hand side of (2.6) holds. For the left hand side, by the monotonicity of and , we have The proof is completed.

Also, it is well known the Green function for the boundary value problem is where . Let clearly, Now, we define an integral operator by and, then, by (2.9), we have

In order to obtain existence of positive solutions to problem (1.1), we will consider the existence of positive solutions to the following modified problem

Lemma 2.4. Let . Then, we can transform (1.1) into (2.15). Moreover, if is a solution of problem (2.15), then the function is a positive solution of problem (1.1).

Proof. It follows from (2.9) that , put and into (1.1), we can transform (1.1) into (2.15).Conversely, if is a solution of (2.15), let , we have , thus is a solution of (1.1). The proof of Lemma 2.4 is completed.

In the rest of the paper, we always suppose that the following assumptions hold. (B1) is continuous and satisfies (B2) is continuous. (B3) satisfies global Crathéodory’s condition and

Remark 2.5. It follows from (B1), (B3) and from the monotonicity of that there exists such that So for convenience, in the rest of this paper, we define serval constants as follows:

Lemma 2.6. Assume (B3) is satisfied. Then, the boundary value problem has a unique solution which satisfies

Proof. First, solves the BVP (2.20), and it is the unique solution of the BVP (2.20), since with boundary conditions only has a trivial solution. Finally, it follows from (2.6) and (B3) that (2.22) holds.

Define a modified function for any by We consider the following approximating problem

Lemma 2.7. If for any is a positive solution of the BVP (2.25), then is a positive solution of the singular perturbed differential equation (1.1).

Proof. In fact, if is a positive solution of the BVP (2.25) such that for any , then, from (2.25) and the definition of , we have Let , then , which implies that Thus, (2.26) becomes (2.15), that is, is a positive solution of the differential equation (2.15). By Lemma 2.4, is a positive solution of the singular perturbed differential equation (1.1). This completes the proof of Lemma 2.7.

Thus, the BVP (2.25) is equivalent to the integral equation Hence, we will look for fixed points , for the mapping defined on by

The basic space used in this paper is , where is a real number set. Obviously, the space is a Banach space if it is endowed with the norm as follows: for any . Let where is defined by (2.7), then is a cone of .

Lemma 2.8. Assume that (B1)–(B3) hold. Then, is well defined. Furthermore, is a completely continuous operator.

Proof. For any fixed , there exists a constant such that . And then, On the other hand, since satisfies global Carathéodory’s condition, we have . Accordingly, in (2.29) is continuous on , and, by (2.32), where This implies that the operator is well defined.Next, for any , by (2.6), we have On the other hand, from (2.6), we also have So which yields that .At the end, according to the Ascoli-Arzela Theorem, using standard arguments, one can show is a completely continuous operator.

3. Main Results

Theorem 3.1. Suppose (B1)–(B3) hold. In addition, assume that the following conditions are satisfied. (S1)There exists a constant such that for any , where and are defined by (2.7) and (2.19), respectively.(S2) There exists a constant such that, for any ,, where are defined by (2.19).(S3)Then, the singular perturbed differential equation (1.1) has at least two positive solutions , and there exist two positive constants such that , for any .

Proof. Let . Then, for any , we have It follows from (S1) that Therefore, On the other hand, let and . Then, for any , noticing and (2.22), we have So by (3.7), for any , we have It follows from (S2), (3.8), and (2.6) that, for any , So we have Next, let us choose such that Then, for the above , by (S3), there exists such that, for any and for any , Let take then .Now let and . Then, for any , we have which implies that By Lemma 1.1, has two fixed points such that .It follows from that As for (3.18), we also find a positive constant such that Let , then By Lemma 2.7, we know that the singular perturbed differential equation (1.1) has at least two positive solutions satisfying for some positive constants . The proof of Theorem 3.1 is completed.

Theorem 3.2. Suppose (B1)–(B3) hold. In addition, assume that the following conditions are satisfied. (S4) There exists a constant such that, for any , where and are defined by (2.7) and (2.19), respectively.(S5) There exists a constant such that, for any , where are defined by (2.19) and is defined by (S4). (S6) Then, the singular perturbed differential equation (1.1) has at least two positive solutions , and there exist two positive constants such that , for any .

Proof. Firstly, let . Then, for any , by (2.22), we have So, by (3.26), for any , we have It follows from (S4), (3.27), and (2.6) that, for any , Therefore, we have Next, by (S5), we have and Let . Then, for any , we have It follows from (S5) that Therefore, On the other hand, choose a large enough real number such that From (S6), there exists such that, for any , Take then . Let , for any and for any , we have So, for any , by (3.35), (3.37), we have Thus, By Lemma 1.1, has two fixed points such that .Noticing that we have As for (3.41), we also can find a positive constant such that Let , then By Lemma 2.7, we know that the singular perturbed differential equation (1.1) has at least two positive solutions satisfying for some positive constants . The proof of Theorem 3.2 is completed.

An example Consider the following singular perturbed boundary value problem where Then, the BVP (3.45) has at least two positive solutions and such that

Proof. In fact, let then The corresponding Green’s functions can be written by and, for , we have and

Now, take , then we have On the other hand, since then (B1)–(B3) are satisfied.

Choose , then and, for any , So the condition (S1) is satisfied.

On the other hand, we take , then , and, for any ,, we have this implies that the condition (S2) holds. Next, we have Thus, (S3) also holds. By Theorem 3.1, the BVP (3.45) has at least two positive solutions and . Since thus

Acknowledgment

The authors were supported financially by the National Natural Science Foundation of China (11071141) and the Natural Science Foundation of Shandong Province of China (ZR2010AM017).

References

R. Ma and H. Wang, “On the existence of positive solutions of fourth-order ordinary differential equations,” Applicable Analysis, vol. 59, no. 1–4, pp. 225–231, 1995.

J. Schröder, “Fourth order two-point boundary value problems; estimates by two-sided bounds,” Nonlinear Analysis: Theory, Methods & Applications, vol. 8, no. 2, pp. 107–114, 1984.

R. P. Agarwal and Y. M. Chow, “Iterative methods for a fourth order boundary value problem,” Journal of Computational and Applied Mathematics, vol. 10, no. 2, pp. 203–217, 1984.

R. P. Agarwal, “On fourth order boundary value problems arising in beam analysis,” Differential and Integral Equations, vol. 2, no. 1, pp. 91–110, 1989.

R. Ma, “Positive solutions of fourth-order two-point boundary value problems,” Annals of Differential Equations, vol. 15, no. 3, pp. 305–313, 1999.

D. O'Regan, “Solvability of some fourth (and higher) order singular boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 161, no. 1, pp. 78–116, 1991.

C. P. Gupta, “Existence and uniqueness results for the bending of an elastic beam equation at resonance,” Journal of Mathematical Analysis and Applications, vol. 135, no. 1, pp. 208–225, 1988.

C. P. Gupta, “Existence and uniqueness theorems for the bending of an elastic beam equation,” Applicable Analysis, vol. 26, no. 4, pp. 289–304, 1988.

Y. S. Yang, “Fourth-order two-point boundary value problems,” Proceedings of the American Mathematical Society, vol. 104, no. 1, pp. 175–180, 1988.

A. R. Aftabizadeh, “Existence and uniqueness theorems for fourth-order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 116, no. 2, pp. 415–426, 1986.

X. G. Zhang and L. L. Liu, “Eigenvalue of fourth-order m-point boundary value problem with derivatives,” Computers and Mathematics with Applications, vol. 56, no. 1, pp. 172–185, 2008.

M. A. del Pino and R. F. Manásevich, “Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition,” Proceedings of the American Mathematical Society, vol. 112, no. 1, pp. 81–86, 1991.

B. Liu, “Positive solutions of fourth-order two point boundary value problems,” Applied Mathematics and Computation, vol. 148, no. 2, pp. 407–420, 2004.

Y. Sun, “Necessary and sufficient condition for the existence of positive solutions of a coupled system for elastic beam equations,” Journal of Mathematical Analysis and Applications, vol. 357, no. 4, pp. 77–88, 2009.

J. R. Graef, C. Qian, and B. Yang, “A three point boundary value problem for nonlinear fourth order differential equations,” Journal of Mathematical Analysis and Applications, vol. 287, no. 1, pp. 217–233, 2003.

L. Liu, X. Zhang, and Y. Wu, “Positive solutions of fourth-order nonlinear singular Sturm-Liouville eigenvalue problems,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1212–1224, 2007.

D. G. Zill and M. R. Cullen, Differential Equations with Boundary-Value Problems, Brooks/Cole, 5th edition, 2001.

W. S. Loud, “Periodic solutions of a perturbed autonomous system,” Annals of Mathematics Second Series, vol. 70, pp. 490–529, 1959.

Y. Xia, Z. Huang, and M. Han, “Existence of almost periodic solutions for forced perturbed systems with piecewise constant argument,” Journal of Mathematical Analysis and Applications, vol. 333, no. 2, pp. 798–816, 2007.

A. Kh. Khanmamedov, “On the existence of a global attractor for the wave equation with nonlinear strong damping perturbed by nonmonotone term,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3372–3385, 2008.

S. Wu and S. Gan, “Errors of linear multistep methods for singularly perturbed Volterra delay-integro-differential equations,” Mathematics and Computers in Simulation, vol. 79, no. 10, pp. 3148–3159, 2009.

O. Makarenkov and P. Nistri, “Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1401–1417, 2008.

D. Liu and Y. Yang, “Doubly perturbed neutral diffusion processes with Markovian switching and Poisson jumps,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1141–1146, 2010.

C. Clavero, J. L. Gracia, and F. J. Lisbona, “An almost third order finite difference scheme for singularly perturbed reaction-diffusion systems,” Journal of Computational and Applied Mathematics, vol. 234, no. 8, pp. 2501–2515, 2010.

M. Cui and F. Geng, “A computational method for solving third-order singularly perturbed boundary-value problems,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 896–903, 2008.

D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.

L. H. Erbe and R. M. Mathsen, “Positive solutions for singular nonlinear boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 46, no. 7, pp. 979–986, 2001.