Abstract and Applied Analysis

Volume 2013 (2013), Article ID 527082, 6 pages

http://dx.doi.org/10.1155/2013/527082

Research Article

Multiple Solutions for a Second-Order Impulsive Sturm-Liouville Equation

Jingli Xie1,2 and Zhiguo Luo²

¹College of Mathematics and Statistics, Jishou University, Jishou, Hunan 416000, China

²Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China

Received 29 March 2013; Revised 27 May 2013; Accepted 18 June 2013

Academic Editor: Gennaro Infante

Copyright © 2013 Jingli Xie and Zhiguo Luo. 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.

We study the existence of solutions to a boundary value problem of a second-order impulsive Sturm-Liouville equation with a control parameter . By employing some existing critical point theorems, we find the range of the control parameter in which the boundary value problem admits at least three solutions. It is also shown that, under certain conditions, there exists an interval of the control parameter in which the boundary value problem possesses infinitely many solutions. Some examples are given to demonstrate the main results in this paper.

1. Introduction

The aim of this paper is to investigate the existence of multiple solutions of the following Neumann boundary value problem with impulsive Sturm-Liouville type equation:

where , , , , , and is a positive real parameter. , where and denote the right and left limits, respectively, of at , .

In the last few years, the existence of multiple solutions to Neumann problems has been widely investigated [1–5]. But little research has focused on the existence of multiple solutions for impulsive Sturm-Liouville equations whose right-hand side nonlinear term is depending on a parameter . Processes subject to sudden changes in their states are modeled by the impulsive differential equations and have been investigated in various fields of science and technology. In the motion of spacecraft, one has to consider instantaneous impulses at a position with jump discontinuities in velocity, but no change in the position [6, 7]. This motivates us to consider problem (1).

In the literature, tools employed to establish the existence of solutions of impulsive differential equations include fixed point theorems, the upper and lower solutions method, the degree theory, critical point theory, and variational methods. See, for example, [8–20]. In this paper, our focus is on the existence of solutions of problem (1) with being the parameter. The problem is first transformed into the existence of critical points of some variational structure. Then with the help of critical point theory, results on the existence of at least three solutions and infinitely many solutions are established.

The rest of this paper is organized as follows. In Section 2 we present some preliminary results. Our main results and their proofs are given in Section 3.

2. Preliminaries

Throughout we assume that and satisfy

Take and define

For the norm in , we put

We have the following relation.

Lemma 1. Let . Then

Proof. For any , it follows from the mean value theorem that

for some . Hence, for , using Hölder inequality and (2), we have

Define a functional as

where

with

Note that is Fréchet differentiable at any , and for any , we have

Lemma 2. If is a weak solution of problem (1), then is a classical solution of problem (1).

Proof. For any function , we get

By the regularity theory, the weak solution is a classical solution of problem (1).

Next we show that a critical point of the functional is a solution of problem (1).

Lemma 3. If is a critical point of , then is a classical solution of problem (1).

Proof. If is a critical point of , by (12) and Lemma 2, we have is a classical solution of problem (1)

For , we define

3. Main Results

3.1. Existence of At Least Three Solutions

In this section we derive conditions under which problem (1) admits at least three solutions. For this purpose, we introduce the following assumptions.

(H1) Assume that there exists a positive constant such that for each

(H2) Assume that there exist positive constants , , and , such that

Let and let with given in (16). Clearly, . For constants , , we define

Theorem 4. Assume that (H1), (H2) are satisfied. If there exist two positive constants , satisfying , and

then, for each , problem (1) admits at least three solutions.

Proof. By Lemma 3, it suffices to show that the functional defined in (8) has at least three critical points. We prove this by verifying the conditions given in [21, Theorem  3.2]. Note that defined in (9) is a nonnegative Gâteaux differentiable, coercive, and sequentially weakly lower semicontinuous functional and its Gâteaux derivative admits a continuous inverse on . Moreover, defined in (10) is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Set

Note that . It then follows from (H1) that

Then, we have

For satisfying , by Lemma 1, one has

which implies that

Hence

So, one has

Making use of (19), we obtain

By (H1) and (H2), when , we easily obtain that the functional is coercive.

Thus all conditions in [21, Theorem  3.2] are verified, and hence for each , the functional admits at least three critical points. Consequently, problem (1) admits at least three solutions.

Example 5. Consider the boundary value problem

where

Here, , , , and . Note that (H1), (H2) are satisfied. Moreover, we have , , , , and

Choose , . Direct calculations give

Therefore it follows from Theorem 4 that (28) admits at least three solutions in provided that .

3.2. Existence of Infinitely Many Solutions

In this section, we derive some conditions under which problem (1) admits infinitely many distinct solutions. To this end, we need the following assumption.

(H3) Assume that

Set

Theorem 6. Assume that (H3) is satisfied. If

holds, then for each , problem (1) has an unbounded sequence of solutions in .

Proof. We apply [22, Theorem  2.1] to show that the functional defined in (8) has an unbounded sequence of critical points.

We first show that . Let be a sequence of positive numbers such that as and

For any positive integer , we let . For satisfying , similar to the proof of Theorem 4, one can show that

which implies that

Note that ; thus we have

which, together with (15), gives us

This shows that . For any fixed , it follows from [22, Theorem  2.1] that either has a global minimum or there is a sequence of critical points (local minima) of such that .

Next we show that the functional has no global minimum for . Since , we can choose a constant such that, for each ,

Thus, there exists such that

Define as follows:

This yields

Then

Note that . Thus the functional has no lower bound and hence it has no global minimum and the proof is complete.

Let

Theorem 7. Assume that (H3) is satisfied. If

holds, then, for each , problem (1) has a sequence of nonzero solutions in , which weakly converges to .

Proof. The proof is similar to that of Theorem 6 by showing that and is not a local minimum of the functional .

Example 8. Consider the boundary value problem

where .

Here, , , , and . Hence we have , , , , and

so (H3) is satisfied. Moreover, we have

Therefore, condition (34) holds and Theorem 6 applies: for , problem (47) admits an unbounded sequence of solutions in .

Example 9. Consider the boundary value problem

where

In this example, , , , and . Hence we have , , , , and

Hence, one has

Therefore (46) holds. Owing to Theorem 7, when problem (50) admits a sequence of pairwise distinct classical solutions strongly converging at 0 in .

Acknowledgments

The authors are very grateful to the referees for their valuable comments and suggestions, which greatly improved the presentation of this paper. This work is supported by Hunan Provincial Natural Science Foundation of China (no. 11JJ3012).

References

G. Bonanno and G. D'Aguì, “A Neumann boundary value problem for the Sturm-Liouville equation,” Applied Mathematics and Computation, vol. 208, no. 2, pp. 318–327, 2009.

G. Dai, “Infinitely many solutions for a Neumann-type differential inclusion problem involving the p(x)-Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 6, pp. 2297–2305, 2009.

Y. Sun, Y. J. Cho, and D. O'Regan, “Positive solutions for singular second order Neumann boundary value problems via a cone fixed point theorem,” Applied Mathematics and Computation, vol. 210, no. 1, pp. 80–86, 2009.

Y. Tian and W. Ge, “The existence of solutions for a second-order discrete Neumann problem with a p-Laplacian,” Journal of Applied Mathematics and Computing, vol. 26, no. 1-2, pp. 333–340, 2008.

Y.-W. Zhang and H.-X. Li, “Positive solutions of a second-order Neumann boundary value problem with a parameter,” Bulletin of the Australian Mathematical Society, vol. 86, no. 2, pp. 244–253, 2012.

D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, Harlow, UK, 1993.

V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific Publishing, Teaneck, NJ, USA, 1989.

R. P. Agarwal and D. O'Regan, “A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 433–439, 2005.

R. P. Agarwal, D. Franco, and D. O'Regan, “Singular boundary value problems for first and second order impulsive differential equations,” Aequationes Mathematicae, vol. 69, no. 1-2, pp. 83–96, 2005.

G. Bonanno, “A critical point theorem via the Ekeland variational principle,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 5, pp. 2992–3007, 2012.

J. Chu and J. J. Nieto, “Impulsive periodic solutions of first-order singular differential equations,” Bulletin of the London Mathematical Society, vol. 40, no. 1, pp. 143–150, 2008.

C. De Coster and P. Habets, “Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results,” in Non-Linear Analysis and Boundary Value Problems for Ordinary Differential Equations (Udine), F. Zanolin, Ed., vol. 371 of CISM Courses and Lectures, pp. 1–78, Springer, Vienna, Austria, 1996.

M. Galewski and S. Głab, “On the discrete boundary value problem for anisotropic equation,” Journal of Mathematical Analysis and Applications, vol. 386, no. 2, pp. 956–965, 2012.

J. J. Nieto and D. O'Regan, “Variational approach to impulsive differential equations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 680–690, 2009.

D. Qian and X. Li, “Periodic solutions for ordinary differential equations with sublinear impulsive effects,” Journal of Mathematical Analysis and Applications, vol. 303, no. 1, pp. 288–303, 2005.

J. Shen and W. Wang, “Impulsive boundary value problems with nonlinear boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 4055–4062, 2008.

Y. Tian and W. Ge, “Applications of variational methods to boundary-value problem for impulsive differential equations,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 51, no. 2, pp. 509–527, 2008.

H. Zhang and Z. Li, “Variational approach to impulsive differential equations with periodic boundary conditions,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 67–78, 2010.

X. Zhang and X. Tang, “Subharmonic solutions for a class of non-quadratic second order Hamiltonian systems,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 113–130, 2012.

Z. Zhang and R. Yuan, “An application of variational methods to Dirichlet boundary value problem with impulses,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 155–162, 2010.

G. Bonanno and P. Candito, “Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities,” Journal of Differential Equations, vol. 244, no. 12, pp. 3031–3059, 2008.

G. Bonanno and G. M. Bisci, “Infinitely many solutions for a boundary value problem with discontinuous nonlinearities,” Boundary Value Problems, vol. 2009, Article ID 670675, 20 pages, 2009.