Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis

Volume 2014, Article ID 968792, 6 pages

http://dx.doi.org/10.1155/2014/968792
Research Article

Parameter Dependence of Positive Solutions for Second-Order Singular Neumann Boundary Value Problems with Impulsive Effects

Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

Received 21 March 2014; Accepted 7 May 2014; Published 19 May 2014

Academic Editor: Juan J. Nieto

Copyright © 2014 Xuemei Zhang. 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 author considers the Neumann boundary value problem = and establishes the dependence results of the solution on the parameter , which cover equations without impulsive effects and are compared with some recent results by Nieto and O’Regan.

1. Introduction

Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. The theory and applications of impulsive differential equations have been emerging as an important area of investigation in recent years [16]. There is a vast literature on the existence of solutions by using different methods such as bifurcation theory [7, 8], fixed point theorems in cones [912], the method of lower and upper solutions [13, 14], and the theory of critical point theory and variational methods [1519]. We remark that on second-order impulsive differential equations with parameter only a few results have been obtained; see, for instance, [9, 2022]. To the best of our knowledge, these papers only consider the existence of positive solutions. However, the corresponding results for the dependence of the solution on the parameter for second-order impulsive differential equations are not investigated until now. In this paper, we try to solve this kind of problem.

Consider the Neumann boundary value problems where is a constant, is a parameter, , is a nonnegative measurable function on , on any open subinterval in , which may be singular at and/or , (where is fixed positive integer) are fixed points with , and , where and represent the right-hand limit and left-hand limit of at , respectively. In addition, , , and satisfy(H1) ;(H2) , , where , .

Some special cases of (1) have been investigated. For example, Nieto and O’Regan [17] studied problem (1) with and for . By using variational methods and critical point theory, the authors proved the existence of solutions of problem (1).

For ease of exposition, we set

Our main results are as follows.

Theorem 1. Assume that and hold. Then the following two conclusions hold:(H3)if , , and , , , then for every problem (1) has a positive solution satisfying ;(H4)if , , and , , , then for every problem (1) has a positive solution satisfying .

Remark 2. Assume that and hold. Furthermore, suppose that or , , in or or , , in . Then the conclusions of Theorem 1 also hold.

Remark 3. It follows from the conditions of Theorem 1 that we develop some ideas of Guo and Lakshmikantham essentially; for detail, see Theorem in [23].

Remark 4. For simplicity we only consider Neumann boundary conditions since all the results obtained in this paper can also be adapted with minor changes to the other boundary conditions.

2. Preliminaries

Let , , and Then is a real Banach space with norm where , .

A function is called a solution of problem (1) if it satisfies (1).

In our main results, we will make use of the following lemmas.

Lemma 5. If and hold, then problem (1) has a unique solution given by where

Proof. The proof is similar to that of Lemma 2.4 in [24].

By (6), we can prove that has the following property:

Define a cone in by where It is easy to see is a closed convex cone of .

Define an operator by

From (10), we know that is a solution of problem (1) if and only if is a fixed point of operator .

Lemma 6. Suppose that and hold. Then and is completely continuous.

Proof. For , it follows from (7) and (10) that It follows from (7), (10), and (11) that Thus, .

Next, by similar arguments of Lemmas 5 and 6 [12] one can prove that is completely continuous. So it is omitted, and the theorem is proved.

To obtain positive solutions of problem (1), the following fixed point theorem in cones is fundamental, which can be found in [23, page 94].

Lemma 7. Let be a cone in a real Banach space . Assume , are bounded open sets in with , . If is completely continuous such that either (a) , , and , , or(b) , , and , ,then has at least one fixed point in .

3. Proofs of the Main Results

For convenience we introduce some notations where , , and is a constant.

Proof of Theorem 1. We need only prove this theorem under condition since the proof is similar when holds, provided the proper adjustments are made.

If , , then there exist and such that where satisfies Then for we have where It follows from (17) that If , , then there exist and such that where satisfies

Let . Thus, when we have and then we get This yields

Hence, for given condition of Lemma 7 is satisfied of operator , which implies that has a fixed point in .

It remains to prove as . In fact, if not, there would exist a number and a sequence such that Furthermore, the sequence contains a subsequence that converges into a number , where . For simplicity, suppose that itself converges into   .

If , then for sufficiently large ( ), and therefore which contradicts .

If , then for sufficiently large   ( ), and therefore it follows from that for any there exists such that and hence it follows from (10) that Since is arbitrary, we have in contradiction to . Therefore, as and the proof is complete.

Remark 8. Comparing with Nieto and O’Regan [17], the main features of this paper are as follows.(i)The parameter is considered.(ii)The parameter dependence of the solution is available.(iii) , not for .

4. An Example

To illustrate how our main results can be used in practice we present an example.

Example 9. Consider the following boundary value problems Evidently, is the trivial solution of problem (29).

Conclusion. Problem (29) has at least one positive solution for any .

Proof. Problem (29) can be regarded as a problem of the form (1), where

It follows from the definition of , , and that and hold, and is singular at and . By calculating, we have Then, the condition of Theorem 1 holds. Hence, by Theorem 1, the conclusion follows, and the proof is complete.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are indebted to the referee’s suggestions. These have greatly improved this paper. This work is sponsored by the Project NSFC (11301178) and the Fundamental Research Funds for the Central Universities (2014MS58).

References

  1. M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  2. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Publishing, River Edge, NJ, USA, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  3. V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  4. S. Gao, L. Chen, J. J. Nieto, and A. Torres, “Analysis of a delayed epidemic model with pulse vaccination and saturation incidence,” Vaccine, vol. 24, pp. 6037–6045, 2006. View at Publisher · View at Google Scholar
  5. A. F. B. A. Prado, “Bi-impulsive control to build a satellite constellation,” Nonlinear Dynamics and Systems Theory, vol. 5, no. 2, pp. 169–175, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. W. Li, Q. Li, X. Liu, and F. Cui, “Principal and nonprincipal solutions of impulsive dynamic equations with applications,” The Journal of Applied Analysis and Computation, vol. 2, no. 4, pp. 431–440, 2012. View at Google Scholar · View at MathSciNet
  7. Y. Liu and D. O'Regan, “Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 1769–1775, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. R. Ma, B. Yang, and Z. Wang, “Positive periodic solutions of first-order delay differential equations with impulses,” Applied Mathematics and Computation, vol. 219, no. 11, pp. 6074–6083, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. X. Hao, L. Liu, and Y. Wu, “Positive solutions for second order impulsive differential equations with integral boundary conditions,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 101–111, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. Feng and D. Xie, “Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations,” Journal of Computational and Applied Mathematics, vol. 223, no. 1, pp. 438–448, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. X. Zhang, M. Feng, and W. Ge, “Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 233, no. 8, pp. 1915–1926, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Yan, “Existence of positive periodic solutions of impulsive functional differential equations with two parameters,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 854–868, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. X. Liu and D. Guo, “Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space,” Computers & Mathematics with Applications, vol. 38, no. 3-4, pp. 213–223, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. D. Guo, “Extremal solutions for nth-order impulsive integro-differential equations on the half-line in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 65, no. 3, pp. 677–696, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. D. Guo, “Variational approach to a class of impulsive differential equations,” Boundary Value Problems, vol. 2014, article 37, 2014. View at Publisher · View at Google Scholar
  16. J. Zhou and Y. Li, “Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 71, no. 7-8, pp. 2856–2865, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. J. Xiao, J. J. Nieto, and Z. Luo, “Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 426–432, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Y. Tian and W. Ge, “Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 1, pp. 277–287, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Feng, X. Li, and C. Xue, “Multiple positive solutions for impulsive singular boundary value problems with integral boundary conditions,” International Journal of Open Problems in Computer Science and Mathematics, vol. 2, no. 4, pp. 546–561, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. J. Sun, H. Chen, and L. Yang, “The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 73, no. 2, pp. 440–449, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. P. Ning, Q. Huan, and W. Ding, “Existence result for impulsive differential equations with integral boundary conditions,” Abstract and Applied Analysis, vol. 2013, Article ID 134691, 9 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, NewYork, NY, USA, 1988. View at MathSciNet
  24. X. Liu and Y. Li, “Positive solutions for Neumann boundary value problems of second-order impulsive differential equations in Banach spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 401923, 14 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet