AIMS Mathematics, 2020, 5(6): 6189-6210. doi: 10.3934/math.2020398.

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem

1 College of Mathematics, Hunan University, Changsha, Hunan 410082, PR China
2 Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada

In this article, we consider the existence of upper and lower solutions to a second-order random impulsive differential equation. We first study the solution form of the corresponding linear impulsive system of the second-order random impulsive differential equation. Based on the form of the solution, we define the resolvent operator. Then, we prove that the fixed point of this operator is the solution to the equation. Finally, we construct the sum of two monotonic iterative sequences and prove that they are convergent. Thus, we conclude that the system has upper and lower solutions.
  Figure/Table
  Supplementary
  Article Metrics

Keywords random impulse differential equation; upper and lower solutions; monotonic iterative sequences; boundary value problem

Citation: Zihan Li, Xiao-Bao Shu, Fei Xu. The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem. AIMS Mathematics, 2020, 5(6): 6189-6210. doi: 10.3934/math.2020398

References

  • 1. X. Yan, D. Peng, X. Lv, et al. Recent progess in impulsive contral systems, Spinger., 155 (2007), 244-268.
  • 2. I. M. Stamova, G. T. Stamov, Lyapunov-Razumikhin method for impulsive functional differential equations and applications to the population dynamics, J. Comput. Appl. Math., 130 (2001), 163-171.    
  • 3. C. Li, J. Sun, R. Sun, Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects, J. Franklin. I., 347 (2010), 1186-1198.    
  • 4. C. Li, J. Shi, J. Sun, Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks, Nonlinear Anal. Theor., 74 (2011), 3099-3111.    
  • 5. L. Shen, J. Sun, Q. Wu, Controllability of linear impulsive stochastic systems in Hilbert spaces, Automatica., 49 (2013), 1026-1030.    
  • 6. Y. Tang, X. Xing, H. R. Karimi, et al. Tracking control of networked multiagent systems under new characterizations of impulses and its Applications in robotic systems, IEEE Trans. Ind. Electron., 63 (2016), 1299-1307.
  • 7. D. Guo, X. Liu, Multiple positive solutions of boundary-value problems for impulsive differential equations, Nonlinear Anal., 25 (1995), 327-337.    
  • 8. T. Jankowski, Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments, Appl. Math. Comput., 197 (2008), 179-189.
  • 9. Y. H. Lee, X. Liu, Study of singular boundary value problems for second order impulsive differential equations, J. Math. Anal. Appl., 331 (2007), 159-176.    
  • 10. M. Yao, A. Zhao, J. Yan, Anti-Periodic boundary value problems of second-order impulsive differential equations, Comput. Math. Appl., 59 (2010), 3617-3629.    
  • 11. Y. Zhao, H. Chen, Multiplicity of solutions to two-point boundary value problems for second-order impulsive differential equations, Appl. Math. Comput., 206 (2008), 925-931.
  • 12. S. Wu, X. Guo, S. Lin, Existence and uniqueness of solutions to random impulsive differential systems, Acta Math. Appl. Sin., 22 (2006), 627-632.    
  • 13. S. Wu, X. Meng, Boundedness of nonlinear differantial systems with impulsive effect on random moments, Acta Math. Appl. Sin., 20 (2004), 147-154.
  • 14. A. Anguraj, S. Wu, A. Vinodkumar, The existence and exponential stability of semilinear functional differential equations with random impulses under nonuniqueness, Nonlinear Anal. Theory Methods Appl., 74 (2011), 331-342.    
  • 15. K. Z. Wang, A new existence result for nonlinear first-order anti-periodic boundary value problems, Appl. Math. Lett., 21 (2008), 1149-1154.    
  • 16. Z. G. Luo, J. H. Shen, J. J. Nieto, Antiperiodic boundary value problem for first-order impulsive ordinary differential equations, Comput. Math. Appl., 49 (2005), 253-261.    
  • 17. W. Ding, Y. P. Xing, M. A. Han, Anti-periodic boundary value problems for first order impulsive functional differential equations, Appl. Math. Comput., 186 (2007), 45-53.
  • 18. B. Ahmad, J. J. Nieto, Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions, Nonlinear Anal., 69 (2008), 3291-3298.    
  • 19. S. Li, L. Shu, X. B. Shu, et al. Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays, Stochastic., 27 (2018), 857-872.
  • 20. S. Zhang, J. Sun, Stability analysis of second-order differential systems with Erlang distribution random impulses, Adv. Difference Equ., 4 (2013), 10.
  • 21. A. Vinodkumar, K. Malar, M. Gowrisankar, et al. Existence, uniqueness and stability of random impulsive fractional differential equations, J. Acta Math. Sci., 36 (2016), 428-442.
  • 22. V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore., 1989.
  • 23. Y. K. Chang, Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos. Soliton Fract., 33 (2007), 1601-1609.    
  • 24. E. K. Lee, Y. H. Lee, Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations, Appl. Math. Comput., 158 (2004), 745-759.
  • 25. S. Zhang, W. Jiang, The existence and exponential stability of random impulsive fractional differential equations, Adv. Difference Equ., 404 (2018), 17.
  • 26. A. Anguraj, A. Vinodkumar, Existence, uniqueness and stability results of random impulsive semilinear differential systems, Nonlinear Anal. Hybrid Syst., 4 (2010), 475-483.
  • 27. P. Niu, X. B. Shu, Y. Li, The existence and hyers-ulam stability for second order random impulsive differential equations, Dyn. Syst. Appl., 28 (2019), 673-690.
  • 28. Y. Guo, X. B. Shu, Y. Li, et al. The existence and Hyers-Ulam stability of solution for impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1 < β < 2, Bound. Value Probl., 18 (2018).
  • 29. R. A. Admas, Sbolev Spaces, Academic Press., 1975.
  • 30. J. Sun, D. Guo, Functional methods for nonlinear ordinary differential equations, Shandong science and technology press., 4, (2005).
  • 31. X. B. Shu, Y. J. Shi, A study on the mild solution of impulsive fractional evolution equations, Appl. Math. Comput., 273 (2016), 465-476.
  • 32. X. B. Shu, Y. Z. Lai, Y. M. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal., 74 (2011), 2003-2011.    
  • 33. X. B. Shu, F. Xu, Upper and lower solution method for fractional evolution equations with order 1 < α < 2, J. Korean Math. Soc., 51 (2014), 1123-1139.
  • 34. R. P. Agarwal, M. Benchohra, S. Hamani, et al. Upper and lower solutions method for impulsive differential equations involving the Caputo fractional derivative, Mem. Differ. Equ. Math. Phys., 53 (2011), 1-12.
  • 35. P. Cheng, F. Q. Deng, F. Q. Yao, Almost sure exponential stability and stochastic stabilization of stochastic differential systems with impulsive effects, Nonlinear Anal.: Hybrid Syst., 27 (2018), 13-28.    
  • 36. L. G. Xu, Z. L. Dai, D. H. He, Exponential ultimate boundedness of impulsive stochastic delay differential equations, Appl. Math. Lett., 85 (2018), 70-76.    
  • 37. L. G. Xu, The pth moment exponential ultimate boundedness of impulsive stochastic differential systems, Appl. Math. Lett., 42 (2015), 22-29.    
  • 38. D. H. He, L. G. Xu, Ultimate boundedness of nonautonomous dynamical complex networks under impulsive control, IEEE Trans. Circuits Syst. II., 62 (2015), 997-1001.    
  • 39. L. X. Shu, X. B. Shu, Q. X. Zhu, et al. Existence and exponential stability of mild solutions for second-order neutral stochastic functional differential equation with random impulses, J. Appl. Anal. Comput., 2020.
  • 40. X. B. Shu, L. X. Shu, F. Xu, A new study on the mild solution for impulsive fractional evolution equations, Dynam. Systems Appl., 2020.

 

Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved