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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Existence and Exponential Stability for Random Impulsive Differential Evolution Equations

Discontinuity, Nonlinearity, and Complexity 11(4) (2022) 599--612 | DOI:10.5890/DNC.2022.12.003

K. Ravikumar, K. Banupriya, S. Varshini, K. Ramkumar

Department of Mathematics, PSG College of Arts and Science, Coimbatore-641014, India

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Abstract

The paper is devoted to the study of existence and exponential stability of mild solutions of random impulsive differential evolution equations. The results are obtained using Leray-Schauder alternative fixed point theorem. Furthermore Exponential stability of the mild solution is established with certain sufficient conditions. An application is provided to illustrate the theory.

References

  1. [1]  Shu, X.B. and Shi, Y. (2016), A study on the mild solution of impulsive fractional evolution equations, Appl. Math. Comput., 273, 465-476.
  2. [2]  Suganya, S., Mallika Arjunan, M., and Trujillo, J.J. (2015), Existence results for an impulsive fractional integrodifferential equation with state-dependent delay, Appl. Math. Comput., 266, 54-69.
  3. [3]  Zhang, S. and Jiang, W. (2014), Exponential stability for a stochastic delay neutral network with impulses., Adv. Differ. Equ, 2014, 250.
  4. [4]  Bianov, D.D. and Simeonov, P.S. (1987), Differentiability of solutions of systems with impulsive effect with respect to initial data and parameter, Bull. Inst. Math. Acad. Sci., 15, 251-269.
  5. [5]  Lakshmikantham, V., Bianov, D.D., and Simeonov, P.S. (1989), Theory of impulsive differential equations, World scientific, Singapore.
  6. [6]  Abro, K.A. and Atangana, A. (2020), Role of non-integer and integer order differentiations on the relaxation phenomena of viscoelastic fluid, Physica Scripta, 95(3), 035228.
  7. [7]  Atangana, A. and Qureshi, S. (2019), Modeling attractors of chaotic dynamical systems with fractal fractional operators, Chaos, Solitons $\&$ Fractals, 123, 320-337.
  8. [8]  Hernndez, E. (2004), Existence results for partial neutral functional integrodifferential equations with unbounded delay, Journal of mathematical Analysis and Applications, 292, 194-210.
  9. [9]  Hernndez, E., Rabello, M., and Henriquez, H.R. (2007), Existence of solutions for impulsive partial neutral functional differential equations, Journal of Mathematical Analysis and Applications, 331, 1135-1158.
  10. [10]  Itho, S. (1979), Random fixed point theorems with an application to random differential equations in Banach spaces, Journal of Mathematical Analysis and Applications, 67, 261-273.
  11. [11]  Gowrisankar, M., Mohankumar, P., and Vinodkumar, A. (2014), Stability results of random impulsive semilinear differential equations, Actams, 34B(4), 1055-1071.
  12. [12]  Anguraj, A. and Vinodkumar, A. (2010), Existence, uniqueness and stability results of random impulsive semilinear differential systems, Nonlinear Analysis: Hybrid systems, 4, 475-483.
  13. [13]  Vinodhkumar, A. (2014), Existence results for random impulsive neutral differential inclusions with lower semicontinuous, Int. J. Computing Science and Mathematics, 5.
  14. [14]  Radhakrishnan, B. and Balachandran, K. (2011), Controllability of impulsive neutral functional evolution integrodifferential systems with infinite delay, Nonlinear Analysis: Hybrid Systems, 5 655-670.
  15. [15]  Radhakrishnan, B., Tamilarasi, M., and Anukokila, P. (2018), Existence, uniqueness and stability results for semilinear integrodifferential non-local evolution equations with random impulse, Filomat, 32(19), 6615-6626.
  16. [16]  Zhang, S. and Jiang, W. (2018), The existence and exponential stability of random impulsive fractional differential equations, Advances in Difference Equations, 2018.
  17. [17]  Akhmetov, M.U. and Zafer, A. (2000), Stability of zero solution of Impulsive differential equations by the Lyapanov second method, Journal of Mathematical Analysis and Applications, 248, 69-82.
  18. [18]  Rogovchenko, Yu.V. (1997), Impulsive evolution systems: main results and new trends, Dynamics of Continuous, Discrete and Impulsive Systems, 3, 57-88.
  19. [19]  Samoilneko, A.M. and Perestyuk, N.A. (1995), Impulsive differential equations, World Scientific, Singapore.
  20. [20]  Li, C., Sun, J., and Sun, R. (2010), Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects, Journal of the Franklin Institute Engineering and Applied mathematics, 347, 1186-1198.
  21. [21]  Luo, Z. and Shen, J. (2001), Stability results for impulsive functional differential equations with infinite delays, ournal of Computational and Applied Mathematics, 131, 55-64.
  22. [22]  Wu, S.J., Guo, Y., and Zhou, Y. (2006), p-moment stability of functional differential equations with random impulses, Computers and Mathematics with Applications, 52, 1683-1694.
  23. [23]  Luo, Z. and Shen, J. (2003), Impulsive stabilization of functional differential equations with infinite delays, Applied Mathematics Letters, 16, 695-701.