Some characterization results of nonlocal special random impulsive evolution differential equations. (English) Zbl 1478.34018
Summary: In this paper, we present existence and uniqueness of special random impulsive differential evolution equations with nonlocal condition in Hilbert spaces. Moreover we study the stability results for the same evolution equations. Existence and uniqueness results are proved using Banach fixed point theorem where as stability results using fixed point approach and semi group theory. Finally we give some applications of the nonlocal impulsive differential equations as well as evolution equations, which shows the importance of our theoretical results.
MSC:
| 34A37 | Ordinary differential equations with impulses |
| 34G20 | Nonlinear differential equations in abstract spaces |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
References:
| [1] | 1. Zavalishchin, A. (1994), Impulse dynamic systems and applications to mathematical economics, Dynamical Systems of Applications, 3, 443-449. · Zbl 0805.34009 |
| [2] | Zavalishchin, A. (1994), Impulse dynamic systems and applications to mathematical economics, Dynamical Systems of Applications, 3, 443-449. · Zbl 0805.34009 |
| [3] | 2. Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), Theory of Impulsive Differential Equations, World Scientific, Singapore. · Zbl 0719.34002 |
| [4] | Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), Theory of Impulsive Differential Equations, World Scientific, Singapore. · Zbl 0719.34002 |
| [5] | 3. Liu, X.Z. and Guo, D.J.(1995), Initial value problems for first order impulsive integro differential equations in Banach spaces, Communications on Applied Nonlinear Analysis, 2, 65-83. · Zbl 0858.34068 |
| [6] | Liu, X.Z. and Guo, D.J.(1995), Initial value problems for first order impulsive integro differential equations in Banach spaces, Communications on Applied Nonlinear Analysis, 2, 65-83. · Zbl 0858.34068 |
| [7] | 4. Wu, S.J. and Meng, X.Z. (2004), Boundedness of nonlinear differential systems with impulsive effect on random moments, Acta Mathematica Applicatae Sinica, 20(1), 147-154. · Zbl 1078.34512 |
| [8] | Wu, S.J. and Meng, X.Z. (2004), Boundedness of nonlinear differential systems with impulsive effect on random moments, Acta Mathematica Applicatae Sinica, 20(1), 147-154. · Zbl 1078.34512 |
| [9] | 5. Iwankievicz, R. and Nielsen, S.R.K. (1992), Dynamic response of non-linear systems to Poisson distributed random impulses, Journal of Sound Vibration, 156, 407-423. |
| [10] | Iwankievicz, R. and Nielsen, S.R.K. (1992), Dynamic response of non-linear systems to Poisson distributed random impulses, Journal of Sound Vibration, 156, 407-423. |
| [11] | 6. Sanz-Serna, J.M. and Stuart, A.M. (1999), Ergodicity of dissipative differential equations subject to random impulses, Journal of Differential Equations, 155, 262-284. · Zbl 0934.34047 |
| [12] | Sanz-Serna, J.M. and Stuart, A.M. (1999), Ergodicity of dissipative differential equations subject to random impulses, Journal of Differential Equations, 155, 262-284. · Zbl 0934.34047 |
| [13] | Tatsuyuki, K., Takashi, K., and Satoshi, S. (1998), Drift motion of granules in chara cells induced by random impulses due to the myosinactin interaction, Physica A, 248(1-2), 21-27. |
| [14] | 7. Tatsuyuki, K., Takashi, K., and Satoshi, S. (1998), Drift motion of granules in chara cells induced by random impulses due to the myosinactin interaction, Physica A, 248(1-2), 21-27. |
| [15] | 8. Freedman, E., Liu, X., and Wu., J. (1991), Comparison principles for impulsive parabolic equations with applications to models of single species growth, Journal of Australian Mathematical Socity, Series B, 32, 382-400. · Zbl 0881.35006 |
| [16] | Freedman, E., Liu, X., and Wu., J. (1991), Comparison principles for impulsive parabolic equations with applications to models of single species growth, Journal of Australian Mathematical Socity, Series B, 32, 382-400. · Zbl 0881.35006 |
| [17] | Guo, D.J. and Liu, X.Z. (1993), Extremal solutions of nonl inear impulsive integro-differential equations in Banach spaces, Journal of Mathematics Analysis and Applications, 177, 538-553. · Zbl 0787.45008 |
| [18] | 9. Guo, D.J. and Liu, X.Z. (1993), Extremal solutions of nonl inear impulsive integro-differential equations in Banach spaces, Journal of Mathematics Analysis and Applications, 177, 538-553. · Zbl 0787.45008 |
| [19] | 10. Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York. · Zbl 0516.47023 |
| [20] | Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York. · Zbl 0516.47023 |
| [21] | 11. Abinaya, S. and Sayooj, A.J. (2019), Neutral impulsive stochastic differential equations driven by fractional brownian motion with poisson jump and nonlocal conditions, Global Journal of Pure and Applied Mathematics, 15(3), 305-321. |
| [22] | Abinaya, S. and Sayooj, A.J. (2019), Neutral impulsive stochastic differential equations driven by fractional brownian motion with poisson jump and nonlocal conditions, Global Journal of Pure and Applied Mathemat-ics, 15(3), 305-321. |
| [23] | 12. Hernan, H.R., Poblete, V., and Pozo, J.C. (2014), Mild solutions of non-autonomous second order problems with nonlocal initial conditions, Journal of Mathematics Analysis and Applications, 412, 1064-1083. · Zbl 1317.34144 |
| [24] | Hernan, H.R., Poblete, V., and Pozo, J.C. (2014), Mild solutions of non-autonomous second order problems with nonlocal initial conditions, Journal of Mathematics Analysis and Applications, 412, 1064-1083. · Zbl 1317.34144 |
| [25] | 13. Yang, H., Agarwal, R.P., and Liung, Y. (2017), Controllabi-lity for a class of Integro-differential equation envo-lving nonlocal Initial conditions, International Journal of Control, 99(12). |
| [26] | Yang, H., Agarwal, R.P., and Liung, Y. (2017), Controllabi-lity for a class of Integro-differential equation envo-lving nonlocal Initial conditions, International Journal of Control, 99(12). |
| [27] | Jose, S.A. and Usha, V. (2018), Existence and uniqueness of solutions for special random impulsive differential equation, Journal of Applied Science and Computations, 5(10), 14-23. |
| [28] | 14. Jose, S.A. and Usha, V. (2018), Existence and uniqueness of solutions for special random impulsive differential equation, Journal of Applied Science and Computations, 5(10), 14-23. |
| [29] | Jose, S.A. and Usha, V. (2018), Existence of solutions for special random impulsive differential equation with nonlocal conditions, International Journal of Computer Science and Engineering, 6(10), 549-554. |
| [30] | 15. Jose, S.A. and Usha, V. (2018), Existence of solutions for special random impulsive differential equation with nonlocal conditions, International Journal of Computer Science and Engineering, 6(10), 549-554. |
| [31] | Guo, D.J. (2000), Multiple positive solutions for first order nonlinear integrodifferential equations in Ban ach spaces, Nonlinear Analysis, 53, 183-195. · Zbl 1026.45013 |
| [32] | 16. Guo, D.J. (2000), Multiple positive solutions for first order nonlinear integrodifferential equations in Ban ach spaces, Nonlinear Analysis, 53, 183-195. · Zbl 1026.45013 |
| [33] | 17. Jahanshahi, M. and Darabadi, M. (2016), An Analytic Solution for a nonlocal Initial boundary value problem including a partial Initial a partial differential equation with variable coefficients, Bulltin of Iranian World Scientific Mathematical Socity, 42(2), 315-326. · Zbl 1373.35068 |
| [34] | Jahanshahi, M. and Darabadi, M. (2016), An Analytic Solution for a nonlocal Initial boundary value problem including a partial Initial a partial differential equation with variable coefficients, Bulltin of Iranian World Scientific Mathematical Socity, 42(2), 315-326. · Zbl 1373.35068 |
| [35] | Liu, J.H. (1999), Nonlinear impulsive evolution equations, Dynamics of Continuous, Discrete and Impulsive Systems, 6, 77-85. · Zbl 0932.34067 |
| [36] | 18. Liu, J.H. (1999), Nonlinear impulsive evolution equations, Dynamics of Continuous, Discrete and Impulsive Systems, 6, 77-85. · Zbl 0932.34067 |
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