ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Some Characterization Results of Nonlocal Special Random Impulsive Differential Evolution Equations

Journal of Applied Nonlinear Dynamics 10(4) (2021) 695--707 | DOI:10.5890/JAND.2021.12.009

Sayooj Aby Jose$^{1,2}$ , Ashitha Tom$^3$, S. Abinaya$^4$, Weera Yukunthorn$^5$

$^{1}$ Ramanujan Centre for Higher Mathematics, Alagappa University, Tamil Nadu, India

$^{2}$ Department of Mathematics, Alagappa University, Tamil Nadu, India

$^{3}$ Department of Mathematics, Madurai Kamaraj University, Tamil Nadu, India

$^{4}$ Department of Mathematics, Rathinam College of Arts $&$ Science, Tamil Nadu, India

$^{5}$ Faculty of Science and Technology, Kanchanaburi Rajabhat University, Thailand

Abstract

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.

References

1.  [1] Zavalishchin, A. (1994), Impulse dynamic systems and applications to mathematical economics, Dynamical Systems of Applications, 3, 443-449.
2.  [2] Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), Theory of Impulsive Differential Equations, World Scientific, Singapore.
3.  [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.
4.  [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.
5.  [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.
6.  [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.
7.  [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.
8.  [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.
9.  [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.
10.  [10] Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York.
11.  [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.
12.  [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.
13.  [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).
14.  [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.
15.  [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.
16.  [16] Guo, D.J. (2000), Multiple positive solutions for first order nonlinear integrodifferential equations in Ban ach spaces, Nonlinear Analysis, 53, 183-195.
17.  [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.
18.  [18] Liu, J.H. (1999), Nonlinear impulsive evolution equations, Dynamics of Continuous, Discrete and Impulsive Systems, 6, 77-85.