Existence and approximation of solution to stochastic fractional integro-differential equation with impulsive effects
- 190 Downloads
In this paper, we study a stochastic fractional integro-differential equation with impulsive effects in separable Hilbert space. Using a finite dimensional subspace, semigroup theory of linear operators and stochastic version of the well-known Banach fixed point theorem is applied to show the existence and uniqueness of an approximate solution. Next, these approximate solutions are shown to form a Cauchy sequence with respect to an appropriate norm, and the limit of this sequence is then a solution of the original problem. Moreover, the convergence of Faedo–Galerkin approximation of solution is shown. In the last, we have given an example to illustrate the applications of the abstract results.
KeywordsAnalytic semigroup Banach fixed point theorem Faedo–Galerkin approximations Hilbert space Stochastic fractional integro-differential equation with impulsive effects Mild solution
Mathematics Subject Classification34G20 34K30 34K40 34K50 35K90
The authors would like to thank the editor and the reviewers for their valuable comments and suggestions. The work of the first author is supported by the “Ministry of Human Resource and Development, India under Grant Number:MHR-02-23-200-44”.
- 18.Gard, T.C.: Introduction to Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 114. Dekker, New York (1988)Google Scholar
- 30.Oksendal, B.: Stochastic Differential Equations, 5th edn. Springer, Berlin (2002)Google Scholar
- 39.Yan, Z., Lu, F.: On approximate controllability of fractional stochastic neutral integro-differential inclusions with infiite delay. Appl. Anal. 2014, 23 (2014)Google Scholar