The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay

By Yajing Li, Yejuan Wang. Journal of Differential Equations, 2019

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Abstract We first prove the existence, uniqueness and continuous dependence of mild solutions to stochastic delay evolution equations with a Caputo fractional derivative: D t α C y ( t ) = A y ( t ) + f ( t , y t ) + g ( t , y t ) d W ( t ) d t , 1 2 α 1 . Then, we investigate the asymptotic behavior of mild solutions to fractional stochastic delay evolution equations of the form D t α C y ( t ) = A y ( t ) + I t 1 − α f ( t , y t ) + [ I t 1 − α g ( t , y t ) ] d W ( t ) d t , 0 α 1 . In particular, the existence of a global forward attracting set in the mean-square topology is established. A general theorem on the existence of mild solutions is obtained by using α-order fractional resolvent operator theory and the Schauder fixed point theorem.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

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