The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay
The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay
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DOI：
10.1016/j.jde.2018.09.009
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发表时间：
2019-03
影响因子：
2.4
通讯作者：
Yajing Li;Yejuan Wang
中科院分区：
文献类型：
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作者：
Yajing Li;Yejuan Wang
首先证明了具有Caputo分数阶导数的随机时滞演化方程D t α C y (t)= ay (t)+ f (t, y t)+ g (t, y t) D W (t) D t, 1,2 < α< 1的温和解的存在唯一性和连续依赖性。然后，研究了一类分数阶随机时滞演化方程的温和解的渐近性质：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。特别地，在均方拓扑中证明了全局正吸引集的存在性。利用α阶分数解算子理论和Schauder不动点定理，得到了温和解存在的一般定理。
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.