The existence and exponential behavior of solutions to time fractional stochastic delay evolution inclusions with nonlinear multiplicative noise and fractional noise

Yajing Li; Yejuan Wang

doi:10.3934/dcdsb.2020027

Article Contents

2020, Volume 25, Issue 7: 2665-2697. Doi: 10.3934/dcdsb.2020027

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The existence and exponential behavior of solutions to time fractional stochastic delay evolution inclusions with nonlinear multiplicative noise and fractional noise

Yajing Li^1,2, and
Yejuan Wang^1, ,

1.
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China
2.
College of Science, Northwest A & F University Yangling, Shaanxi 712100, China

^* Corresponding author: Yejuan Wang
^* Corresponding author: Yejuan Wang

Received: March 2019

Early access: April 2020

Published: July 2020

This work was supported by NSF of China (Grant Nos. 41875084, 11801452, 11571153), the Fundamental Research Funds for the Central Universities under Grants No. lzujbky-2018-ot03 and lzujbky-2018-it58, the Natural Science Basic Research Plan in Shaanxi Province of China under Grant No. 2019JQ-196, and the Doctor Fund of Northwest A & F University under Grant No. 2452018017

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Abstract

This article is devoted to study time fractional stochastic evolution inclusions with infinite delays driven by a nonlinear multiplicative noise and a fractional Brownian motion with Hurst parameter $ H\in(\frac{1}{2},1) $. First of all, we investigate the local and global existence of mild solutions to such evolution inclusions by using the fractional resolvent operator theory and some new results on the measure of noncompactness for the stochastic integral term. Further, we prove that every mild solution decays exponentially to zero in the sense of mean-square topology.

Keywords:

Fractional stochastic evolution inclusion,
infinite delay,
measure of noncompactness,
exponential decay,
Caputo fractional derivative,
nonlinear multiplicative noise,
fractional Brownian motion.

Mathematics Subject Classification: Primary: 34A08, 34K09, 34K50, 35K58.

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