Khasminskii-Type Theorem and LaSalle-Type Theorem for Stochastic Evolution Delay Equations

TL;DR

This paper extends classical theorems to stochastic evolution delay equations, proving existence, uniqueness, and stability of solutions without requiring linear growth conditions, supported by illustrative examples.

Contribution

It introduces a Khasminskii-Type Theorem and a LaSalle-Type Theorem for stochastic evolution delay equations under less restrictive conditions, expanding theoretical understanding.

Findings

Proved existence and uniqueness of solutions without linear growth condition.

Established a stochastic LaSalle-type theorem for stability analysis.

Provided examples demonstrating the applicability of the theories.

Abstract

In this paper we study the well-known Khasminskii-Type Theorem, i.e. the existence and uniqueness of solutions of stochastic evolution delay equations, under local Lipschitz condition, but without linear growth condition. We then establish one stochastic LaSalle-type theorem for asymptotic stability analysis of strong solutions. Moreover, several examples are established to illustrate the power of our theories.