Entire Solutions of Stochastic Unbounded Delay Evolution Variational Inequalities Driven by Tempered Fractional Noise with an Exponential Dichotomy

The aim of this paper is to study a stochastic unbounded delay evolution variational inequality which consists of a stochastic unbounded delay evolution equation driven by tempered fractional noise with exponential dichotomy and a stochastic variational inequality. First, the existence and uniqueness of the mild solution on \( {\mathbb {R}} \) are established for the linear stochastic evolution equation overcoming the challenges posed by the exponential dichotomy of the evolution family \( \left\{ S(t,s)\right\} _{t\ge s} \) generated by the family of closed, densely defined linear operators A(t) in (1). Then after giving the equivalent form of the stochastic variational inequality defined on \( {\mathbb {R}} \), the existence and uniqueness of the mild solution on \( {\mathbb {R}} \) are proved for the stochastic unbounded delay evolution variational inequality (1) by using the Banach fixed point theorem instead of the iteration method and convergence analysis. Notably, due to the nontrivial exponential dichotomy of the evolution family \( \left\{ S(t,s)\right\} _{t\ge s} \), the stability can not be established for the nonlinear stochastic evolution variational inequality (1) and even for the linear stochastic evolution equation (5). Moreover, we show the exponential stability of the nontrivial equilibrium solution for the stochastic unbounded delay evolution variational inequality (1) but under the assumption that the evolution family \( \left\{ S(t,s)\right\} _{t\ge s} \) is exponential stable. Finally, the stochastic reaction diffusion variational inequality is considered as an example of application.