On almost periodic mild solutions for stochastic functional differential equations. (English) Zbl 1244.34100

Cao, Junfei; Yang, Qigui; Huang, Zaitang

Nonlinear Anal., Real World Appl. 13, No. 1, 275-286 (2012).

Summary: The class of stochastic functional differential equations given by \[ \begin{aligned} dx(t) &= (Ax(t)+ F(t, x(t), xt))\,dt+ G(t,x(t), x_t)\circ d\omega(t),\quad t\in [0,T],\\ x(t) &= \varphi(t)\quad\text{for }t\in [-\sigma,0],\end{aligned} \] is investigated. Under some suitable assumptions, the existence and stability of quadraticmean almost periodic mild solutions for the equations are discussed by means of semigroups of operators and fixed point method. Moreover, an example is given to illustrate our results.

MSC:

34K50	Stochastic functional-differential equations
34K14	Almost and pseudo-almost periodic solutions to functional-differential equations
34K20	Stability theory of functional-differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

stochastic functional differential equations; quadratic-mean almost periodic solution; stability

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