Existence theorems of random solutions to stochastic functional integral equations (Q1110913)

The author proves the existence and uniqueness theorems for the stochastic functional integral equation \[ x(t)=h(t,x(t))+\int^{t}_{0}f(t,s,\lambda,x(s),x(\cdot))ds+\int^{t}_{0} g(t,s,\lambda,x(s),x(\cdot))d\beta (s), \] \[ x(t)=I(t),\quad t\in [- \tau_ 0,0],\quad I(0)=h(0,I(0)), \] where f, g depends on x(\(\sigma)\) for \(\sigma \in [-\tau_ 0,s]\), but independent in x(\(\sigma)\) for \(\sigma >s\); \(\beta\) (t) is a martigale.