On the existence and asymptotic behavior of the random solutions of the random integral equation with advancing argument (Q2785632)

We present new existence theorems for a class of integral equations with advancing argument. We shall deal with a random integral equation of type: NEWLINE\[NEWLINEx(t,w)= h(t,w)+ \int^{t +S(t)}_0 k(t,\tau,w) f\bigl(\tau,x_\tau (w)\bigr)d \tau, \quad (t,w) \in\mathbb{R}^+ \times \Omega, NEWLINE\]NEWLINE where (i) \((\Omega, {\mathcal A},P)\) is a complete probability space, (ii) \(x= x(t,w)\) denotes an unknown random function. Our method is based on the notion of a measure of noncompactness in Banach spaces and the fixed point theorem of Darbo type.