Existence of solution for multi-valued integral equations (Q1775328)

The authors give conditions under which there is a global solution to the integral equation \[ x(t) \in a(t) + \int_{T_0}^t G(s,t)F(s,x(s))\, ds,\quad t\in [T_0, T), \] in a Banach space \(X\) where \(F\) is a set-valued and \(G(s,t)\) is a linear mapping. It is assumed that for all \(x\in X\) one has \(F(s,x) \subset \alpha(s)(1+\| x\| )K\) where \(K\) is a weakly compact set and \(\alpha\) is locally integrable. The authors also prove a result on the continuous dependence of the solution on a parameter.