In this paper one existence theorem for the differential inclusion (1)

$\dot{x}\in F(t,x)$,

$x\left({t}_{0}\right)={x}_{0}$ in a separable Banach space is proved. The multifunction F has nonempty, compact, convex values and satisfies the Caratheodory-type conditions. The proof uses the Ky Fan fixed point theorem and some properties of integral of multifunctions. Next the continuous dependence of solutions to (1) on the right-hand side is studied where the convergence of

${F}_{n}$ to F is unerstand in Kuratowski-Mosco sense. At the end one theorem on convergence of the sets of fixed points of some sequence of multifunctions, say

$\left\{{F}_{n}\right\}$, (with Lipschitz constants smaller than 1) to the set of fixed points of the limit of

${F}_{n}$ in a Banach space with Frechet-differentiable norm is proved. A theorem of the same type is proved for the set of integrable selectors of a sequence of multifunctions.