On the topological dimension of the solutions sets for some classes of operator and differential inclusions

In the present paper, we give the lower estimation for the topological dimension of the fixed points set of a condensing continuous multimap in a Banach space. The abstract result is applied to the fixed point set of the multioperator of the form $ = S _F$ where $_F$ is the superposition multioperator generated by the Carathéodory type multifunction F and S is the shift of a linear injective operator. We present sufficient conditions under which this set has the infinite topological dimension. In the last section of the paper, we consider the applications of the solutions sets for Cauchy and periodic problems for semilinear differential inclusions in a Banach space.