Topological degree of condensing multi-valued perturbations of the $(S)_+$-class maps and its applications

Applications of topological characteristics of nonlinear (one-valued and multi-valued) maps are well-known efficient tools for the investigation of solvability for various problems of the theory of differential equations and of optimal control theory. In this paper, a construction of one such characteristic is proposed: this is the degree of condensing multi-valued perturbations of maps of class $(S)_+$. Principal properties of the characteristic are studied. The considered characteristic is applied for the investigation of a class of controllable systems.