On a Class of Condensing Multivalued Maps

Abstract

We define a new class of $(Q,\beta)$-condensing multivalued maps, where $\beta$ is a vector measure of noncompactness and $Q$ is a bounded positive linear operator whose spectral radius $r(Q)<1.$ Some properties of such multivalued maps are studied. In particular, it is shown that the topological degree theory and classical fixed point theorems can be extended to this class of multimaps. An analog of the theorem on a diagonal representation of a condensing multimap is proved. It is demonstrated that the translation multioperator along trajectories of a system of semilinear differential inclusions in Banach spaces is $(Q,\mathcal{X})$-condensing, where $\mathcal{X}$ is the vector Hausdorff measure of noncompactness.