DEMONSTRATIO MATHEMATICA Vol. XXVI NO 1 1993 Z. Dzedzej, B.D. Gelman DIMENSION OF THE SOLUTION SET FOR DIFFERENTIAL INCLUSIONS Introduction Our paper is naturally divided into two short sections. In the first part we make some observations on results obtained by J.Saint Raymond in [7], [8] concerning topological dimen- sion of a fixed point set of convex-valued contraction. Slightly modifying the proofs, we generalize his theorems to the case of arbitrary closed convex subsets of a Banach space. In the second section we apply the previous result to the solution set of a Cauchy problem with set-valued right-hand side. We prove that it has an infinite dimension if only va- lues of the right-hand side function are at least one- dimensional. 1. Fixed points of set-valued contractions Let E be a Banach space and X its closed, convex subset. We denote by Cv(X) (resp. Kv(X)) the space of all closed, con- vex and bounded (resp. compact) subsets of X with Hausdorff metric. We say that F : X —» Cv(X) is a contraction if there exists a constant k, 0 a k < 1, such that h(F(x), F (y)) a k||x - y||, where h is the Hausdorff metric. By dim X we will denote the topological (covering) dimension of the space X. We are interested in the following problem: when dim F (x) fc n for each x e X implies that dim Fix F a n ? Two partial answers to this questions are given in [7], [8]. Let us start with the following result. (1.1) Remark. Let F : X —> Cv(X) be a contraction. Then there exists a bounded closed and convex subset B c X such that Fix F c B and B is F-invariant, i. e. F : B —» Cv(B) .