Topological Approach to Differential Inclusions on Closed Subset of ℝn

The present paper is a survey of the current results concerning the existence problem, topological characterization of the set of solutions and periodic solutions of differential inclusions on subsets of euclidean spaces. First, based on [5, 15, 17, 19, 21], we give a topological approach to these questions on whole ℝn or on balls in ℝn. It is presented in Sects. 5 and 6. Later, in Sects. 7, 8 and 9 we give some new results. Namely, we are trying to tackle the mentioned problems on compact subsets of euclidean spaces called by us sets with property p. Let us remark that in particular convex sets and smooth manifolds with boundary or without boundary have property p. Note that our results obtained in Sects. 7, 8 and 9 provide an application and generalization of respective results given in [3, 7, 13, 20, 21 and 25]. We would like to add that in our considerations, we use the topological degree methods only. Finally let us remark that instead of the topological degree the analytical methods in the theory of differential inclusions are well developed (cf. [3] and [30]).