## ABSTRACT

The theorems of Leray-Schauder type, also called continuation theorems, represent a powerful existence tool in studying operator equations and inclusions (of particular interest is the theory of nonlinear differential equations). Roughly speaking, by means of a continuation theorem we can obtain a solution of a given equation if we start from one of the solutions of a simpler equation. To be more explicit, let us consider two nonempty sets E and 8, a proper subset, B of O and a map rl : Z i O . Suppose that we are'interested in the solvability of the inclusion

r, (X) E B. (1.1) The main idea of any continuation method for (1.1) consists in joining this inclusion to a 'simpler' one,

by means of an 'homotopy' q : E X [0, l] + 8 in such a way that

q ( . ,O) = ro and q ( . , l ) = I'l.