Leray-Schauder results for multivalued nonlinear contractions defined on closed subsets of a Fréchet space

In this paper, we present new fixed point results for nonlinear contractions (both single and multivalued) defined on subsets X (which may have empty interior) of a Fréchet space E. Some results for single-valued maps were presented in [2, 3] and the approach in these papers was based on constructing a specific map Fn (for each n∈N= {1,2, . . .}) whose fixed points converge to a fixed point of the original operator F. In the approach in this paper, the maps {Fn}n∈N only need to satisfy a closure property and are specified in a completely different way. The advantage of this approach is that multivalued maps can also be discussed. Our theory is based on results in Banach spaces and on viewing a Fréchet space E as a projective limit of a sequence of Banach spaces {En}n∈N. For the remainder of this section, we present some definitions and some known facts. Let (X ,d) be a metric space and S a nonempty subset of X . For x ∈ X , let d(x,S) = inf y∈S d(x, y). Also diamS = sup{d(x, y) : x, y ∈ S}. We let B(x,r) denote the open ball in X centered at x of radius r and by B(S,r) we denote ⋃ x∈S B(x,r). For two nonempty subsets S1 and S2 of X , we define the generalized Hausdorff distance H to be