Fixed Points of Cone Compression and Expansion Multimaps Defined on Frechet Spaces: the Projective Limit Approach

This paper presents cone compression and expansion fixed point results of KrasnoselskiiPetryshyn type formultimaps between Fréchet spaces. Two approaches have recently been presented in the literature, both of which are based on the fact that a Fréchet space can be viewed as a projective limit of a sequence of Banach spaces {En}n∈N (here N= {1,2, . . .}). Both approaches are based on constructing maps Fn defined on subsets of En whose fixed points converge to a fixed point of the original operator F. In the first approach [6, 7], for each n∈N a specificmap Fn is discussed; whereas in the second approach [2–4], themaps {Fn}n∈N only need to satisfy a closure-type property. Both approaches have advantages and disadvantages over the other [1]. In this paper, we combine the advantages of both approaches to present a very general fixed point result. Existence in Section 2 is based on the following result of Petryshyn [14, Theorem 3].