Fixed Point Theory for Admissible Type Maps with Applications

In this paper, assuming a natural sequentially compact conditionwe establish new fixed point theorems for Urysohn type maps between Fréchet spaces. In Section 2 we present new LeraySchauder alternatives, Krasnoselskii and Lefschetz fixed point theory for admissible type maps. The proofs rely on fixed point theory in Banach spaces and viewing a Fréchet space as the projective limit of a sequence of Banach spaces. Our theory is partly motivated by a variety of authors in the literature see 1–6 and the references therein . Existence in Section 2 is based on a Leray-Schauder alternative for Kakutani maps see 4, 5, 7 for the history of this result which we state here for the convenience of the reader.