Fixed points of asymptotically linear maps in ordered Banach spaces

Abstract

The purpose of this paper is to extend some fixed point theorems of M. A. Krasnosel’skii for asymptotically linear completely continuous maps leaving invariant a cone in a Banach space to strict set-contractions. The proofs will be based on the fixed point index and, as a by-product, we obtain new and simplified proofs of Krasnosel’skii’s theorems also. Moreover, even in the case of completely continuous maps, some of our results improve the known theorems. Throughout this paper all vector spaces will be over the reals. Let E be a normed vector space. A subset P of E is said to be a cone if it is closed, convex, invariant under multiplication by nonnegative real numbers, and if P n (-P) = (0). Each cone P induces an antisymmetric, reflexive, transitive ordering in E by defining: x > y if and only if x y E P. This ordering is compatible with the linear structure, i.e., 01 E R + := [0, m) and x > 0 imply 01x 3 0 and, for every x E E, x > y implies x + x > y + z, and it is compatible with the topology, i.e., xj 3 0 and xj -+ x imply x > 0. Let E be a normed vector space (.resp. a Banach space), and let P be a cone in E. Then the pair (E, P) is called an ordered normed vector space (resp. ordered Banach space) with positive cone P if E is given the ordering induced by P. The elements x E p : = P\(O) are said to be positive and we write x > 0 if x E P. -A cone is said to be total if E = P P andgenerating if E = P P. It is said to be normal if there exists 6 2 1 such that, for every pair % Y E p, II x II G 6 II x + Y /ILet E, F be normed vector spaces. Then we denote by L(E, F) the normed vector space of all continuous linear operators u: E -+ F.