Fixed Point Theorems for Mappings Satisfying Inwardness Conditions

Let A" be a normed linear space and let K be a convex subset of X. The inward set, I¡((x), of x relative to K is defined as follows: I^(x) = {x + c(u x):c > 1, u e K). A mapping T:K —► X is said to be inward if Tx S I/ç(x) for each x e K, and weakly inward if Tx belongs to the closure of If¿(x) for each x e K. In this paper a characterization of weakly inward mappings is given in terms of a condition arising in the study of ordinary differential equations. A general fixed point theorem is proved and applied to derive a generalization of the Contraction Mapping Principle in a complete metric space, and then applied together with the characterization of weakly inward mappings to obtain some fixed point theorems in Banach spaces. 0. Introduction. Let X be a topological vector space, KG X, and T a mapping of K into X. An inwardness condition on T is one which asserts that, in some sense, T maps points x of K "toward" K, or more precisely into the set generated by rays emanating from jc and passing through other points of K. Such conditions are always weaker than the assumption that T map the boundary of K, dK, into K. They have been formulated in a variety of ways and imposed by several authors recently in connection with studies both in fixed point theory and in certain differential equations. Our purpose in this paper is to illustrate how different types of inwardness assumptions are related, and to prove several new fixed point theorems in which these concepts play a role. Before stating precise definitions we give a brief review of some of the previous work in this area. The study of inward mappings originated with the investigations of B. Halpern in his 1965 doctoral thesis [7] where he obtained a generalization of the Schauder-Tychonov Theorem, a result he and Bergman further generalized in 1968 [9]. Since then many results have appeared in the literature concerning inward and weakly inward mappings in Halpern's sense, for both single and multivalued mappings (cf. [3], [6], [8], [9], [14], [16]-[19]). Another type of inwardness assumption was used by H. Brezis [1] in Received by the editors May 1, 1974 and, in revised form, October 16, 1974. AMS (MOS) subject classifications (1970). Primary 47H10; Secondary 54H25.