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Fixed Point Theorems for Mappings Satisfying Inwardness Conditions

James Caristi
Transactions of the American Mathematical Society
Vol. 215 (Jan., 1976), pp. 241-251
DOI: 10.2307/1999724
Stable URL: http://www.jstor.org/stable/1999724
Page Count: 11
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Abstract

Let X be a normed linear space and let K be a convex subset of X. The inward set, IK(x), of x relative to K is defined as follows: IK(x) = {x + c(u - x): c ⩾ 1, u ∈ K}. A mapping $T: K \longrightarrow X$ is said to be inward if Tx ∈ IK(x) for each x ∈ K, and weakly inward if Tx belongs to the closure of IK(x) for each x ∈ 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.

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