Fixed point theorems for mappings satisfying inwardness conditions

Author:
James Caristi

Journal:
Trans. Amer. Math. Soc. **215** (1976), 241-251

MSC:
Primary 47H10

DOI:
https://doi.org/10.1090/S0002-9947-1976-0394329-4

MathSciNet review:
0394329

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Abstract: Let *X* be a normed linear space and let *K* be a convex subset of *X*. The inward set, , of *x* relative to *K* is defined as follows: . A mapping is said to be inward if for each , and weakly inward if *Tx* belongs to the closure of for each . 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1976-0394329-4

Keywords:
Fixed point theorems,
contraction and nonexpansive mappings,
inward and weakly inward conditions,
complete metric space,
Banach space

Article copyright:
© Copyright 1976
American Mathematical Society