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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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, $ {I_K}(x)$, of x relative to K is defined as follows: $ {I_K}(x) = \{ x + c(u - x):c \geqslant 1,u \in K\} $. A mapping $ T:K \to X$ is said to be inward if $ Tx \in {I_K}(x)$ for each $ x \in K$, and weakly inward if Tx belongs to the closure of $ {I_K}(x)$ for each $ x \in 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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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