## Fixed point theorems for mappings satisfying inwardness conditions

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- by James Caristi
- Trans. Amer. Math. Soc.
**215**(1976), 241-251 - DOI: https://doi.org/10.1090/S0002-9947-1976-0394329-4
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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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*Fixed point theorems for outward maps*, Doctoral Thesis, Univ. of California, Los Angeles, Calif., 1965.

*Fixed point theorems for multivalued non-compact inward maps*(to appear).

*Existence comparison and asymptotic behavior of solutions of ordinary differential equations in finite and infinite dimensional Banach spaces*(to appear). —,

*Nonexistence of periodic solutions of differential equations and applications to zeros of nonlinear operators*(to appear).

## Bibliographic Information

- © Copyright 1976 American Mathematical Society
- 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