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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Generalized contractions and fixed point theorems


Author: Chi Song Wong
Journal: Proc. Amer. Math. Soc. 42 (1974), 409-417
MSC: Primary 54H25
MathSciNet review: 0331358
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Abstract: Let T be a self-mapping on a complete metric space (X, d). Then T has a fixed point if there exist self-mappings $ {\alpha _1},{\alpha _2},{\alpha _3},{\alpha _4},{\alpha _5}$ on $ [0,\infty ]$ such that (a) $ {\alpha _1}(t) + {\alpha _2}(t) + {\alpha _3}(t) + {\alpha _4}(t) + {\alpha _5}(t) < t$ for $ t > 0$, (b) each $ {\alpha _1}$ is upper semicontinuous from the right, (c)

$\displaystyle d(T(x),T(y)) \leqq {a_1}d(x,T(x)) + {a_2}d(y,T(y)) + {a_3}d(x,T(y)) + {a_4}d(y,T(x)) + {a_5}d(x,y)$

for all pairs of distinct x, y in X, where $ {\alpha _i} = {\alpha _i}(d(x,y))/d(x,y)$. Related results are obtained for two mappings and mappings on a bounded convex subset of a uniformly convex Banach space.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0331358-4
PII: S 0002-9939(1974)0331358-4
Keywords: Asymptotic center, commuting family, contraction, error control, uniform convex Banach space, upper semicontinuity, weakly compact convex set
Article copyright: © Copyright 1974 American Mathematical Society