Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54H25

Retrieve articles in all journals with MSC: 54H25

Additional Information

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