Generalized contractions and fixed point theorems

Wong, Chi Song

doi:10.1090/S0002-9939-1974-0331358-4

Generalized contractions and fixed point theorems
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Proc. Amer. Math. Soc. 42 (1974), 409-417 Request permission

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