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International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 9, Pages 531-536
http://dx.doi.org/10.1155/S0161171202007524

A fixed point theorem for mappings satisfying a general contractive condition of integral type

Viale Martiri della Libertà 20, Macerata 62100, Italy

Received 30 April 2001

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We analyze the existence of fixed points for mappings defined on complete metric spaces (X,d) satisfying a general contractive inequality of integral type. This condition is analogous to Banach-Caccioppoli's one; in short, we study mappings f:XX for which there exists a real number c]0,1[, such that for each x,yX we have 0d(fx,fy)φ(t)dtc0d(x,y)φ(t)dt, where φ:[0,+[[0,+] is a Lebesgue-integrable mapping which is summable on each compact subset of [0,+[, nonnegative and such that for each ε>0, 0εφ(t)dt>0.