International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 9, Pages 531-536
doi: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 A. Branciari. 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:X→X for which there exists a real number
c∈]0,1[, such that for each x,y∈X we have
∫0d(fx,fy)φ(t)dt≤c∫0d(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.