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

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.