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 mappingsf:X→Xfor which there exists a real numberc∈]0,1[, such that for eachx,y∈Xwe have∫0d(fx,fy)φ(t)dt≤c∫0d(x,y)φ(t)dt, whereφ:[0,+∞[→[0,+∞]is a Lebesgue-integrable mapping which is summable on each compact subset of[0,+∞[, nonnegative and such that for eachε>0,∫0εφ(t)dt>0.