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 , 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, .