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A fixed point theorem for mappings satisfying a general contractive condition of integral type. (English) Zbl 0993.54040
Summary: 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 \(f : X \rightarrow X\) for which there exists a real number \(c \in ]0,1[\), such that for each \(x, y \in X\) we have \[ \int_0^{d(fx,fy)} \varphi(t) dt \leq c \int_0^{d(x,y)}\varphi(t) dt, \] where \(\varphi : [0,+\infty[ \rightarrow [0,+\infty]\) is a Lebesgue-integrable mapping which is summable on each compact subset of \([0, +\infty[\), nonnegative and such that for each \(\varepsilon>0, \int_0^\varepsilon\varphi(t) dt > 0\).

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
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