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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 $\varepsilon>0, \int_0^\varepsilon\varphi(t) dt > 0$.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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