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

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