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Maps satisfying generalized contractive conditions of integral type for which \(F(T)=F(T^n)\). (English) Zbl 1161.54024
Let \(\varphi:\mathbb{R}_+\to \mathbb{R}_+\) be a Lebesgue integrable function with \(\Phi(t):=\int_0^t\varphi(s)\,ds> 0\), \(\forall t> 0\); and fix some \(\lambda\) in \([0,1)\). Then, take a selfmap \(T\) of the complete metric space \((X,d)\) fulfilling an integral contractivity condition like: \(\Phi(d(Tx,T^2x))\leq \lambda \Phi(d(x,Tx))\), \(\forall x\in X\). Sufficient conditions are given in order that \(F(T)=F(T^n)\), for each \(n\in \mathbb{N}\).

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