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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