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A fixed-point theorem for integral type inequality depending on another function. (English) Zbl 1216.54015

Let \((X,d)\) be a complete metric space, \(0 \leq \alpha < 1\), and \(S, T\) be selfmaps of \(X\) such that \(T\) is injective, continuous, and subsequentially convergent. Suppose that there exists \(x \in X\) such that \(\int_0^{d(Tsy,TS^2y)}\varphi(t)\,dt \leq \alpha\int_0^{d(T,TSy)}\varphi(t)\,dt\) for each \(y\) in the orbit of \(x\), where \(\varphi := [0, +\infty) \to [0, +\infty)\) is a Lebesgue integrable mapping which is summable, nonnegative, and such that \(\int_0^{\varepsilon}\varphi(t)\,dt > 0 \) for each \(\varepsilon > 0\). Then the authors show that
(i) \(\lim_nTS^nx = Tq\),
(ii) \(\int_0^{d(Tq,TS^nx)}\varphi(t)\,dt \leq \alpha_n\int_0^{d(Tq,Tx)}\varphi(t)\,dt\), and
(iii) \(q\) is a fixed point of \(S\) if and only if \(G(x) := d(TSx, Tx)\) is \(S_T\)-orbitally lower semicontinuous at \(q\).

MSC:

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