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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 in topological spaces
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