## 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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