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Let us consider two nonempty subsets A and B of a uniformly convex Banach space X. Let $$T:A\cup B\rightarrow A\cup B$$ T : A ∪ B → A ∪ B be a mapping such that $$T(A)\subseteq A,\,T(B)\subseteq B$$ T ( A ) ⊆ A , T ( B ) ⊆ B and there is a sequence $$\{k_n\}$$ { k n } in $$[1,\infty )$$ [ 1 , ∞ ) , with $$k_n\rightarrow 1$$ k n → 1 , satisfying $$\Vert T^nx-T^ny\Vert \le k_n\Vert x-y\Vert $$ ‖ T n x - T n y ‖ ≤ k n ‖ x - y ‖ , for all $$x\in A$$ x ∈ A and $$y\in B$$ y ∈ B . We investigate sufficient conditions for the existence of fixed points x in A and y in B in such a way that the distance between x and y is optimum in some sense. Our main result provides a natural and simple proof for a particular case of Rajesh and Veeramani (Numer Funct Anal Optim 37:80–91, 2016) fixed point theorem for asymptotically relatively nonexpansive mappings.
The Journal of Analysis – Springer Journals
Published: Dec 8, 2017
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