Abstract
In this paper, we study strong and △-convergence for a newly defined two-step iteration process involving two asymptotically nonexpansive mappings in the intermediate sense which is wider than the class of asymptotically nonexpansive mappings in the setting of \(\operatorname{CAT}(0)\) spaces. Our results generalize, unify and extend many known results from the existing literature.
Highlights
A metric space (X, d) is a CAT( ) space if it is geodesically connected and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane
Fixed point theory in CAT( ) spaces was first studied by Kirk
Where and throughout the paper {αn}, {βn} are sequences such that ≤ αn, βn ≤, for all n ≥. They studied the modified S-iteration process for asymptotically quasinonexpansive mappings in a CAT( ) space and established some strong convergence results under some suitable conditions which generalize some results of Khan and Abbas [ ]
Summary
A metric space (X, d) is a CAT( ) space if it is geodesically connected and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane. – αn)Tnxn ⊕ αnT βn)xn ⊕ βnT nxn, n where and throughout the paper {αn}, {βn} are sequences such that ≤ αn, βn ≤ , for all n ≥ They studied the modified S-iteration process for asymptotically quasinonexpansive mappings in a CAT( ) space and established some strong convergence results under some suitable conditions which generalize some results of Khan and Abbas [ ]. Consider K to be a nonempty closed convex subset of a complete CAT( ) space X and S, T : K → K to be two asymptotically nonexpansive mappings in the intermediate sense with F = F(S) ∩ F(T) = ∅. ) involving two asymptotically nonexpansive mappings in the intermediate sense and investigate the existence and convergence theorems for the above said mappings and iteration scheme in the framework of CAT( ) spaces. The converse does not hold as the following example shows
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