Viscosity approximation methods for monotone inclusion and fixed point problems in CAT(0) space
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Viscosity approximation methods for monotone inclusion and fixed point problems in CAT(0) space Bashir Ali1
· Auwalu Ali Alasan2
Received: 18 December 2019 / Accepted: 18 September 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In this paper, we introduce a new Viscosity algorithm for approximating an element in the intersection of the set of common solutions of monotone inclusion problems and common fixed points of family of asymptotically quasi-non expansive mappings in a complete CAT(0) space. Strong convergence theorem is proved which improved and generalized recently announced results in the literature. Keywords CAT(0) space · Common fixed point · Monotone operators · Resolvent operators · Asymptotically quasi-nonexpansive mapping Mathematics Subject Classification 47H09 · 47J25
1 Introduction Let (X , d) be a metric space, x, y ∈ X and d(x, y) = l. A geodesic path from x to y is an isometry c : [0, l] → c([0, l]) ⊂ X such that c(0) = x, c(l) = y. The image of a geodesic path between two points is called a geodesic segment. A Metric space (X , d) is called a geodesic space if every two points of X are joined by a geodesic segment. A geodesic triangle, represented by (x, y, z), in a geodesic space consists of three points x, y, z and three geodesic segments joining each pair of the points. A comparison triangle of a geodesic triangle (x, y, z), denoted by (x, y, z) or (x, y, z), is a triangle in the Euclidean plane R2 such that d(x, y) = dR2 (x, y), d(x, z) = dR2 (x, z) and d(y, z) = dR2 (y, z). A geodesic segment joining two points x, y in a geodesic space X is denoted by [x, y] and every point z ∈ [x, y] is represented by αx ⊕ (1 − α)y where α ∈ [0, 1], that is, [x, y] := {αx ⊕ (1 − α)y : α ∈ [0, 1]} (see, [2]). A geodesic space is called a C AT (0) space if for every geodesic triangle and its comparison triangle , the following inequality is
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Bashir Ali [email protected] Auwalu Ali Alasan [email protected]
1
Department of Mathematical Sciences, Bayero University, Kano, Nigeria
2
Department of Basic Studies, Kano State Polytechnic, Kano, Nigeria
123
B. Ali, A. A. Alasan
satisfied: d(x, y) ≤ dR2 (x, y) ∀x, y ∈ , x, y ∈ . A complete C AT (0) space is called Hadamard space. Examples of C AT (0) spaces include Euclidean spaces Rn , Hilbert spaces, R-trees, Hilbert ball equipped with hyperbolic metric. For more details on these spaces, see for example, [4,14,22]. Definition 1.1 Let X be a metric space. Let {xn }∞ n=1 be any bounded sequence in X . For x ∈ X , set r (x, {xn }) := lim supd(xn , x), then n →∞
i. the asymptotic radius of the sequence {xn } ⊆ X denoted by r ({xn }) is defined by r ({xn }) = inf r ({xn }, x). x∈X
ii. the asymptotic center of {xn } ⊆ X is a set A({xn }) = {z ∈ X : r (z, {xn }) = r ({xn })}. A sequence {xn } ⊆ X is said to -converge to x if every subsequence {x n k } of {xn } satisfies the condition that A({xn k }) = {x}. That is to say a sequence {xn } in X -converges to a point x ∈
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