Common fixed point results in CAT(0) spaces

Common fixed points for some generalized multivalued nonexpansive mappings in uniformly convex metric spaces

“…In this paper, we extend this result to the general setting of uniformly convex metric spaces in the sense of Goebel and Reich [24]. Nevertheless, condition (E) of T can be weakened to the strongly demiclosedness of I -T. Since uniformly convex Banach spaces and CAT(0) spaces are uniformly convex metric spaces, then our results extend and improve the results in [4,25,26,23,27] and many others.…”

“…By using the argument in the proof of Theorem 3.4, we can also obtain the following result that is an extension of [ [4], Theorem 3.8]. Recall that a point z X is said to be a center of the mapping t : K X if for each x K, d(z, t(x)) ≤ d(z, x).…”

“…Motivated by Suzuki's results, Garcia-Falset et al [23] introduced two generalizations of condition (C), namely, conditions (E) and (C l ) and studied both the existence of fixed points and the asymptotic behavior of mappings satisfying such conditions. Recently, Abkar and Eslamian [4] proved that if K is a nonempty bounded closed convex subset of a complete CAT(0) space X, t : K K is a single-valued quasi-nonexpansive mapping and T : K KC(K) is a multivalued mapping satisfying conditions (E) and (C l ) for some l (0, 1) such that t and T are weakly commuting, then there exists a point z K such that z = t(z) T(z). In this paper, we extend this result to the general setting of uniformly convex metric spaces in the sense of Goebel and Reich [24].…”

“…He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed and many of papers have appeared (see e.g., [4][5][6][7][8][9][10][11][12][13][14] and the references therein). It is worth mentioning that fixed point theorems in CAT(0) spaces (specially in ℝ -trees) can be applied to graph theory, biology, and computer science (see e.g., [15][16][17][18][19][20]).…”

On strong and Δ -convergence of modified S-iteration for uniformly continuous total asymptotically nonexpansive mappings in CAT(κ) spaces

“…Later on, many authors generalized the notion of CAT(κ) given in [18,19], mainly focusing on CAT(0) spaces (see e.g., [1,9,10,12,16,20,22,29,26,30]). The results of a CAT(0) space can be applied to any CAT(κ) space with κ ≤ 0 since any CAT(κ) space is a CAT(κ ) space for every κ ≥ κ (see in [5]).…”