Proximal Point Algorithms for a Hybrid Pair of Nonexpansive Single-Valued and Multi-Valued Mappings in Geodesic Metric Spaces

Convergence Theorems of Modified Proximal Algorithms for Asymptotical Quasi-nonexpansive Mappings in CAT(0) Spaces

“…where T is a single-valued nonexpansive mapping, S is a multi-valued nonexpansive mapping, and {λ n } is a sequence such that λ n ≥ λ > 0 for all n ≥ 1 and some λ. Inspired by the above work, in this paper, we come up with a new modified algorithm, which improved and extended the results [7].…”

“…Definition2.2 [7] A single-valued mapping T : D → D is said to be semicompact if for any sequence {x n } in D such that lim n→∞ d(x n , T x n ) = 0, there exists a subsequence {x n i } of {x n } such that {x n i } converges strongly to p ∈ D. The set of fixed points of T is denoted by F (T ), that is, F (T ) = {x ∈ D : x = T x}.…”

“…Definition2.4 [7] Let {x n } be a bounded sequence in a CAT (0) space X. For x ∈ X, we define a mapping r(•, {x n }) :…”

Proximal point algorithm for nonlinear multivalued type mappings in Hadamard spaces

“…Shimizu and Takahashi proved the existence of fixed points for nonexpansive multivalued mappings in convex metric spaces; that is, every multivalued mapping T : X → C ( X ) has a fixed point in a bounded, complete, and uniformly convex metric space ( X , d ), where C ( X ) is the family of all compact subsets of X . For several algorithms are presented for finding fixed points of multivalued, refer to Suantai et al and Shimizu et al…”

“…Shimizu and Takahashi 14 proved the existence of fixed points for nonexpansive multivalued mappings in convex metric spaces; that is, every multivalued mapping T ∶ X → C(X) has a fixed point in a bounded, complete, and uniformly convex metric space (X, d), where C(X) is the family of all compact subsets of X. For several algorithms are presented for finding fixed points of multivalued, refer to Suantai et al [15][16][17] and Shimizu et al 14 Let CB(D) be the collection of all nonempty and closed bounded subsets and (D) be the collection of all nonempty proximal bounded and closed subsets of D, respectively. Let the Hausdorff distance on CB(D) be defined by…”