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Approximating fixed points of nonexpansive nonself mappings in CAT(0) spaces. (English) Zbl 1188.54021
Summary: Suppose that K is a nonempty closed convex subset of a complete CAT(0) space X with the nearest point projection P from X onto K. Let T:KX be a nonexpansive nonself mapping with F(T):={xK:Tx=x}. Suppose that {x n } is generated iteratively by x 1 K, x n+1 =P((1-α n )x n α n TP[(1-β n )x n β n Tx n ]), n1, where {α n } and {β n } are real sequences in [ε,1-ε] for some ε(0,1). Then {x n } Δ-converges to some point x* in F(T). This is an analog of a result in Banach spaces of N. Shahzad [Nonlinear Anal., Theory Methods Appl. 61, No. 6(A), 1031–1039 (2005; Zbl 1089.47058)] and extends a result of S. Dhompongsa and B. Panyanak [Comput. Math. Appl. 56, No. 10, 2572–2579 (2008; Zbl 1165.65351)] to the case of nonself mappings.
54H25Fixed-point and coincidence theorems in topological spaces
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties