Mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces
Let \(C\) be a nonempty subset of \((M,d)\). A mapping \(T: C\to C\) is said to be asymptotic pointwise nonexpansive if for any \(x\in C\), there exists a sequence numbers \(k_n(x)\) for all \(n\geq 1\) such that \[ d(t^n(x),T^n(y))\leq k_n(x)\cdot d(x, y) \] for any \(y\in C\), and \(\lim_{n\to\infty} k_n(x)= 1\). In the present paper the authors prove the following results: Theorem. Let \((M,d)\) be a complete hyperbolic metric space which is 2-uniformly convex, \(C\subset M\) be nonempty closed convex bounded and \(T: C\to C\) be asymptotic pointwise nonexpansive. Then the fixed point set \(\text{Fix}(T)\) is nonempty closed and convex. Additionaly, if \(\sum_n (k_n(x)- 1)< \infty\) for any \(x\in C\) and there exist two real numbers \(a\), \(b\) such that \(0< a\leq t_n\leq b< 1\), \(n\geq 1\), then for any \(p\in\text{Fix}(T)\) and the Mann iteration process: \(x_1\in C\), \(x_{n+1}= t_nT^n(x_n)\oplus(1- t_n) x_n\), \(n\geq 1\), \(\lim_{n\to\infty} d(p,x_n)\) exists.
scientific article; zbMATH DE number 6110006