Existence theorems for generalized asymptotically nonexpansive mappings in uniformly convex metric spaces (Q2850731)

Let \(C\) be a nonempty closed subset of a metric space \((X,d)\). A mapping \(T: C\to C\) is called a generalized asymptotically nonexpansive mapping if there exist sequences \(\{k_n\}\subset[0,1)\) and \(\{s_n\}\subset[0,\infty)\) with \(\lim_n k_n= 1\), \(\lim_n s_n= 0\) such that \(d(T^nx, T^ny)\leq k_nd(x, y)+ s_n\) for all \(x,y\in C\) and \(n\in \mathbb{N}\). The nonexpansive mappings, asymptotically nonexpansive mappings, and asymptotically nonexpansive mappings in the intermediate sense are all generalized asymptotically nonexpansive mappings.NEWLINENEWLINE In the present paper, the authors prove, among others, the following result.NEWLINENEWLINE Let \(C\) be a nonempty bounded closed convex subset of complete uniformly convex metric space \((X,d,W)\) (introduced by \textit{T. Shimizu} and \textit{W. Takahashi} [Topol. Methods Nonlinear Anal. 8, No.~1, 197--203 (1996; Zbl 0902.47049)]). If \(T: C\to C\) is a generalized asymptotically nonexpansive mapping whose graph \(G(T)= \{(x, y)\in C\times C: y= Tx\}\) is closed, then the set of fixed points \(F(T)= \{x\in C: x= Tx\}\) is nonempty closed and convex.