New iterative schemes for asymptotically quasi-nonexpansive nonself-mappings in Banach spaces
Suppose \(X\) to be a real normed linear space, \(K\) a nonempty closed convex subset of \(X\) and \(F(T)=\{x\in K: Tx=x\}\), \(T:K\rightarrow K\). The main result of the paper is given in the following theorem: Let \(X\) be a uniformly convex space, and \(K\) a nonempty closed convex set in \(X\). Let \(T_{1},T_{2}:K\rightarrow X\) be uniformly \((L,\alpha)\)-Lipschitzian, asymptotically quasi-nonexpansive nonself-mappings with sequences \(\{ r_{n}^{(i)}\}\) such that \(\sum _{n=1}^{\infty} r_{n}^{(i)} <\infty\) for all \(i=1,2\) and satisfying the following conditions: {\parindent8mm \begin{itemize}\item[(i)] There exists a non-decreasing function \(f:[0,\infty )\rightarrow [0,\infty ),f(0)=0, f(r)>0 \) for all \(r\in (0,\infty )\) such that \[ \| x-T_{1}x\| \geq f(d(x,F)) \quad \text{or}\quad \| x-T_{2}x\| \geq f(d(x,F)) \quad\forall x\in K \] where \(F=F(T_{1})\cap F(T_{2})\neq \emptyset\). \item[(ii)] \(\sum _{n=1}^{\infty} \gamma_{n} <\infty, \quad \sum _{n=1}^{\infty} \gamma_{n}' <\infty ;\) \item[(iii)] \(\{ \beta_{n}\},\{ \beta_{n}'\}\) are two sequences in \( [\epsilon,1-\epsilon] \text{ for some }\epsilon >0.\) \end{itemize}} Then \(\{x_{n}\}\) converges strongly to the common fixed point of \(T_{1}\) and \(T_{2}\). In a second theorem condition (i) is replaced by the demicompactness of \(T_{1}\) or \(T_{2}\).
asymptotically quasi-nonexpansive nonself-mappings
scientific article; zbMATH DE number 5677339