Asymptotic behavior of nonexpansive mappings and some geometric properties in Banach spaces (Q1083680)

Let X be a real Banach space; \(C\subset X\) a closed convex subset and T:C\(\to C\) a nonexpansive mapping. Let \(A\subset X\times X\) be an accretive operator satisfying \(R(I+\lambda A)\supset \overline{D(A)}\) for all \(\lambda >0\) and \(J_{\lambda}=(I+\lambda A)^{-1}\) be its resolvent. Main results. 1) Let the sequence \(\{x_ n\}_{n\geq 0}\) be defined by \(x_{n+1}=c_ nTx_ n+(1-c_ n)x_ n\), where \(x_ 0\in C\), \(0<c_ n\leq 1\) and \(a_ n=\sum^{n}_{i=0}c_ i\to 0\) as \(n\to \infty\). Then there exists \(f\in X^*\), \(\| f\| =1\) such that for any \(x,x_ 0\in C:\lim_{n\to \infty}f(T^ nx)/n=\lim_{n\to \infty}\| T^ nx\| /n=\inf_{y\in C}\| Ty-y\| =\lim_{n\to \infty}f(x_{n+1})/a_ n)=\lim_{n\to \infty}\| x_{n+1}\| /a_ n.\) 2) Let the sequence \(\{x_ n\}_{n\geq 0}\) be defined by \(x_{n+1}=J_{c_ n}x_ n\), where \(x_ 0\in \overline{D(A)}\) and \(\{c_ n\}_{n\geq 0}\) is a positive sequence such that \(a_ n=\sum^{n}_{i=0}c_ i\to \infty\) as \(n\to \infty\). Then there exists \(f\in X^*\), \(\| f\| =1\) such that for any \(x,x_ 0\in \overline{D(A)}:\lim_{n\to \infty}f(J^ n_ 1x)/n=\lim_{n\to \infty}\| J^ n_ 1x\| /n=d(0,R(A))=\lim_{n\to \infty}f(x_{n+1})/a_ n=\lim_{n\to \infty}\| x_{n+1}\| /a_ n.\) The author investigates also some conditions equivalent to the reflexivity and strict convexity of a Banach space.