Locally almost nonexpansive mappings (Q2782931)

This article deals with a locally almost nonexpansive selfmapping \(T\) defined on a weakly compact convex subset \(C\) in a Banach space \(X\) satisfying Opial's property. A mapping \(T:C\to X\) is called locally almost nonexpansive if for all \(x\in C\) and \(\varepsilon>0\) there exists a weak neighborhood \(N(x,\varepsilon)\) in \(C\) such that \(\|Tu-Tv\|\leq\|u-v\|+ \varepsilon\) \((u,v\in N(x, \varepsilon))\). The main results are the following: if \(T\) is locally almost nonexpansive, then \(I-T\) is demiclosed (i.e., sequentially closed as a mapping from \(C\) with the weak topology into \(C\) with the strong topology); in addition, if \(T^nx-T^{n+1}x\) weakly converges to 0 then every weak cluster point of \(T^nx\) is a fixed point of \(T\). Fixed points of multivalued locally almost nonexpansive mappings are also studied.