Τ and Weak Demicloseness Principle and Asymptotic Behavior for Asymptotically Nonexpansive Type Mappings

The purpose of this paper is to provide the new demicloseness principle– τ (weakly) demicloseness principle. We prove that if X is a Banach space with locally uniformly τ -Opial condition, where τ is a Hausdorff topology on X, C is a nonempty τ compact subset of X, and T : C → C is a asymptotically nonexpansive type mapping. If {xα} is a net in C which converges to x in the sense of τ topology and if the net {xα − T mxα} converges to zero in the sense of τ topology for each m ∈ N , then x−Tx = 0. We also give the weakly demicloseness theorem in a Banach space with Opial property. This result is to be used to study convergence theorem for almost-orbits of asymptotically nonexpansive type mappings in a Banach space. 1.INTRODUCTION Let C be a nonempty subset of a Banach space X and let T : C 7→ C be a mapping. Then T is said to be asymptotically nonexpansive [9] if there exists a sequence {kn} of real number with limn→∞ kn = 1 such that ‖Tnx− Tny‖ ≤ kn‖x− y‖ for allx, y ∈ C and n ∈ N (1.1) If (1.1) is valid for all kn ≡ 1, T is said to be nonexpansive. More generally T is of asymptotically nonexpansive type [9] if lim sup n→∞ sup y∈C (‖Tnx− Tny‖ − ‖x− y‖) ≤ 0 for every x ∈ C. 1991 Mathematics Subject Classification. 47H09, 47H10, 47H20.