Asymptotic Expansion Non-self Mapping Classes In Banach Space Noor - Type Iterative Convergence Theorem

Fixed point iteration processes for asymptotically nonexpansive mappings in Banach spaces are important parts of functional analysis. The theory of fixed points originated from Banach’s principle of contraction mappings. A classical iteration for a nonexpansive mapping was first introduced by Mann. Ishikawa introduced an iteration procedure for approximating fixed points of pseudo-contractive compact mapping in real Hilbert spaces, which is often cited as Ishikawa iteration process. In2000, Noor investigated a three-step iterative scheme and studied the approximate solutions of variation inclusion. Later, Xu and Noor introduced a three-step scheme to approximate fixed point of asymptotically nonexpansive mappings in Banach spaces. In2005, Suantai investigated the convergence criteria of Noor iterations for asymptotically nonexpansive mappings. In2006, Plubtieng, Wangkeeree and Punpaeng studied on the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings, Thianwan and Suantai obtained convergence criteria of three-step iteration with errors for nonexpansive nonself-mappings at the same time. In2009, Thianwan constructed a new iteration scheme for approximating common fixed points of two asymptotically nonexpansive nonself-mappings and proved the strong and weak convergence theorems for such schemes in a uniformly convex Banach space.In this paper, we introduce a new three-step Noor iterative scheme for three asymptotically nonexpansive nonself mappings in a uniformly convex Banach space. Firstly, strong convergence theorems are established for this Noor iterative scheme of three asymptotically nonexpansive nonself mappings which satisfy the condition (A) that is weaker than semicompactness and complete continuity. Secondly, we give the weak convergence theorems for such iterative scheme in the case that the space satisfies Opial’s condition or whose dual space has Kadec-Klee property. Let C be a nonempty bounded closed convex subset of uniformly convex Banach space X and P:Xâ†'C be a nonexpansive retraction from X to C.Let T1,T2,T3: Câ†'X be three asymptotically nonexpansive nonself mappings with F≠φ and the sequences{kn})(?)[1,+∞) satisfying defined by:x1∈C and where{αn(i)),{βn(i)),{γn(i)}are in[0,1]with0<p≤α(i),βn(i)≤g<1,αn(i)+βn(i)+γn(i)=1, and{un(i)}are bounded sequences in X,i=1,2,3.(i)If the family of {Ti}i-13satisfies condition(A),then{xn},{yn},{zn)converge strongly to a common fixed point of{Ti}i-13.(ii)If X satisfies the Opial’s condition or its dual X’has the Kadec-Klee property, then{xn},{yn},{zn}converge weakly to a common fixed point of{Ti}i=3The results obtained in this paper not only extend and improve the main results in [3,29,34],but also are new even in the case of nonexpansive nonself mappings and in the case that the space has a Frechet differentiable norm.