Banach Space Of Asymptotically Non Semigroup Iterative Sequence Of Strong Convergence Theorem

The theory of fixed points is an important research topic of functional analysis and it is widely used in many subjects such as differential equations,nonlinear analysis and mathematical economics. The theory of fixed points of contractions established the foundation of the theory of fixed points. One classical method to study fixed points is to use contractions to approximate directly or approximate by iterations of nonexpansive mappings.In 2000, A. Moudafi proposed a viscosity approximation method of selecting a particular fixed point of a nonexpansive mapping in Hilbert spaces. In 2002, Gang Li and Brailey Sims showed that every semigroup of asymptocally nonexpansive type mappings has at least a common fixed point under certain appropriate conditions in Banach space with uniformly normal structure. In 2004, H. K. Xu studied the viscosity approximation methods, which is proposed by Moudafi, for nonexpansive mappings in uniformly smooth Banach space. In 2006, N. Shahzad and A. Udomene studied the convergence of the implicit iteration process and the explicit iteration process in a real Banach space with uniformly Ga?teaux differentiable norm and uniform normal structure, and proved that both the implicit iteration process and the explicit iteration process converge strongly to a fixed point, which is the unique solution to a variational inequality.In this paper, under the framework of Banach space with uniformly Ga?teaux differentiable norm and uniform normal structure, we show the convergence of the implicit iteration process and the explicit iteration process for asymptocally nonexpansive semigroup.First we investigate the convergence of the implicit iteration process: Let E be a Banach space with a uniformly Ga?teaux differentiable norm and uniform normal structure. Let K be a nonempty closed convex subset of E and ?= {T(t): t≥0}be a uniformly asymptotically regular asymptotically nonexpansive semigroup on K. If f : K→K is a contraction and a sequence {αn} ? (0, 1)satifies lni→m∞αn= 0 and {t n} ? (0, +∞)with lni→m∞tn=∞, n∈large enough, there exists a unique x n∈K such that x n =αn f(x n ) + (1 ?αn )T(t n )xn, and {x n} converges strongly to a fixed point p∈F( ?) , which is also the unique solution of the variational inequality (I ? f)p, j(p ? x ? )≤0, ? x ?∈F(? ) .Then we investigate the convergence of the explicit iteration process: Let E be a Banach space with a uniformly Ga?teaux differentiable norm and uniform normal structure. Let K be a nonempty closed convex subset of E and ?= {T(t): t≥0}be a uniformly asymptotically regular asymptotically nonexpansive semigroup on K. If f : K→K is a contraction and a sequence {αn} ? (0, 1)satifies lni→m∞αn= 0 and {t n} ? (0, +∞)with lni→m∞tn=∞, y 0∈K, the explicit iteration process y n+1 =αn f(y n ) + (1 ?αn )T(t n )yn, n≥1, satisfies nli→m∞y n ? T(h)y n= 0, ? h≥0, then {y n} converges strongly to a fixed point p∈F( ?) , which is also the unique solution of the variational inequality (I ? f)p, j(p ? x ? )≤0, ?x ?∈F(? ) .Our results improve and generalize some results of [6] and [12].