In this paper, we
introduce some new classes of the totally quasi-G-asymptotically nonexpansive
mappings and the totally quasi-G-asymptotically nonexpansive semigroups. Then,
with the generalized f-projection operator, we prove some strong convergence
theorems of a new modified Halpern type hybrid iterative algorithm for the
totally quasi-G-asymptotically nonexpansive semigroups in Banach space. The
results presented in this paper extend and improve some corresponding ones by
many others.

Abstract:
We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others. 1. Introduction Throughout this paper we denoted by and the set of all positive integers and all positive real numbers, respectively. Let be a real Banach space, and let be a nonempty closed convex subset of . A mapping of into itself is said to be nonexpansive if for each . We denote by the set of fixed points of . We know that is nonempty if is bounded; for more detail see [1]. A one-parameter family from of into itself is said to be a nonexpansive semigroup on if it satisfies the following conditions: (i) ;(ii) for all ;(iii)for each the mapping is continuous;(iv) for all and . We denote by the set of all common fixed points of , that is, . We know that is nonempty if is bounded; see [2]. Recall that a self-mapping is a contraction if there exists a constant such that for each . As in [3], we use the notation to denote the collection of all contractions on , that is, . Note that each has a unique fixed point in . In the last ten years, the iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [3–5]. Let be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let be a strongly positive bounded linear operator on : that is, there is a constant with property A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space : where is the fixed point set of a nonexpansive mapping on and is a given point in . In 2003, Xu [3] proved that the sequence generated by converges strongly to the unique solution of the minimization problem (1.2) provided that the sequence satisfies certain conditions. Using the viscosity approximation method, Moudafi [6] introduced the iterative process for nonexpansive mappings (see [3, 7] for further developments in both Hilbert and Banach spaces) and proved that if is a real Hilbert space, the sequence generated by the following algorithm: where is a contraction mapping with constant and satisfies certain conditions, converges strongly to a fixed point of in which is unique solution of the variational inequality: In 2006, Marino and Xu [8] combined the iterative method (1.3)

Throughout
this paper, we introduce a new hybrid iterative algorithm for finding a common
element of the set of common fixed points of a finite family of uniformly
asymptotically nonexpansive semigroups and the set of solutions of an
equilibrium problem in the framework of Hilbert spaces. We then prove the strong
convergence theorem with respect to the proposed iterative algorithm. Our
results in this paper extend and improve some recent known results.

Abstract:
Let be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from to . Let be a nonexpansive semigroup on such that , and is a contraction on with coefficient . Let be -strongly accretive and -strictly pseudocontractive with and a positive real number such that . When the sequences of real numbers and satisfy some appropriate conditions, the three iterative processes given as follows: , , , , and , converge strongly to , where is the unique solution in of the variational inequality , . Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others. 1. Introduction Let be a real Banach space. A mapping of into itself is said to be nonexpansive if for each . We denote by the set of fixed points of . A mapping is called -contraction if there exists a constant such that for all . A family of mappings of into itself is called a nonexpansive semigroup on if it satisfies the following conditions: (i) for all ; (ii) for all ; (iii) for all and ; (iv) for all , the mapping is continuous. We denote by the set of all common fixed points of , that is, In [1], Shioji and Takahashi introduced the following implicit iteration in a Hilbert space where is a sequence in and is a sequence of positive real numbers which diverges to . Under certain restrictions on the sequence , Shioji and Takahashi [1] proved strong convergence of the sequence to a member of . In [2], Shimizu and Takahashi studied the strong convergence of the sequence defined by in a real Hilbert space where is a strongly continuous semigroup of nonexpansive mappings on a closed convex subset of a Banach space and . Using viscosity method, Chen and Song [3] studied the strong convergence of the following iterative method for a nonexpansive semigroup with in a Banach space: where is a contraction. Note however that their iterate at step is constructed through the average of the semigroup over the interval . Suzuki [4] was the first to introduce again in a Hilbert space the following implicit iteration process: for the nonexpansive semigroup case. In 2002, Benavides et al. [5], in a uniformly smooth Banach space, showed that if satisfies an asymptotic regularity condition and fulfills the control conditions , , and , then both the implicit iteration process (1.5) and the explicit iteration process (1.6), converge to a same point of . In 2005, Xu [6] studied the strong convergence of the implicit iteration process (1.2) and (1.5) in a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping.

Abstract:
Let be a nonempty closed convex subset of a reflexive Banach space with a weakly continuous dual mapping, and let be an infinite countable family of asymptotically nonexpansive mappings with the sequence satisfying for each , , and for each . In this paper, we introduce a new implicit iterative scheme generated by and prove that the scheme converges strongly to a common fixed point of , which solves some certain variational inequality.

Abstract:
In this paper we prove the weak and strong convergence of the implicit iterative process with errors to a common fixed point of a finite family $\{T_j\}_{i=1}^N$ of asymptotically quasi $I_j-$nonexpansive mappings as well as a family of $\{I_j\}_{j=1}^N$ of asymptotically quasi nonexpansive mappings in the framework of Banach spaces.

Abstract:
Let K be a nonempty closed convex subset of a reflexive Banach space E with a weakly continuous dual mapping, and let {Ti}i=1 ￠ be an infinite countable family of asymptotically nonexpansive mappings with the sequence {kin} satisfying kin ￠ ‰ ￥1 for each i=1,2, ￠ € |, n=1,2, ￠ € |, and limn ￠ ’ ￠ kin=1 for each i=1,2, ￠ € |. In this paper, we introduce a new implicit iterative scheme generated by {Ti}i=1 ￠ and prove that the scheme converges strongly to a common fixed point of {Ti}i=1 ￠ , which solves some certain variational inequality.

Abstract:
Strong convergence theorems are obtained from modified Halpern iterative scheme for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups, respectively. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Kim, Xu, and some others.

Abstract:
Strong convergence theorems are obtained from modified Halpern iterative scheme for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups, respectively. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Kim, Xu, and some others.

Abstract:
We introduce a new general system of variational inclusions in Banach spaces and propose a new iterative scheme for finding common element of the set of solutions of the variational inclusion with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point of nonexpansive semigroups in a uniformly convex and 2-uniformly smooth Banach space. Furthermore, strong convergence theorems are established under some certain control conditions. As applications, finding a common solution for a system of variational inequality problems and minimization problems is given. 1. Introduction In the theory of variational inequalities and variational inclusions, the development of an efficient and implementable iterative algorithm is interesting and important. The important generalization of variational inequalities called variational inclusions, have been extensively studied and generalized in different directions to study a wide class of problems arising in optimization, nonlinear programming, finance, economics, and applied sciences. Variational inequalities are being used as a mathematical programming tool in modeling a wide class of problems arising in several branches of pure and applied mathematics. Several numerical techniques for solving variational inequalities and the related optimization problem have been considered by many authors. Throughout this paper, we denoted by and the set of all positive integers and all positive real numbers, respectively. Let be a real Banach space and be its dual space. Let denote the unit sphere of . is said to be uniformly convex if for each , there exists a constant such that for all , The norm on is said to be Gateaux differentiable if the limit exists for each , and in this case is smooth. Moreover, we say that the norm is said to have a uniformly Gateaux differentiable if the above limit is attained uniformly for all and in this case is said to be uniformly smooth. We define a function , called the modulus of smoothness of , as follows: It is know that is uniformly smooth if and only if . Let be a fixed real number . A Banach space is said to be -uniformly smooth if there exists a constant such that for all . From [1], we know the following property. Let be a real number with and let be a Banach space. Then, is -uniformly smooth if and only if there exists a constant such that The best constant in the above inequality is called the -uniformly smoothness constant of (see [1] for more details). Let be a real Banach space and the dual space of . Let denote the pairing between and .