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In this paper we prove a fixed point theorem for asymptotically nonexpansive mappings acting in modular spaces. This result generalises the 1972 fixed point theorem by K. Goebel and W.A. Kirk. In the process, we extend several other results (including the Milman--Pettis theorem) from the class of Banach spaces to the larger class of regular modular spaces. [ABSTRACT FROM AUTHOR]

In this paper we present the following two results: 1.- A characterization of the renorming invariant family of asymptotically nonexpansive mappings defined on a convex, closed and bounded set of a Banach space; 2.- A comparison of the renorming invariant family of asymptotically nonexpansive mappings with the renorming invariant family of nonexpansive mappings. Additionally, a series of examples are shown for general and particular cases. [ABSTRACT FROM AUTHOR]

In this paper we establish the following general theorem. Let S be a semitopological reversible semigroup and let Y be a linear subspace of ℓ ∞ (S) which contains the constants and is left translation invariant. Suppose that Y has a left invariant mean. Let (E , ‖ · ‖ E) be a uniformly convex Banach space and let C be a nonempty, bounded, closed and convex subset of E. Assume that C has nonempty interior, is locally uniformly rotund and T = T s s ∈ S is an asymptotically nonexpansive semigroup which acts on C. Assume also that T has a unique fixed point x ~ and that, in addition, this point x ~ lies on the boundary ∂ C of C. Then { u (s) } s ∈ S converges strongly to x ~ for each almost orbit u ∈ A O Y (T) . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the convergence theorems in CAT (κ) spaces with k > 0 for total asymptotically nonexpansive mappings which are essentially wider than nonexpansive mappings, asymptotically nonexpansive mapping, and asymptotically nonexpansive mappings in the intermediate sense. Our results generalize, unify, and improve several comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

In a real Hilbert space, let the GSVI and CFPP represent a general system of variational inclusions and a common fixed point problem of countable nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new inertial subgradient extragradient rule we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP as constraints. Some strong convergence theorems for the proposed algorithms are established under some mild assumptions. Our results improve and extend some corresponding results in the earlier and very recent literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a new viscosity iterative algorithm for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping and the set of solutions of a new variational inequality problem involving inverse-strongly monotone operators in Hilbert spaces. We prove some strong convergence theorems under some suitable assumptions imposed on the parameters by using a modified extragradient method. The results obtained in this paper may be an improvement of many recent ones in this fields. [ABSTRACT FROM AUTHOR]

ln this paper, a new approach for finding common element of the set of solutions of the variational inequality problem for accretive mappings and the set of fixed points for asymptotically nonexpansive mappings is introduced and studied. Consequently, strong convergence results for finite families of asymptotically nonexpansive mappings and variational inequality problems are established in the setting of uniformly convex Banach space and 2-uniformly smooth Banach space. Furthermore, we prove that a slight modification of our novel scheme could be applied in finding common element of solution of variational inequality problems in Hilbert space. Our results improve, extend and generalize several recently announced results in literature. [ABSTRACT FROM AUTHOR]

We introduce a one-step implicit iterative method for two finite families of asymptotically nonexpansive mappings in a hyperbolic space and use it to approximate common fixed points of these families. The results presented in this paper are new in the setting of hyperbolic spaces. On top, these are generalizations of several results in literature from Banach spaces to hyperbolic spaces. At the end of the paper, we give an example to validate our results. [ABSTRACT FROM AUTHOR]

In a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we introduce hybrid inertial subgradient extragradient algorithms with line-search process for solving a variational inequality problem (VIP) with pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. The proposed algorithms are based on inertial subgradient extragradient method with line-search process, hybrid steepest-descent method, and viscosity approximation method. Under mild conditions, we prove strong convergence of the proposed algorithms to an element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set. [ABSTRACT FROM AUTHOR]

In this paper, we propose an iteration algorithm for finding a split common fixed point of an asymptotically nonexpansive mapping in the frameworks of two real Banach spaces. Under some suitable conditions imposed on the sequences of parameters, some strong convergence theorems are proved, which also solve some variational inequalities that are closely related to optimization problems. The results here generalize and improve the main results of other authors. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study some iterative algorithms for finding a common element of the set of solutions of systems of variational inequalities for inverse-strongly accretive mappings and the set of fixed points of an asymptotically nonexpansive mapping in uniformly convex and 2-uniformly smooth Banach space or uniformly convex and q-uniformly smooth Banach space. Strong convergence theorems are obtained under suitable conditions. We also give some numerical examples to support our main results. The results obtained in this paper improve and extend the recent ones announced by many others in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in solving minimization problem and common fixed point problem of finite family consisting asymptotically nonexpansive mappings in Hadamard spaces. For finding common solutions of their problems, we introduce a new modified proximal point algorithm involving convex combination technique. Under suitable conditions, Δ-convergence and strong convergence theorems are proved. Furthermore, we provide some numerical examples which show the efficiency and implementation of the proposed algorithm as well as show computational overview by comparing it with previously known algorithms. [ABSTRACT FROM AUTHOR]

Our focus in this paper is on introducing an iterative scheme based on the generalized implicit method and viscosity approximation method with Meir-Keeler contraction for solving a fixed point problem of an asymptotically nonexpansive mapping in the framework of Banach spaces. The strong convergence results of the proposed iterative scheme are established. An application based on convex minimization problem is derived from our main result. The numerical applicability and efficiency of our results are demonstrated using some examples. Our results extend, generalize and unify the previously known results given in the literature. [ABSTRACT FROM AUTHOR]

In this paper, two different classes of mappings namely, uniformly continuous asymptotically nonexpansive and uniformly continuous asymptotically demicontractive mappings are considered on the general modified Noor iteration process with errors and proved to converge strongly to the fixed point of uniformly continuous asymptotically demicontractive mappings in uniformly smooth Banach spaces. The new result can be viewed as an improvement to a multitude of results in the fixed point theory especially those of Xu and Noor [8], Owojori and Imoru [5] and also the results of Owojori [6]. [ABSTRACT FROM AUTHOR]

In this paper, we establish the generalized viscosity implicit rules of asymptotically nonexpansive mappings in CAT(0) spaces. The strong convergence theorems of the implicit rules proposed are proved under certain assumptions imposed on the control parameters. The results presented in this paper improve and extend some recent corresponding results announced. [ABSTRACT FROM AUTHOR]

In this paper, a new modified proximal point algorithm involving fixed point of Cesàro type mean of asymptotically nonexpansive mappings in CAT(0) spaces is proposed. We also introduce a new iterative scheme. Under suitable conditions, the ▵-convergence and the strong convergence to a common element of the set of minimizers of a convex function and the set of fixed points of the Cesàro type mean of asymptotically nonexpansive mapping in CAT(0) space are proved. The results presented in the paper are new. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert spaces. The strong convergence of this viscosity method is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown that the limit solves an additional variational inequality. Applications to nonlinear variational inclusion problem, nonlinear Volterra integral equations, variational inequality problem and hierarchical minimization problems are included. The results presented in the paper extend and improve some recent results announced in the current literature. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to introduce a general iteration process for the finite families of total asymptotically nonexpansive mappings and generalized nonexpansive multivalued mappings in hyperbolic space which include Hadamard manifold and CAT(0) space as special cases. Some important properties of the new defined iterative process are analyzed. Finally, Δ-convergence and strong convergence for the iterative process in hyperbolic space are established. The results presented in this paper extend and improve some existing results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the generalized viscosity implicit rules of one asymptotically nonexpansive mapping in Hilbert spaces. We obtain some strong convergence theorems under certain assumptions imposed on the parameters. We also apply our main results to solve mixed equilibrium problem in Hilbert spaces. A numerical example is also given to support our main results. The results obtained in this paper improve and extend many recent ones in this field. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to introduce a modified viscosity implicit rule for finding a common element of the set of solutions of variational inequalities for two inverse-strongly monotone operators and the set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces. Some strong convergence theorems are obtained under some suitable assumptions imposed on the parameters. As an application, we give an algorithm to solve fixed point problems for nonexpansive mappings, variational inequality problems and equilibrium problems in Hilbert spaces. Finally, we give one numerical example to illustrate our convergence analysis. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of periodic points of asymptotically nonexpansive mappings defined on non-convex domains in uniformly convex Banach spaces. We also introduce the notion of firmly asymptotically nonexpansive mappings and prove the existence of fixed points of such mappings with non-convex domains. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce an iterative algorithm to approximate a common solution of a generalized equilibrium problem and a fixed point problem for an asymptotically nonexpansive mapping in a real Hilbert space. We prove that the sequences generated by the iterative algorithm converge strongly to a common solution of the generalized equilibrium problem and the fixed point problem for an asymptotically nonexpansive mapping. The results presented in this paper extend and generalize many previously known results in this research area. Some applications of main results are also provided.

In this paper, a new mixed type iteration process for approximating a common fixed point of two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings is constructed. We then establish a strong convergence theorem under mild conditions in a uniformly convex hyperbolic space. The results presented here extend and improve some related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a new modified proximal point algorithm involving fixed point iterates of asymptotically nonexpansive mappings in $\operatorname {CAT}(0)$ spaces is proposed and the existence of a sequence generated by our iterative process converging to a minimizer of a convex function and a common fixed point of asymptotically nonexpansive mappings is proved. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the concept of total asymp- totically nonexpansive mappings and to prove some Δ-convergence theorems of it- eration processes with errors to approximating a common fixed point for four total asymptotically nonexpansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results announced in the current lit- erature. [ABSTRACT FROM AUTHOR]

In this paper, we study asymptotically quasi-φ- nonexpansive mappings in the intermediate sense and Ky Fan inequalities. A convergence theorem is established in a strictly convex and uniformly smooth Banach space. The results presented in the paper improve and extend some recent results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iterative sequence for solving common problems which consist of split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in the framework of Hilbert spaces and prove some strong convergence theorems of the generated sequence $\{x_{n}\}$ by the shrinking projection method. Our results improve and extend the previous results given in the literature. [ABSTRACT FROM AUTHOR]

Some results extending the Goebel–Kirk fixed point theorem for asymptotically nonexpansive mappings are presented. [ABSTRACT FROM AUTHOR]

In this paper, we construct a modified Ishikawa iterative process to approximate common fixed points of two multivalued asymptotically nonexpansive mappings and prove some convergence theorems in uniformly convex hyperbolic spaces. [ABSTRACT FROM AUTHOR]

ln this paper, we introduce a new class of mapping called asymptotically demicontractive multivalued mapping in the setting of a real Hilbert space. Furthermore, a new iteration scheme was constructed, and it was proved that our algorithm converges strongly to the common element of solutions of an equilibrium problem and the set of common fixed points of two finite families of type-one asymptotically demicontractive multivalued mappings without any sum conditions imposed on the finite family of the control sequences. Also, we provided a numerical example to demonstratethe implementablity of our proposed iteration technique. Our results improve, extend and generalize many recently announced results in the current

In this paper we present the following two results: 1.- A characterization of the renorming invariant family of asymptotically nonexpansive mappings defined on a convex, closed and bounded set of a Banach space; 2.- A comparison of the renorming invariant family of asymptotically nonexpansive mappings with the renorming invariant family of nonexpansive mappings. Additionally, a series of examples are shown for general and particular cases.

Abstract The purpose of this paper is to study the convergence theorems in CAT ( κ ) $\operatorname{CAT} (\kappa )$ spaces with k > 0 $k > 0$ for total asymptotically nonexpansive mappings which are essentially wider than nonexpansive mappings, asymptotically nonexpansive mapping, and asymptotically nonexpansive mappings in the intermediate sense. Our results generalize, unify, and improve several comparable results in the existing literature.

In this paper, a shrinking projection algorithm based on the extragradient iteration method for finding a common element of the set of common fixed points of a finite family of asymptotically nonexpansive mappings and a generalized nonexpansive set-valued mapping and the set of solutions of equilibrium problem for pseudomonotone and Lipschitz-type continuous bifunctions is introduced and investigated in Hilbert spaces. Moreover, the strong convergence of the sequence generated by the proposed algorithm is derived under some suitable assumptions. These results are new and develop some recent results in this field. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of fixed points for a class of Lipschitzian type mappings and asymptotic pointwise Lipschitz type mappings in hyperbolic 2- uniformly convex metric spaces. In the same space setting, we deal the problem of approximation of fixed points via modified Mann iteration process. Our result generalizes and extends the corresponding results of Dehaish et al. [7], Goebel and Kirk [8], Kirk and Xu [18] and Sahu et al. [24] and many others in this direction. [ABSTRACT FROM AUTHOR]

In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms with linesearch process for finding a common solution of the VIP and the CFPP by using a subgradient approach. These fully absorb hybrid steepest-descent ideas, viscosity iteration ideas, and composite Mann-type iterative ideas. With suitable conditions on real parameters, it is shown that the sequences generated our algorithms converge to a common solution in norm, which is a unique solution of a hierarchical variational inequality (HVI). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate convergence of a hybrid Halpern process for fixed point and equilibrium problems. Strong convergence theorems of common solutions are established in a strictly convex and uniformly smooth Banach space which also has the Kadec-Klee property. [ABSTRACT FROM AUTHOR]

The generalized viscosity implicit rules of nonexpansive asymptotically mappings in Hilbert spaces are considered. The strong convergence theorems of the rules are proved under certain assumptions imposed on the sequences of parameters. An application of it in the convex minimization problem is considered. The results presented in this paper improve and extend some recent corresponding results in the literature.

In this paper, we propose a generalized viscosity implicit iterative method for asymptotically non-expansive mappings in Banach spaces. The strong convergence theorem of this algorithm is proved, which solves the variational inequality problem. Moreover, we provide some applications to zero-point problems and equilibrium problems. Further, a numerical example is given to illustrate our convergence analysis. The results generalize and improve corresponding results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a multiple hybrid implicit iteration method for finding a solution for a monotone variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities, and a common fixed point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in Hilbert spaces. Strong convergence of the proposed method to the unique solution of the problem is established under some suitable assumptions. [ABSTRACT FROM AUTHOR]

In a real Hilbert space, let the VIP, GSVI, HVI, and CFPP denote a variational inequality problem, a general system of variational inequalities, a hierarchical variational inequality, and a common fixed-point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping, respectively. We design two Mann implicit composite subgradient extragradient algorithms with line-search process for finding a common solution of the CFPP, GSVI, and VIP. The suggested algorithms are based on the Mann implicit iteration method, subgradient extragradient method with line-search process, and viscosity approximation method. Under mild assumptions, we prove the strong convergence of the suggested algorithms to a common solution of the CFPP, GSVI, and VIP, which solves a certain HVI defined on their common solutions set. [ABSTRACT FROM AUTHOR]

Abstract In a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively.