Family Of Nonexpansive Mappings Research Articles

Strong Convergence for Generalized Equilibrium Problems, Fixed Point Problems and Relaxed Cocoercive Variational Inequalities

We introduce a new iterative scheme for finding the common element of the set of solutions of the generalized equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of the variational inequality problems for a relaxed -cocoercive and -Lipschitz continuous mapping in a real Hilbert space. Then, we prove the strong convergence of a common element of the above three sets under some suitable conditions. Our result can be considered as an improvement and refinement of the previously known results.

An extragradient method for relaxed cocoercive variational inequality and equilibrium problems

The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of a countable family of nonexpansivemappings, the set of an equilibrium problem and the set of solutions of the variational inequality problem for a relaxed cocoercive and Lipschitz continuous mapping in Hilbert spaces. Then, we show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions, which are connected with Yao, Liou, Yao, Takahashi and many others.

Iterative algorithms for variational inequality and equilibrium problems with applications

In this paper, we introduce an iterative method for finding a common element of the set of solutions of equilibrium problems, of the set of variational inequalities and of the set of common fixed points of an infinite family of nonexpansive mappings in the framework of real Hilbert spaces. Strong convergence of the proposed iterative algorithm is obtained. As an application, we utilize the main results which improve the corresponding results announced in Chang et al. (Nonlinear Anal, 70:3307---3319, 2009), Colao et al. (J Math Anal Appl, 344:340---352, 2008), Plubtieng and Punpaeng (Appl Math Comput, 197:548---558, 2008) to study the optimization problem.

Iterative algorithm by using the hybrid method in mathematical programming for solving variational inequality problems and equilibrium problems

In this paper, we introduce an iterative scheme by a new hybrid method for finding a common element of the set of fixed points of a countable family of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α -inverse-strongly monotone mappings in a real Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets, which solves some fixed point problems, variational inequality problems and equilibrium problems by using the hybrid method in mathematical programming which connected with optimization problems. The results are connected with Kumam’s results [11, 12], Shinzato and Takahashi’s result [19], Tada and Takahashi’s result [23] and Takahashi’s et al. result [26] and many others.

Two extragradient methods for generalized mixed equilibrium problems, nonexpansive mappings and monotone mappings

In this paper, we introduce iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of an infinite (a finite) family of nonexpansive mappings and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. We obtain some weak convergence theorems for the sequences generated by these processes in Hilbert spaces. The results in this paper generalize, extend and unify some well-known weak convergence theorems in the literature.

An Implicit Iteration for a Countable Family of Nonexpansive Mappings in Banach Spaces

We introduce an implicit sequence for an infinite family of nonexpansive mappings in a uniformly convex Banach space. We prove weak and strong convergence theorems for finding a common fixed point of the mappings. Our results not only include Plubtieng et al. (Numer. Funct. Anal. Optim. 2007; 28:737–749), Kikkawa and Takahashi (Ann. Univ. Mariae Curie-Skłodowska Sect. A 2004; 58:69–78), Kimura and Takahashi (Set-Valued Anal. 2008; 16:597–619) as special cases but also are established under the weaker assumptions.

Strong convergence theorems by hybrid methods for a finite family of nonexpansive mappings and inverse-strongly monotone mappings

In this paper, we introduce and study an iterative scheme by a hybrid method for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping in a real Hilbert space. Then, we prove that the iterative sequence converges strongly to a common element of the three sets. Using this result, we consider the problem of finding a common fixed point of a finite family of nonexpansive mappings and a strictly pseudocontractive mapping and the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of zeros of an inverse-strongly monotone mapping. The results obtained in this paper extend and improve the several recent results in this area.

Hybrid Iteration Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings in Banach Spaces

Let E be a real uniformly convex Banach space, and let{Ti : i ∈ I} be N nonexpansive mappings from E into itself with F = {x ∈ E : Tix = x, i ∈ I} ≠ ϕ, where I = {1, 2, …, N}. From an arbitrary initial point x1 ∈ E, hybrid iteration scheme {xn} is defined as follows: xn+1 = αnxn + (1 − αn)(Tnxn − λn+1μA(Tnxn)), n ≥ 1, where A : E → E is an L‐Lipschitzian mapping, Tn = Ti, i = n(mod N), 1 ≤ i ≤ N, μ > 0, {λn}⊂[0, 1), and {αn}⊂[a, b] for some a, b ∈ (0, 1). Under some suitable conditions, the strong and weak convergence theorems of {xn} to a common fixed point of the mappings {Ti : i ∈ I} are obtained. The results presented in this paper extend and improve the results of Wang (2007) and partially improve the results of Osilike, Isiogugu, and Nwokoro (2007).

A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings

LetXbe a real uniformly convex Banach space andCa closed convex nonempty subset ofX. Let{Ti}i=1rbe a finite family of nonexpansive self-mappings ofC. For a givenx1∈C, let{xn}and{xn(i)},i=1,2,…,r, be sequences definedxn(0)=xn,xn(1)=an1(1)T1xn(0)+(1-an1(1))xn(0),xn(2)=an2(2)T2xn(1)+an1(2)T1xn+(1-an2(2)-an1(2))xn,…, xn+1=xn(r)=anr(r)Trxn(r-1)+an(r-1)(r)Tr-1xn(r-2)+⋯+an1(r)T1xn+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))xn,n≥1, whereani(j)∈[0,1]for allj∈{1,2,…,r},n∈ℕandi=1,2,…,j. In this paper, weak and strong convergence theorems of the sequence{xn}to a common fixed point of a finite family of nonexpansive mappingsTi (i=1,2,…,r)are established under some certain control conditions.

Some results on non-expansive mappings and relaxed cocoercive mappings in Hilbert spaces

In this article, we introduce a new iterative scheme to investigate the problem of finding a common element of the set of common fixed points of a finite family of non-expansive mappings and the set of solutions of a variational inequality problem for a relaxed cocoercive mapping by viscosity approximate methods. Our results improve and extend the recent ones announced by Iiduka and Tahakshi [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005), pp. 341–350] and many others.

STRONG CONVERGENCE THEOREMS BY THE VISCOSITY APPROXIMATION METHOD FOR A COUNTABLE FAMILY OF NONEXPANSIVE MAPPINGS

In this paper, we extend Moudafi’s result in a Hilbert space to that in a Banach spaces. Then, we introduce implicit and explicit sequences for an infinite family of nonexpansive mappings in Banach spaces and prove strong convergence theorems for finding a common fixed point of the family of mappings.

Convergence of path and an iterative method for families of nonexpansive mappings

Let E be a real q-uniformly smooth Banach space with q ≥ 1 + d q . Let K be a closed, convex and nonempty subset of E. Let be a family of nonexpansive self-mappings of K. For arbitrary fixed δ ∈ (0, 1) define a family of nonexpansive maps by S i := (1 − δ)I + δT i where I is the identity map of K. Let Assume either at least one of the T i 's is demicompact or E admits weakly sequentially continuous duality map. It is proven that the fixed point sequence {z t n } converges strongly to a common fixed point of the family where and {t n } is a sequence in (0, 1), satisfying appropriate conditions. As an application, it is proven that the iterative sequence {x n } defined by: x 0 ∈ K, converges strongly to a common fixed point of the family where {α n } and {σ i,n } are sequences in (0, 1) satisfying appropriate conditions.

An Extragradient Approximation Method for Equilibrium Problems and Fixed Point Problems of a Countable Family of Nonexpansive Mappings

AbstractWe introduce a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. This main theorem extends a recent result of Yao et al. (2007) and many others.

Strong Convergence of a Modified Iterative Algorithm for Mixed-Equilibrium Problems in Hilbert Spaces

The purpose of this paper is to study the strong convergence of a modified iterative scheme to find a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a relaxed cocoercive mapping, as well as the set of solutions of a mixed-equilibrium problem. Our results extend recent results of Takahashi and Takahashi (2007), Marino and Xu (2006), Combettes and Hirstoaga (2005), Iiduka and Takahashi (2005), and many others.

Convergence Theorems of Fixed Points for a Finite Family of Nonexpansive Mappings in Banach Spaces

We modify the normal Mann iterative process to have strong convergence for a finite family nonexpansive mappings in the framework of Banach spaces without any commutative assumption. Our results improve the results announced by many others.

Approximation Methods for a Common Fixed Point of a Finite Family of Nonexpansive Mappings

Let K be a nonempty closed and convex subset of a real Banach space E. Let T: K → E be a continuous pseudocontractive mapping and f:K → E a contraction, both satisfying weakly inward condition. Then for t ∈ (0, 1), there exists a sequence {y t } ⊂ K satisfying the following condition: y t = (1 − t)f(y t ) + tT(y t ). Suppose further that {y t } is bounded or F(T) ≠ ∅ and E is a reflexive Banach space having weakly continuous duality mapping J ϕ for some gauge ϕ. Then it is proved that {y t } converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, an explicit iteration process that converges strongly to a common fixed point of a finite family of nonexpansive mappings and hence to a solution of a certain variational inequality is constructed.

Approximation of Fixed Points of Nonexpansive Mappings and Solutions of Variational Inequalities

Let be a real -uniformly smooth Banach space with constant , . Let and be a nonexpansive map and an -strongly accretive map which is also -Lipschitzian, respectively. Let be a real sequence in that satisfies the following condition: and . For and , define a sequence iteratively in by , , . Then, converges strongly to the unique solution of the variational inequality problem (search for such that for all , where . A convergence theorem related to finite family of nonexpansive maps is also proved.

Iterative approximation to common fixed points of a countable family of nonexpansive mappings

We proved several strong convergence results by using the conception of a uniformly asymptotically regular sequence {T n } of nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping J ϕ(l p (1 < p < ∞) has a weakly continuous duality map with gauge ϕ(t) = t p−1. The results presented develop and complement the corresponding ones by Song, Y. and Chen, R., 2007 [Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces. Nonlinear Analysis, 66, 591–603], Song, Y., Chen, R. and Zhou, H., 2007 [Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces. Nonlinear Analysis, 66, 1016–1024] and O'Hara, J.G., Pillay, P. and Xu, H.K., 2006 [Iterative approaches to convex feasibility problem in Banach Space. Nonlinear Analysis, 64, 2022–2042], O'Hara, J.G., Pillay, P. and Xu, H.K., 2003 [Iterative approaches to fineding nearest common fixed point of nonexpansive mappings in Hilbert spaces. Nonlinear Analysis, 54, 1417–1426] and Jung, J.S., 2005 [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications, 302, 509–520] and many other existing literatures.

Implicit Iterations of Two Finite Families for Nonexpansive Mappings in Banach Spaces

The purpose of this paper is to study the weak and strong convergence of an implicit iteration process to a common fixed point for two finite families of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Xu and Ori, Numer. Funct. Anal. Optim. 2001; 22:767–773, Zhou and Chang, Numer. Fund. Anal. Optim. 2002; 23:911–921, Chidume and Shahzad, Nonlinear. Anal. 2005; 62: 1149–1156.

Strong Convergence to Common Fixed Points of a Finite Family of Nonexpansive Mappings

We suggest and analyze an iterative algorithm for a finite family of nonexpansive mappings . Further, we prove that the proposed iterative algorithm converges strongly to a common fixed point of .