An Alternative Regularization Method for Equilibrium Problems and Fixed Point of Nonexpansive Mappings

We introduce a new regularization iterative algorithm for equilibrium and fixed point problems of nonexpansive mapping. Then, we prove a strong convergence theorem for nonexpansive mappings to solve a unique solution of the variational inequality and the unique sunny nonexpansive retraction. Our results extend beyond the results of S. Takahashi and W. Takahashi (2007), and many others.

Modified Noor's Extragradient Method for Solving Generalized Variational Inequalities in Banach Spaces

Motivated and inspired by Korpelevich's and Noor's extragradient methods, we suggest an extragradient method by using the sunny nonexpansive retraction which has strong convergence for solving the generalized variational inequalities in Banach spaces.

Iterative Approximation of Common Fixed Points of Two Nonself Asymptotically Nonexpansive Mappings

Suppose thatKis nonempty closed convex subset of a uniformly convex and smooth Banach spaceEwithPas a sunny nonexpansive retraction andF:=F(T1)∩F(T2)={x∈K:T1x=T2x=x}≠∅. LetT1,T2:K→Ebe two weakly inward nonself asymptotically nonexpansive mappings with respect toPwith two sequences{kn(i)}⊂[1,∞)satisfying∑n=1∞(kn(i)-1)<∞(i=1,2), respectively. For any givenx1∈K, suppose that{xn}is a sequence generated iteratively byxn+1=(1-αn)(PT1)nyn+αn(PT2)nyn,yn=(1-βn)xn+βn(PT1)nxn,n∈N, where{αn}and{βn}are sequences in[a,1-a]for somea∈(0,1). Under some suitable conditions, the strong and weak convergence theorems of{xn}to a common fixed point ofT1andT2are obtained.

Path Convergence and Approximation of Common Zeroes of a Finite Family ofm-Accretive Mappings in Banach Spaces

LetEbe a real Banach space which is uniformly smooth and uniformly convex. LetKbe a nonempty, closed, and convex sunny nonexpansive retract ofE, whereQis the sunny nonexpansive retraction. IfEadmits weakly sequentially continuous duality mappingj, path convergence is proved for a nonexpansive mappingT:K→K. As an application, we prove strong convergence theorem for common zeroes of a finite family ofm-accretive mappings ofKtoE. As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings fromKtoEunder certain mild conditions.

Strong convergence of approximation fixed points for nonexpansive nonself-mapping

LetCbe a closed convex subset of a uniformly smooth Banach spaceE, andT:C→Ea nonexpansive nonself-mapping satisfying the weakly inwardness condition such thatF(T)≠∅, andf:C→Ca fixed contractive mapping. Fort∈(0,1), the implicit iterative sequence{xt}is defined byxt=P(tf(xt)+(1−t)Txt), the explicit iterative sequence{xn}is given byxn+1=P(αnf(xn)+(1−αn)Txn), whereαn∈(0,1)andPis a sunny nonexpansive retraction ofEontoC. We prove that{xt}strongly converges to a fixed point ofTast→0, and{xn}strongly converges to a fixed point ofTasαnsatisfying appropriate conditions. The results presented extend and improve the corresponding results of Hong-Kun Xu (2004) and Yisheng Song and Rudong Chen (2006).

Construction of sunny nonexpansive retractions in Banach spaces

Let  be a commutative family of nonexpansive self-mappings of a closed convex subset C of a uniformly smooth Banach space X such that the set of common fixed points is nonempty. It is shown that if a certain regularity condition is satisfied, then the sunny nonexpansive retraction from C to F can be constructed in an iterative way.