Abstract:
Viscosity approximation methods for nonexpansive nonself-mappings are studied. Let C be a nonempty closed convex subset of Hilbert space H, P a metric projection of H onto C and let T be a nonexpansive nonself-mapping from C into H. For a contraction f on C and {tn}⊆(0,1), let xn be the unique fixed point of the contraction x↦tnf(x)

Abstract:
We introduce two general hybrid iterative approximation methods (one implicit and one explicit) for finding a fixed point of a nonexpansive mapping which solving the variational inequality generated by two strongly positive bounded linear operators. Strong convergence theorems of the proposed iterative methods are obtained in a reflexive Banach space which admits a weakly continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Marino and Xu (2006), Wangkeeree et al. (in press), and Ceng et al. (2009).

Abstract:
We 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.

Abstract:
Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E*, C a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E, and T:C ￠ ’E a non-expansive nonself-mapping with F(T) ￠ ‰ ￠ …. In this paper, we study the strong convergence of two sequences generated by xn+1= ±nx+(1 ￠ ’ ±n)(1/n+1) ￠ ‘j=0n(PT)jxn and yn+1=(1/n+1) ￠ ‘j=0nP( ±ny+(1 ￠ ’ ±n)(TP)jyn) for all n ￠ ‰ ￥0, where x,x0,y,y0 ￠ C, { ±n} is a real sequence in an interval [0,1], and P is a sunny non-expansive retraction of E onto C. We prove that {xn} and {yn} converge strongly to Qx and Qy, respectively, as n ￠ ’ ￠ , where Q is a sunny non-expansive retraction of C onto F(T). The results presented in this paper generalize, extend, and improve the corresponding results of Matsushita and Kuroiwa and many others.

Abstract:
Let be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from to , a nonempty closed convex subset of which is also a sunny nonexpansive retract of , and a non-expansive nonself-mapping with . In this paper, we study the strong convergence of two sequences generated by and for all , where , is a real sequence in an interval , and is a sunny non-expansive retraction of onto . We prove that and converge strongly to and , respectively, as , where is a sunny non-expansive retraction of onto . The results presented in this paper generalize, extend, and improve the corresponding results of Matsushita and Kuroiwa and many others.

Abstract:
We 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.

Abstract:
We consider a hybrid projection algorithm based on the shrinking projection method for two families of quasi--nonexpansive mappings. We establish strong convergence theorems for approximating the common element of the set of the common fixed points of such two families and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. As applications, at the end of the paper we first apply our results to consider the problem of finding a zero point of an inverse-strongly monotone operator and we finally utilize our results to study the problem of finding a solution of the complementarity problem. Our results improve and extend the corresponding results announced by recent results.

Abstract:
We introduce an iterative scheme for finding a common element of the set of fixed points of a -strictly pseudocontractive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping, and the set of solutions of the mixed equilibrium problem in a real Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we first apply our results to study the optimization problem and we next utilize our results to study the problem of finding a common element of the set of fixed points of two families of finitely -strictly pseudocontractive mapping, the set of solutions of the variational inequality, and the set of solutions of the mixed equilibrium problem. The results presented in the paper improve some recent results of Kim and Xu (2005), Yao et al. (2008), Marino et al. (2009), Liu (2009), Plubtieng and Punpaeng (2007), and many others.

Abstract:
We introduce an iterative scheme by the viscosity approximation to find the set of solutions of the generalized system of relaxed cocoercive quasivariational inclusions and the set of common fixed points of an infinite family of strictly pseudocontractive mappings problems in Hilbert spaces. We suggest and analyze an iterative scheme under some appropriate conditions imposed on the parameters; we prove that another strong convergence theorem for the above two sets is obtained. The results presented in this paper improve and extend the main results of Li and Wu (2010) and many others. 1. Introduction and Preliminaries Let be a real Hilbert space with inner product and norm being denoted by and , respectively, and let be a nonempty closed convex subset of . Recall that is the metric projection of onto ; that is, for each there exists the unique point in such that A mapping is called nonexpansive if and the mapping is called a contraction if there exists a constant such that A point is a fixed point of provided . We denote by the set of fixed points of ; that is, . If is bounded, closed and convex and is a nonexpansive mappings of into itself, then is nonempty (see [1]). Recall that a mapping is said to be (i)monotone if (ii) -Lipschitz continuous if there exists a constant such that if , then is a nonexpansive, (iii)pseudocontractive if (iv) -strictly pseudocontractive if there exists a constant such that it is obvious that is a nonexpansive if and only if is a -strictly pseudocontractive, (v) -strongly monotone if there exists a constant such that (vi) -inverse-strongly monotone (or -cocoercive) if there exists a constant such that if , then is called that firmly nonexpansive; it is obvious that any -inverse-strongly monotone mapping is monotone and (1/ )-Lipschitz continuous, (vii)relaxed -cocoercive if there exists a constant such that (viii)relaxed -cocoercive if there exists two constants such that it is obvious that any the -strongly monotonicity implies to the relaxed -cocoercivity. Recall that a set-valued mapping is called monotone if for all and imply . A monotone mapping is maximal if the graph of of is not properly contained in the graph of any other monotone mappings. The existence common fixed points for a finite family of nonexpansive mappings has been considered by many authers (see [2–5] and the references therein). In this paper, we study the mapping defined by where is nonnegative real sequence in , for all , form a family of infinitely nonexpansive mappings of into itself. It is obvious that is nonexpansive from into itself, such a

Abstract:
Suppose that C is a nonempty closed convex subset of a real uniformly convex Banach space X. Let T:C→C be an asymptotically quasi-nonexpansive mapping. In this paper, we introduce the three-step iterative scheme for such map with error members. Moreover, we prove that if T is uniformly L-Lipschitzian and completely continuous, then the iterative scheme converges strongly to some fixed point of T.