New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces

Recently, Ceng, Guu and Yao introduced an iterative scheme by viscosity-like approximation method to approximate the fixed point of nonexpansive mappings and solve some variational inequalities in Hilbert space (see Ceng et al. (2009) [9]). Takahashi and Takahashi proposed an iteration scheme to solve an equilibrium problem and approximate the fixed point of nonexpansive mapping by viscosity approximation method in Hilbert space (see Takahashi and Takahashi (2007) [12]). In this paper, we introduce an iterative scheme by viscosity approximation method for finding a common element of the set of a countable family of nonexpansive mappings and the set of an equilibrium problem in a Hilbert space. We prove the strong convergence of the proposed iteration to the unique solution of a variational inequality.     

Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces

The purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of two quasi-?$?$-nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Our results improve and extend the corresponding results announced by many others.     

A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications

The purpose of this paper is to prove by using a new hybrid method a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem, the set of solutions for a variational inequality problem and the set of common fixed points for a pair of relatively nonexpansive mappings in a Banach space. As applications, we utilize our results to obtain some new results for finding a solution of an equilibrium problem, a fixed point problem and a common zero-point problem for maximal monotone mappings in Banach spaces.     

A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for α$\alpha$-inverse-strongly monotone mappings in a 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 utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings. The results presented in the paper extend and improve some recent results of Yao and Yao [Y.Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2) (2007) 1551–1558], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings, Appl. Math. Comput. (2007) doi:10.1016/j.amc.2007.07.075], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for Equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006) 506–515], Su, Shang and Qin [Y.F. Su, M.J. Shang, X.L. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. (2007) doi:10.1016/j.na.2007.08.045] and Chang, Cho and Kim [S.S. Chang, Y.J. Cho, J.K. Kim, Approximation methods of solutions for equilibrium problem in Hilbert spaces, Dynam. Systems Appl. (in print)].     

A hybrid iterative scheme for mixed equilibrium problems and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions of a mixed equilibrium problem (MEP) and the set of common fixed points of finitely many nonexpansive mappings in a real Hilbert space. First, by using the well-known KKM technique we derive the existence and uniqueness of solutions of the auxiliary problems for the MEP. Second, by virtue of this result we introduce a hybrid iterative scheme for finding a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings. Furthermore, we prove that the sequences generated by the hybrid iterative scheme converge strongly to a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings.     

Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications

In this paper, we introduce a general iterative scheme for finding a common element of the set of common solutions of generalized equilibrium problems, the set of common fixed points of a family of infinite non-expansive mappings. Strong convergence theorems are established in a real Hilbert space under suitable conditions. As some applications, we consider convex feasibility problems and equilibrium problems. The results presented improve and extend the corresponding results of many others.     

An implicit iterative scheme for monotone variational inequalities and fixed point problems

In this paper we introduce an implicit iterative scheme for finding a common element of the set of common fixed points of N$N$ nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The implicit iterative scheme is based on two well-known methods: extragradient and approximate proximal. We obtain a weak convergence theorem for three sequences generated by this implicit iterative scheme. On the basis of this theorem, we also construct an implicit iterative process for finding a common fixed point of N+1$N+1$ mappings, such that one of these mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other N$N$ mappings are nonexpansive.     

Fixed point solutions of generalized equilibrium problems for nonexpansive mappings

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a generalized equilibrium problem in a real Hilbert space. Then, strong convergence of the scheme to a common element of the two sets is proved. As an application, problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem is solved. Moreover, solution is given to the problem of finding a common element of fixed points set of nonexpansive mappings and the set of solutions of a variational inequality problem.     

Strong convergence theorem by a new hybrid method for equilibrium problems and variational inequality problems

In this paper, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of the solutions of the variational inequality problem by using a new hybrid method. We obtain a new result for finding a solution of an equilibrium problem and the solutions of the variational inequality problem.     

Hybrid pseudo-viscosity approximation schemes for systems of equilibrium problems and fixed point problems of infinite family and semigroup of non-expansive mappings

In this paper, we introduce hybrid pseudo-viscosity approximation schemes with strongly positive bounded linear operators for finding a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of an infinite family and left amenable semigroup of non-expansive mappings in the frame work of Hilbert spaces. Our goal is to prove a result of strong convergence for hybrid pseudo-viscosity approximation schemes to approach a solution of systems of equilibrium problems which is also a common fixed point of an infinite family and left amenable semigroup of non-expansive mappings. The results presented in this paper can be treated as an extension and improvement of the corresponding results announced by Ceng et al. [L.C. Ceng, Q.H. Ansari, and J.C. Yao, Hybrid pseudo-viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many non-expansive mappings, Nonlinear Analysis 4 (2010) 743-754] and many others.     

A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces

The purpose of this paper is to study the strong convergence of a general iterative scheme to find a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of variational inequality for a relaxed cocoercive mapping and the set of solutions of an equilibrium problem. Our results extend the recent results of Takahashi and Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515], Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43–52], Combettes and Hirstoaga [P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 486–491], Iiduka and Takahashi, [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and many others.     

Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points

In this paper, we present a new multi-step iterative method. We prove the strong convergence of the method to a common fixed point of a finite number of nonexpansive mappings that also solves a suitable equilibrium problem.     

Path convergence,approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces

Let E$E$ be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K$K$ be a nonempty closed convex subset of E$E$, and let T:K?E$T:K?E$ be a continuous pseudocontraction which satisfies the weakly inward condition. For f:K?K$f:K?K$ any contraction map on K$K$, and every nonempty closed convex and bounded subset of K$K$ having the fixed point property for nonexpansive self-mappings, it is shown that the path x→_xt,t∈[0,1)$x\to x_t,t\in [0,1)$, in K$K$, defined by _xt=tT_xt+(1−t)f(_xt)$x_t=tT{x}_t+(1-t)f\left(x_t\right)$ is continuous and strongly converges to the fixed point of T$T$, which is the unique solution of some co-variational inequality. If, in particular, T$T$ is a Lipschitz pseudocontractive self-mapping of K$K$, it is also shown, under appropriate conditions on the sequences of real numbers {_αn},{_μn}$\left\{{\alpha }_n\right\},\left\{{\mu }_n\right\}$, that the iteration process: _z1∈K$z_1\in K$, _zn+1=_μn(_αnT_zn+(1−_αn)_zn)+(1−_μn)f(_zn),n∈N$z_{n+1}={\mu }_n({\alpha }_{n}T{z}_n+(1-{\alpha }_n)z_n)+(1-{\mu }_n)f\left(z_n\right),n\in \mathbb{N}$, strongly converges to the fixed point of T$T$, which is the unique solution of the same co-variational inequality. Our results propose viscosity approximation methods for Lipschitz pseudocontractions.     

A new iteration process for finite families of generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings

A new iteration process is introduced and proved to converge strongly to a common fixed point for a finite family of generalized Lipschitz nonlinear mappings in a real reflexive Banach space E$E$ with a uniformly Gâteaux differentiable norm if at least one member of the family is pseudo-contractive. It is also proved that a slight modification of the process converges to a common zero for a finite family of generalized Lipschitz accretive operators defined on E$E$. Results for nonexpansive families are obtained as easy corollaries. Finally, the new iteration process and the method of proof are of independent interest.     

An explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings

Let H be a Hilbert space and C be a nonempty closed convex subset of H, {T_i}_i∈N be a family of nonexpansive mappings from C into H, G_i:C×C→R be a finite family of equilibrium functions (i∈{1,2,…,K}), A be a strongly positive bounded linear operator with a coefficient and -Lipschitzian, relaxed (μ,ν)-cocoercive map of C into H. Moreover, let , {α_n} satisfy appropriate conditions and ; we introduce an explicit scheme which defines a suitable sequence as follows:

     

Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications

In this paper, we prove a strong convergence theorem by the hybrid method for a countable family of relatively nonexpansive mappings in a Banach space. We also establish a new control condition for the sequence of mappings {T_n} which is weaker than the control condition in Lemma 3.1 of Aoyama et al. [K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350-2360]. Moreover, we apply our results for finding a common fixed point of two relatively nonexpansive mappings in a Banach space and an element of the set of solutions of an equilibrium problem in a Banach space, respectively. Our results are applicable to a wide class of mappings.     

Strong convergence theorems for a finite family of nonexpansive mappings and semigroups via the hybrid method

Strong convergence theorems are obtained for a finite family of nonexpansive mappings and semigroups by the hybrid method.     

A contraction proximal point algorithm with two monotone operators

It is a known fact that the method of alternating projections introduced long ago by von Neumann fails to converge strongly for two arbitrary nonempty, closed and convex subsets of a real Hilbert space. In this paper, a new iterative process for finding common zeros of two maximal monotone operators is introduced and strong convergence results associated with it are proved. If the two operators are subdifferentials of indicator functions, this new algorithm coincides with the old method of alternating projections. Several other important algorithms, such as the contraction proximal point algorithm, occur as special cases of our algorithm. Hence our main results generalize and unify many results that occur in the literature.