Infinite Family Of Nonexpansive Mappings Research Articles

Strong convergence theorems for a family non-expansive mappings and application to generalized mixed equilibrium problems systems

In this paper, we introduce an iterative scheme for finding common fixed point of an infinite family of non-expansive mappings in Hilbert space. Furthermore, we prove that the proposed iterative scheme converges strongly to a common fixed point. As an application, we succeed in applying our algorithm to find a solution of generalized mixed equilibrium problems systems. Our results improve and extend the corresponding results announced by W. Takahashi, Y. Takeuchi, R. Kubota (J. Math. Anal. Appl. 341:276–286, 2008); S. Takahashi, W. Takahashi (Nonlinear Anal. 69:1025–1033, 2008); N. Petrot, K. Wattanawitoon, P. Kumam (Nonlinear Anal. Hybrid Syst., 2010, doi:10.1016/j.nahs.2010.03.008) and some others in references.

A general iterative method for addressing mixed equilibrium problems and optimization problems

In this paper, we introduce a new general iterative method for finding a common element of the set of solutions of a mixed equilibrium problem (MEP), the set of fixed points of an infinite family of nonexpansive mappings { T n } n = 1 ∞ and the set of solutions of variational inequalities for a ξ -inverse-strongly monotone mapping in Hilbert spaces. Furthermore, we establish the strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under some suitable conditions, which solves some optimization problems. Our results extend and improve the recent results of Yao et al. [Y. Yao, M.A. Noor, S. Zainab, Y.C. Liou, Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl. 354 (2009) 319–329; Y. Yao, M. A. Noor, Y.C. Liou, On iterative methods for equilibrium problems, Nonlinear Anal. 70 (1) (2009) 479–509] and many others.

Modified hybrid steepest-descent methods for variational inequalities and fixed points

Assume that C is a nonempty closed convex subset of a Hilbert space H and B : C → H is a strongly monotone mapping. Assume also that F is the intersection of the common fixed points of an infinite family of nonexpansive mappings on C and the set of solutions of a system of equilibrium problems. We devise a modified hybrid steepest-descent method which generates a sequence ( x n ) from an arbitrary initial point x 0 ∈ H . The sequence ( x n ) is shown to converge in norm to the unique solution of the variational inequality V I ( B , F ) under suitable conditions.

A new hybrid iterative algorithm for variational inequalities

Let H be a real Hilbert space. Let F : H → H be a strongly monotone and Lipschitzian mapping. Let { T n } n = 1 ∞ : H → H be an infinite family of non-expansive mappings with common fixed points set ⋂ n = 1 ∞ Fix ( T n ) ≠ ∅ . We devise an iterative algorithm y n = x n - λ n F ( x n ) , x n + 1 = ( 1 - α n ) y n + α n W n y n , n ⩾ 0 , where { λ n } is a sequence in ( 0 , ∞ ) , { α n } is a sequence in (0,1) and W n is the W-mapping. We prove that the sequence { x n } converges in norm to x ∗ ∈ ⋂ n = 1 ∞ Fix ( T n ) which is the unique solution of the following variational inequality 〈 Fx ∗ , x - x ∗ 〉 ⩾ 0 , ∀ x ∈ ⋂ n = 1 ∞ Fix ( T n ) .

Strong convergence of the modified Ishikawa iterative method for infinitely many nonexpansive mappings in Banach spaces

In this paper, we introduce a new modified Ishikawa iterative process for computing fixed points of an infinite family nonexpansive mapping in the framework of Banach spaces. Then, we establish the strong convergence theorem of the proposed iterative scheme under some mild conditions which solves a variational inequality. The results obtained in this paper extend and improve on the recent results of Qin et al. [Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces, Journal of Computational and Applied Mathematics 230 (1) (2009) 121–127], Cho et al. [Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces, Computers and Mathematics with Applications 56 (2008) 2058–2064] and many others.

New Hybrid Iterative Schemes for an Infinite Family of Nonexpansive Mappings in Hilbert Spaces

We propose some new iterative schemes for finding common fixed point of an infinite family of nonexpansive mappings in a Hilbert space and prove the strong convergence of the proposed schemes. Our results extend and improve ones of Nakajo and Takahashi (2003).

A New Iterative Method for Solving Equilibrium Problems and Fixed Point Problems for Infinite Family of Nonexpansive Mappings

We introduce a new iterative scheme for finding a common element of the solutions sets of a finite family of equilibrium problems and fixed points sets of an infinite family of nonexpansive mappings in a Hilbert space. As an application, we solve a multiobjective optimization problem using the result of this paper.

A New Iterative Method for Finding Common Solutions of a System of Equilibrium Problems, Fixed-Point Problems, and Variational Inequalities

We introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed point sets of an infinite family of nonexpansive mappings, and the solution set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove strong convergence theorem. The results in this paper unify and generalize some well-known results in the literature.

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 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 γ ̄ and B λ -Lipschitzian, relaxed ( μ , ν ) -cocoercive map of C into H . Moreover, let { r k , n } k = 1 K , { α n } satisfy appropriate conditions and F ≔ ( ∩ k = 1 K E P ( G k ) ) ∩ V I ( C , B ) ∩ ( ∩ n ∈ N F i x ( T n ) ) ≠ 0̸ ; we introduce an explicit scheme which defines a suitable sequence as follows: x n + 1 = α n γ f ( x n ) + ( 1 − α n A ) W n P C ( I − s n B ) S r 1 , n 1 S r 2 , n 2 ⋯ S r K , n K x n ∀ n ∈ N and { x n } strongly converges to x ∗ ∈ F which satisfies the variational inequality 〈 ( A − γ f ) x ∗ , x − x ∗ 〉 ≥ 0 for all x ∈ F . The results presented in this paper mainly extend and improve a recent result of Colao [V. Colao, An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings, Nonlinear Analysis (2009), doi:10.1016/j.na.2009.01.115] and Qin [X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis 69 (2008) 3897–3909].

Some results of variational inclusion problems and fixed point problems with applications

This paper introduces a general iterative algorithm to approximate a common element in the solution set of quasi-variational inclusion problems and the common fixed point set of an infinite family of nonexpansive mappings. It is proven that the iterative sequences generated in the proposed iterative algorithm converge strongly to some common element in the framework of the real Hilbert spaces.

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.

A general iterative method for solving equilibrium problems, variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings

In this paper, we introduce and analyze a new general iterative scheme by the viscosity approximation method for finding the common element of the set of equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings and the set solutions of the variational inequality problems for an ξ-inverse-strongly monotone mapping in Hilbert spaces. We show that the sequence converge strongly to a common element of the above three sets under some parameters controlling conditions. The result extends and improves a recent result of Chang et al. (Nonlinear Anal. 70:3307–3319, 2009) and many others.

Strong convergence of the iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems of an infinite family of nonexpansive mappings

In this paper, we introduce an iterative scheme based on the extragradient approximation method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a mixed equilibrium problem, and the set of solutions of the variational inequality problem for a monotone L-Lipschitz continuous mapping in a real Hilbert space. Then, the strong convergence theorem is proved under some parameters controlling conditions. Applications to optimization problems are given. The results obtained in this paper improve and extend the recent ones announced by Wangkeeree [R. Wangkeeree, An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings, Fixed Point Theory and Applications (2008) 17. doi:10.1155/2008/134148. Article ID 134148], Kumam and Katchang [P. Kumam, P. Katchang, A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings, Nonlinear Anal. Hybrid Syst. (2009) doi:10.1016/j.nahs.2009.03.006] and many others.

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.

An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems

In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a general system of variational inequalities for a cocoercive mapping in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets. Our results extend and improve the corresponding results of Ceng, Wang, and Yao [L.C. Ceng, C.Y. Wang, J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res. 67 (2008) 375–390], Ceng and Yao [L.C. Ceng, J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. doi:10.1016/j.cam.2007.02.022], 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] and many others.

A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings

In this paper, we introduce a new viscosity approximation scheme based on the extragradient method 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 of nonexpansive mappings and the set of solutions to the variational inequality for a monotone, Lipschitz continuous mapping. Several convergence results for the sequences generated by these processes in Hilbert spaces were derived.

A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems

In this paper, we introduce and study a new hybrid iterative method for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of variational inequalities for a ξ -Lipschitz continuous and relaxed ( m , v ) -cocoercive mappings in Hilbert spaces. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm which solves some optimization problems under some suitable conditions. Our results extend and improve the recent results of Yao et al. [Y. Yao, M.A. Noor, S. Zainab and Y.C. Liou, Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl (2009). doi:10.1016/j.jmaa.2008.12.005] and Gao and Guo [X. Gao and Y. Guo, Strong convergence theorem of a modified iterative algorithm for Mixed equilibrium problems in Hilbert spaces, J. Inequal. Appl. (2008). doi:10.1155/2008/454181] and many others.

An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems

In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of some variational inequality. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets. Our results extend and improve the corresponding results of Zeng and Yao [L.C. Zeng and J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008) 186–201], Takahashi and Takahashi [S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331(2007), 506–515] and many others.

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

Let H be a Hilbert space. Consider on H a sequence of nonexpansive mappings { T n } with common fixed points, a finite family of equilibrium functions { G i } i = 1 , … , K , a contraction f with coefficient 0 < α < 1 and a strongly positive linear bounded operator A with coefficient γ ¯ > 0 . Let 0 < γ < γ ¯ / α . Assuming there are common equilibrium points of the family { G i } i = 1 , … , K which are also fixed points for { T n } , we define a suitable sequence which strongly converges to the unique such point which also satisfies the variational inequality 〈 ( A − γ f ) x ∗ , x − x ∗ 〉 ≥ 0 for all the x in the intersection of the equilibrium points and the common fixed points of the sequence { T n } .