Mann viscosity Tseng type method for inclusion and fixed point problems in Hilbert spaces

We propose a modified version of the classical Cesáro means method endowed with the hybrid shrinking projection method to solve the split equilibrium and fixed point problems (SEFPP) in Hilbert spaces. One of the main reasons to equip the classical Cesáro means method with the shrinking projection method is to establish strong convergence results which are often required in infinite-dimensional functional spaces. As a consequence, the convergence analysis is carried out under mild conditions on the underlying shrinking Cesáro means method. We emphasize that the results accounted in this manuscript can be considered as an improvement and generalization of various existing exciting results in this field of study.

In this paper, we introduce a new general alternative regularization algorithm for solving split equilibrium and fixed point problems in real Hilbert spaces. The proposed method does not require a prior estimate of the norm of the bounded linear operator nor a fixed stepsize for its convergence. Instead, we employ a line search technique and prove a strong convergence result for the sequence generated by the algorithm. A numerical experiment is given to show that the proposed method converges faster in terms of number of iteration and CPU time of computation than some existing methods in the literature.

<p style='text-indent:20px;'>In this paper, we present extension of a class of split variational inequality problem and fixed point problem due to Lohawech et al. (J. Ineq Appl. 358, 2018) to a class of multiple sets split variational inequality problem and common fixed point problem (CMSSVICFP) in Hilbert spaces. Using the Halpern subgradient extragradient theorem of variational inequality problems, we propose a parallel Halpern subgradient extragradient CQ-method with adaptive step-size for solving the CMSSVICFP. We show that a sequence generated by the proposed algorithm converges strongly to the solution of the CMSSVICFP. We give a numerical example and perform some preliminary numerical tests to illustrate the numerical efficiency of our method.</p>

A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings

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

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

Implicit and explicit viscosity methods for finding common solutions of equilibrium and hierarchical fixed points are presented. These methods are used to solve systems of equilibrium problems and variational inequalities where the involving operators are complements of nonexpansive mappings. The results here are situated on the lines of the research of the corresponding results of Moudafi [Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Probl. 23 (2007), pp. 1635–1640; Weak convergence theorems for nonexpansive mappings and equilibrium problems, to appear in JNCA], Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-points problems, Fixed Point Theory Appl. Art ID 95453 (2006), 10 pp.; Strong convergence of an iterative method for hierarchical fixed point problems, Pac. J. Optim. 3 (2007), pp. 529–538; Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems, to appear in JNCA], Yao and Liou [Weak and strong convergence of Krasnosel'skiĭ–Mann iteration for hierarchical fixed point problems, Inverse Probl. 24 (2008), 015015 8 pp.], S. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006), pp. 506–515], Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, preprint.], Combettes and Hirstoaga [Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), pp. 117–136] and Plubtieng and Pumbaeang [A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007), pp. 455–469.].

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.

An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings

This paper proposes a modified iterative algorithm using a viscosity approximation method with a weak contraction. The purpose is to find a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of a finite family of equilibrium problems that is also a solution to a variational inequality. Under suitable conditions, some strong convergence theorems are established in the framework of Hilbert spaces. The results presented in the paper improve and extend the corresponding results of Colao et al. (Colao, V., Acedo, G. L., and Marino, G. 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 Anal.71, 2708–2715 (2009)), Plubtieng and Punpaeng (Plubtieng, S. and Punpaeng, R. A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl.336, 455–469 (2007)), Colao et al. (Colao, V., Marino, G., and Xu, H. K. An iterative method for finding common solutions of equilibrium problem and fixed point problems. J. Math. Anal. Appl.344, 340–352 (2008)), Yao et al. (Yao, Y., Liou, Y. C., and Yao, J. C. Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Application2007, Article ID 64363 (2007) DOI 10.1155/2007/64363), and others.

In this paper, a monotone inclusion problem and a fixed point problem of nonexpansive mappings are investigated based on a Mann-type iterative algorithm with mixed errors. Strong convergence theorems of common elements are established in the framework of Hilbert spaces. MSC:47H05, 47H09, 47J25.

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

The purpose of this paper is to introduce an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a k−strict pseudo-contraction non-self mapping in Hilbert space. By the viscosity approximation algorithms, under suitable conditions , some strong convergence theorems for approximating to this common elements are proved. The results presented in the paper extend and improve some recent results of Marino and Xu [G.Marino,H.K.Xu, Weak and strong convergence theorems for k−strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–349], Zhou [H.Zhou, Convergence theorems of fixed Points for k−strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 69 (2008) 456–462], 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], Ceng,Homidan,etc [L. C. Ceng, S.A.Homidan, Q.H.Ansari, J. C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math. 223 (2009) 967–974].

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

The purpose of this paper is to present an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with Lipschitzian and strongly monotone operator, fixed point of nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert spaces. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper improve and extend the main results of M. Tian [A general iterative algorithem for nonexpansive mappings in Hilbert space, Nonlinear Analysis, 73(2010)689-694] and Plubtemg and Sripard [A viscosity approximation method for finding common solution of variational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces, Fixed Point Theory and Applications, vol.2009, Article ID 567147,20 pages].