Search

Let C be a closed, convex and nonempty subset of a q-uniformly smooth Banach space X. Let be a strict pseducontractive inward mapping. We consider the boundary point map depending on C and T defined by , for all. Then for a suitable step-by-step construction of the control coefficients by using the function , we show the convergence of the Mann-Dotson algorithm to a fixed point of T. We obtain strong convergence if and weak convergence if. [ABSTRACT FROM AUTHOR]

A matrix approach to approximating solutions of variational inequalities in Hilbert spaces is introduced. This approach uses two matrices: one for iteration process and the other for regularization. Ergodicity and convergence (both weak and strong) are studied. Our methods combine new or well-known iterative methods (such as the original Mann’s method) with regularized processes involved regular matrices in the sense of Toeplitz. [ABSTRACT FROM PUBLISHER]

In this article, we study the approximation of common zeros of non-self inverse strongly monotone operators defined on a closed convex subsetCof a Hilbert spaceH. For a non-self family of operators, we introduce an iterative algorithm without relying on projections. Approximation of common fixed points for finite families of non-self strict pseudo-contractions in the sense of Browder-Petryshyn is also obtained. The novelty of our algorithm is that the coefficients are not given a priori and no assumptions are made on them, but they are constructed step by step in a natural way. [ABSTRACT FROM PUBLISHER]

In this article, we propose a modified Korpelevich's method for solving variational inequalities. Under some mild assumptions, we show that the suggested method converges strongly to the minimum-norm solution of some variational inequality in an infinite-dimensional Hilbert space. [ABSTRACT FROM PUBLISHER]

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 Mainge [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'skii˘-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.]. [ABSTRACT FROM AUTHOR]