Ergodic approximations via matrix regularization approach

AbstractIn this paper we use a matrix approach to approximate solutions of variational inequalities in Hilbert spaces. The methods studied combine new or well-known iterative methods (as the original Mann method) with regularized processes that involve regular matrices in the sense of Toeplitz. We obtain ergodic type results and convergence

Krasnoselskii-Mann method for non-self mappings

AbstractLet H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If $T:C\to H$ T : C → H is a non-self and non-expansive mapping, we can define a map $h:C\to\mathbb{R}$ h : C → R by $h(x):=\inf\{\lambda\geq 0:\lambda x+(1-\lambda)Tx\in C\}$ h ( x ) : = inf { λ ≥ 0 : λ x + ( 1 − λ ) T x ∈ C } . Then, for a fixed $x_{0}\in C$ x 0 ∈ C and for $\alpha_{0}:=\max\{1/2, h(x_{0})\}$ α 0 : = max { 1 / 2 , h ( x 0 ) } , we define the Krasnoselskii-Mann algorithm $x_{n+1}=\alpha _{n}x_{n}+(1-\alpha_{n})Tx_{n}$ x n + 1 = α n x n + ( 1 − α n ) T x n , where $\alpha_{n+1}=\max\{\alpha_{n},h(x_{n+1})\}$ α n + 1 = max { α n , h ( x n + 1 ) } . We will prove both weak and strong convergence results when C is a strictly convex set and T is an inward mapping

Viscosity Method for Hierarchical Fixed Point Problems with an Infinite Family of Nonexpansive Nonself-Mappings

A viscosity method for hierarchical fixed point problems is presented to solve variational inequalities, where the involved mappings are nonexpansive nonself-mappings. Solutions are sought in the set of the common fixed points of an infinite family of nonexpansive nonself-mappings. The results generalize and improve the recent results announced by many other authors

On solving variational inequalities defined on fixed point sets of multivalued mappings in Banach spaces

AbstractWe are concerned with the problem of solving variational inequalities which are defined on the set of fixed points of a multivalued nonexpansive mapping in a reflexive Banach space. Both implicit and explicit approaches are studied. Strong convergence of the implicit method is proved if the space satisfies Opial's condition and has a duality map weakly continuous at zero, and the strong convergence of the explicit method is proved if the space has a weakly continuous duality map. An essential assumption on the multivalued nonexpansive mapping is that the mapping be single valued on its nonempty set of fixed points

Convergence Theorems for Hierarchical Fixed Point Problems and Variational Inequalities

This paper deals with a modifed iterative projection method for approximating a solution of hierarchical fixed point problems for nearly nonexpansive mappings. Some strong convergence theorems for the proposed method are presented under certain approximate assumptions of mappings and parameters. As a special case, this projection method solves some quadratic minimization problem. It should be noted that the proposed method can be regarded as a generalized version of Wang et.al. [15], Ceng et. al. [14], Sahu [4] and many other authors.Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1403.321

Some Weak Convergence Theorems for a Family of Asymptotically Nonexpansive Nonself Mappings

A one-step iteration with errors is considered for a family of asymptotically nonexpansive nonself mappings. Weak convergence of the purposed iteration is obtained in a Banach space

Some Results on the Approximation of Solutions of Variational Inequalities for Multivalued Maps on Banach Spaces

AbstractMultivalued $$*$$ ∗ -nonexpansive mappings are studied in Banach spaces. The demiclosedness principle is established. Here we focus on the problem of solving a variational inequality which is defined on the set of fixed points of a multivalued $$*$$ ∗ -nonexpansive mapping. For this purpose, we introduce two algorithms approximating the unique solution of the variational inequality