Vittorio Colao

ON AN IMPLICIT HIERARCHICAL FIXED POINT APPROACH TO VARIATIONAL INEQUALITIES

Abstract: Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-point problems, Fixed Point Theory Appl. (2006), Art. ID 95453, 10pp] and Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math. 13(6) (2009)] studied an implicit viscosity method for approximating solutions of variational inequalities by solving hierarchical fixed point problems. The approximate solutions are a net (xs,t) of two parameters s,tE (0,1), and under certain conditions, the iterated lim t→0lim s→0xs,t exists in the norm topology. Moudafi, Maingé and Xu stated the problem of convergence of (xs,t) as (s,t)→(0,0) jointly in the norm topology. In this paper we further study the behaviour of the net (xs,t); in particular, we give a negative answer to this problem.

Strong convergence of the modified Mann iterative method for strict pseudo-contractions

In this paper, we introduce a modified Mann iterative process for approximating a common fixed point of a finite family of strict pseudo-contractions in Hilbert spaces. We establish the strong convergence theorem of the general iteration scheme under some mild conditions. Our results extend and improve the recent ones announced by many others.

KRASNOSELSKI–MANN ITERATION FOR HIERARCHICAL FIXED POINTS AND EQUILIBRIUM PROBLEM

We give an explicit Krasnoselski–Mann type method for finding common solutions of the following system of equilibrium and hierarchical fixed points: { G(x, y)≥ 0, ∀y ∈ C, find x ∈ Fix(T ) such that 〈x − f (x), x − x〉 ≥ 0, ∀x ∈ Fix(T ), where C is a closed convex subset of a Hilbert space H , G : C × C→R is an equilibrium function, T : C→ C is a nonexpansive mapping with Fix(T ) its set of fixed points and f : C→ C is a ρ-contraction. Our algorithm is constructed and proved using the idea of the paper of [Y. Yao and Y.-C. Liou, ‘Weak and strong convergence of Krasnosel’skiı̆–Mann iteration for hierarchical fixed point problems’, Inverse Problems 24 (2008), 501–508], in which only the variational inequality problem of finding hierarchically a fixed point of a nonexpansive mapping T with respect to a ρ-contraction f was considered. The paper follows the lines of research of corresponding results of Moudafi and Théra. 2000 Mathematics subject classification: primary 47H09, 47H10; secondary 58E35.

Strong convergence theorems for approximating common fixed points of families of nonexpansive mappings and applications

An implicit algorithm for finding common fixed points of an uncountable family of nonexpansive mappings is proposed. A new inexact iteration method is also proposed for countable family of nonexpansive mappings. Several strong convergence theorems based on our main results are established in the setting of Banach spaces. Both algorithms are applied for finding zeros of accretive operators and for solving convex minimization, split feasibility and equilibrium problems.

Alternative iterative methods for nonexpansive mappings, rates of convergence and applications

Alternative iterative methods for a nonexpansive mapping in a Banach space are proposed and proved to be convergent to a common solution to a fixed point problem and a variational inequality. We give rates of asymptotic regularity for such iterations using proof-theoretic techniques. Some applications of the convergence results are presented.

On some auxiliary mappings generated by nonexpansive and strictly pseudo-contractive mappings

Abstract Starting by a finite family of mappings, we define the concept of procedure with Lipschitzian dependence of the coefficients. We give seven concrete examples of such procedures and prove the strong convergence of two viscosity methods.

Viscosity methods for common solutions for equilibrium and hierarchical fixed point problems

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 Krasnoselskiĭ–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.].

A general inexact iterative method for monotone operators, equilibrium problems and fixed point problems of semigroups in Hilbert spaces

AbstractLet H be a real Hilbert space. Consider on H a nonexpansive family T={T(t):0≤t<∞} with a common fixed point, a contraction f with the coefficient 0 < α < 1, and a strongly positive linear bounded self-adjoint operator A with the coefficient γ̄>0. Assume that 0<γ<γ̄/α and that S={St:0≤t<∞} is a family of nonexpansive self-mappings on H such that F(T)⊆F(S) and has property (A) with respect to the family . It is proved that the following schemes (one implicit and one inexact explicit): xt=btγfxt+I-btAStxt and x0∈H,xn+1=αnγfxn+βnxn+1-βnI-αnAStnxn+en,n≥0 converge strongly to a common fixed point x*∈F(T), where F(T) denotes the set of common fixed point of the nonexpansive semigroup. The point x* solves the variational in-equality 〈(γf −A)x*, x−x*〉 ≤ 0 for all x∈F(T). Various applications to zeros of monotone operators, solutions of equilibrium problems, common fixed point problems of nonexpansive semigroup are also presented. The results presented in this article mainly improve the corresponding ones announced by many others.2010 Mathematics Subject Classification: 47H09; 47J25.

On the Approximation of Zeros of Non-Self Monotone Operators

ABSTRACT In this article, we study the approximation of common zeros of non-self inverse strongly monotone operators defined on a closed convex subset C of a Hilbert space H. 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.