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

Colao Vittorio, Marino Giuseppe, Sahu Daya, Sahu

Description

Let 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 γ Ì„ > 0 . Assume that 0 < γ < γ Ì„ / α and that S = { S t : 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): x t = b t γ f x t + I - b t A S t x t and x 0 ∈ H , x n + 1 = α n γ f x n + β n x n + 1 - β n I - α n A S t n x n + e n , 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 .

Sujets

• Contraction
• Variational inequality

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 676
• Langue: English

Extrait

Colao et al . Fixed Point Theory and Applications 2012, 2012 :83 http://www.fixedpointtheoryandapplications.com/content/2012/1/83

R E S E A R C H Open Access A general inexact iterative method for monotone operators, equilibrium problems and f ı xed point problems of semigroups in Hilbert spaces Vittorio Colao 1* , Giuseppe Marino 1 and Daya Ram Sahu 2

* Correspondence: colao@mat. Abstract unical.it 1 Dipartimento di Matematica, Let H be a real Hilbert space. Consider on H a nonexpansive family Universita della Calabria, Arcavacata T = { T t : 0 ≤ t < ∞ with a common fixed point, a contraction f with the di Rende (Cs) 87036, Italy coefficient 0 < a < 1, and a strongly positive linear bounded self-adjoint operator A Full list of author information is available at the end of the article with the coefficient ¯ > 0 . Assume that 0 < γ < γ ¯ α and that S = { S ( t ) : 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): x t = b t γ ( x t ) + ( I − b t A ) S ( t ) x t and x 0 ∈ H , x n +1 = α n γ ( x n ) + β n x n + (( 1 − β n ) I − α n A ) S ( t n ) x n + e n , n ≥ 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 〈 ( g f − 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. Keywords: nonexpansive semigroup, common fixed point, contraction, variational inequality

1. Introduction Let H be a real Hilbert space and T be a nonlinear mapping with the domain D ( T ). A point x Î D ( T ) is a fixed point of T provided Tx = x . Denote by F ( T ) the set of fixed points of T ; that is, F ( T ) = { x Î D ( T ): Tx = x }. Recall that T is said to be nonexpansive if || Tx − T || ≤ || x − || , ∀ x , ∈ D ( T ) Recall that a family T = { T s : s ≥ 0 of mappings from H into itself is called a one- parameter nonexpansive semigroup if it satisfies the following conditions: (i) T (0) x = x , ∀ x Î H ; (ii) T ( s + t ) x = T ( s ) T ( t ) x , ∀ s , t ≥ 0 and ∀ x Î H ;

© 2012 Colao et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.