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

作者：Colao Vittorio^*; Marino Giuseppe; Sahu Daya Ram

来源：Fixed Point Theory and Applications, 2012, 83.

DOI：10.1186/1687-1812-2012-83

• nonexpansive semigroup
• common fixed point
• contraction
• variational inequality

Let H be a real Hilbert space. Consider on H a nonexpansive family T = {T(t) : 0 %26lt;= t %26lt; infinity} with a common fixed point, a contraction f with the coefficient 0 %26lt; alpha %26lt; 1, and a strongly positive linear bounded self-adjoint operator A with the coefficient %26lt;(gamma)over bar%26gt; 0. Assume that 0 %26lt; gamma %26lt; (gamma) over bar/alpha and that S = {S (t) : 0 %26lt;= t %26lt; infinity} is a family of nonexpansive self-mappings on H such that F(T) subset of F(S) and T has property (A) with respect to the family S. It is proved that the following schemes (one implicit and one inexact explicit): %26lt;br%26gt;x(t) = b(t)gamma f (x(t)) + (I - b(t)A) S (t) x(t) %26lt;br%26gt;and %26lt;br%26gt;x(0) is an element of H, x(n+1) = alpha(n)gamma f (x(n)) + beta(n)x(n) + ((1 - beta(n)) I - alpha(n)A) S (tn) x(n) + e(n), n %26gt;= 0 %26lt;br%26gt;converge strongly to a common fixed point x* is an element of F(T), where F(T) denotes the set of common fixed point of the nonexpansive semigroup. The point x* solves the variational in-equality %26lt;(gamma f -A)x*, x-x*%26gt; %26lt;= 0 for all x is an element of 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.

• 出版日期2012