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

Fixexd point theory and applications

Daya Ram Sahu¹  Giuseppe Marino²  Vittorio Colao² 

[1] Department of Mathematics, Banaras Hindu University, Varanasi, India;Dipartimento di Matematica, Universita della Calabria, Arcavacata di Rende (Cs), Italy

关键词: nonexpansive semigroup;    common fıxed point;    contraction;    variational inequality;   

DOI  :  10.1186/1687-1812-2012-83

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

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 γ̄ >0. Assume that 0<γ<γ̄ /α and that S={S(t):0≤t<∞} is a family of nonexpansive self-mappings on H such that F(T)⊆F(S) andhas property (A) with respect to the family. It is proved that the following schemes (one implicit and one inexact explicit):xt=btγf(xt)+(I−btA)S(t)xtandx0∈H,xn+1=αnγf(xn)+βnxn+((1−βn)I−αnA)S(tn)xn+en,n≥0converge 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.

【 授权许可】

CC BY