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Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces. (English) Zbl 1210.47080

A continuous semigroup T(s), s0, in a Hilbert space H is called nonexpansive if T(s)x-T(s)yx-y for arbitrary x,y in H. For a bifunction G:H×S, EP(G) denotes the set of equilibrium points, i.e., all x such that G(x,y)0 for all yH. The present paper studies iterative methods of finding solutions to a variational problem among equilibrium points, which are fixed points of T.

The authors present two iterative constructions, which they call implicit and explicit. In the implicit iterative construction, a pair of continuous sequences x t , u t H is defined as solution to

G(u t ,y)+1 r t y-u t ,u t -x t 1,yH

and

x t =tγf(x t )+(I-tA)1 λ t 0 λ t T(s)u t ds·

Here, 0<t<1 and A is a strongly positive bounded linear operator.

The explicit iterations are defined by the system

x n+1 =α n γf(x n )+(I-α n A)1 s n 0 s n t(s)u n ds

and

G(u n ,y)+1 r n y-u n ,u n -x n 0,yH·

It is demonstrated that, with an appropriate definition of the coefficients r t ,λ t ,α n ,s n , the implicit sequences x t ,u t , and the the explicit sequences x n , u n , both converge strongly to the unique solution to variational inequality

(γf-A)z,p-z0

on the set of fixed points of the semigroup T belonging to EP(G).

No examples illustrate the theoretical results.


MSC:
47H20Semigroups of nonlinear operators
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
47J20Inequalities involving nonlinear operators
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