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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 $T\left(s\right)$, $s\ge 0$, in a Hilbert space $H$ is called nonexpansive if $\parallel T\left(s\right)x-T\left(s\right)y\parallel \le \parallel x-y\parallel$ for arbitrary $x,y$ in $H$. For a bifunction $G:H×S\to ℝ$, $EP\left(G\right)$ denotes the set of equilibrium points, i.e., all $x$ such that $G\left(x,y\right)\ge 0$ for all $y\in H$. 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 ${x}_{t},$ ${u}_{t}\in H$ is defined as solution to

$G\left({u}_{t},y\right)+\frac{1}{{r}_{t}}〈y-{u}_{t},{u}_{t}-{x}_{t}〉\ge 1,\phantom{\rule{1.em}{0ex}}\forall y\in H$

and

${x}_{t}=t\gamma f\left({x}_{t}\right)+\left(I-tA\right)\frac{1}{{\lambda }_{t}}{\int }_{0}^{{\lambda }_{t}}T\left(s\right){u}_{t}\phantom{\rule{0.166667em}{0ex}}ds·$

Here, $0 and $A$ is a strongly positive bounded linear operator.

The explicit iterations are defined by the system

${x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right)\frac{1}{{s}_{n}}{\int }_{0}^{{s}_{n}}t\left(s\right){u}_{n}\phantom{\rule{0.166667em}{0ex}}ds$

and

$G\left({u}_{n},y\right)+\frac{1}{{r}_{n}}〈y-{u}_{n},{u}_{n}-{x}_{n}〉\ge 0,\phantom{\rule{1.em}{0ex}}\forall y\in H·$

It is demonstrated that, with an appropriate definition of the coefficients ${r}_{t},{\lambda }_{t},{\alpha }_{n},{s}_{n}$, 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

$〈\left(\gamma f-A\right)z,p-z〉\le 0$

on the set of fixed points of the semigroup $T$ belonging to $EP\left(G\right)$.

No examples illustrate the theoretical results.

##### MSC:
 47H20 Semigroups of nonlinear operators 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions) 47J20 Inequalities involving nonlinear operators
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