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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 $T(s)$, $s\geq0$, in a Hilbert space $H$ is called nonexpansive if $\|T(s)x-T(s)y\|\leq\|x-y\|$ for arbitrary $x,y$ in $H$. For a bifunction $G:H\times S\rightarrow\mathbb{R}$, $EP(G)$ denotes the set of equilibrium points, i.e., all $x$ such that $G(x,y)\geq0$ 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 $$G(u_t, y) +\frac{1}{r_t}\langle y-u_t, u_t-x_t\rangle\geq 1, \quad \forall y\in H$$ and $$x_t = t\gamma f(x_t)+(I-tA)\frac1{\lambda_t}\int_0^{\lambda_t} T(s)u_t\,ds.$$ Here, $0<t<1$ and $A$ is a strongly positive bounded linear operator. The explicit iterations are defined by the system $$x_{n+1} = \alpha_n\gamma f(x_n)+(I-\alpha_n A)\frac1{s_n}\int_0^{s_n}t(s)u_n\, ds$$ and $$G(u_n, y)+\frac1{r_n}\langle y-u_n, u_n-x_n\rangle \geq 0, \quad \forall y\in H.$$ 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 $$\langle (\gamma f-A)z, p-z \rangle \leq0$$ on the set of fixed points of the semigroup $T$ belonging to $EP(G)$. 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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