## 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 involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 47J20 Variational and other types of inequalities involving nonlinear operators (general)
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