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.


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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