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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 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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