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Krasnoselski–Mann iteration for hierarchical fixed points and equilibrium problem. (English) Zbl 1165.47050

Following A.Moudafi and M.Théra [“Proximal and dynamical approaches to equilibrium problems”, Lect.Notes Econ.Math.Syst.477, 187–201 (1999; Zbl 0944.65080)], the present authors employ an explicit Krasnoselskij–Mann-type scheme for the variational inequality problem (VIP) and equilibrium problem, given as follows: \(x_{0} \in C; \quad G(u_{n},y) + \frac{1}{r_{n}} \langle y-u_{n}, u_{n}-x_{n} \rangle \geq 0\) for all \(y \in C; \;x_{n+1}= (1- \alpha _{n} )x_{n} + \alpha _{n} (\lambda _{n}f(x_{n}) + (1 - \lambda _{n})Tu_{n})\), \(n \geq 1\). Here, \(C\) is a closed convex subset of a real Hilbert space, \(T\) is a nonexpansive mapping of \(C\) into itself such that \(\text{Fix}(T)\neq\emptyset \) and \(f:C\to C\) be a \(\rho\)-contraction; \(\{x_n\}_{n=0}^{\infty}\subset C\); and \(G\) is a bifunction on \(C\times C\) to \(\mathbb{R}\).
They study a particular case of two equilibrium functions, one induced by a contractive VIP and one satisfying Condition (1), namely: \(G(x^{*},y)\geq 0\) for all \(y \in C\); find \(x^{*}\in\text{Fix}(T)\) with \(\langle x^{*}-f(x^{*}),x-x^{*}\rangle\geq 0\) for all \(x \in\text{Fix}(T)\), to find common solutions of Condition (1).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
58E35 Variational inequalities (global problems) in infinite-dimensional spaces

Citations:

Zbl 0944.65080
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References:

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