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Zbl 1220.47125
Suantai, Suthep; Petrot, Narin
Existence and stability of iterative algorithms for the system of nonlinear quasi-mixed equilibrium problems. (English)
[J] Appl. Math. Lett. 24, No. 3, 308-313 (2011). ISSN 0893-9659

Let $\mathcal H$ be a real Hilbert space and the maps $\Phi_1, \Phi_2: \mathcal H \times \mathcal H \rightarrow \mathcal H$ satisfy $\Phi_i (x,x) = 0,$ $i = 1,2$; let $T_1, T_2: \mathcal H \times \mathcal H \rightarrow \mathcal H$ be nonlinear maps, and $C_1, C_2: \mathcal H \multimap \mathcal H$ be multimaps with nonempty convex closed values. The authors consider the problem of finding $(x^*,y^*) \in \mathcal H \times \mathcal H$ such that $x^* \in C_1(x^*),$ $y^* \in C_2 (y^*)$ and $$\cases \Phi_1(x^*,z) + (T_1(x^*,y^*), z - x^*) \geq 0, &\forall z \in C_1(x^*), \\ \Phi_2(y^*,z) + (T_2(x^*,y^*), z - y^*) \geq 0, &\forall z \in C_2(y^*). \endcases$$ They prove existence and uniqueness results and describe convergence and stability of a Mann type iterative algorithm.
[Valerii V. Obukhovskij (Voronezh)]

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)
47J20 Inequalities involving nonlinear operators
47H04 Set-valued operators
47H05 Monotone operators (with respect to duality)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: system of nonlinear quasi-mixed equilibrium problems; $\nu $-strongly monotone; ($\tau, \sigma)$-Lipschitz mapping; stability analysis; iterative algorithm; Mann type iterative algorithm

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