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Existence and stability of iterative algorithms for the system of nonlinear quasi-mixed equilibrium problems. (English) Zbl 1220.47125
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.

47J25Iterative procedures (nonlinear operator equations)
47J20Inequalities involving nonlinear operators
47H04Set-valued operators
47H05Monotone operators (with respect to duality) and generalizations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
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