Abstract
In this paper, we introduce two general iterative methods (one implicit method and one explicit method) for finding a solution of a general system of variational inequalities (GSVI) with the constraints of finitely many generalized mixed equilibrium problems and a fixed point problem of a continuous pseudocontractive mapping in a Hilbert space. Then we establish strong convergence of the proposed implicit and explicit iterative methods to a solution of the GSVI with the above constraints, which is the unique solution of a certain variational inequality. The results presented in this paper improve, extend, and develop the corresponding results in the earlier and recent literature.
Highlights
Let C be a nonempty closed convex subset of a real Hilbert space H with inner product ·, · and induced norm ·
general system of variational inequalities (GSVI) (1.3) and each generalized mixed equilibrium problem both are transformed into fixed point problems of nonexpansive mappings
), which is the unique solution of a certain variational inequality
Summary
Let C be a nonempty closed convex subset of a real Hilbert space H with inner product ·, · and induced norm ·. Alofi et al [8] introduced two composite iterative algorithms based on the composite iterative methods in Ceng et al [9] and Jung [10] for solving the problem of GSVI (1.2) They showed strong convergence of the proposed algorithms to a common solution of these two problems. In the meantime, inspired by Ceng et al [1], Jung [12] introduced a general system of variational inequalities (GSVI) for two continuous monotone mappings F1 and F2 of finding (x∗, y∗) ∈ C × C such that. Φi, Bi) of finitely many generalized mixed equilibrium problems and the fixed point set of a continuous pseudocontractive mapping T. GSVI (1.3) and each generalized mixed equilibrium problem both are transformed into fixed point problems of nonexpansive mappings. ), which is the unique solution of a certain variational inequality
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