Iterative algorithm for solutions to new system of generalized mixed implicit equilibrium

doi:10.1007/s10483-013-1657-x

Abstract

Abstract: A new system of generalized mixed implicit equilibrium problems (SGMIEP) involving nonmonotone set-valued mappings is introduced and studied in real reflexive Banach spaces. First, an auxiliary mixed equilibrium problem (AMEP) is introduced. The existence and the uniqueness of the solutions to the AMEP are proved under quite mild assumptions without any coercive conditions. Next, by using the solution mapping of the AMEP, a system of generalized equation problems (SGEP) is considered, and its equivalence with the SGMIEP is shown. By using the SGEP, a new iterative algorithm for solving the SGMIEP is proposed and analyzed. The strong convergence of the iterative sequences generated by the algorithm is proved under suitable conditions. These results are new, which unify and generalize some recent results in this field.

Key words: system of generalized mixed implicit equilibrium problems (SGMIEP), auxiliary mixed equilibrium problem (AMEP), system of generalized equation problems (SGEP), iterative algorithm, reflexive Banach space, nonlinear system, boundary value problem, diagonalization, singular perturbation

DING Xie-Ping. Iterative algorithm for solutions to new system of generalized mixed implicit equilibrium. Applied Mathematics and Mechanics (English Edition), 2013, 34(1): 113-126.

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