Existence and algorithms for the bilevel new generalized mixed equilibrium problems in Banach spaces

In this paper, we introduce and analyze the new generalized mixed equilibrium problems (NGMEP) and the bilevel new generalized mixed equilibrium problems (BNGMEP) in Banach spaces. First, by using a minimax inequality, some new existence theorems of the solution and the behavior of solution set for the NGMEP and the BNGMEP are obtained in both arbitrary Banach spaces with the coercive conditions and a reflexive Banach space without the coercive conditions. Next, by using auxiliary principle technique, some new iterative algorithms for solving the NGMEP and the BNGMEP are suggested and analyzed. The strong convergence of the iterative sequences generated by the algorithms are also proved in both arbitrary Banach spaces with the coercive conditions and a reflexive Banach space without the coercive conditions. These results are new and generalize some recent results in this field.