Semicontinuity of the solution multifunctions of the parametric generalized operator equilibrium problems Original Research Article

In this paper, we first introduce the parametric form of the generalized operator equilibrium problem and generalized operator quasi-equilibrium problem due to Kum and Kim [S. Kum, W.K. Kim, On generalized operator quasi-equilibrium problems, J. Math. Anal. Appl. 345 (2008) 559–565]. Next, we consider the upper and lower semicontinuity properties for the parametric generalized operator equilibrium problem (simply, PGOEP) and the parametric generalized operator quasi-equilibrium problem (simply, PGOQEP). Our semicontinuity properties of the solution sets for (PGOEP) and (PGOQEP) can be suitable for the stability properties of many kinds of nonlinear problems, e.g., the operator equilibrium problem, operator quasi-equilibrium problem, operator variational-like inequality problem, operator variational inequality problem, operator minimization problem and vector equilibrium problem.