The existence and the stability of solutions for equilibrium problems with lower and upper bounds

In this paper, we study a class of equilibrium problems with lower and upper bounds. We obtain some existence results of solutions for equilibrium problems with lower and upper bounds by employing some classical fixed-point theorems. We investigate the stability of the solution sets for the problems, and establish sufficient conditions for the upper semicontinuity, lower semicontinuity and continuity of the solution set mapping $S:Lambda_1 imesLambda_2 o2^{X}$ in a Hausdorff topological vector space, in the case where a set $K$ and a mapping $f$ are perturbed respectively by parameters $lambda$ and $mu.$