Well-Posedness Under Relaxed Semicontinuity for Bilevel Equilibrium and Optimization Problems with Equilibrium Constraints

Abstract

Bilevel equilibrium and optimization problems with equilibrium constraints are considered. We propose a relaxed level closedness and use it together with pseudocontinuity assumptions to establish sufficient conditions for well-posedness and unique well-posedness. These conditions are new even for problems in one-dimensional spaces, but we try to prove them in general settings. For problems in topological spaces, we use convergence analysis while for problems in metric cases we argue on diameters and Kuratowski’s, Hausdorff’s, or Istrǎtescu’s measures of noncompactness of approximate solution sets. Besides some new results, we also improve or generalize several recent ones in the literature. Numerous examples are provided to explain that all the assumptions we impose are very relaxed and cannot be dropped.