Levitin-Polyak Well-posedness for Split Equilibrium Problems

Abstract

The notion of well-posedness has drawn the attention of many researchers in nonlinear analysis in connection with problems where the exact solution is unknown or may be costly to compute. Well-posedness guarantees the convergence of a sequence of approximate solutions obtained by iterative methods to the exact solution of the given problem. Motivated by these facts, we extend here the concept of Levitin-Polyak well-posedness to split equilibrium problems in infinite-dimensional Hilbert spaces. We establish, in particular, a metric characterization of Levitin-Polyak wellposedness by perturbations. We also show an equivalence between Levitin-Polyak well-posedness by perturbations for split equilibrium problems and the existence and uniqueness of their solutions under certain finite-dimensional assumptions on the mappings involved.