The Nonlinear Scalarization And Stability Of Vector Equilibrium Problems

The stability analysis for vector equilibrium problems is mainly to study the continuity of perturbed solution mappings and it is an important topic in vector optimization. The research of H?lder or Lipschitz continuity to solution mappings for parametric vector equilibrium problems is very interesting. We know that the scalarization method uses linear(nonlinear) scalarizing functions to turn vector optimization problems into scalar ones. The researches show that, the nonlinear scalarizing approaches by virtue of nonlinear scalarization functions are efficient methods to study H?lder continuity of parametric vector equilibrium problems.Nonlinear scalarization functions as scalarization tools play key roles in the research of vector optimization problems. Among them the well-known Gerstewitz(Tammer) function and(Hiriart-Urruty) oriented distance function are widely used for studying vector optimization problems. In this thesis, firstly, some useful properties of the Gerstewitz scalarizing function are discussed, such as its globally Lipschitz property, concavity and monotonicity. As an application of these properties, verifiable sufficient conditions for H?lder continuity of approximate solutions to parametric generalized vector equilibrium problems are established via Gerstewitz scalarizations.Secondly, a new kind of monotonicity hypothesis is proposed and then new results for H?lder continuity of the unique solution to a parametric generalized vector quasiequilibrium problem are established via nonlinear scalarization, with and without using the free-disposal condition. The globally Lipschitz property together with other useful properties of the Gerstewitz function are fully exploited for proving. Especially, our approach does not impose any convexity condition on the considered model. Moreover, by virtue of the oriented distance function, new results of H?lder continuity to parametric generalized vector quasiequilibrium problems are also established.