Theoretical Research On Generalized Vector Quasi-equilibrium Problems Under Noncompact Conditions

The generalized vector quasi-equilibrium problem has always been very important in the field of optimization research.At the same time,many equilibrium problems can be regarded as their sub-problems or can be transformed with each other,such as the vector quasi-equilibrium problem,the extreme value Ky Fan inequality problem,the generalized implicit vector quasi-variational inequality problem,the generalized implicit vector variational inequality problem,and so on In this paper,we mainly study the existence and stability of solutions to generalized vector quasi-equilibrium problems and related equilibrium problems under noncompact conditions.First,several kinds of equilibrium problem models including the generalized vector quasi-equilibrium problem are introduced,and with the help of the generalized Browder fixed point theorem,propose the existence results of the solutions of the generalized vector quasi-equilibrium problem and sub-problems under the noncompact condition and give the proof,thus weakening the compactness condition to the noncompact condition,generalizes the previous research results.Second,based on the existence theorem of solutions for generalized vector quasi equilibrium problems and related problems under noncompact conditions,the solution space and solution mapping for generalized vector quasi equilibrium problems and related equilibrium problems under noncompact conditions are defined,and it is proved that the solution mapping is upper semicontinuous and compact under noncompact conditions We also prove the existence theorems of essential connected regions for generalized vector quasi equilibrium problems and sub problems under non compact conditions From the existence theorem of solutions for generalized vector quasi equilibrium problems and related problems under non compact conditions,combining the definitions of Hadamard well-posedness,Tikhonov well-posedness and Levitin-Polyak well-posedness,we prove that the solutions of generalized vector quasi-equilibrium problems and sub-problems are well-posedness under noncompact conditions.Third,the existence results and stability results of the solutions of the generalized vector quasi-equilibrium problem are generalized to the related problems that can be transformed with each other.