A Class Of Existence And Stability On The Solution For Parametric Generalized Vector Variational-Like Inequalities

Generalized vector variational-like inequality problem is a generalization of variational inequality-like problem,which is an important theoretical basis and tool for studying optimization theory and nonlinear analysis,and is widely used in transportation,finance,economics,mathematical physics,engineering science,etc.In particular,the properties of solution sets are widely studied,such as the existence,stability,and well-posedness of solutions.In practical problems,we study the existence and stability of solution sets with parametric generalized vector variational-like inequalities and related problems due to uncertainties that lead to changes in results,especially when there are parametric perturbations.Firstly,according to the generalized vector variational-like inequality problem,this chapter proposes the parametric generalized vector variational-like inequality problem,when the set value mapping and the constraint set are perturbed by parameters,respectively,according to the definition of the gap function,the gap function of the problem is obtained,and it is proved that the gap function is lower semicontinuous.The existence of the understanding is proved by KKM mapping and set correlation properties,and then the solution mapping is understood to be upper semi-continuous and has closed values according to the correlation properties of the set.According to the proof of the lower semi-continuity of the gap function,it is understood that the mapping is Hausdorff continuous.Finally,relevant inferences are drawn and examples are given to verify the conclusions.Secondly,on the basis of the generalized vector variational-like inequality problem,this chapter adds a vector value mapping and also obtains the parametric generalized vector quasi variational-like inequality problem when the set value map and constraint set are perturbed by parameters,respectively.According to the definition of the gap function,the problem of the lower semi-continuity of the gap function is proved.Different from the previous hypothetical conditions,the existence of the understanding is proved by KKM mapping and set correlation properties,and then the solution mapping is understood to be upper semi-continuous and closed according to the correlation property of sets,then the solution mapping of the parametric generalized vector quasi variational-like inequality problem is Hausdorff continuous according to the gap function and related properties,and finally the conclusion is verified by example.Thirdly,this chapter proposes a series of models of related parameter inclusive problems,different from the previous two chapters,in this chapter we only discuss the parametric generalized vector equilibrium problem when the constraint set is perturbed by parameters,and obtain the gap function of the parametric generalized vector equilibrium problem according to the definition of the gap function,and under certain assumptions,it is proved that the nonempty closure,upper semi-continuity and Hausdorff continuity of the mapping are understood.