Abstract
This paper aims to present some uniqueness and well-posedness results for vector equilibrium problems (for short, VEPs). We first construct a complete metric space M consisting of VEPs satisfying some conditions. Using the method of set-valued analysis, we prove that there exists a dense everywhere residual subset Q of M such that each VEP in Q has a unique solution. Moreover, we introduce and obtain the generalized Hadamard well-posedness and generic Hadamard well-posedness of VEPs by considering the perturbations of both vector-valued functions and feasible sets. As an application, we provide a representation theorem for the solution set to each VEP in M.
Highlights
The vector equilibrium problem is a natural generalization of the equilibrium problem for the vector-valued function
It is well known that the vector equilibrium problem is a unified model of several fundamental mathematical problems, namely, the vector optimization problem, the vector variational inequality, the vector complementarity problem, the multiobjective game, the vector network equilibrium problem etc
The reason why results on uniqueness are so few is due to the fact that except for a few types of mathematical problems, most of the mathematical problems cannot guarantee the uniqueness of the solution
Summary
The vector equilibrium problem (for short, VEP) is a natural generalization of the equilibrium problem for the vector-valued function. The study of the generic uniqueness of solutions to VEPs has an essential difficulty: that the values of different vector-valued functions are incomparable. To overcome such a difficulty is one of the main tasks in this paper. Fang et al [ ] investigated the well-posedness of equilibrium problems; Kimura et al [ ] studied the parametric well-posedness for vector equilibrium problems; Bianchi et al [ ] introduced and studied two types of well-posedness for vector equilibrium problems; Li and Li [ ] investigated the Levitin-Polyak well-posedness of vector equilibrium problems with variable domination structures; Salamon [ ] analyzed the Hadamard well-posedness of parametric vector equilibrium problems; Peng et al [ ] investigated several types of Levitin-Polyak well-posedness of generalized vector equilibrium problems Most of these works considered the perturbation of the parameters in the vector-valued functions. Let us recall some definitions and lemmas about set-valued mappings; for more details see [ ].
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