On the Hadamard Well-Posedness of Generalized Mixed Variational Inequalities in Banach Spaces

Lu-Chuan Ceng,Yeong-Cheng Liou,Long He,Ching-Feng Wen,Yun-Ling Cui,Hui-Ying Hu,Yongqiang Fu

doi:10.1155/2021/9527107

Abstract

We introduce a new concept of Hadamard well-posedness of a generalized mixed variational inequality in a Banach space. The relations between the Levitin–Polyak well-posedness and Hadamard well-posedness for a generalized mixed variational inequality are studied. The characterizations of Hadamard well-posedness for a generalized mixed variational inequality are established.

Highlights

In [1], Tykhonov first introduced the well-posedness of a minimization problem, which means that it has a unique minimizer and every minimizing sequence converges to the unique minimizer. ere are two concepts of well-posedness which are Tykhonov well-posedness [1] and Hadamard wellposedness [2]
Relations between Levitin–Polyak well-posedness and Hadamard well-posedness of a general mixed variational inequality were presented. ey established some characterizations of Hadamard well-posedness for a general mixed variational inequality
Motivated and inspired by the research work going on this field, we introduce a new concept of Hadamard wellposedness for a generalized mixed variational inequality in a Banach space

Summary

Introduction

In [1], Tykhonov first introduced the well-posedness of a minimization problem, which means that it has a unique minimizer and every minimizing sequence converges to the unique minimizer. ere are two concepts of well-posedness which are Tykhonov well-posedness [1] and Hadamard wellposedness [2]. Variational inequality (VI) has been extensively studied due to the facts that it has many potential applications and that it is closely related to a differentiable minimization problem. In 2013, Li and Xia [12] introduced the concept of Hadamard well-posedness of a general mixed variational inequality in Banach spaces. Relations between Levitin–Polyak well-posedness and Hadamard well-posedness of a general mixed variational inequality were presented. Ey established some characterizations of Hadamard well-posedness for a general mixed variational inequality. Motivated and inspired by the research work going on this field, we introduce a new concept of Hadamard wellposedness for a generalized mixed variational inequality in a Banach space. The relations between the Levitin–Polyak well-posedness and Hadamard well-posedness for a generalized mixed variational inequality are studied. Extend, and develop the earlier and recent ones announced by some others, e.g., Ceng and Yao [7] and Li and Xia [12, 20]

Preliminaries

Well-Posedness

Metric Characterization and Conditions for Hadamard Well-Posedness