Generalized well-posedness results for a class of new mixed variational inequalities. (English) Zbl 1509.49007
Summary: This paper is devoted to investigate a generalized type of mixed variational inequality (GMVI) in Banach space. Under the general assumptions, we first apply Minty’s approach to deliver an equivalent result for GMVI and provide the existence condition of the solutions of GMVI. Then, the concepts of the strong and the weak well-posedness are introduced in the generalized sense, which are applied to discuss the essential relation between the metric characterizations and the generalized strong well-posedness as well as the weak well-posedness for GMVI. Moreover, the theorems are established to determine the generalized strong and the weak well-posedness of GMVI, respectively. Furthermore, we consider a family of approximating problems corresponding to GMVI, which are dominated by the perturbation parameter \(\varepsilon\), and a critical convergence result is obtained. Finally, an example is given to illustrate our main results.
MSC:
| 49J40 | Variational inequalities |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 35J87 | Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators |
| 49J53 | Set-valued and variational analysis |
Keywords:
mixed variational inequality; generalized well-posedness; approximating sequence; perturbation; convergenceReferences:
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