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**Metric characterizations of \(\alpha\)-well-posedness for a system of mixed quasivariational-like inequalities in Banach spaces.**
*(English)*
Zbl 1235.49017

Summary: The purpose of this paper is to investigate the problems of the well-posedness for a system of mixed quasivariational-like inequalities in Banach spaces. First, we generalize the concept of \(\alpha\)-well-posedness to the system of mixed quasivariational-like inequalities, which includes symmetric quasi-equilibrium problems as a special case. Second, we establish some metric characterizations of \(\alpha\)-well-posedness for the system of mixed quasivariational-like inequalities. Under some suitable conditions, we prove that the \(\alpha\)-well-posedness is equivalent to the existence and uniqueness of solution for the system of mixed quasivariational-like inequalities. The corresponding concept of \(\alpha\)-well-posedness in the generalized sense is also considered for the system of mixed quasivariational-like inequalities having more than one solution. The results presented in this paper generalize and improve some known results in the literature.

### MSC:

49J40 | Variational inequalities |

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\textit{L. C. Ceng} and \textit{Y. C. Lin}, J. Appl. Math. 2012, Article ID 264721, 22 p. (2012; Zbl 1235.49017)

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