A generalized distance and enhanced Ekeland’s variational principle for vector functions |
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Authors: | Phan Quoc Khanh Dinh Ngoc Quy |
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Affiliation: | a Department of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Viet Nam b Department of Mathematics, Cantho University, Cantho, Viet Nam |
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Abstract: | We propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation. This result is shown to contain probably a large part of existing versions of Ekeland’s variational principle (EVP). We introduce the notion of a weak τ-function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced EVP for various settings, under relaxed lower semicontinuity assumptions. These principles conclude the existence not only of p-strict minimizers of p-perturbations of the considered vector function, but also p-sharp and p-strong minimizers. Our results are proved to be stronger than the classical EVP and many generalizations in the literature, even in the usual one-dimensional case, by numerous corollaries and examples. We include equivalent formulations of our enhanced EVP as well. |
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Keywords: | 58E30 58E47 65K10 49J53 90C48 |
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