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Set-valued pseudo-metric families and Ekeland's variational principles in fuzzy metric spaces

机译：模糊度量空间中的集值伪度量族和Ekeland的变分原理

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摘要

In this paper, we introduce a set-valued pseudo-metric family on a fuzzy metric space and the notion of compatibility between the set-valued pseudo-metric family and the original fuzzy metric. By means of this notion, we prove a general set-valued EVP, where the perturbation involves a set-valued pseudo-metric family compatible with the original fuzzy metric. From the general EVP, we deduce several particular EVPs, which extend the EVPs in Qiu (2013) [36] and in Gutierrez et al. (2008) [20] to fuzzy metric spaces. By using set-valued pseudo-metric families and using the unified approach for approximate solutions introduced by Gutierrez, Jimenez and Novo, we deduce a general version of set-valued EVP based on (C, epsilon)-efficient solutions in fuzzy metric spaces, where C is a coradiant set contained in the order cone. By choosing two specific versions of the coradiant set C in the general version of EVP, we obtain several particular set-valued EVPs for epsilon-efficient solutions in the sense of Nemeth and of Dentcheva and Helbig, respectively. These EVPs improve and generalize the related known results. (C) 2016 Elsevier B.V. All rights reserved.
机译：在本文中，我们介绍了一个在模糊度量空间上的集值伪度量族以及集值伪度量族与原始模糊度量之间的兼容性概念。通过这个概念，我们证明了一个通用的设定值EVP，其中扰动涉及与原始模糊度量兼容的设定值伪度量族。根据一般的EVP，我们推论出几个特定的​​EVP，在Qiu（2013）[36]和Gutierrez等人中扩展了EVP。 （2008）[20]模糊度量空间。通过使用集值伪度量族，并使用古铁雷斯，希门尼斯和诺沃引入的近似解的统一方法，我们推导出了基于模糊度量空间中基于（C，epsilon）有效解的集值EVP的通用版本，其中C是阶锥中包含的共辐射集。通过在EVP的常规版本中选择共辐射集C的两个特定版本，我们分别获得了Nemeth以及Dentcheva和Helbig的几个特定的​​具有值的EVP，用于ε高效解。这些EVP改进并推广了相关的已知结果。 （C）2016 Elsevier B.V.保留所有权利。

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