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P-distances, q-distances and a generalized Ekeland’s variational principle in uniform spaces

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Abstract

In this paper, we attempt to give a unified approach to the existing several versions of Ekeland’s variational principle. In the framework of uniform spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance (particularly, a p-distance), which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi’s nonconvex minimization theorem, a generalized Ekeland’s variational principle and a generalized Caristi’s fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland’s variational principle, we deduce a number of particular versions of Ekeland’s principle, which include many known versions of the principle and their improvements.

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Authors and Affiliations

  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, P. R. China

    Jing Hui Qiu & Fei He

  2. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, P. R. China

    Fei He

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  1. Jing Hui Qiu
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  2. Fei He
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Correspondence to Jing Hui Qiu.

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Supported by National Natural Science Foundation of China (Grant No. 10871141)

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Qiu, J.H., He, F. P-distances, q-distances and a generalized Ekeland’s variational principle in uniform spaces. Acta. Math. Sin.-English Ser. 28, 235–254 (2012). https://doi.org/10.1007/s10114-011-0629-z

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  • DOI: https://doi.org/10.1007/s10114-011-0629-z

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