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The best approximation theorems and variational inequalities for discontinuous mappings in Banach spaces

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Abstract

We discuss Ky Fan’s theorem and the variational inequality problem for discontinuous mappings f in a Banach space X. The main tools of analysis are the variational characterizations of the metric projection operator and the order-theoretic fixed point theory. Moreover, we derive some properties of the metric projection operator in Banach spaces. As applications of our best approximation theorems, three fixed point theorems for non-self maps are established and proved under some conditions. Our results are generalizations and improvements of various recent results obtained by many authors.

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

    LiShan Liu & DeZhou Kong

  2. College of Information Science and Engineering, Shandong Agricultural University, Taian, 271018, China

    DeZhou Kong

  3. Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA, 6845, Australia

    LiShan Liu & YongHong Wu

Authors
  1. LiShan Liu
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  2. DeZhou Kong
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  3. YongHong Wu
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Correspondence to LiShan Liu.

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Liu, L., Kong, D. & Wu, Y. The best approximation theorems and variational inequalities for discontinuous mappings in Banach spaces. Sci. China Math. 58, 2581–2592 (2015). https://doi.org/10.1007/s11425-015-5020-6

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  • DOI: https://doi.org/10.1007/s11425-015-5020-6

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