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Some Generalizations of Fixed Point Theorems of Caristi Type and Mizoguchi–Takahashi Type Under Relaxed Conditions

  • Wei-Shih DuEmail author
Article
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Abstract

In this paper, we first study the approximate fixed point property for hybrid Caristi type and Mizoguchi–Takahashi type mappings on metric spaces. We give some new generalizations of Mizoguchi–Takahashi’s fixed point theorem and Caristi’s fixed point theorem under new relaxed conditions which are quite original in the existing literature. We present new generalized Ekeland’s variational principle, generalized Takahashi’s nonconvex minimization theorem and nonconvex maximal element theorem for uniformly below sequentially lower semicontinuous from above functions and essential distances. Their equivalence relationships are also established.

Keywords

\({{\mathcal {M}}}{{\mathcal {T}}}\)-function (\({\mathcal {R}}\)-function) Essential distance \({{\mathcal {M}}}{{\mathcal {T}}}(\lambda )\)-function Approximate fixed point property Caristi’s fixed point theorem Mizoguchi–Takahashi’s fixed point theorem Below sequentially lower semicontinuous from above Uniformly below sequentially lower semicontinuous from above 

Mathematics Subject Classification

47H04 47H10 54H25 54C60 

Notes

Acknowledgements

This research was supported by Grant No. MOST 107-2115-M-017-004-MY2 of the Ministry of Science and Technology of the Republic of China.

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Copyright information

© Sociedade Brasileira de Matemática 2018

Authors and Affiliations

  1. 1.Department of MathematicsNational Kaohsiung Normal UniversityKaohsiungTaiwan

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