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.
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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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Du, WS. Some Generalizations of Fixed Point Theorems of Caristi Type and Mizoguchi–Takahashi Type Under Relaxed Conditions. Bull Braz Math Soc, New Series 50, 603–624 (2019). https://doi.org/10.1007/s00574-018-0117-5
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DOI: https://doi.org/10.1007/s00574-018-0117-5
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