Mizoguchi-Takahashi local contractions to Feng-Liu contractions

Pallab Maiti

India

Indian Institute of Technology Bhilai image/svg+xml

Submitted: 2023-04-26

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Accepted: 2024-04-16

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Published: 2024-10-01

DOI: https://doi.org/10.4995/agt.2024.19619

Copyright (c) 2024 Pallab Maiti, Asrifa Sultana

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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

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Keywords:

fixed points, set-valued map, metrically convex metric space, uniformly local contractions

Supporting agencies:

Ministry of Human Resource Development, India

Science and Engineering Research Board (SERB), India

Abstract:

In this article, we establish that any uniformly local Mizoguchi-Takahashi contraction is actually a set-valued contraction due to Feng and Liu on a metrically convex complete metric space. Through an example, we demonstrate that this result need not hold on any arbitrary metric space. Furthermore, when the metric space is compact, we derive that any Mizoguchi-Takahashi local contraction and Nadler local contraction are equivalent. Moreover, a result related to invariant best approximation is established.

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References:

M. A. Al-Thagafi and N. Shahzad, Noncommuting selfmaps and invariant approximations, Nonlinear Analysis 64 (2006), 2778-2786. https://doi.org/10.1016/j.na.2005.09.015

M. Berinde and V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl. 326 (2007), 772-782. https://doi.org/10.1016/j.jmaa.2006.03.016

M. Edelstein, An extension of Banach's contraction principle, Proc. Amer. Math. Soc. 12 (1961), 7-10. https://doi.org/10.1090/S0002-9939-1961-0120625-6

A. A. Eldred, J. Anuradha, and P. Veeramani, On the equivalence of the Mizoguchi-Takahashi fixed point theorem to Nadler's theorem, Appl. Math. Lett. 22 (2009), 1539-1542. https://doi.org/10.1016/j.aml.2009.03.022

Y. Feng and S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006), 103-112. https://doi.org/10.1016/j.jmaa.2005.12.004

T. Kamran, Coincidence and fixed points for hybrid strict contractions, J. Math. Anal. Appl. 299 (2004), 235-241. https://doi.org/10.1016/j.jmaa.2004.06.047

Z. D. Mitrović, V. Parvaneh, N. Mlaiki, N. Hussain, and Stojan Radenović, On some new generalizations of Nadler contraction in b-metric spaces, Cogent Mathematics and Statistics 7 (2020), 871-882. https://doi.org/10.1080/25742558.2020.1760189

N. Mizoguchi and W. Takahashi, Fixed point theorems for multi-valued mappings on complete metric space, J. Math. Anal. Appl. 141 (1989), 177-188. https://doi.org/10.1016/0022-247X(89)90214-X

B. Mohammadi, V. Parvaneh, H. Aydi and H. Isik, Extended Mizoguchi-Takahashi type fixed point theorems and their application, Mathematics 7 (2019), 575. https://doi.org/10.3390/math7070575

S. B. Nadler, Multivalued contraction mappings, Pacific J. Math. 30 (1969), 475-488. https://doi.org/10.2140/pjm.1969.30.475

V. Parvaneh, B. Mohammadi, M. De La Sen, E. Alizadeh, and H. K. Nashine, On existence of solutions for some nonlinear fractional differential equations via Wardowski-Mizoguchi-Takahashi type contractions, International Journal of Nonlinear Analysis and Applications 12 (2021), 893-902.

H. K. Pathak and N. Hussain, Common fixed points for Banach operator pairs with applications, Nonlinear Analysis 69 (2008), 2788-2802. https://doi.org/10.1016/j.na.2007.08.051

B. Samet, C. Vetro, and P. Vetro, Fixed point theorem for α-ψ contractive type mappings, Nonlinear Analysis 75 (2012), 2154-2165. https://doi.org/10.1016/j.na.2011.10.014

W. Sintunavarat and P. Kumam, Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition, Applied Mathematics Letters 2 (2009), 1877-1881. https://doi.org/10.1016/j.aml.2009.07.015

A. Sultana and V. Vetrivel, Fixed points of Mizoguchi-Takahashi contraction on a metric space with a graph and applications, J. Math. Anal. Appl. 417 (2014), 336-344. https://doi.org/10.1016/j.jmaa.2014.03.015

A. Sultana and X. Qin, On the equivalence of the Mizoguchi-Takahashi locally contractive map and the Nadler's locally contractive map, Numer. Funct. Anal. Optim. 40 (2019), 1964-1971. https://doi.org/10.1080/01630563.2019.1658603

T. Suzuki, Mizoguchi-Takahashi's fixed point theorem is a real generalization of Nadler's, J. Math. Anal. Appl. 340 (2008), 752-755. https://doi.org/10.1016/j.jmaa.2007.08.022

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