On two new contractions and discontinuity on fixed points

Mi Zhou; Naeem Saleem; Xiao-lan Liu; Nihal Özgür; Mi Zhou; Naeem Saleem; Xiao-lan Liu; Nihal Özgür

doi:10.3934/math.2022095

AIMS Mathematics

2022, Volume 7, Issue 2: 1628-1663. doi: 10.3934/math.2022095

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Research article

On two new contractions and discontinuity on fixed points

1.
School of Science and Technology, University of Sanya, Sanya, Hainan, 572000, China
2.
Department of Mathematics, University of Management and Technology, Lahore, Pakistan
3.
College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, China
4.
Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationlization and Internet of Things, Zigong, Sichuan, 643000, China
5.
South Sichuan Center for Applied Mathematics, Zigong, Sichuan, 643000, China
6.
Department of Mathematics, Balıkesir University, 10145 Balıkesir, Turkey

Received: 12 September 2021 Accepted: 21 October 2021 Published: 01 November 2021
MSC : 47H10, 54H25

This paper deals with a well known open problem raised by Kannan (Bull. Calcutta Math. Soc., 60: 71–76, 1968) and B. E. Rhoades (Contemp. Math., 72: 233–245, 1988) on the existence of general contractions which have fixed points, but do not force the continuity at the fixed point. We propose some new affirmative solutions to this question using two new contractions called $ (\psi, \varphi) $-$ \mathcal{A} $-contraction and $ (\psi, \varphi) $-$ \mathcal{A^{\prime}} $-contraction inspired by the results of H. Garai et al. (Applicable Analysis and Discrete Mathematics, 14(1): 33–54, 2020) and P. D. Proinov (J. Fixed Point Theory Appl. (2020) 22: 21). Some new fixed point and common fixed point results in compact metric spaces and also in complete metric spaces are proved in which the corresponding contractive mappings are not necessarily continuous at their fixed points. Moreover, we show that new solutions to characterize the completeness of metric spaces. Several examples are provided to verify the validity of our main results.
Citation: Mi Zhou, Naeem Saleem, Xiao-lan Liu, Nihal Özgür. On two new contractions and discontinuity on fixed points[J]. AIMS Mathematics, 2022, 7(2): 1628-1663. doi: 10.3934/math.2022095

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Abstract

This paper deals with a well known open problem raised by Kannan (Bull. Calcutta Math. Soc., 60: 71–76, 1968) and B. E. Rhoades (Contemp. Math., 72: 233–245, 1988) on the existence of general contractions which have fixed points, but do not force the continuity at the fixed point. We propose some new affirmative solutions to this question using two new contractions called $ (\psi, \varphi) $-$ \mathcal{A} $-contraction and $ (\psi, \varphi) $-$ \mathcal{A^{\prime}} $-contraction inspired by the results of H. Garai et al. (Applicable Analysis and Discrete Mathematics, 14(1): 33–54, 2020) and P. D. Proinov (J. Fixed Point Theory Appl. (2020) 22: 21). Some new fixed point and common fixed point results in compact metric spaces and also in complete metric spaces are proved in which the corresponding contractive mappings are not necessarily continuous at their fixed points. Moreover, we show that new solutions to characterize the completeness of metric spaces. Several examples are provided to verify the validity of our main results.

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