A fixed point result of weakly contractive operators in generalized metric spaces

Mohammed Shehu Shagari; Faryad Ali; Trad Alotaibi; Shazia Kanwal; Akbar Azam; Mohammed Shehu Shagari; Faryad Ali; Trad Alotaibi; Shazia Kanwal; Akbar Azam

doi:10.3934/math.2022969

AIMS Mathematics

2022, Volume 7, Issue 9: 17603-17611. doi: 10.3934/math.2022969

Previous Article Next Article

Research article Special Issues

A fixed point result of weakly contractive operators in generalized metric spaces

1.
Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh 11623, Saudi Arabia
3.
Department of Mathematics and Statistics, Taif University, Saudi Arabia
4.
Department of Mathematics, Government College University, Faisalabad, Pakistan
5.
Department of Mathematics, Grand Asian University, Sialkot, 7KM, Pasrur Road, Sialkot 51310, Pakistan

Received: 03 July 2022 Revised: 21 July 2022 Accepted: 24 July 2022 Published: 01 August 2022
MSC : 47H10, 54H25, 47H04

In this note, by using basic properties of the recently introduced concepts of generalized metric spaces, new conditions for the existence of a fixed point for weakly type contractive operator which sends a closed subset into the ambient space under consideration are examined. Our obtained result extends and unifies its corresponding ideas in metric and modular spaces. A comparative non-trivial example is provided to show the novelty and preeminence of our proposed notion.
- fixed point,
- generalized metric space,
- modular space,
- weakly contractive
Citation: Mohammed Shehu Shagari, Faryad Ali, Trad Alotaibi, Shazia Kanwal, Akbar Azam. A fixed point result of weakly contractive operators in generalized metric spaces[J]. AIMS Mathematics, 2022, 7(9): 17603-17611. doi: 10.3934/math.2022969

Related Papers:

Abstract

In this note, by using basic properties of the recently introduced concepts of generalized metric spaces, new conditions for the existence of a fixed point for weakly type contractive operator which sends a closed subset into the ambient space under consideration are examined. Our obtained result extends and unifies its corresponding ideas in metric and modular spaces. A comparative non-trivial example is provided to show the novelty and preeminence of our proposed notion.

References

[1]	T. Abdeljawad, Fixed points for generalized weakly contractive mappings in partial metric spaces, Math. Comp. Model., 54 (2011), 2923–2927. https://doi.org/10.1016/j.mcm.2011.07.013 doi: 10.1016/j.mcm.2011.07.013
[2]	Y. I. Alberand, S. Guerre-Delabrere, Principle of weakly contractive maps in Hilbert spaces, In: I. Gohberg, Y. Lyubich, New results in operator theory and its applications, Operator Theory: Advances and Applications, Basel, 98 (1997), 7–22. https://doi.org/10.1007/978-3-0348-8910-0_2
[3]	S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181.
[4]	A. Betiuk-Pilarska, T. B. Domınguez, Fixed points for nonexpansive mappings and generalized nonexpansive mappings on Banach lattices, Pure Appl. Func. Anal., 1 (2016), 343–359.
[5]	F. S. de Blasi, J. Myjak, S. Reich, A. J. Zaslavski, Generic existence and approximation of fixed points for nonexpansive set-valued maps, Set-Valued Anal., 17 (2009), 97–112. https://doi.org/10.1007/s11228-009-0104-5 doi: 10.1007/s11228-009-0104-5
[6]	P. Chakraborty, B. S. Choudhury, Locally weak version of the contraction mapping principle, Math. Notes, 109 (2021), 859–866. https://doi.org/10.1134/S0001434621050199 doi: 10.1134/S0001434621050199
[7]	W. Chistyakov, Modular metric spaces generated by $F$-modulars, Folia Math., 14 (2008), 3–25.
[8]	V. V. Chistyakov, Modular metric spaces, I: Basic concepts, Nonlinear Anal.: Theory Methods Appl., 72 (2010), 1–14. https://doi.org/10.1016/j.na.2009.04.057 doi: 10.1016/j.na.2009.04.057
[9]	P. N. Dutta, B. S. Choudhury, A generalization of contraction principle in metricspaces, Fixed Point Theory Appl., 2008 (2008), 406368. https://doi.org/10.1155/2008/406368 doi: 10.1155/2008/406368
[10]	A. Gholidahaneh, S. Sedghi, V. Parvaneh, Some fixed point results for Perov-Ćirić-Prešić-type $F$-contractions with application, J. Funct. Spaces, 2020 (2020), 1464125. https://doi.org/10.1155/2020/1464125 doi: 10.1155/2020/1464125
[11]	A. Gholidahneh, S. Sedghi, O. Ege, Z. D. Mitrovic, M. de la Sen, The Meir-Keeler type contractions in extended modular b-metric spaces with an application, AIMS Math., 6 (2021), 1781–1799. https://doi.org/10.3934/math.2021107 doi: 10.3934/math.2021107
[12]	M. A. Khamsi, W. M. Kozlowski, S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal.: Theory Methods Appl., 14 (1990), 935–953. https://doi.org/10.1016/0362-546X(90)90111-S doi: 10.1016/0362-546X(90)90111-S
[13]	S. Koshi, T. Shimogaki, On $F$-norms of quasi-modular spaces, J. Fac. Sci., Hokkaido Univ. Ser. I, Math., 15 (1961), 202–218.
[14]	W. M. Kozlowski, An introduction to fixed point theory in modular function spaces, In: S. Almezel, Q. Ansari, M. Khamsi, Topics in fixed point theory, Springer, 2014,159–222. https://doi.org/10.1007/978-3-319-01586-6_5
[15]	R. Kubota, W. Takahashi, Y. Takeuchi, Extensions of Browder's demiclosedness principle and Reich's lemma and their applications, Pure Appl. Func. Anal., 1 (2016), 63–84.
[16]	Y. Lim, Solving the nonlinear matrix equation $X = Q+ \sum\limits_{i = 1}^{m}M^_iX^{\delta_i}M^_i$ via acontraction principle, Linear Algebra Appl., 430 (2009), 1380–1383. https://doi.org/10.1016/j.laa.2008.10.034 doi: 10.1016/j.laa.2008.10.034
[17]	S. Mazur, W. Orlicz, On some classes of linear spaces, Stud. Math., 17 (1958), 97–119.
[18]	Z. D. Mitrovic, S. Radenovic, H. Aydi, A. A. Altasan, C. Ozel, On two new approaches in modular spaces, Ita. J. Pure Appl. Math., 41 (2019), 679–690.
[19]	S. S. Mohammed, A. Azam, Integral type contractions of soft set-valued maps with application to neutral differential equation, AIMS Math., 5 (2019), 342–358. https://doi.org/10.3934/math.2020023 doi: 10.3934/math.2020023
[20]	S. S. Mohammed, S. Rashid, K. M. Abualnaja, A. Monairah, On non-linear fuzzy set-valued $\Theta$-contraction with applications, AIMS Math., 6 (2021), 10431–10448. https://doi.org/10.3934/math.2021605 doi: 10.3934/math.2021605
[21]	J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, Vol. 1034, Springer-Verlag, 1983. https://doi.org/10.1007/BFb0072210
[22]	H. Nakano, Modulared semi-ordered linear spaces, Maruzen Company, 1950.
[23]	S. Reich, A. J. Zaslavski, On a class of generalized nonexpansive mappings, Mathematics, 8 (2020), 1085. https://doi.org/10.3390/math8071085 doi: 10.3390/math8071085
[24]	S. Reich, J. Z. Alexander, A fixed point result in generalized metric spaces, J. Anal., 2022. https://doi.org/10.1007/s41478-022-00412-2 doi: 10.1007/s41478-022-00412-2
[25]	B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal.: Theory Methods Appl., 47 (2001), 2683–2693. https://doi.org/10.1016/S0362-546X(01)00388-1 doi: 10.1016/S0362-546X(01)00388-1
[26]	A. A. Taleb, E. Hanebaly, A fixed point theorem and its applications to integral equations in modular function spaces, Proc. Amer. Math. Soc., 128 (1999), 419–426. https://doi.org/10.1090/S0002-9939-99-05546-X doi: 10.1090/S0002-9939-99-05546-X
[27]	Z. Xue, G. Lv, Remarks of fixed point for $(\psi, \phi)$-weakly contractive mappings, J. Math., 2021 (2021), 5561165. https://doi.org/10.1155/2021/5561165 doi: 10.1155/2021/5561165
[28]	S. Yamamuro, On conjugate spaces of Nakano spaces, Trans. Amer. Math. Soc., 90 (1959), 291–311. https://doi.org/10.2307/1993206 doi: 10.2307/1993206

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)