Another Meir-Keeler-type nonlinear contractions

Daniel Francis; Godwin Amechi Okeke; Aviv Gibali; Daniel Francis; Godwin Amechi Okeke; Aviv Gibali

doi:10.3934/math.2025349

AIMS Mathematics

2025, Volume 10, Issue 4: 7591-7635. doi: 10.3934/math.2025349

Previous Article Next Article

Theory article Special Issues

Another Meir-Keeler-type nonlinear contractions

1.
Department of Mathematics, Collage of Physical and Applied Sciences, Michael Okpara University of Agriculture Umudike, P.M.B 7267, Abia State, Nigeria
2.
Functional Analysis and Optimization Research Group Laboratory (FANORG), Department of Mathematics, School of Physical Sciences, Federal University of Technology, Owerri, P.M.B. 1526 Imo State, Nigeria
3.
Department of Applied Mathematics, Holon institute of Technology, Holon, Israel

Received: 28 January 2025 Revised: 16 March 2025 Accepted: 20 March 2025 Published: 01 April 2025
MSC : 47H09, 47H10, 47H30

This paper deals with the concept of $ \alpha $-$ \hat{v} $-$ A $-$ B $-$ C $-Meir-Keeler type nonlinear contractions, a new class mappings within the of modular extended $ b $-metric spaces. We establish common unique fixed-point theorems that generalize, unify, and extend several key results in modular $ b $-metric and modular extended $ b $-metric spaces. These theorems bridge the gap between classical and contemporary fixed-point theories, showcasing broader applicability in nonlinear analysis. To ensure clarity and practical relevance, a detailed example is presented, further validating the theoretical findings. This work provides some level of understanding of the space under investigation and sets the stage for future developments in this evolving domain.
- existence,
- uniqueness,
- metric modular,
- $ b $-metric space,
- $ \alpha $-$ \hat{v} $-$ A $-$ B $-$ C $ Meir-Keeler,
- nonlinear contractions
Citation: Daniel Francis, Godwin Amechi Okeke, Aviv Gibali. Another Meir-Keeler-type nonlinear contractions[J]. AIMS Mathematics, 2025, 10(4): 7591-7635. doi: 10.3934/math.2025349

Related Papers:

Abstract

This paper deals with the concept of $ \alpha $-$ \hat{v} $-$ A $-$ B $-$ C $-Meir-Keeler type nonlinear contractions, a new class mappings within the of modular extended $ b $-metric spaces. We establish common unique fixed-point theorems that generalize, unify, and extend several key results in modular $ b $-metric and modular extended $ b $-metric spaces. These theorems bridge the gap between classical and contemporary fixed-point theories, showcasing broader applicability in nonlinear analysis. To ensure clarity and practical relevance, a detailed example is presented, further validating the theoretical findings. This work provides some level of understanding of the space under investigation and sets the stage for future developments in this evolving domain.

References

[1]	A. A. N. Abdou, Fixed points of Kannan maps in modular metric spaces, AIMS Math., 5 (2020), 6395–6403. https://doi.org/10.3934/math.2020411 doi: 10.3934/math.2020411
[2]	M. Abtahi, Fixed point theorems for Meir-Keeler type contractions in metric spaces, arXiv Preprint, 2016, 1–11. https://doi.org/10.48550/arXiv.1604.01296
[3]	U. Aksoya, E. Karapinara, I. M. Erhana, V. Rakocevic, Meir-Keeler type contractions on modular metric spaces, Filomat, 32 (2018), 3697–3707. https://doi.org/10.2298/FIL1810697A doi: 10.2298/FIL1810697A
[4]	U. Aksoy, E. Karapinar, I. M. Erhan, Fixed point theorems in complete modular metric spaces and an application to anti-periodic boundary value problems, Filomat, 31 (2017), 5475–5488. https://doi.org/10.2298/FIL1717475A doi: 10.2298/FIL1717475A
[5]	U. Aksoy, E. Karapinar, I. M. Erhan, Fixed points of generalized alpha-admissible contractions on $b$-metric spaces with an application to boundary value problems, J. Nonlinear Convex A., 17 (2016), 1095–1108.
[6]	A. S. S. Alharbi, H. H. Alsulami, E. Karapinar, On the power of simulation and admissible functions in metric fixed point theory, J. Funct. Space., 2017 (2017). https://doi.org/10.1155/2017/2068163
[7]	H. H. Alsulami, S. Gulyaz, I. M. Erhan, Fixed points of $\alpha$-admissible Meir-Keeler contraction mappings on quasi-metric spaces, J. Inequal. Appl., 84 (2015). https://doi.org/10.1186/s13660-015-0604-9
[8]	M. Arshad, E. Ameer, E. Karapinar, Generalized contractions with triangular alpha-orbital admissible mapping on Branciari metric spaces, J. Inequal. Appl., 63 (2016), 22–34. https://doi.org/10.1186/s13660-016-1010-7 doi: 10.1186/s13660-016-1010-7
[9]	B. Azadifar, G. Sadeghi, R. Saadati, C. Park, Integral type contractions in modular metric spaces, J. Inequal. Appl., 483 (2013), 65–79. https://doi.org/10.1186/1029-242X-2013-483 doi: 10.1186/1029-242X-2013-483
[10]	S. Barootkoob, E. Karapinar, H. Lakzian, A. Chanda, Extensions of Meir-Keeler contraction via $w$-distances with an application, Kragujev. J. Math., 46 (2022), 533–547.
[11]	E. Canzoneri, P. Vetroa, Fixed points for asymptotic contractions of integral Meir-Keeler type, J. Nonlinear Sci. Appl., 5 (2012), 126–132.
[12]	C. M. Chen, E. Karapinar, D. O'regan, On $(\alpha$-$\phi)$-Meir-Keeler contractions on partial Hausdorff metric spaces, U. Politeh. Buch. Ser. A, 80 (2018), 101–110.
[13]	V. V. Chistyakov, A fixed point theorem for contractions in metric modular spaces, arXiv Preprint, 2011, 65–92.
[14]	V. V. Chistyakov, Metric modular spaces, I basic concepts, Nonlinear Anal.-Theor., 72 (2010), 1–14. https://doi.org/10.1016/j.na.2009.04.057
[15]	L. B. Ciric, N. Cakic, M. Rajovic, J. S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory A., 11 (2008), 131294.
[16]	L. B. Ciric, On contraction type mappings, Math. Balk., 1 (1971), 52–57.
[17]	P. Debnath, Z. D. Mitrović, S. Y. Cho, Common fixed points of Kannan, Chatterjea and Reich type pairs of self-maps in a complete metric space, São Paulo J. Math. Sci., 15 (2021), 383–391. https://doi.org/10.1007/s40863-020-00196-y doi: 10.1007/s40863-020-00196-y
[18]	P. Debnath, Banach, Kannan, Chatterjea, and Reich-type contractive inequalities for multivalued mappings and their common fixed points, Math. Method. Appl. Sci., 45 (2021), 1587–1596. https://doi.org/10.1002/mma.7875 doi: 10.1002/mma.7875
[19]	M. E. Ege, C. Alaca, Some results for modular $b$-metric spaces and an application to system of linear equations, Azerbaijan J. Math., 8 (2018), 3–14.
[20]	N. Gholamian, M. Khanehgir, Fixed points of generalized Meir-Keeler contraction mappings in $b$-metric-like spaces, Fixed Point Theory A., 34 (2016). https://doi.org/10.1186/s13663-016-0507-6
[21]	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 (2020), 1781–1799. https://doi.org/10.3934/math.2021107 doi: 10.3934/math.2021107
[22]	S. Gulyaz, E. Karapinar, I. M. Erhan, Generalized $\alpha$-Meir-Keeler contraction mappings on Branciari $b$-metric spaces, Filomat, 17 (2017), 5445–5456. https://doi.org/10.2298/FIL1717445G doi: 10.2298/FIL1717445G
[23]	X. He, M. Song, D. Chen, Common fixed points for weak commutative mappings on a multiplicative metric space, Fixed Point Theory A., 48 (2014), 1–23. https://doi.org/10.1186/1687-1812-2014-48 doi: 10.1186/1687-1812-2014-48
[24]	M. Jleli, E. Karapinar, B. Samet, A best proximity point result in modular spaces with the Fatou property, Abstr. Appl. Anal., 2013, 329451. https://doi.org/10.1155/2013/329451
[25]	G. Jungck, Commuting mappings and fixed points, Am. Math. Mon., 83 (1976), 261–263. https://doi.org/10.1080/00029890.1976.11994093 doi: 10.1080/00029890.1976.11994093
[26]	E. Karapinar, S. Czerwik, H. Aydi, $(\alpha$-$\psi)$-Meir-Keeler contraction mappings in generalized $b$-metric spaces, J. Funct. Space., 2018 (2018), 3264620. https://doi.org/10.1155/2018/3264620 doi: 10.1155/2018/3264620
[27]	E. Karapinar, B. Samet, D. Zhang, Meir-Keeler type contractions on JS-(metric) spaces and related fixed point theorems, J. Fixed Point Theory A., 20 (2018), 1–18. https://doi.org/10.1007/s11784-018-0544-3 doi: 10.1007/s11784-018-0544-3
[28]	E. Karapinar, P. Kumam, P. Salimi, On $\alpha$-$\psi$-Meir-Keeler contractive mappings, Fixed Point Theory A., 2013, 1–12.
[29]	K. N. V. V. V. Prasad, A. K. Singh, Meir-Keeler type contraction via rational expression, Acta Math. Univ. Comen., 1 (2020), 19–25.
[30]	P. Kumam, W. Sintunavarat, S. Sedghi, N. Shobkolaei, Common fixed point of two $R$-weakly commuting mappings in $b$-metric spaces, J. Funct. Space., 2015 (2015), 350840. https://doi.org/10.1155/2015/350840 doi: 10.1155/2015/350840
[31]	M. Maiti, T. K. Pal, Generalization of two fixed point theorems, B. Calcutta Math. Soc., 70 (1978), 57–61.
[32]	A. Meir, E. Keeler, A theorem on contraction mapping, J. Math. Anal. Appl., 28 (1969), 326–329.
[33]	M. Neog, P. Debnath, S. Radenović, New extension of some common fixed point theorems in complete metric spaces, Fixed Point Theory, 20 (2019), 567–580. https://doi.org/10.24193/fpt-ro.2019.2.37 doi: 10.24193/fpt-ro.2019.2.37
[34]	G. A. Okeke, D. Francis, A. Gibali, On fixed point theorems for a class of $\alpha$-$\hat{v}$-Meir-Keeler-type contraction mapping in modular extended $b$-metric spaces, J. Anal., 2022 (2022), 1–22. https://doi.org/10.1007/s41478-022-00403-3 doi: 10.1007/s41478-022-00403-3
[35]	D. Panthi, Some common fixed point theorems satisfying Meir-Keeler type contractive conditions, Open J. Discrete Math., 8 (2018), 35–47. https://doi.org/10.4236/ojdm.2018.82004 doi: 10.4236/ojdm.2018.82004
[36]	S. Park, B. E. Rhoades, Meir-Keeler type contractive conditions, Math. Japonica, 26 (1981), 13–20.
[37]	V. Parvaneh, S. J. H. Ghoncheh, Fixed points of $(\psi, \varphi)_{\Omega}$-contractive mappings in ordered $p$-metric spaces, Global Anal. Discrete Math., 4 (2020), 15–29.
[38]	O. Popescu, Some new fixed point theorems for $\alpha$-Geraghty contraction type maps in metric spaces, Fixed Point Theory A., 2014 (2014). https://doi.org/10.1186/1687-1812-2014-190
[39]	I. H. N. Rao, K. P. R. Rao, Generalization of fixed point theorems of Meir-Keleer type, Indian J. Pure Ap. Mat., 16 (1985), 1249–1262.
[40]	B. Samet, C. Vetro, P. Vetro, Fixed point theorems for $\alpha$-$\psi$-contractive type mapping, Nonlinear Anal., 75 (2012), 2154–2165. https://doi.org/10.1016/j.na.2011.10.014 doi: 10.1016/j.na.2011.10.014
[41]	S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. I. Math., 32 (1982), 149–153.
[42]	Y. R. Sharma, Fixed point and weak commuting mapping, Int. J. Res. Eng. Technol., 2 (2013), 1–4.

Reader Comments

Your name:*

Email:*
© 2025 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)