Two fixed point theorems in complete metric spaces

Huaping Huang; Bessem Samet; Huaping Huang; Bessem Samet

doi:10.3934/math.20241478

AIMS Mathematics

2024, Volume 9, Issue 11: 30612-30637. doi: 10.3934/math.20241478

Previous Article Next Article

Research article Special Issues

Two fixed point theorems in complete metric spaces

Huaping Huang ^{1
,
,},
Bessem Samet ²

1.
School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404020, China
2.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

Received: 27 August 2024 Revised: 22 October 2024 Accepted: 22 October 2024 Published: 28 October 2024
MSC : 47H10, 54E40, 33B15

Two new classes of self-mappings defined on a complete metric space $ (M, d) $ are introduced. The first one, called the class of $ p $-contractions with respect to a family of mappings, includes mappings $ F: M\to M $ satisfying a contraction involving a finite number of mappings $ S_i: M\times M\to M $. The second one, called the class of $ (\psi, \Gamma, \alpha) $-contractions, includes mappings $ F: M\to M $ satisfying a contraction involving the famous ratio $ \psi\left(\frac{\Gamma(t+1)}{\Gamma(t+\alpha)}\right) $, where $ \psi:[0, \infty)\to [0, \infty) $ is a function, $ \Gamma $ is the Euler Gamma function, and $ \alpha\in (0, 1) $ is a given constant. For both classes, under suitable conditions, we establish the existence and uniqueness of fixed points of $ F $. Our results are supported by some examples in which the Banach fixed point theorem is inapplicable. Moreover, the paper includes some interesting questions related to our work for further studies in the future. These questions will push forward the development of fixed point theory and its applications.
- complete metric space,
- weakly Picard continuous,
- $ p $-contraction,
- $ (\psi, \Gamma, \alpha) $-contraction,
- fixed point,
- Euler Gamma function
Citation: Huaping Huang, Bessem Samet. Two fixed point theorems in complete metric spaces[J]. AIMS Mathematics, 2024, 9(11): 30612-30637. doi: 10.3934/math.20241478

Related Papers:

Abstract

Two new classes of self-mappings defined on a complete metric space $ (M, d) $ are introduced. The first one, called the class of $ p $-contractions with respect to a family of mappings, includes mappings $ F: M\to M $ satisfying a contraction involving a finite number of mappings $ S_i: M\times M\to M $. The second one, called the class of $ (\psi, \Gamma, \alpha) $-contractions, includes mappings $ F: M\to M $ satisfying a contraction involving the famous ratio $ \psi\left(\frac{\Gamma(t+1)}{\Gamma(t+\alpha)}\right) $, where $ \psi:[0, \infty)\to [0, \infty) $ is a function, $ \Gamma $ is the Euler Gamma function, and $ \alpha\in (0, 1) $ is a given constant. For both classes, under suitable conditions, we establish the existence and uniqueness of fixed points of $ F $. Our results are supported by some examples in which the Banach fixed point theorem is inapplicable. Moreover, the paper includes some interesting questions related to our work for further studies in the future. These questions will push forward the development of fixed point theory and its applications.

References

[1]	S. Banach, Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales, Fund. Math., 3 (1922), 133–181. https://doi.org/10.4064/fm-3-1-133-181 doi: 10.4064/fm-3-1-133-181
[2]	R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 10 (1968), 71–76. https://doi.org/10.2307/2316437
[3]	S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulg. Sci., 25 (1972), 727–730.
[4]	S. Reich, Kannan's fixed point theorem, Boll. Unione. Mat. Ital., 4 (1971), 1–11.
[5]	S. Reich, Some remarks concerning contraction mappings, Can. Math. Bull., 14 (1971), 121–124. https://doi.org/10.4153/CMB-1971-024-9 doi: 10.4153/CMB-1971-024-9
[6]	L. Ćirić, A generalization of Banach's contraction principle, P. Am. Math. Soc., 45 (1974), 267–273. https://doi.org/10.1090/S0002-9939-1974-0356011-2 doi: 10.1090/S0002-9939-1974-0356011-2
[7]	V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9 (2004), 43–53.
[8]	F. Khojasteh, S. Shukla, S. Radenović, A new approach to the study of fixed point theorems via simulation functions, Filomat, 29 (2015), 1189–1194. https://doi.org/10.2298/FIL1506189K doi: 10.2298/FIL1506189K
[9]	J. Górnicki, Various extensions of Kannan's fixed point theorem, J. Fix. Point Theory Appl., 20 (2018), 1–12. https://doi.org/10.1007/s11784-018-0500-2 doi: 10.1007/s11784-018-0500-2
[10]	E. Petrov, Fixed point theorem for mappings contracting perimeters of triangles, J. Fix. Point Theory Appl., 25 (2023). https://doi.org/10.1007/s11784-023-01078-4
[11]	C. M. Pǎcurar, O. Popescu, Fixed point theorem for generalized Chatterjea type mappings, Acta Math. Hung., 2024 (2024). https://doi.org/10.1007/s10474-024-01455-6
[12]	A. N. Branga, I. M. Olaru, Generalized contractions and fixed point results in spaces with altering metrics, Mathematics, 10 (2022). https://doi.org/10.3390/math10214083
[13]	S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5–11. Available from: http://dml.cz/dmlcz/120469.
[14]	Z. Djedid, S. Al-Sharif, M. Al-Khaleel, J. Jawdat, On solutions of differential and integral equations using new fixed point results in cone $E_b$-metric spaces, Part. Differ. Equ. Appl. Math., 8 (2023). https://doi.org/10.1016/j.padiff.2023.100559
[15]	Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7 (2006), 289–297.
[16]	I. Altun, A. Erduran, Two fixed point results on $F$-metric spaces, Topol. Algebra Appl., 10 (2022), 61–67. https://doi.org/10.1515/taa-2022-0114 doi: 10.1515/taa-2022-0114
[17]	V. Ozturk, S. Radenović, Hemi metric spaces and Banach fixed point theorem, Appl. Gen. Topol., 25 (2024), 175–182. https://doi.org/10.4995/agt.2024.19780 doi: 10.4995/agt.2024.19780
[18]	F. Khojasteh, H. Khandani, Scrutiny of some fixed point results by $S$-operators without triangular inequality, Math. Slovaca, 70 (2020), 467–476. https://doi.org/10.1515/ms-2017-0364 doi: 10.1515/ms-2017-0364
[19]	V. Rakočević, K. Roy, M. Saha, Wardowski and Ćirić type fixed point theorems over non-triangular metric spaces, Quaest. Math., 45 (2022), 1759–1769. https://doi.org/10.2989/16073606.2021.1970044 doi: 10.2989/16073606.2021.1970044
[20]	M. Berzig, First results in suprametric spaces with applications, Mediterr. J. Math., 19 (2022), 1–18. https://doi.org/10.1007/s00009-022-02148-6 doi: 10.1007/s00009-022-02148-6
[21]	M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, New York, 1992.
[22]	I. V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to $\pi^{-1}$, J. Math. Anal. Appl., 442 (2016), 404–434. https://doi.org/10.1016/j.jmaa.2016.04.032 doi: 10.1016/j.jmaa.2016.04.032
[23]	L. Gordon, A stochastic approach to the gamma function, Am. Math. Mon., 101 (1994), 858–865. https://doi.org/10.1080/00029890.1994.11997039 doi: 10.1080/00029890.1994.11997039
[24]	S. S. Dragomir, Operator inequalities of the Jensen, Cěbysěv and Grüss type, Springer Briefs in Mathematics, Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-1521-3

Reader Comments

Your name:*

Email:*
© 2024 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)