On a class of fixed points for set contractions on partial metric spaces with a digraph

Talat Nazir; Zakaria Ali; Shahin Nosrat Jogan; Manuel de la Sen; Talat Nazir; Zakaria Ali; Shahin Nosrat Jogan; Manuel de la Sen

doi:10.3934/math.2023065

AIMS Mathematics

2023, Volume 8, Issue 1: 1304-1328. doi: 10.3934/math.2023065

Previous Article Next Article

Research article Special Issues

On a class of fixed points for set contractions on partial metric spaces with a digraph

1.
Department of Mathematical Sciences, University of South Africa, Florida 0003, South Africa
2.
Institute of Research and Development of Processes, University of the Basque Country, Leioa 48940, Bizkaia, Spain

Received: 27 July 2022 Revised: 07 October 2022 Accepted: 10 October 2022 Published: 19 October 2022
MSC : 47H04, 47H07, 47H10

We investigate the existence of fixed point problems on a partial metric space. The results obtained are for set contractions in the domain of sets and the pattern for the partial metric space is constructed on a directed graph. Essentially, our main strategy is to employ generalized $ \phi $-contractions in order to prove our results, where the fixed points are investigated with a graph structure. Moreover, we state and prove the well-posedness of fixed point based problems of the generalized $ \phi $-contractive operator in the framework of a partial metric space. We illustrate the main results in this manuscript by providing several examples.
- fixed point,
- set-contraction,
- partial metric space,
- graph,
- graph contraction
Citation: Talat Nazir, Zakaria Ali, Shahin Nosrat Jogan, Manuel de la Sen. On a class of fixed points for set contractions on partial metric spaces with a digraph[J]. AIMS Mathematics, 2023, 8(1): 1304-1328. doi: 10.3934/math.2023065

Related Papers:

Abstract

We investigate the existence of fixed point problems on a partial metric space. The results obtained are for set contractions in the domain of sets and the pattern for the partial metric space is constructed on a directed graph. Essentially, our main strategy is to employ generalized $ \phi $-contractions in order to prove our results, where the fixed points are investigated with a graph structure. Moreover, we state and prove the well-posedness of fixed point based problems of the generalized $ \phi $-contractive operator in the framework of a partial metric space. We illustrate the main results in this manuscript by providing several examples.

References

[1]	M. Abbas, M. R. Alfuraidan, A. R. Khan, T. Nazir, Fixed point results for set-contractions on metric spaces with a directed graph, Fixed Point Theory Appl., 2015 (2015), 14. https://doi.org/10.1186/s13663-015-0263-z doi: 10.1186/s13663-015-0263-z
[2]	M. Abbas, M. R. Alfuraidan, T. Nazir, M. Rashed, Common fixed points of multivalued $F$-contractions on metric spaces with a directed graph, Carpathian J. Math., 32 (2016), 1–12.
[3]	M. Abbas, B. Ali, Fixed point of Suzuki-Zamfirescu hybrid contractions in partial metric spaces via partial Hausdorff metric, Fixed Point Theory Appl., 2013 (2013), 1–21. https://doi.org/10.1186/1687-1812-2013-21 doi: 10.1186/1687-1812-2013-21
[4]	M. Abbas, V. Parvaneh, A. Razani, Periodic points of $T$-Ciric generalized contraction mappings in ordered metric spaces, Georgian Math. J., 19 (2012), 597–610. https://doi.org/10.1515/gmj-2012-0036 doi: 10.1515/gmj-2012-0036
[5]	S. M. A. Aleomraninejad, S. Rezapour, N. Shahzad, Some fixed point results on a metric space with a graph, Topol. Appl., 159 (2012), 659–663. https://doi.org/10.1016/j.topol.2011.10.013 doi: 10.1016/j.topol.2011.10.013
[6]	N. Assad, W. Kirk, Fixed point theorems for set-valued mappings of contractive type, Pacific J. Math., 43 (1972), 553–562. http://dx.doi.org/10.2140/pjm.1972.43.553 doi: 10.2140/pjm.1972.43.553
[7]	I. Altun, A. Erduran, Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory Appl., 2011 (2011), 508730. https://doi.org/10.1155/2011/508730 doi: 10.1155/2011/508730
[8]	I. Altun, F. Sola, H. Simsek, Generalized contractions on partial metric spaces, Topol. Appl., 157 (2010), 2778–2785. https://doi.org/10.1016/j.topol.2010.08.017 doi: 10.1016/j.topol.2010.08.017
[9]	H. Aydi, M. Abbas, C. Vetro, Partial Hausdorff metric and Nadler's fixed point theorem on partial metric spaces, Topol. Appl., 159 (2012), 3234–3242. https://doi.org/10.1016/j.topol.2012.06.012 doi: 10.1016/j.topol.2012.06.012
[10]	S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations inté grales, Fund. Math., 3 (1922), 133–181.
[11]	F. Bojor, Fixed point of $\varphi $-contraction in metric spaces endowed with a graph, Ann. Univ. Craiova Ser. Mat. Inform., 37 (2010), 85–92. https://doi.org/10.52846/ami.v37i4.374 doi: 10.52846/ami.v37i4.374
[12]	C. Chifu, G. Petruşel, Generalized contractions in metric spaces endowed with a graph, Fixed Point Theory Appl., 2012 (2012), 1–9. https://doi.org/10.1186/1687-1812-2012-161 doi: 10.1186/1687-1812-2012-161
[13]	N. Hussain, J. R. Roshan, V. Parvaneh, A. Latif, A unification of $G$-metric, partial metric, and $b$-metric spaces, Abstr. Appl. Anal., 2014 (2014), 180698. https://doi.org/10.1155/2014/180698 doi: 10.1155/2014/180698
[14]	J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), 1359–1373. https://doi.org/10.1090/S0002-9939-07-09110-1 doi: 10.1090/S0002-9939-07-09110-1
[15]	J. Jachymski, I. Jóźwik, Nonlinear contractive conditions: A comparison and related problems, Banach Center Publ., 1 (2007), 123–146. https://doi.org/10.4064/bc77-0-10 doi: 10.4064/bc77-0-10
[16]	A. Latif, T. Nazir, M. A. Kutbi, Common fixed point results for class of set-contraction mappings endowed with a directed graph, RACSAM, 113 (2019), 3207–3222. https://doi.org/10.1007/s13398-019-00689-2 doi: 10.1007/s13398-019-00689-2
[17]	A. Latif, V. Parvaneh, P. Salimi, A. E. Al-Mazrooei, Various Suzuki type theorems in $b$-metric spaces, J. Nonlinear Sci. Appl., 8 (2015), 363–377. http://dx.doi.org/10.22436/jnsa.008.04.09 doi: 10.22436/jnsa.008.04.09
[18]	G. Lin, X. Cheng, Y. Zhang, A parametric level set based collage method for an inverse problem in elliptic partial differential equations, J. Comput. Appl. Math., 340 (2018), 101–121. https://doi.org/10.1016/j.cam.2018.02.008 doi: 10.1016/j.cam.2018.02.008
[19]	S. G. Matthews, Partial metric topology, Ann. New York Acad. Sci., 728 (1994), 183–197. https://doi.org/10.1111/j.1749-6632.1994.tb44144.x
[20]	Z. Mustafa, J. R. Roshan, V. Parvaneh, Z. Kadelburg, Fixed point theorems for weakly $T$-Chatterjea and weakly $T$-Kannan contractions in $b$-metric spaces, J. Inequal. Appl., 2014 (2014), 46. https://doi.org/10.1186/1029-242X-2014-46 doi: 10.1186/1029-242X-2014-46
[21]	A. Nicolae, D. O'Regan, A. Petruşel, Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph, J. Georgian Math. Soc., 18 (2011), 307–327. https://doi.org/10.1515/gmj.2011.0019 doi: 10.1515/gmj.2011.0019
[22]	J. J. Nieto, R. Rodriguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223–239. http://dx.doi.org/10.1007/s11083-005-9018-5 doi: 10.1007/s11083-005-9018-5
[23]	A. Pansuwan, W. Sintunavarat, V. Parvaneh, Some fixed point theorems for ($\alpha, \theta, k$)-contractive multi-valued mappings with some applications, Fixed Point Theory Appl., 2015 (2015), 132. https://doi.org/10.1186/s13663-015-0385-3 doi: 10.1186/s13663-015-0385-3
[24]	V. Parvaneh, N. Hussain, Z. Kadelburg, Generalized wardowski type fixed point theorems via $\alpha$-admissible $FG$-contractions in $b$-metric spaces, Acta Math. Sci., 36 (2016), 1445–1456. https://doi.org/10.1016/S0252-9602(16)30080-7 doi: 10.1016/S0252-9602(16)30080-7
[25]	A. M. C. Ran, M. C. Reuring, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Am. Math. Soc., 132 (2004), 1435–1443. https://doi.org/10.1090/S0002-9939-03-07220-4 doi: 10.1090/S0002-9939-03-07220-4
[26]	S. Romaguera, M. Schellekens, Partial metric monoids and semivaluation spaces, Topol. Appl., 153 (2005), 948–962. https://doi.org/10.1016/j.topol.2005.01.023 doi: 10.1016/j.topol.2005.01.023
[27]	S. Romaguera, A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl., 2010 (2010), 493298. https://doi.org/10.1155/2010/493298 doi: 10.1155/2010/493298
[28]	J. R. Roshan, N. Shobkolaei, S. Sedghi, V. Parvaneh, S. Radenovic, Common fixed point theorems for three maps in discontinuous $G_b$-metric spaces, Acta Math. Sci., 34 (2014), 1643–1654. https://doi.org/10.1016/S0252-9602(14)60110-7 doi: 10.1016/S0252-9602(14)60110-7
[29]	M. P. Schellekens, The correspondence between partial metrics and semivaluations, Theor. Comput. Sci., 315 (2004), 135–149.
[30]	O. Valero, On Banach fixed point theorems for partial metric spaces, Appl. Gen. Top., 6 (2005), 229–240. https://doi.org/10.1016/j.tcs.2003.11.016 doi: 10.1016/j.tcs.2003.11.016
[31]	Y. Zhang, B. Hofmann, Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems, Inverse Probl. Imag., 15 (2021), 229–256. https://doi.org/10.3934/ipi.2020062 doi: 10.3934/ipi.2020062
[32]	Q. Zhang, Y. Song, Fixed point theory for generalized $\varphi $-weak contractions, Appl. Math. Lett., 22 (2009), 75–78. https://doi.org/10.1016/j.aml.2008.02.007 doi: 10.1016/j.aml.2008.02.007

Reader Comments

your name:*

Email:*
© 2023 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)