On common fixed point results in bicomplex valued metric spaces with application

Asifa Tassaddiq; Jamshaid Ahmad; Abdullah Eqal Al-Mazrooei; Durdana Lateef; Farha Lakhani; Asifa Tassaddiq; Jamshaid Ahmad; Abdullah Eqal Al-Mazrooei; Durdana Lateef; Farha Lakhani

doi:10.3934/math.2023278

AIMS Mathematics

2023, Volume 8, Issue 3: 5522-5539. doi: 10.3934/math.2023278

Previous Article Next Article

Research article

On common fixed point results in bicomplex valued metric spaces with application

1.
Department of Basic Sciences and Humanities, College of Computer and Information Sciences, Majmaah University, Al Majmaah 11952, Saudi Arabia
2.
Department of Mathematics, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia
3.
Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
4.
Department of Mathematics, College of Science, Taibah University, Madina 41411, Saudi Arabia
5.
School of Computing, University of Leeds, Leeds, United Kingdom

Received: 11 September 2022 Revised: 21 November 2022 Accepted: 04 December 2022 Published: 19 December 2022
MSC : 46S40, 54H25, 47H10

Metric fixed-point theory has become an essential tool in computer science, communication engineering and complex systems to validate the processes and algorithms by using functional equations and iterative procedures. The aim of this article is to obtain common fixed point results in a bicomplex valued metric space for rational contractions involving control functions of two variables. Our theorems generalize some famous results from literature. We supply an example to show the originality of our main result. As an application, we develop common fixed point results for rational contractions involving control functions of one variable in the context of bicomplex valued metric space.
- bicomplex valued metric space,
- common fixed point,
- control functions,
- rational expressions
Citation: Asifa Tassaddiq, Jamshaid Ahmad, Abdullah Eqal Al-Mazrooei, Durdana Lateef, Farha Lakhani. On common fixed point results in bicomplex valued metric spaces with application[J]. AIMS Mathematics, 2023, 8(3): 5522-5539. doi: 10.3934/math.2023278

Related Papers:

Abstract

Metric fixed-point theory has become an essential tool in computer science, communication engineering and complex systems to validate the processes and algorithms by using functional equations and iterative procedures. The aim of this article is to obtain common fixed point results in a bicomplex valued metric space for rational contractions involving control functions of two variables. Our theorems generalize some famous results from literature. We supply an example to show the originality of our main result. As an application, we develop common fixed point results for rational contractions involving control functions of one variable in the context of bicomplex valued metric space.

References

[1]	M. Camelo, D. Papadimitriou, L. Fàbrega, P. Vilà, Geometric routing with word-metric spaces, IEEE Commun. Lett., 18 (2014), 2125–2128. https://doi.org/10.1109/LCOMM.2014.2364213 doi: 10.1109/LCOMM.2014.2364213
[2]	K. J. Lippert, R. Cloutier, Cyberspace: a digital ecosystem, Systems, 9 (2021), 48. https://doi.org/10.3390/systems9030048 doi: 10.3390/systems9030048
[3]	M. Y. Khachay, Y. Y. Ogorodnikov, Efficient approximation of the capacitated vehicle routing problem in a metric space of an arbitrary fixed doubling dimension, Dokl. Math., 102 (2020), 324–329. https://doi.org/10.1134/S1064562420040080 doi: 10.1134/S1064562420040080
[4]	S. K. Panda, A. Tassaddiq, R. P. Agarwal, A new approach to the solution of non-linear integral equations via various $F_Be$-contractions, Symmetry, 11 (2019), 206 https://doi.org/10.3390/sym11020206 doi: 10.3390/sym11020206
[5]	A. Tassaddiq, S. Kanwal, S. Perveen, R. Srivastava, Fixed points of single-valued and multi-valued mappings in sb-metric spaces, J. Inequal. Appl., 2022 (2022), 85. https://doi.org/10.1186/s13660-022-02814-z doi: 10.1186/s13660-022-02814-z
[6]	A. Shoaib, S. Kazi, A. Tassaddiq, S. S Alshoraify, T. Rasham, Double controlled quasi-metric type spaces and some results, Complexity, 2020 (2020), 3460938. https://doi.org/10.1155/2020/3460938 doi: 10.1155/2020/3460938
[7]	A. Tassaddiq, General escape criteria for the generation of fractals in extended Jungck–Noor orbit, Math. Comput. Simul., 196 (2022), 1–14. https://doi.org/10.1016/j.matcom.2022.01.003 doi: 10.1016/j.matcom.2022.01.003
[8]	D. Li, A. A. Shahid, A. Tassaddiq, A.Khan, X. Guo, M. Ahmad, CR iteration in generation of antifractals with s-convexity, IEEE Access, 8 (2020), 61621–61630. https://doi.org/10.1109/ACCESS.2020.2983474 doi: 10.1109/ACCESS.2020.2983474
[9]	C. Zou, A. Shahid, A. Tassaddiq, A. Khan, M. Ahmad, Mandelbrot sets and Julia sets in Picard-Mann orbit, IEEE Access, 8 (2020), 64411–64421. https://doi.org/10.1109/ACCESS.2020.298468 doi: 10.1109/ACCESS.2020.298468
[10]	A. Tassaddiq, M. Tanveer, M. Azhar, W. Nazeer, S. Qureshi, A four step feedback iteration and its applications in fractals, Fractal Fract., 6 (2022), 662. https://doi.org/10.3390/fractalfract6110662 doi: 10.3390/fractalfract6110662
[11]	A. Tassaddiq, M. S. Shabbir, Q. Din, H. Naaz, Discretization, bifurcation, and control for a class of predator-prey interactions, Fractal Fract., 6 (2022), 31. https://doi.org/10.3390/fractalfract6010031 doi: 10.3390/fractalfract6010031
[12]	A. Tassaddiq, M. S. Shabbir, Q. Din, K. Ahmad, S. Kazi, A ratio-dependent nonlinear predator-prey model with certain dynamical results, IEEE Access, 8 (2020), 195074–195088. https://doi.org/10.1109/ACCESS.2020.3030778 doi: 10.1109/ACCESS.2020.3030778
[13]	M. S. Shabbir, Q. Din, K. Ahmad, A. Tassaddiq, A. H. Soori, M. A. Khan, Stability, bifurcation, and chaos control of a novel discrete-time model involving Allee effect and cannibalism, Adv. Differ. Equ., 2020 (2020), 379. https://doi.org/10.1186/s13662-020-02838-z doi: 10.1186/s13662-020-02838-z
[14]	M. S. Shabbir, Q. Din, R. Alabdan, A. Tassaddiq, K. Ahmad, Dynamical complexity in a class of novel discrete-time predator-prey interaction with cannibalism, IEEE Access, 8 (2020), 100226–100240. https://doi.org/10.1109/ACCESS.2020.2995679 doi: 10.1109/ACCESS.2020.2995679
[15]	N. Hussain, H. Işık, M. Abbas, Common fixed point results of generalized almost rational contraction mappings with an application, J. Nonlinear Sci. Appl., 9 (2016), 2273–2288. http://dx.doi.org/10.22436/jnsa.009.05.30 doi: 10.22436/jnsa.009.05.30
[16]	H. Işık, V. Parvaneh, B. Mohammadi, I. Altun, Common fixed point results for generalized Wardowski type contractive multi-valued mappings, Mathematics, 7 (2019), 1130. https://doi.org/10.3390/math7111130 doi: 10.3390/math7111130
[17]	H. Işık, W. Sintunavarat, An investigation of the common solutions for coupled systems of functional equations arising in dynamic programming, Mathematics, 7 (2019), 977. https://doi.org/10.3390/math7100977 doi: 10.3390/math7100977
[18]	H. Işık, Existence of a common solution to systems of integral equations via fixed point results, Open Math., 18 (2020), 249–261. https://doi.org/10.1515/math-2020-0024 doi: 10.1515/math-2020-0024
[19]	C. Segre, Le rappresentazioni reali delle forme complesse a gli enti iperalgebrici, Math. Ann., 40 (1892), 413–467. https://doi.org/10.1007/BF01443559 doi: 10.1007/BF01443559
[20]	G. B. Price, An introduction to multicomplex spaces and functions, CRC Press, 1991. https://doi.org/10.1201/9781315137278
[21]	A. Azam, B. Fisher, M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optim., 32 (2011), 243–253.
[22]	G. A. Okeke, Iterative approximation of fixed points of contraction mappings in complex valued Banach spaces, Arab J. Math. Sci., 25 (2019), 83–105. https://doi.org/10.1016/j.ajmsc.2018.11.001 doi: 10.1016/j.ajmsc.2018.11.001
[23]	J. Choi, S. K. Datta, T. Biswas, N. Islam, Some fixed point theorems in connection with two weakly compatible mappings in bicomplex valued metric spaces, Honam Math. J., 39 (2017), 115–126. https://doi.org/10.5831/HMJ.2017.39.1.115 doi: 10.5831/HMJ.2017.39.1.115
[24]	I. H. Jebril, S. K. Datta, R. Sarkar, N. Biswas, Common fixed point theorems under rational contractions for a pair of mappings in bicomplex valued metric spaces, J. Interdiscip. Math., 22 (2019), 1071–1082. https://doi.org/10.1080/09720502.2019.1709318 doi: 10.1080/09720502.2019.1709318
[25]	M. S. Abdullahi, A. Azam, Multivalued fixed points results via rational type contractive conditions in complex valued metric spaces, J. Int. Math. Virtual Inst., 7 (2017), 119–146
[26]	A. Azam, J. Ahmad, P. Kumam, Common fixed point theorems for multi-valued mappings in complex-valued metric spaces, J. Inequal. Appl., 2013 (2013), 578. https://doi.org/10.1186/1029-242X-2013-578 doi: 10.1186/1029-242X-2013-578
[27]	A. J. Gnanaprakasam, S. M. Boulaaras, G. Mani, B. Cherif, S. A. Idris, Solving system of linear equations via bicomplex valued metric space, Demonstr. Math., 54 (2021), 474–487. https://doi.org/10.1515/dema-2021-0046 doi: 10.1515/dema-2021-0046
[28]	Z. Gu, G. Mani, A. J. Gnanaprakasam, Y. Li, Solving a system of nonlinear integral equations via common fixed point theorems on bicomplex partial metric space, Mathematics, 9 (2021), 1584. https://doi.org/10.3390/math9141584 doi: 10.3390/math9141584
[29]	I. Beg, S. K. Datta, D. Pal, Fixed point in bicomplex valued metric spaces, Int. J. Nonlinear Anal. Appl., 12 (2021), 717–727. https://doi.org/10.22075/IJNAA.2019.19003.2049 doi: 10.22075/IJNAA.2019.19003.2049
[30]	R. Tabassum, M. S. Shagari, A. Azam, O. M. Kalthum S. K. Mohamed, A. A. Bakery, Intuitionistic fuzzy fixed point theorems in complex valued $b$ -metric spaces with applications to fractional differential equations, J. Funct. Spaces, 2022 (2022), 1–17. https://doi.org/10.1155/2022/2261199 doi: 10.1155/2022/2261199

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)