Common fixed and coincidence point theorems for nonlinear self-mappings in cone $ b $-metric spaces using $ \varphi $-mapping

Mingliang Song; Dan Liu; Mingliang Song; Dan Liu

doi:10.3934/era.2023245

Electronic Research Archive

2023, Volume 31, Issue 8: 4788-4806. doi: 10.3934/era.2023245

Previous Article Next Article

Research article

Common fixed and coincidence point theorems for nonlinear self-mappings in cone $ b $-metric spaces using $ \varphi $-mapping

Mingliang Song ^,,
Dan Liu

School of Mathematical Sciences, Jiangsu Second Normal University, Nanjing 210013, China

Received: 19 March 2023 Revised: 14 June 2023 Accepted: 27 June 2023 Published: 07 July 2023

In this paper, by means of a mapping $ \varphi\in\Phi(P, P_1) $, some new common fixed and coincidence point theorems for four and six nonlinear self-mappings in cone $ b $-metric spaces are established, respectively. Also, some examples are given to prove the effectiveness of our results. And with some remarks stating that our results complement and sharply improve some related results in the literature.
- coincidence point,
- common fixed point,
- $ \varphi $-mapping,
- nonlinear contraction,
- cone $ b $-metric space
Citation: Mingliang Song, Dan Liu. Common fixed and coincidence point theorems for nonlinear self-mappings in cone $ b $-metric spaces using $ \varphi $-mapping[J]. Electronic Research Archive, 2023, 31(8): 4788-4806. doi: 10.3934/era.2023245

Related Papers:

Abstract

In this paper, by means of a mapping $ \varphi\in\Phi(P, P_1) $, some new common fixed and coincidence point theorems for four and six nonlinear self-mappings in cone $ b $-metric spaces are established, respectively. Also, some examples are given to prove the effectiveness of our results. And with some remarks stating that our results complement and sharply improve some related results in the literature.

References

[1]	G. Jungck, Common fixed points for noncontinuous nonself maps on non-metric spaces, Far East J. Math. Sci., 4 (1996), 199–215.
[2]	G. Jungck, B. E. Rhoades, Fixed point for set valued functions without continuity, Indian J. Pure Appl. Math., 29 (1998), 227–238.
[3]	G. Jungck, Common fixed points for commuting and compatible maps on compacta, Proc. Amer. Math. Soc., 103 (1988), 977–983. http://doi.org/10.1090/S0002-9939-1988-0947693-2 doi: 10.1090/S0002-9939-1988-0947693-2
[4]	G. Jungck, S. Radenović, S. Radojević, V. Rakočević, Common fixed point theorems for weakly compatible pairs on cone metric spaces, Fixed Point Theory Appl., 57 (2009), 643840. http://doi.org/10.1155/2009/643840 doi: 10.1155/2009/643840
[5]	R. P. Pant, Common fixed points of noncommuting mappings, J. Math. Anal. Appl., 188 (1994), 436–440. http://doi.org/10.1006/jmaa.1994.1437 doi: 10.1006/jmaa.1994.1437
[6]	I. Beg, M. Abbas, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed Point Theory Appl., 7 (2006), 74503. https://doi.org/10.1155/FPTA/2006/74503 doi: 10.1155/FPTA/2006/74503
[7]	M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric space, J. Math. Anal. Appl., 341 (2008), 416–420. https://doi.org/10.1016/j.jmaa.2007.09.070 doi: 10.1016/j.jmaa.2007.09.070
[8]	S. Radenović, Common fixed points under contractive conditions in cone metric spaces, Comput. Math. Appl., 58 (2009), 1273–1278. https://doi.org/10.1016/j.camwa.2009.07.035 doi: 10.1016/j.camwa.2009.07.035
[9]	S. K. Malhotra, S. Shukla, R. Sen, Some coincidence and common fixed point theorems for Prešić-Reich type mappings in cone metric spaces, Rend. Semin. Mat. Univ. Politec. Torino., 70 (2012), 247–260.
[10]	G. Song, X. Sun, Y. Zhao, G. Wang, New common fixed point theorems for maps on cone metric spaces, Appl. Math. Lett., 23 (2010), 1033–1037. https://doi.org/10.1016/j.aml.2010.04.032 doi: 10.1016/j.aml.2010.04.032
[11]	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., 46 (2014), 1–14. https://doi.org/10.1186/1029-242X-2014-46 doi: 10.1186/1029-242X-2014-46
[12]	A. Petrusel, G. Petrusel, J. C. Yao, Fixed point and coincidence point theorems in $b$-metric spaces with applications, Appl. Anal. Discrete Math., 11 (2017), 199–215. https://doi.org/10.2298/AADM1701199P doi: 10.2298/AADM1701199P
[13]	J. R. Roshan, N. Shobkolaei, S. Sedghi, M. Abbas, Common fixed point of four maps in $b$-metric spaces, Hacettepe J. Math. Stat., 43 (2014), 613–624.
[14]	S. Aleksić, H. Huang, Z. D. Mitrović, S. Radenović, Remarks on some fixed point results in $b$-metric spaces, J. Fixed Point Theory Appl., 20 (2018), 1–17. https://doi.org/10.1007/s11784-018-0626-2 doi: 10.1007/s11784-018-0626-2
[15]	N. Hussain, M. H. Shah, KKM mappings in cone $b$-metric spaces, Comput. Math. Appl., 62 (2011), 1677–1684. https://doi.org/10.1016/j.camwa.2011.06.004 doi: 10.1016/j.camwa.2011.06.004
[16]	H. Huang, S. Xu, Fixed point theorems of contractive mappings in cone $b$-metric spaces and applications, Fixed Point Theory Appl., 112 (2013), 1–10. https://doi.org/10.1186/1687-1812-2013-112 doi: 10.1186/1687-1812-2013-112
[17]	J. R. Roshan, N. Shobkolaei, S. Sedghi, V. Parvaneh, S. Radenović, 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
[18]	A. Z. Rezazgui, A. A. Tallafha, W. Shatanawi, Common fixed point results via $\mathcal{A}_{\vartheta} $-$\alpha$-contractions with a pair and two pairs of self-mappings in the frame of an extended quasi $b$-metric space, AIMS Math., 8 (2023), 7225–7241. https://doi.org/10.3934/math.2023363 doi: 10.3934/math.2023363
[19]	J. Ahmad, A. E. Al-Mazrooei, H. Aydi, M. D. L. Sen, Rational contractions on complex-valued extended $b$-metric spaces and an application, AIMS Math., 8 (2023), 3338–3352. https://doi.org/10.3934/math.2023172 doi: 10.3934/math.2023172
[20]	M. Abbas, B. E. Rhoades, T. Nazir, Common fixed points for four maps in cone metric spaces, Appl. Math. Comput., 216 (2010), 80–86. https://doi.org/10.1016/j.amc.2010.01.003 doi: 10.1016/j.amc.2010.01.003
[21]	Y. Han, S. Xu, New common fixed point results for four maps on cone metric spaces, Appl. Math., 2 (2011), 1114–1118. https://doi.org/10.4236/am.2011.29153 doi: 10.4236/am.2011.29153
[22]	M. Rangamma, K. Prudhvi, Common fixed points under contractive conditions for three maps in cone metric spaces, Bull. J. Math. Anal. Appl., 4 (2012), 174–180.
[23]	S. K. Malhotra, S. Shukla, R. Sen, Some coincidence and common fixed point theorems in cone metric spaces, Bull. J. Math. Anal. Appl., 4 (2012), 64–71.
[24]	A. K. Dubey, U. Mishra, Coincidence point and fixed point results for three self mapping in cone metric spaces, Int. J. Nonlinear Anal., 5 (2014), 104–110. http://doi.org/10.22075/IJNAA.2014.142 doi: 10.22075/IJNAA.2014.142
[25]	L. Liu, F. Gu, Common fixed point theorems for six self-maps in $b$-metric spaces with nonlinear contractive conditions, J. Nonlinear Sci. Appl., 9 (2016), 5909–5930. http://doi.org/10.22436/jnsa.009.12.02 doi: 10.22436/jnsa.009.12.02
[26]	L. G. Huang, X. Zhang, Cone metric space and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468–1476. https://doi.org/10.1016/j.jmaa.2005.03.087 doi: 10.1016/j.jmaa.2005.03.087
[27]	X. Zhang, Common fixed point theorem of Lipschitz type mappings on cone metric space, Acta Math. Sin., 53 (2010), 1139–1148.

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)