Fixed point results for nonlinear contractions of Perov type in abstract metric spaces with applications

Shaoyuan Xu; Yan Han; Suzana Aleksić; Stojan Radenović; Shaoyuan Xu; Yan Han; Suzana Aleksić; Stojan Radenović

doi:10.3934/math.2022817

AIMS Mathematics

2022, Volume 7, Issue 8: 14895-14921. doi: 10.3934/math.2022817

Previous Article Next Article

Research article Special Issues

Fixed point results for nonlinear contractions of Perov type in abstract metric spaces with applications

1.
School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, Guangdong 521041, China
2.
Faculty of Information Technology, Macau University of Science and Technology, Macau, China
3.
School of Mathematics and Statistics, Zhaotong University, Zhaotong, Yunnan 657000, China
4.
Faculty of Science, University of Kragujevac, Radoja Domanovica 12, 34000 Kragujevac, Serbia
5.
Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia

Received: 05 April 2022 Revised: 29 May 2022 Accepted: 31 May 2022 Published: 10 June 2022
MSC : 54H25, 47H10

In this paper, we present some common fixed point results for $ g $-quasi-contractions of Perov type in cone $ b $-metric spaces without the assumption of continuity. Besides, by constructing a non-expansive mapping from a real Banach algebra $ \mathcal{A} $ to $ \mathcal{B}(\mathcal{A}) $, the space of all of its bounded linear operators, we explore the relationship between the results for the mappings of Perov type on cone metric (cone $ b $-metric) spaces and that for the corresponding mappings on cone metric (cone $ b $-metric) spaces over Banach algebras. As consequences, without the assumption of normality, we obtain common fixed point theorems for generalized $ g $-quasi-contractions with the spectral radius $ r(\lambda) $ of the $ g $-quasi-contractive constant vector $ \lambda $ satisfying $ r(\lambda)\in [0, \frac{1}{s}) $ (where $ s\ge 1 $) in the setting of cone $ b $-metric spaces over Banach algebras. In addition, we also get some fixed point theorems for nonlinear contractions of Perov type in the setting of cone normed spaces. The main results generalize, extend and unify several well-known comparable results in the literature. Finally, we apply our main results to some nonlinear equations.
- cone $ b $-metric spaces (over Banach algebras),
- cone normed space,
- non-normal cones,
- nonlinear contractions of Perov type,
- fixed points
Citation: Shaoyuan Xu, Yan Han, Suzana Aleksić, Stojan Radenović. Fixed point results for nonlinear contractions of Perov type in abstract metric spaces with applications[J]. AIMS Mathematics, 2022, 7(8): 14895-14921. doi: 10.3934/math.2022817

Related Papers:

Abstract

In this paper, we present some common fixed point results for $ g $-quasi-contractions of Perov type in cone $ b $-metric spaces without the assumption of continuity. Besides, by constructing a non-expansive mapping from a real Banach algebra $ \mathcal{A} $ to $ \mathcal{B}(\mathcal{A}) $, the space of all of its bounded linear operators, we explore the relationship between the results for the mappings of Perov type on cone metric (cone $ b $-metric) spaces and that for the corresponding mappings on cone metric (cone $ b $-metric) spaces over Banach algebras. As consequences, without the assumption of normality, we obtain common fixed point theorems for generalized $ g $-quasi-contractions with the spectral radius $ r(\lambda) $ of the $ g $-quasi-contractive constant vector $ \lambda $ satisfying $ r(\lambda)\in [0, \frac{1}{s}) $ (where $ s\ge 1 $) in the setting of cone $ b $-metric spaces over Banach algebras. In addition, we also get some fixed point theorems for nonlinear contractions of Perov type in the setting of cone normed spaces. The main results generalize, extend and unify several well-known comparable results in the literature. Finally, we apply our main results to some nonlinear equations.

References

[1]	M. Fréchet, La notion d'écart et le calcul fonctionnel, C. R. Math. Acad. Sci. Paris, 140 (1905), 772–774.
[2]	D. R. Kurepa, Tableaux ramifi'es d'ensembles espaces pseudo-distanci'es, C. R. Acad. Sci. Paris, 198 (1934), 1563–1565.
[3]	B. Rzepecki, On fixed point theorems of Maia type, Publ. Inst. Math., 28 (1980), 179–186.
[4]	S. D. Lin, A common fixed point theorem in abstract spaces, Indian J. Pure Appl. Math., 18 (1987), 685–690.
[5]	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
[6]	I. Beg, M. Abbas, T. Nazır, Generalized cone metric spaces, J. Nonlinear Sci. Appl., 1 (2010), 21–31.
[7]	I. Beg, A. Azam, M. Arshad, Common fixed points for maps on topological vector space valued cone metric spaces, Int. J. Math. Math. Sci., 2009 (2009). https://doi.org/10.1155/2009/560264
[8]	A. I. Perov, On Cauchy problem for a system of ordinary differential equations, Priblizhen. Metody Reshen. Difer. Uravn., 2 (1964), 115–134.
[9]	A. Szilard, A note on Perov's fixed point theorem, Fixed Point Theory A., 4 (2003), 105–108.
[10]	M. Cvetković, V. Rakočević, Quasi-contraction of Perov type, Appl. Math. Comput., 235 (2014), 712–722. https://doi.org/10.1016/j.amc.2014.02.065 doi: 10.1016/j.amc.2014.02.065
[11]	M. Cvetković, V. Rakočević, Extersions of Perov theorem, Carpathian J. Math., 31 (2015), 181–188.
[12]	M. Cvetković, V. Rakočević, Common fixed point results for mappings of Perov type, Math. Nachr., 288 (2015), 1873–1890. https://doi.org/10.1002/mana.201400098 doi: 10.1002/mana.201400098
[13]	S. Xu, Ć. Dolićanin, S. Radenović, Some remarks on Perov's fixed point theorem, J. Adv. Math. Stud., 9 (2018), 361–369.
[14]	S. Radenović, F. Vetro, Some remarks on Perov type mappings in cone metric spaces, Mediterr. J. Math., 14 (2017), 1–16. https://doi.org/10.1007/s00009-017-1039-y doi: 10.1007/s00009-017-1039-y
[15]	F. Vetro, S. Radenović, Some results of Perov type in rectangular cone metric spaces, J. Fixed Point Theory A., 20 (2018), 1–16. https://doi.org/10.1007/s11784-018-0520-y doi: 10.1007/s11784-018-0520-y
[16]	M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, 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
[17]	S. Radenović, V. Rakočević, S. Rezapour, Common fixed points for $(g, f)$ type maps in cone metric spaces, Appl. Math. Comput., 218 (2011), 480–491. https://doi.org/10.1016/j.amc.2011.05.088 doi: 10.1016/j.amc.2011.05.088
[18]	S. H. Cho, J. S. Bae, Common fixed points theorems for mappings satisfying property $(E, A)$ on cone metric spaces, Math. Comput. Modell., 53 (2011), 945–951. https://doi.org/10.1016/j.mcm.2010.11.002 doi: 10.1016/j.mcm.2010.11.002
[19]	D. Ilić, V. Rakočević, Quasi-contraction on a cone metric space, Appl. Math. Lett., 22 (2009), 728–731. https://doi.org/10.1016/j.aml.2008.08.011 doi: 10.1016/j.aml.2008.08.011
[20]	Z. M. Fadail, G. S. Rad, V. Ozturk, S. Radenović, Some remarks on coupled, tripled and n-tupled fixed points theorems in ordered abstract metric spaces, Far East J. Math. Sci., 97 (2015), 809–839. http://dx.doi.org/10.17654/FJMSAug2015_809_839 doi: 10.17654/FJMSAug2015_809_839
[21]	L. Gajić, V. Rakočević, Quasi-contractions on a nonnormal cone metric space, Funct. Anal. Appl., 46 (2012). https://doi.org/10.1007/s10688-012-0008-2
[22]	N. Hussian, 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
[23]	H. Huang, S. Xu, Fixed point theorems of contractive mappings in cone $b$-metric spaces and applications, Fixed Point Theory A., 2013 (2013), 112. https://doi.org/10.1186/1687-1812-2013-112 doi: 10.1186/1687-1812-2013-112
[24]	L. Shi, S. Xu, Common fixed point theorems for two weakly compatible self-mappings in cone $b$-metric spaces, Fixed Point Theory A., 2013 (2013), 120. https://doi.org/10.1186/1687-1812-2013-120 doi: 10.1186/1687-1812-2013-120
[25]	A. S. Cvetković, M. P. Stanić, S. Dimitrijević, S. Simić, Common fixed point theorems for four mappings on cone metric type spaces, Fixed Point Theory A., 2011, (2011). https://doi.org/10.1155/2011/589725
[26]	L. B. Ćirić, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc., 45 (1974), 267–273. https://doi.org/10.2307/2040075 doi: 10.2307/2040075
[27]	M. H. Shah, S. Simić, N. Hussain, A. Sretenović, S. Radenović, Common fixed points theorems for occasionally weakly compatible pairs on cone metric type spaces, J. Comput. Anal. Appl., 14 (2012), 290–297.
[28]	M. P. Stanić, A. S. Cvetković, S. Simić, S. Dimitrijević, Common fixed point under contractive condition of Ćirić type on cone metric type spaces, Fixed Point Theory A., 2012 (2012), 35. https://doi.org/10.1186/1687-1812-2012-35 doi: 10.1186/1687-1812-2012-35
[29]	W. S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal., 72 (2010), 2259–2261. https://doi.org/10.1016/j.na.2009.10.026 doi: 10.1016/j.na.2009.10.026
[30]	Y. Feng, W. Mao, The equivalence of cone metric spaces and metric spaces, Fixed Point Theor., 2 (2010), 259–264.
[31]	H. Çakallı, A. Sönmez, Ç. Genç, On an equivalence of topological vector space valued cone metric spaces and metric spaces, Appl. Math. Lett., 25 (2012), 429–433. https://doi.org/10.1016/j.aml.2011.09.029 doi: 10.1016/j.aml.2011.09.029
[32]	W. S. Du, E. Karapinar, A note on cone $b$-metric and its related results: Generalizations or equivalence? Fixed Point Theory A., 2013 (2013), 210. https://doi.org/10.1186/1687-1812-2013-210
[33]	P. Kumam, N. V. Dung, V. T. L. Hang, Some equivalence between cone b-metric spaces and b-metric spaces, Abstr. Appl. Anal., 2013 (2013). https://doi.org/10.1155/2013/573740
[34]	H. Liu, S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory A., 2013 (2013), 320. https://doi.org/10.1186/1687-1812-2013-320 doi: 10.1186/1687-1812-2013-320
[35]	H. Liu, S. Xu, Fixed point theorem of quasi-contractions on cone metric spaces with Banach algebras, Abstr. Appl. Anal., 2013 (2013). https://doi.org/10.1155/2013/187348
[36]	W. Rudin, Functional analysis, 2 Eds., McGraw-Hill, 1991.
[37]	S. Radenović, B. E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces, Comput. Math. Appl., 57 (2009), 1701–1707. https://doi.org/10.1016/j.camwa.2009.03.058 doi: 10.1016/j.camwa.2009.03.058
[38]	G. Jungck, S. Radenović, S. Radojević, V. Rakočević, Common fixed point theorems for weakly compatible pairs on cone metric spaces, Fixed Point Theory A., 2009 (2019). https://doi.org/10.1155/2009/643840
[39]	M. Abbas, M. Ali Khan, S. Radenović, Common coupled fixed point theorems in cone metric spaces for w-compatible mappings, Appl. Math. Comput., 217 (2010), 195–202. https://doi.org/10.1016/j.amc.2010.05.042 doi: 10.1016/j.amc.2010.05.042
[40]	S. Xu, S. Radenović, Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory A., 2014 (2014), 102. https://doi.org/10.1186/1687-1812-2014-102 doi: 10.1186/1687-1812-2014-102
[41]	D. llić, V. Rakočević, Common fixed points for maps on cone metric space, J. Math. Anal. Appl., 341 (2008), 876–882. https://doi.org/10.1016/j.jmaa.2007.10.065 doi: 10.1016/j.jmaa.2007.10.065
[42]	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
[43]	K. M. Das, K. Naik, Common fixed point theorems for commuting maps on a metric space, Proc. Am. Math. Soc., 77 (1979), 369–373. https://doi.org/10.1090/S0002-9939-1979-0545598-7 doi: 10.1090/S0002-9939-1979-0545598-7
[44]	M. Asadi, H. Soleimani, S. M. Vaezpour, B. E. Rhoades, On T-stability of picard iteration in cone metric spaces, Fixed Point Theory A., 2009 (2009). https://doi.org/10.1155/2009/751090
[45]	G. D. Birkhoff, Dynamical systems, AMS Coll. Publ., New York, 1927.
[46]	K. Vogtmann, A. Weinstein, Mathematical methods of classical mechanics, Springer, 1978.
[47]	P. P. Zabreiko, K-metric and K-normed spaces: Survey, Collect. Math., 48 (1997), 825–859.
[48]	E. Karapinar, Fixed point theorems in cone Banach spaces, Fixed Point Theory A., 2009 (2009). https://doi.org/10.1155/2009/609281
[49]	E. D. Pascale, L. D. Pascale, Fixed points for some non-obviously contractive operators, Proc. Amer. Math. Soc., 130 (2002), 3249–3254. https://doi.org/10.1090/S0002-9939-02-06704-7 doi: 10.1090/S0002-9939-02-06704-7
[50]	A. Şahin, K. Şamdanli, Some coincidence best proximity point results in $S$-metric spaces, Proc. Inter. Math. Sci., 3 (2021), 75–87. https://doi.org/10.47086/pims.1035385 doi: 10.47086/pims.1035385
[51]	G. V. R. Babu, D. R. Babu, Common fixed points of Geraghty-Suzuki type contraction maps in $b$- metric spaces, Proc. Int. Math. Sci., 2 (2020), 26–47.
[52]	H. Çakallı, On variations of quasi-Cauchy sequences in cone metric spaces, Filomat, 30 (2016), 603–610. https://doi.org/10.2298/FIL1603603C doi: 10.2298/FIL1603603C
[53]	M. Aslantas, H. Şahin, U. Sadullah, Some generalizations for mixed multivalued mappings, Appl. Gen. Topol., 23 (2022), 169–178. https://doi.org/10.4995/agt.2022.15214 doi: 10.4995/agt.2022.15214
[54]	M. Aslantas, H. Şahin, D. Turkoglu, Some Caristi type fixed point theorems, J. Anal., 29 (2020), 1–15. https://doi.org/10.1007/s41478-020-00248-8 doi: 10.1007/s41478-020-00248-8
[55]	Z. M. Fadail, A. Savic, S. Radenović, New distance in cone $S$-metric spaces and common fixed point theorems, J. Math. Comput. Sci., 26 (2022), 368–378. https://doi.org/10.22436/jmcs.026.04.05 doi: 10.22436/jmcs.026.04.05
[56]	S. M. Abusalim, M. Sahar, Z. M. Fadail, New coupled and common coupled fixed point results with generalized $c$-distance on cone $b$-metric spaces, J. Math. Comput. Sci., 25 (2022), 209–218. https://doi.org/10.22436/jmcs.025.03.01 doi: 10.22436/jmcs.025.03.01
[57]	K. A. Singh, M. R. Singh, M. B. Devi, Singh, T. C. Singh, Cone $A_b$-metric space and some coupled fixed point theorems, J. Math. Comput. Sci., 24 (2022), 246–255. https://doi.org/10.22436/jmcs.024.03.06 doi: 10.22436/jmcs.024.03.06
[58]	A. Elhamed, M. Gehad, Fixed point results for $(\beta, \alpha)$-implicit contractions in two generalized b-metric spaces, J. Nonlinear Sci. Appl., 14 (2021), 39–47. https://doi.org/10.22436/jnsa.014.01.05 doi: 10.22436/jnsa.014.01.05

Reader Comments

Your name:*

Email:*
© 2022 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)