Interpolative Chatterjea and cyclic Chatterjea contraction on quasi-partial $b$-metric space

Pragati Gautam; Vishnu Narayan Mishra; Rifaqat Ali; Swapnil Verma; Pragati Gautam; Vishnu Narayan Mishra; Rifaqat Ali; Swapnil Verma

doi:10.3934/math.2021103

AIMS Mathematics

2021, Volume 6, Issue 2: 1727-1742. doi: 10.3934/math.2021103

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Research article

Interpolative Chatterjea and cyclic Chatterjea contraction on quasi-partial $b$-metric space

1.
Department of Mathematics, Kamala Nehru College, University of Delhi, August Kranti Marg, New Delhi 110049, India
2.
Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484887, India
3.
Department of Mathematics, College of Science and Arts, Muhayil, King Khalid University, P. O. Box 9004, Postal Code 61413, Abha, KSA

Received: 24 August 2020 Accepted: 12 November 2020 Published: 27 November 2020
MSC : 46T99, 47H10, 47H09, 54H25

The fixed point results for Chatterjea type contraction in the setting of Complete metric space exists in literature. Taking this approach forward Karapinar gave the concept of cyclic Chatterjea contraction mappings. Fan also worked on these cyclic mappings in a new setting of quasi-partial b-metric space. Motivated by the work of these researchers, we have introduced the notion of $qp_{b}$-cyclic Chatterjea contractive mappings and established fixed point results on them. The aim of this paper is to use an interpolative approach in the framework of quasi-partial b-metric space and to prove existence and uniqueness of fixed point theorem for $qp_{b}$-interpolative Chatterjea contraction mappings. The results are affirmed with applications based on them.
- quasi-partial $b$-metric space,
- fixed point,
- Chatterjea contraction,
- cyclic mapping,
- $qp_b$-cyclic Chatterjea contraction mapping,
- interpolation
Citation: Pragati Gautam, Vishnu Narayan Mishra, Rifaqat Ali, Swapnil Verma. Interpolative Chatterjea and cyclic Chatterjea contraction on quasi-partial $b$-metric space[J]. AIMS Mathematics, 2021, 6(2): 1727-1742. doi: 10.3934/math.2021103

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Abstract

The fixed point results for Chatterjea type contraction in the setting of Complete metric space exists in literature. Taking this approach forward Karapinar gave the concept of cyclic Chatterjea contraction mappings. Fan also worked on these cyclic mappings in a new setting of quasi-partial b-metric space. Motivated by the work of these researchers, we have introduced the notion of $qp_{b}$-cyclic Chatterjea contractive mappings and established fixed point results on them. The aim of this paper is to use an interpolative approach in the framework of quasi-partial b-metric space and to prove existence and uniqueness of fixed point theorem for $qp_{b}$-interpolative Chatterjea contraction mappings. The results are affirmed with applications based on them.

References

[1]	S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math., 3 (1922), 133-181. doi: 10.4064/fm-3-1-133-181
[2]	I. A. Bakhtin, The contraction principle in quasi metric spaces, Funct. Anal., 30 (1989), 26-37.
[3]	S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav., 1 (1993), 5-11.
[4]	S. Shukla, Partial b-metric spaces and fixed point theorems, Mediterr. J. Math., 11 (2014), 703-711. doi: 10.1007/s00009-013-0327-4
[5]	S. G. Matthews, Partial metric topology, Ann. N. Y. Acad. Sci., 728 (1994), 183-197. doi: 10.1111/j.1749-6632.1994.tb44144.x
[6]	R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71-76.
[7]	S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci., 25 (1972), 727-730.
[8]	W. A. Kirk, P. S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying cyclic contractive conditions, Fixed Point Theory, 4 (2003), 79-89.
[9]	E. Karapnar, Fixed point theory for cyclic weak ψ-contraction, Appl. Math. Lett., 24 (2011), 822-825. doi: 10.1016/j.aml.2010.12.016
[10]	X. Fan, Fixed point theorems for cyclic mappings in quasi-partial b-metric spaces, J. Nonlinear Sci. Appl., 9 (2016), 2175-2189. doi: 10.22436/jnsa.009.05.22
[11]	E. Karapınar, Revisiting the Kannan type contractions via interpolation, Adv. Theory Nonlinear Anal. Appl., 2 (2018), 85-87.
[12]	E. Karapınar, R. P. Agarwal, H. Aydi, Interpolative Reich-Rus-Ćirić type contractions on partial metric spaces, Mathematics, 6 (2018), 1-7.
[13]	E. Karapınar, O. Alqahtani, H. Aydi, On interpolative Hardy-Rogers type contractions, Symmetry, 11 (2019), 1-7.
[14]	Y. U. Gaba, E. Karapınar, A new approach to the interpolation contraction, Axioms, 8 (2019), 1-4.
[15]	H. Aydi, C. M. Chen, E. Karapınar, Interpolative Ćirić-Riech-Rus type contractions via the Brainciari distance, Mathematics, 7 (2019), 1-7.
[16]	E. Karapınar, R. P. Agarwal, Interpolative Rus-Riech-Ćirić type contractions via the simulation function, An. St. Univ. Ovidius Constant, 27 (2019), 137-152..
[17]	H. Aydi, E. Karapınar, A. F. Rol$\mathop {\rm{d}}\limits^` $an López de Hierro, ω-Interpolative Ćirić-Riech-Rus type contractions, Mathematics, 7 (2019), 1-8.
[18]	L. B. Ćirić, On contraction type mappings, Math. Balkanica, 1 (1971), 52-57.
[19]	L. B. Ćirić, Generalized contractions and fixed-point theorems, Publ. Inst. Math., 12 (1971), 19-26.
[20]	S. Reich, Some remarks concerning contraction mappings, Bull. Can. Math., 14 (1971), 121-124. doi: 10.4153/CMB-1971-024-9
[21]	S. Reich, Fixed point of contractive functions, Boll. Un. Mat. Ital., 4 (1972), 26-42.
[22]	S. Reich, Kannan's fixed point theorem, Boll. Un. Mat. Ital., 4 (1971), 1-11.
[23]	G. E. Hardy, T. D. Rogers, A generalization of a fixed point theorem of reich, Can. Math. Bull., 16 (1973), 201-206. doi: 10.4153/CMB-1973-036-0
[24]	H. Aydi, E. Karapınar, A Meir-Keeler common type fixed point theorem on partial metric spaces, Fixed Point Theory Appl., 2012 (2012), 1-10. doi: 10.1186/1687-1812-2012-1
[25]	L. Ćirić, B. Samet, H. Aydi, C. Vetro, Common fixed points of generalized contractions on partial metric spaces and an application, Appl. Math. Comput., 218 (2011), 2398-2406.
[26]	E. Karapınar, K. P. Chi, T. D. Thanh, A generalization of Ćirić quasi-contractions, Abstr. Appl. Anal., 2012 (2012), 1-9.
[27]	S. G. Krein, J. I. Petunin, E. M. Semenov, Interpolation of linear operators, American Mathematical Society, Providence, RI, USA, 2002.
[28]	E. Karapınar, I. M. Erhan, Fixed point theorem for operators on partial metric space, Appl. Math. Lett., 24 (2011), 1894-1899. doi: 10.1016/j.aml.2011.05.013
[29]	A. Gupta, P. Gautam, Topological structure of quasi-partial b-metric spaces, Int. J. Pure Math. Sci., 17 (2016), 8-18. doi: 10.18052/www.scipress.com/IJPMS.17.8
[30]	K. P. Chi, E. Karapınar, T. D. Thanh, A generalized contraction principle in partial metric space, Math. Comput. Modell., 55 (2012), 1673-1681. doi: 10.1016/j.mcm.2011.11.005
[31]	E. Karapınar, İ. M. Erhan, A. Özt ürk, Fixed point theorems on quasi-partial metric space, Math. Comput. Modell., 57 (2013), 2442-2448. doi: 10.1016/j.mcm.2012.06.036
[32]	A. Gupta, P. Gautam, Quasi-partial b-metric spaces and some related fixed point theorems, Fixed Point Theory Appl., 2015 (2015), 1-12. doi: 10.1186/1687-1812-2015-1
[33]	V. N. Mishra, L. M. Snchez Ruiz, P. Gautam, S. Verma, Interpolative ReichRusĆirić and HardyRogers contraction on quasi-partial b-metric space and related fixed point results, Mathematics, 8 (2020), 1-11.
[34]	A. Gupta, P. Gautam, Some coupled fixed point theorems on quasi-partial b-metric spaces, Int. J. Math. Anal., 9 (2015), 293-306. doi: 10.12988/ijma.2015.412388

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