On The Existence of Common Fixed Point in a Complete Metric Space Using Integral Type Contraction Conditions

Mainak Mitra¹ , Jigmi Dorjee Bhutia² , Kalishankar Tiwary³

1. Department of Mathematics, Raiganj University, Raiganj, India.
2. Department of Mathematics, Kalimpong College, University of North Bengal, Kalimpong, India.
3. Department of Mathematics, Raiganj University, Raiganj, India.

Section:Research Paper, Product Type: Journal-Paper
Vol.9 , Issue.6 , pp.25-35, Dec-2022

Online published on Dec 31, 2022

Copyright © Mainak Mitra, Jigmi Dorjee Bhutia, Kalishankar Tiwary . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
The main objective of this paper is to introduce two new fixed point theorems involving more than one functions. Here in this article we will use some integral type contraction conditions and choose some suitable control functions to establish our result. We’ll further prove show that our result is a generalization of the result of Gupta et. al. which was introduced in 2020. Moreover we’ll see that our result will generate some other contraction conditions that will assure the existence of common fixed points.

Key-Words / Index Term :
Almost Altering Distance, C-Class Function, Common Fixed Point, Compatible Mapping, Integral Type Contraction,Metric Space, Ultra Altering Distance.

References :
[1] M. Akkouchi, “Some fixed and common fixed point theorems connected to a results of MS Khan, M. Swald and S. Sessa”, Demonstratio Mathematica, Vol. 35, pp.129–138, 2002.
[2] A.H. Ansari, “Note on ?-? contractive type mappings and related xed point”, Proceedings of the 2nd Regional Conference on Mathematics And Applications, PNU, Iran, pp.377–380, 2014.
[3] G.V.R. Babu and G.N. Alemayehu, “Point of coincidence and common fixed points of a pair of generalized weakly contractive maps”, Journal of Advanced Research in Pure Mathematics, Vol. 2, pp.89-106,2010.
[4] Stefan Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales”, Fund. math, Vol. 3,pp.133–181,1922,.
[5] A. Branciari, “A fixed point theorem for mappings satisfying a general contractive condition of integral type”, International Journal of Mathematics and Mathematical Sciences, Vol. 29, pp.531–536,2002.
[6] S.K. Chatterjea, “Fixed-point theorems”, Dokladi na Bolgarskata Akademiya na Naukite, Vol. 25, pp.727–,1972.
[7] Lj. B. ?iri?, “A generalization of Banach’s contraction principle”, Proceedings of the American Mathematical society, Vol. 45,pp.267–273, 1974.
[8] Ljubomir B. ?iri?, “A new fixed-point theorem for contractive mappings”., Publications de l’Institut Mathématique. Nouvelle Série, Vol. 29, pp.25–27,1981.
[9] Michael Edelstein, “An extension of Banach’s contraction principle”, Proceedings of the American Mathematical Society, Vol. 12, pp.7–10,1961.
[10] Michael Edelstein, “On fixed and periodic points under contractive mappings”, Journal of the London Mathematical Society, Vol. 1, pp.74–79,1962.
[11] Vishal Gupta and Naveen Mani,”A common fixed point theorem for two weakly compatible mappings satisfying a new contractive condition of integral type”,Mathematical Theory and Modeling, Vol. 1, pp.1–6,2011.
[12] Vishal Gupta and Naveen Mani, “Existence and uniqueness of fixed point for contractive mapping of integral type”, International Journal of Computing Science and Mathematics, Vol. 4, pp.72–83,2013.
[13] Vishal Gupta, A.H. Ansari, and Naveen Mani, “Fixed point theorem for new type of auxiliary functions”, Acta Universitatis Sapientiae, Mathematica, Vol. 12,pp.97–111,2020.
[14] G. Hardy and T. Rogers, “A generalization of a fixed point theorem of Reich”, Canadian Mathematical Bulletin, Vol. 16,pp.201–206,1973.
[15] M. Jeyaraman, V.B. Shakila, "Some Common Fixed Point Theorems in Complex Valued Intuitionistic Fuzzy Metric Spaces," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.1, pp.24-31, 2022.
[16] Gerald Jungck, “Compatible mappings and common fixed points”, International Journal of Mathematics and Mathematical Sciences, Vol. 9,pp.771–779,1986.
[17] R. Kannan, “Some results on fixed points—II”, The American Mathematical Monthly, Vol. 76,pp.405–408, 1969.
[18] Valeriu Popa and Alina Mihaela Patriciu, “A general fixed point theorem for a pair of self mappings with common limit range property in G-metric spaces”, Facta Universitatis. Series: Mathematics and Informatics, Vol. 29, pp.351-370,2015.
[19] Ephraim Rakotch, “A note on contractive mappings”, Proceedings of the American Mathematical Society, pp.459–465, 1962.
[20] B.E. Rhoades, “Two fixed-point theorems for mappings satisfying a general contractive condition of integral type”, International Journal of Mathematics and Mathematical Sciences, Vol. 63, pp.4007–7013,2003.
[21] Bessem Samet and Habib Yazidi, “Fixed point theorems with respect to a contractive condition of integral type”,Rendiconti del Circolo Matematico di Palermo, Vol. 60,pp.181–190,2011.
[22] K.P.R. Sastry, K.K.M. Sarma, P. Krishna Kumari, "A Fixed Point theorem on S_b-metric spaces with a weak S_b-metric," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.5, pp.7-11, 2019.
[23] S. Sessa, B.E. Rhoades and M.S. Khan, “On common fixed points of compatible mappings in metric and Banach spaces”, International Journal of Mathematics and Mathematical Sciences, Vol. 11, pp.375–392,1988.
[24] Kalishankar Tiwary, T Basu and S Sen, “Some common fixed point theorems in complete metric spaces”, Soochow Journal of mathematics, Vol. 21,pp.451–459,1995.