Title: On a Common Fixed Point Theorem in Bicomplex Valued b-metric Space
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-20-00017; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 358-366
Document Type: Research Paper
Author(s): Sanjib Kumar Datta a , Dipankar Pal b , Rakesh Sarkar c , Arghyatanu Manna d
aDepartment of Mathematics, University of Kalyani, P.O.: Kalyani, Dist.: Nadia, PIN-741235, West Bengal, India. Ph. no.-8017105978
bDepartment of Mathematics, Prof. Syed Nurul Hasan College, P.O.: Farakka Barrage, Dist.: Murshidabad, PIN-742212, West Bengal, India. Ph.no.-9153837959
cDepartment of Mathematics, Gour Mahavidyalaya, P.O.: Mangalbari, Dist.: Malda, PIN-732142, West Bengal, India. Ph. no.-9733608224
dMousini Co-operative High School(H.S.), Bagdanga, Fraserganj Coastral, Kakdwip,South 24 Parganas, PIN-743357. Ph.no.-8599972619
Received: 8 July 2020, Accepted: 15 April 2021, Published: 25 April 2021.
Corresponding Author: Sanjib Kumar Datta (Email address: firstname.lastname@example.org)
Full Text: PDF
The main purpose of this paper is to investigate a common fixed point theorem in bicomplex valued b-metric space satisfying some rational inequalities for two pairs of weakly compatible self contracting mappings and to extend the result obtained by Azam et al.. Our result is the generalisation of the result of Mitra and the application of Banach contraction principle. Also we use the concepts of Choi et al. and Datta et al.
Keywords: Common fixed point, Contractive type mapping, Bicomplex valued metric space, Bicomplex valued b-metric spaceReferences:
- A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optim. 32 (3), 243-253, 2011.
- J. Ahmad, A. Azam and S. Saejung, Common fixed point results for contractive mappings in complex valued metric spaces, Fixed point Theory and App. 2014, 1-11, 2014.
- D. Alpay, M. E. Luna-Elizarrarás, M. Shapiro and D. C. Struppa, Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis, Springer Briefs, 2014.
- S. Banach, Sur les operations dans les ensembles abs traits et leur application aux equations integrales, Fund. Math. 3, 133-181, 1922.
- S. Bhatt, S. Chaukiyal and R. C. Dimri, A common fixed point theorem for wealky compatible maps in complex valued metric spaces, Internat. J. Math. Sci. Appl. 1 (3), 1385-1389, 2011.
- S. Bhatt, S. Chaukiyal and R. C. Dimri, Common fixed point of mappings satisfying rational inequality in complex valued metric space, Internat. J. Pure. App. Math. 73 (2), 159-164, 2011.
- H. De. Bie, D. C. Struppa, A. Vajiac and M. Vajiac, The Cauchy-Kowalevski product for bicomplex holomorphic functions, Math. Nachr 285 (10), 1230-1242, 2012.
- F. Colombo, I. Sabadini, D.C. Struppa, A. Vajiac and M. Vajiac, Singularities of functions of one and several bicomplex variables, Ark. Math., 2010.
- B. S. Choudhury, N. Metiya and P. Konar, Fixed point results in partially ordered complex valued metric spaces for rational type expressions, Bangmod Int. J. Math. & Comp. Sci. 1 (1), 55-62, 2015.
- B. S. Choudhury, N. Metiya and P. Konar, Fixed point results for rational type contruction in partially ordered complex valued metric spaces, Bulletin Internat. Math. Virtual Institute 5, 73-80, 2015.
- J. Choi, S. K. Datta, T. Biswas and Md. N. Islam, Some common fixed point theorems in connection with two weakly compatible mappings in bicomplex valued metric spaces, Honam Mathematical J. 39 (1), 115-126, 2017.
- G. Scorza Dragoni, Sulle funzioni olomorfe di una variabile bicomplessa, Reale Accad. d’Italia, Mem. Classe Sci. Nat. Fis. Mat. 5, 597-665, 1934.
- A. K. Dubey, R. Shukla and P. P. Dubey, Some common fixed point theorems for contractive mappings in complex-valued b-metric spaces, Asian J. Math. App. 1, Article ID ama0266, 2015.
- S. K. Datta, D. Pal, N. Biswas and S. Sarkar, On the study of fixed point theorems in bicomplex valued metric spaces, Journal of the Calcutta Mathematical Society 16 (1), 73-94, 2020.
- S. K. Datta, D. Pal, R. Sarkar and J. Saha, Some common fixed point theorems for contracting mappings in bicomplex valued b-metric spaces, Bull. Cal. Math. Soc. 112 (4), 329-354, 2020.
- S. K. Datta, D. Pal, R. Sarkar and J. Saha, Common fixed point theorems in bicomplex valued metric spaces b-metric spaces or rational contraction, Jñānābha 50 (2), 93-105, 2020.
- Jaishree, On Conjugates and modulii of bicomplex numbers, Internat. J. Engrg. Sci. Tech. 4 (6), 2567-2575, 2012.
- I. H. Jebril, S. K. Datta, R. Sarkar and N. Biswas, Common fixed point theorems under rational contractions for a pair of mappings in bicomplex valued metric spaces, Journal of Interdisciplinary Mathematics 22 (7), 1071-1082, 2019.
- A. Kumar, P. Kumar and P. Dixit, Maximum and minimum modulus principle for bicomplex holomorphic functions, Internat. J. Engrg. Tech. 3 (2), 1484-1491, 2011.
- A. A. Mukheimer, Some common fixed point theorems in complex-valued b-metric spaces, Sci. World J. , Article ID 587825, 2014.
- T. Mitra, A common fixed point theorem in complex valued b-metric spaces, Internat. J. Adv. Sci. Tech. Res. 4 (5), 504-548, 2015.
- G. B. Price, An introduction to multicomplex spaces and functions, Marcel Dekker, New York, 1991.
- D. Rochon and M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers, An. Univ. Oradea Fasc. Mat. 11, 71-110, 2004.
- F. Rouzkard and M. Imdad, Some common fixed point theorems on complex valued metric spaces, Computers and Math. App. 64, 1866-1874, 2012.
- K. P. R. Rao, P. Ramga Swamy and J. Rajendra Prasad, A common fixed point theorem in complex-valued b-metric spaces, Bull. Math. & Stat. Res. 1 (1), 1-8, 2013.
- C. Segre, Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici, Math. Ann. 40, 413-467, 1892.
- D. Singh, O. P. Chauhan, N. Singh and V. Joshi, Common fixed point theorems in complex-valued b-metric spaces, J. Math. Comput. Sci. 5 (3), 412-429, 2015.
- R. Tiwari and D. P. Shukla, Six maps with a common fixed point in complex valued metric spaces, Res. J. Pure Algebra 2 (12), 365-369, 2012.
- R. K. Verma and H. K. Pathak, Common fixed point theorems for a pair of mappings in complex-valued metric spaces, J. Math. Sci. & Comp. Sci. 6, 18-26, 2013.