Probabilistic <em>α</em>-min Ciric type contraction results using a control function

Samir Kumar Bhandari; Dhananjay Gopal; Pulak Konar; Samir Kumar Bhandari; Dhananjay Gopal; Pulak Konar

doi:10.3934/math.2020082

AIMS Mathematics

2020, Volume 5, Issue 2: 1186-1198. doi: 10.3934/math.2020082

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Research article

Probabilistic α-min Ciric type contraction results using a control function

1.
Department of Mathematics, Bajkul Milani Mahavidyalaya, P. O–Kismat Bajkul, Dist–Purba Medinipur, Bajkul, West Bengal, 721655, India
2.
Department of Applied Mathematics and Humanities, S. V. National Institute of Technology, Surat, 395007, Gujarat, India
3.
Department of Mathematics, Amity University, Kadampukur, 24PGS(N), Kolkata, West Bengal, 700135, India

Received: 11 October 2019 Accepted: 02 January 2020 Published: 17 January 2020
MSC : 47H10, 54E40, 54H25

The purpose of the paper is to propose some new probabilistic α-minimum Ciric type contraction results. Our results are established on probabilistic generalization of metric spaces or probabilistic metric spaces. The use of class of control fucntion φ which was introduced by Choudhury et al. in 2008 helped us to deduce the result. We also get a corollary. Some illustrative examples are given here. Our results are supported by those examples. Lastly an application of integral equation is given. An important conclusion is also made at the end of the results.
- PM-space,
- Ciric type contraction,
- Cauchy sequence,
- fixed point,
- altering distance function
Citation: Samir Kumar Bhandari, Dhananjay Gopal, Pulak Konar. Probabilistic α-min Ciric type contraction results using a control function[J]. AIMS Mathematics, 2020, 5(2): 1186-1198. doi: 10.3934/math.2020082

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Abstract

The purpose of the paper is to propose some new probabilistic α-minimum Ciric type contraction results. Our results are established on probabilistic generalization of metric spaces or probabilistic metric spaces. The use of class of control fucntion φ which was introduced by Choudhury et al. in 2008 helped us to deduce the result. We also get a corollary. Some illustrative examples are given here. Our results are supported by those examples. Lastly an application of integral equation is given. An important conclusion is also made at the end of the results.

References

[1]	S. K. Bhandari, Probabilistic Ciric type contraction results using drastic t-norm, B. Cal. Math. Soc., 109 (2017), 439-454.
[2]	S. K. Bhandari, B. S. Choudhury, Two unique fixed point results of p-cyclic probabilistic c-contractions using different types of t-norm, J. Int. Math. Vir. Inst., 7 (2017), 147-164.
[3]	S. K. Bhandari, Unique Probablistic p-cyclic c-contraction results using special product T-Norm, B. Cal. Math. Soc., 109 (2017), 55-68.
[4]	S. K. Bhandari, Generalized contraction results on probabilistic 2-metric spaces using a control function, Int. J. Eng. Sci., 7 (2018), 49-55.
[5]	S. S. Chang, B. S. Lee, Y. J. Cho, et al. Generalized contraction mapping principle and differential equations in probabilistic metric spaces, P. Am. Math. Soc., 124 (1996), 2367-2376. doi: 10.1090/S0002-9939-96-03289-3
[6]	B. S. Choudhury, K. P. Das, A new contraction principle in Menger spaces, Acta Math. Sin., 24 (2008), 1379-1386. doi: 10.1007/s10114-007-6509-x
[7]	B. S. Choudhury, K. P. Das, S. K. Bhandari, Two Ciric type probabilistic fixed point theorems for discontinuous mappings, Int. Electron. J. Pure Appl. Math., 5 (2012), 111-126.
[8]	B. S. Choudhury, S. K. Bhandari, Ciric type p-cyclic contraction results for discontinuous mappings, J. Int. Math. Vir. Inst., 4 (2014), 27-42.
[9]	B. S. Choudhury, S. K. Bhandari, P. Saha, A cyclic probabilistic c-contraction results using Hadzic and Lukasiewicz t-norms in Menger spaces, Anal. Theory Appl., 31 (2015), 283-298. doi: 10.4208/ata.2015.v31.n3.6
[10]	B. S. Choudhury, S. K. Bhandari, P. Saha, Unique fixed points of p-cyclic kannan type probabilistic contractions, Boll. Unione Mat. Ital., 10 (2017), 179-189. doi: 10.1007/s40574-016-0073-1
[11]	B. S. Choudhury, S. K. Bhandari, P- cyclic c-contraction result in Menger spaces using a control function, Demonstratio Math., 49 (2016), 213-223.
[12]	B. S. Choudhury, S. K. Bhandari, P. Saha, Probabilistic p-cyclic contractions using different types of t-norms, Random Oper. Stoch. Equ., 26 (2018), 39-52. doi: 10.1515/rose-2018-0004
[13]	P. N. Dutta, B. S. Choudhury, K. P. Das, Some fixed point results in Menger spaces using a control function, Surv. Math. Appl., 4 (2009), 41-52.
[14]	D. Gopal, M. Abbas, C. Vetro, Some new fixed point theorems in Menger PM-spaces with application to Volterra type integral equation, Appl. Math. Comput., 232 (2014), 955-967.
[15]	O. Hadzic, E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, 2001.
[16]	A. F. R. L. de Hierro, M. de la Sen, Some fixed point theorems in Menger probabilistic metric-like spaces, Fixed Point Theory A.,2015 (2015), 176.
[17]	M. S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, B. Aust. Math. Soc., 30 (1984), 1-9. doi: 10.1017/S0004972700001659
[18]	I. Kramosil, J. Michálek, Fuzzy metrics and statistical metric spaces, Kybernetika, 11 (1975), 336-344.
[19]	M. A. Kutbi, D. Gopal, C. Vetro, et al. Further generalization of fixed point theorems in Menger PM-spaces, Fixed Point Theory A., 2015 (2015), 32.
[20]	K. Menger, Statistical metrics, P. Natl. Acad. Sci. USA, 28 (1942), 535-537. doi: 10.1073/pnas.28.12.535
[21]	D. Mihet, Altering distances in probabilistic Menger spaces, Nonlinear Anal., 71 (2009), 2734-2738. doi: 10.1016/j.na.2009.01.107
[22]	S. V. R. Naidu, Some fixed point theorems in metric spaces by altering distances, Czec. Math. J., 53 (2003), 205-212. doi: 10.1023/A:1022991929004
[23]	B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α-ψ-contrantive type mappings, Nonlinear Anal., 75 (2012), 2154-2165. doi: 10.1016/j.na.2011.10.014
[24]	K. P. R. Sastry, G. V. R. Babu, Some fixed point theorems by altering distances between the points, Ind. J. Pure. Appl. Math., 30 (1999), 641-647.
[25]	K. P. R. Sastry, S. V. R. Naidu, G. V. R. Babu, et al. Generalisation of common fixed point theorems for weakly commuting maps by altering distances, Tamkang J. Math., 31 (2000), 243-250.
[26]	B. Schweizer, A. Sklar, Probabilistic Metric Spaces, Elsevier, North-Holland, 1983.
[27]	V. M. Sehgal, A. T. Bharucha-Reid, Fixed point of contraction mappings on probabilistic metric spaces, Theor. Math. Syst., 6 (1972), 97-102. doi: 10.1007/BF01706080
[28]	M. De la Sen, A. Ibeas, On the global stability of an iterative scheme in a probabilistic Menger space, J. Inequal. Appl., 2015 (2015), 243.
[29]	G. Verdoolage, G. Karagounis, A. Murari, et al. Jet-Efda contributors, Modelling fusion data in probabilistic metric spaces: applications to the identification of confinement regimes and plasma disruptions, Fusion Sci. Tech., 62 (2012), 356-365. doi: 10.13182/FST12-A14627

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