GENERALIZED α-MIN SPECIAL TYPE CONTRACTION RESULTS ON 2-MENGER SPACES

  • Samir Kumar Bhandari Department of Mathematics, Bajkul Milani Mahavidyalaya, P.O- Kismat Bajkul, Dist - Purba Medinipur, Bajkul, West Bengal - 721655, India
  • Sumit Chandok School of Mathematics, Thapar Institute of Engineering and Technology, Patiala-147004, Punjab, India
Keywords: 2-Menger spaces, Cauchy sequence, Fixed point, ϕ-function, Altering distance function

Abstract


The goal of this paper is to present some novel probabilistic α-minimum contraction results on probabilistic 2-metric spaces. Our findings are based on probabilistic 2-metric spaces, which are the probabilistic generalisations of 2-metric spaces. An illustrative example backs up our findings.

References

Bhandari, S. K. 2017a, Probabilistic Ciric type contraction results using drastic t-norm, Bull. Cal. Math. Soc., 109(6), pp. 439-454.

Bhandari, S. K. 2017b, Unique Probablistic p-cyclic c-contraction results using special product T-Norm, Bull. Cal. Math. Soc., 109(1), pp. 55-68.

Bhandari, S. K. & Choudhury, B. S. 2017, Two unique fixed point results of p-cyclic probabilistic c-contractions using different types of t-norm, J. Internat. Math. Virt. Inst., 7, pp. 147-164.

Choudhury, B. S. & Bhandari, S. K. 2014, Ciric type p-cyclic contraction results for discontinuous mappings, J. Int. Math. Virt. Inst., 4, pp. 27-42.

Choudhury, B. S. & Bhandari, S. K. 2016, P-cyclic c-contraction result in Menger spaces using a control function, Demonst. Math., 49 (2), pp. 213-223. https://doi.org/10.1515/dema-2016-0018

Choudhury, B. S., Bhandari, S. K. & Saha, P. 2015, A cyclic probabilistic c-contraction results using Hadzic and Lukasiewicz tnorms in Menger spaces, Anal. Theory Appl., 31, pp. 283-298.

Choudhury, B. S. & Das, K. P. 2008, A new contraction principle in Menger spaces, Acta Math. Sinica, 24, pp. 1379-1386.

Choudhury, B. S., Das, K. P. & Bhandari, S. K. 2012, Two Ciric type probabilistic fixed point theorems for discontinuous mappings, Internat. Elect. J. Pure Appl. Math., 5(3), pp. 111-126.

Dutta, P. N., Choudhury, B. S. & Das, K. P. 2009, Some fixed point results in Menger spaces using a control function, Sur. Math. Appl., 4, pp. 41-52.

Gähler, S. 1963, 2-metrische Räume und ihre topologische Struktur, Math. Nachr., 26, pp. 115-148. https://doi.org/10.1002/mana.19630260109

Gähler, S. 1965, Uber die unifromisieberkeit 2-metrischer Räume, Math. Nachr., 28, pp. 235-244.

Gopal, D., Abbas, M. & Vetro, C. 2014, Some new fixed point theorems in Menger PM-spaces with application to Volterra type integral equation, Appl. Math. Comput., 232, pp. 955-967. https://doi.org/10.1016/j.amc.2014.01.135

Hadzic, O. 1994, A fixed point theorem for multivalued mappings in 2-menger spaces, Univ. u Novom Sadu, Zb. Rad. Prirod. Mat. Fak., Ser. Mat., 24, pp. 1-7.

Hadzic, O. & Pap, E. 2001, Fixed Point Theory in Probabilistic Metric Spaces (Kluwer Academic Publishers).

Khan, M., Swaleh, M. & Sessa, S. 1984, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30, pp. 1-9.

Kutbi, M. A., Gopal, D., Vetro, C. & Sintunavarat, W. 2015, Further generalization of fixed point theorems in Menger PM-spaces, Fixed Point Theory Appl., 2015(32). https://doi.org/10.1186/s13663-015-0279-4

Mihet, D. 2009, Altering distances in probabilistic Menger spaces, Nonlinear Anal., 71, pp. 2734-2738.

Schweizer, B. & Sklar, A. 1983, Probabilistic Metric Spaces (Elsevier, North-Holland).

Shi, Y., Ren, L. & Wang, X. 2003, The extension of fixed point theorems for set valued mapping, J. Appl. Math. Comput., 13, pp. 277-286.

Shih-sen, C. & Nan-Jing, H. 1989, On generalized 2-metric spaces and probabilistic 2-metric spaces with applications to fixed point theory, Math. Jap., 34(6), pp. 885-900.

Verdoolage, G., Karagounis, G., Murari, A. et al. 2012, Modelling fusion data in probabilistic metric spaces: applications to the identification of confinement regimes and plasma disruptions, Fusion Sci. Tech., 62(2), pp. 356-365. https://doi.org/10.13182/FST12-A14627

Zeng, W.-Z. 1987, Probabilistic 2-metric spaces, J. Math. Research Expo., 2, pp. 241-245.

Published
2022/06/23
Section
Original Scientific Paper