On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators

Authors

DOI:

https://doi.org/10.4995/agt.2021.15561

Keywords:

Menger PM-spaces, fixed point, almost orbital continuity, non-expansive mapping

Abstract

A Meir-Keeler type fixed point theorem for a family of mappings is proved in Menger probabilistic metric space (Menger PM-space). We establish that completeness of the space is equivalent to fixed point property for a larger class of mappings that includes continuous as well as discontinuous mappings. In addition to it, a probabilistic fixed point theorem satisfying (ϵ - δ) type non-expansive mappings is established.

Downloads

Download data is not yet available.

Author Biographies

Ravindra K. Bisht, National Defence Academy

Department of Mathematics

Vladimir Rakocević, University of Nis

Faculty of Sciences and Mathematics

References

R. K. Bisht and R. P. Pant, A remark on discontinuity at fixed point, J. Math. Anal. Appl. 445 (2017), 1239-1242. https://doi.org/10.1016/j.jmaa.2016.02.053

R. K. Bisht, A probabilistic Meir-Keeler type fixed point theorem which characterizes metric completeness, Carpathain J. Math. 36, no. 2 (2020), 215-222. https://doi.org/10.37193/CJM.2020.02.05

R. K. Bisht and V. Rakočević, Generalized Meir-Keeler type contractions and discontinuity at fixed point, Fixed Point Theory 19, no. 1 (2018), 57-64. https://doi.org/10.24193/fpt-ro.2018.1.06

R. K. Bisht and V. Rakočević, Discontinuity at fixed point and metric completeness, Appl. Gen. Topol. 21, no. 2 (2020), 349-362. https://doi.org/10.4995/agt.2020.13943

Lj. B. Ćirić, On contraction type mappings, Math. Balkanica 1 (1971), 52-57.

T. Hicks and B. E. Rhoades, Fixed points and continuity for multivalued mappings, International J. Math. Math. Sci. 15 (1992), 15-30. https://doi.org/10.1155/S0161171292000024

D. S. Jaggi, Fixed point theorems for orbitally continuous functions, Indian J. Math. 19, no. 2 (1977), 113-119.

G. F. Jungck, Generalizations of continuity in the context of proper orbits and fixed pont theory, Topol. Proc. 37 (2011), 1-15.

A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326-329. https://doi.org/10.1016/0022-247X(69)90031-6

K. Menger, Statistical metric, Proc. Nat. Acad. Sci. USA 28 (1942), 535-537. https://doi.org/10.1073/pnas.28.12.535

A. Pant and R. P. Pant, Fixed points and continuity of contractive maps, Filomat 31, no. 11 (2017), 3501-3506. https://doi.org/10.2298/FIL1711501P

A. Pant, R. P. Pant and M. C. Joshi, Caristi type and Meir-Keeler type fixed point theorems, Filomat 33, no. 12 (2019), 3711-3721. https://doi.org/10.2298/FIL1912711P

A. Pant and R. P. Pant, Fixed points and continuity of contractive maps, Filomat 31, no. 11 (2017), 3501-3506. https://doi.org/10.2298/FIL1711501P

A. Pant, R. P. Pant and W. Sintunavarat, Analytical Meir-Keeler type contraction mappings and equivalent characterizations, RACSAM 37 (2021), 115. https://doi.org/10.1007/s13398-020-00939-8

R. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl. 240 (1999), 284-289. https://doi.org/10.1006/jmaa.1999.6560

R. P. Pant, N. Y. Özgür and N. Tac s, On discontinuity problem at fixed point, Bull. Malays. Math. Sci. Soc. 43, no. 1 (2020), 499-517. https://doi.org/10.1007/s40840-018-0698-6

R. P. Pant, A. Pant, R. M. Nikolić and S. N. Ješić, A characterization of completeness of Menger PM-spaces, J. Fixed Point Theory Appl. 21, (2019) 90. https://doi.org/10.1007/s11784-019-0732-9

R. P. Pant, N. Y. Özgür and N. Taş, Discontinuity at fixed points with applications, Bulletin of the Belgian Mathematical Society-Simon Stevin 25, no. 4 (2019), 571-589. https://doi.org/10.36045/bbms/1576206358

O. Popescu, A new type of contractions that characterize metric completeness, Carpathian J. Math. 31, no. 3 (2015), 381-387. https://doi.org/10.37193/CJM.2015.03.15

B. E. Rhoades, Contractive definitions and continuity, Contemporary Mathematics 72 (1988), 233-245. https://doi.org/10.1090/conm/072/956495

S. Romaguera, w-distances on fuzzy metric spaces and fixed points, Mathematics 8, no. 11 (2020), 1909. https://doi.org/10.3390/math8111909

B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 415-417. https://doi.org/10.2140/pjm.1960.10.313

B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, New York, Elsevier 1983.

V. M. Sehgal and A. T. Bharucha-Reid, Fixed points of contraction mappings in PM-spaces, Math. System Theory 6 (1972), 97-102. https://doi.org/10.1007/BF01706080

P. V. Subrahmanyam, Completeness and fixed points, Monatsh. Math. 80 (1975), 325-330. https://doi.org/10.1007/BF01472580

T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136, no. 5 (2008), 1861-1869. https://doi.org/10.1090/S0002-9939-07-09055-7

N. Taş and N. Y. Özgür, A new contribution to discontinuity at fixed point, Fixed Point Theory 20, no. 2 (2019), 715-728. https://doi.org/10.24193/fpt-ro.2019.2.47

Downloads

Published

2021-10-01

How to Cite

[1]
R. K. Bisht and V. Rakocević, “On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators”, Appl. Gen. Topol., vol. 22, no. 2, pp. 435–446, Oct. 2021.

Issue

Section

Articles