Generalized Gravi-Electromagnetism

A self consistant and manifestly covariant theory for the dynamics of four charges (masses) (namely electric, magnetic, gravitational, Heavisidian) has been developed in simple, compact and consistent manner. Starting with an invariant Lagrangian density and its quaternionic representation, we have obtained the consistent field equation for the dynamics of four charges. It has been shown that the present reformulation reproduces the dynamics of individual charges (masses) in the absence of other charge (masses) as well as the generalized theory of dyons (gravito - dyons) in the absence gravito - dyons (dyons). key words: dyons, gravito - dyons, quaternion PACS NO: 14.80H

Some notes on PD-operator pairs

This paper points out several remarks on the paper of Pathak and Rai[H. K.Pathak and D. Rai, Common fixed point theorems for PD-operator pairs under relaxed con-ditions with applications, Journal of Computational and Applied Mathematics 239 (2013)103-113]. In fact, under contractive conditions (assumed in the above paper), proving existence of common fixed point by assuming the notion of PD-operator is equivalent to proving the existence of common fixed point by assuming the existence of common fixed point

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

[EN] A Meir-Keeler type fixed point theorem for a family of mappings is proved in Mengerprobabilistic metric space (Menger PM-space). We establish that completeness of the space isequivalent to fixed point property for a larger class of mappings that includes continuous as wellas discontinuous mappings. In addition to it, a probabilistic fixed point theorem satisfying (ϵ - δ)type non-expansive mappings is established.Bisht, RK.; Rakocević, V. (2021). On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators. Applied General Topology. 22(2):435-446. https://doi.org/10.4995/agt.2021.15561OJS435446222R. 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.2016.02.053R. 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.2020.02.05R. 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.2018.1.06R. 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.2020.13943Lj. 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/S0161171292000024D. 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)90031-6K. Menger, Statistical metric, Proc. Nat. Acad. Sci. USA 28 (1942), 535-537. https://doi.org/10.1073/pnas.28.12.535A. Pant and R. P. Pant, Fixed points and continuity of contractive maps, Filomat 31, no. 11 (2017), 3501-3506. https://doi.org/10.2298/FIL1711501PA. 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/FIL1912711PA. 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/s13398-020-00939-8R. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl. 240 (1999), 284-289. https://doi.org/10.1006/jmaa.1999.6560R. 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/s40840-018-0698-6R. 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/s11784-019-0732-9R. 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/1576206358O. 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.2015.03.15B. E. Rhoades, Contractive definitions and continuity, Contemporary Mathematics 72 (1988), 233-245. https://doi.org/10.1090/conm/072/956495S. Romaguera, w-distances on fuzzy metric spaces and fixed points, Mathematics 8, no. 11 (2020), 1909. https://doi.org/10.3390/math8111909B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 415-417. https://doi.org/10.2140/pjm.1960.10.313B. 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/BF01706080P. V. Subrahmanyam, Completeness and fixed points, Monatsh. Math. 80 (1975), 325-330. https://doi.org/10.1007/BF01472580T. 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/S0002-9939-07-09055-7N. 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.2019.2.4