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

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

Discontinuity at fixed point and metric completeness

[EN] In this paper, we prove some new fixed point theorems for a generalized class of Meir-Keeler type mappings, which give some new solutions to the Rhoades open problem regarding the existence of contractive mappings that admit discontinuity at the fixed point. In addition to it, we prove that our theorems characterize completeness of the metric space as well as Cantor's intersection property.Bisht, RK.; Rakocevic, V. (2020). Discontinuity at fixed point and metric completeness. Applied General Topology. 21(2):349-362. https://doi.org/10.4995/agt.2020.13943OJS349362212R. M. T. Bianchini, Su un problema di S. Reich riguardante la teoria dei puntifissi, Boll. Un. Mat. Ital. 5 (1972), 103-108.R. K. Bisht and N. Özgür, Geometric properties of discontinuous fixed point set of $(epsilon-delta)$ contractions and applications to neural networks, Aequat. Math. 94 (2020), 847-863. https://doi.org/10.1007/s00010-019-00680-7R. K. Bisht and R. P. Pant, A remark on discontinuity at fixed points, J. Math. Anal. Appl. 445 (2017), 1239-1242. https://doi.org/10.1016/j.jmaa.2016.02.053R. K. Bisht and R. P. Pant, Contractive definitions and discontinuity at fixed point, Appl. Gen. Topol. 18, no. 1 (2017), 173-182. https://doi.org/10.4995/agt.2017.6713R. K. Bisht and V. Rakocevic , 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. Rakocevic , Fixed points of convex and generalized convex contractions, Rend. Circ. Mat. Palermo, II. Ser., 69, no. 1 (2020), 21-28. https://doi.org/10.1007/s12215-018-0386-2S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci. 25 (1972), 15-18.Lj. B. Ciric, On contraction type mapping, Math. Balkanica 1 (1971), 52-57.Lj. B. Ciric, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc. 45, no. 2 (1974), 267-273. https://doi.org/10.2307/2040075X. Ding, J. Cao, X. Zhao and F. E. Alsaadi, Mittag-Leffler synchronization of delayed fractional-order bidirectional associative memory neural networks with discontinuous activations: state feedback control and impulsive control schemes, Proc. Royal Soc. A: Math. Eng. Phys. Sci. 473 (2017), 20170322. https://doi.org/10.1098/rspa.2017.0322M. Forti and P. Nistri, Global convergence of neural networks with discontinuous neuron activations, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50, no. 11 (2003) 1421-1435. https://doi.org/10.1109/TCSI.2003.818614H. Garai, L. K. Dey and Y. J. Cho, On contractive mappings and discontinuity at fixed points, Appl. Anal. Discrete Math. 14 (2020), 33-54. https://doi.org/10.2298/AADM181018007GT. L. Hicks and B. E. Rhoades, A Banach type fixed-point theorem, Math. Japon. 24, (1979/80), 327-330.J. Jachymski, Equivalent conditions and Meir-Keeler type theorems, J. Math. Anal. Appl. 194 (1995), 293-303. https://doi.org/10.1006/jmaa.1995.1299R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76. https://doi.org/10.2307/2316437R. Kannan, Some results on fixed points-II, Amer. Math. Monthly 76 (1969), 405-408. https://doi.org/10.1080/00029890.1969.12000228M. Maiti and T. K. Pal, Generalizations of two fixed point theorems, Bull. Calcutta Math. Soc. 70 (1978), 57-61.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)90031-6L. V. Nguyen, On fixed points of asymptotically regular mappings, Rend. Circ. Mat. Palermo, II. Ser., to appear.X. Nie and W. X. Zheng, On multistability of competitive neural networks with discontinuous activation functions. In: 4th Australian Control Conference (AUCC), (2014) 245-250. https://doi.org/10.1109/AUCC.2014.7358690X. Nie and W. X. Zheng, Multistability of neural networks with discontinuous non-monotonic piecewise linear activation functions and time-varying delays, Neural Networks 65 (2015), 65-79. https://doi.org/10.1016/j.neunet.2015.01.007X. Nie and W. X. Zheng, Dynamical behaviors of multiple equilibria in competitive neural networks with discontinuous nonmonotonic piecewise linear activation functions, IEEE Transactions On Cybernatics 46, no. 3 (2015), 679-693.https://doi.org/10.1109/TCYB.2015.2413212N. Y. Özgür and N. Tas, Some fixed-circle theorems and discontinuity at fixed circle, AIP Conference Proceedings 1926 (2018), 020048. https://doi.org/10.1063/1.5020497N. Y. Özgür and N. Tas, Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc. 42, no. 4 (2019), 1433-1449. https://doi.org/10.1007/s40840-017-0555-zA. 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/FIL1912711PR. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl. 240 (1999), 284-289. https://doi.org/10.1006/jmaa.1999.6560R. P. Pant, Fixed points of Lipschitz type mappings, preprint.R. P. Pant, N. Özgür, N. Tas, A. Pant and M. C. Joshi, New results on discontinuity at fixed point, J. Fixed Point Theory Appl. (2020) 22:39. https://doi.org/10.1007/s11784-020-0765-0R. P. Pant, N. Y. Özgür and N. Tas, On discontinuity problem at fixed point, Bull. Malays. Math. Sci. Soc. 43 (2020), 499-517. https://doi.org/10.1007/s40840-018-0698-6R. P. Pant, N. Y. Özgür and N. Tas}, Discontinuity at fixed points with applications, Bulletin of the Belgian Mathematical Society-Simon Stevin 25, no. 4 (2019), 571-589.M. Rashid, I. Batool and N. Mehmood, Discontinuous mappings at their fixed points and common fixed points with applications, J. Math. 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Fixed point of Lipschitz type mappings

In this paper, we prove some fixed point theorems for Lipschitz type mappings in the setting of metric spaces. Our results open up the unexplored area of fixed points of Lipschitz type mappings for investigation

Contractive definitions and discontinuity at fixed point

[EN] In this paper, we investigate some contractive definitions which are strong enough to generate a fixed point that do not force the mapping to be continuous at the fixed point. Finally, we obtain a fixed point theorem for generalized nonexpansive mappings in metric spaces by employing Meir-Keeler type conditions.Bisht, RK.; Pant, RP. (2017). Contractive definitions and discontinuity at fixed point. Applied General Topology. 18(1):173-182. doi:10.4995/agt.2017.6713.SWORD173182181Bisht, R. K., & Pant, R. P. (2017). A remark on discontinuity at fixed point. Journal of Mathematical Analysis and Applications, 445(2), 1239-1242. doi:10.1016/j.jmaa.2016.02.053Boyd, D. W., & Wong, J. S. W. (1969). On nonlinear contractions. Proceedings of the American Mathematical Society, 20(2), 458-458. doi:10.1090/s0002-9939-1969-0239559-9Jachymski, J. (1995). Equivalent Conditions and the Meir-Keeler Type Theorems. Journal of Mathematical Analysis and Applications, 194(1), 293-303. doi:10.1006/jmaa.1995.1299Kannan, R. (1969). Some Results on Fixed Points--II. The American Mathematical Monthly, 76(4), 405. doi:10.2307/2316437Meir, A., & Keeler, E. (1969). A theorem on contraction mappings. Journal of Mathematical Analysis and Applications, 28(2), 326-329. doi:10.1016/0022-247x(69)90031-6Pant, R. P. (1999). Discontinuity and Fixed Points. Journal of Mathematical Analysis and Applications, 240(1), 284-289. doi:10.1006/jmaa.1999.6560Reich, S. (1971). Some Remarks Concerning Contraction Mappings. Canadian Mathematical Bulletin, 14(1), 121-124. doi:10.4153/cmb-1971-024-9Rhoades, B. E. (1977). A comparison of various definitions of contractive mappings. Transactions of the American Mathematical Society, 226, 257-257. doi:10.1090/s0002-9947-1977-0433430-4Rhoades, B. E. (1988). Contractive definitions and continuity. Contemporary Mathematics, 233-245. doi:10.1090/conm/072/95649

Common Fixed Points for Pairs of Mappings with Variable Contractive Parameters

We establish some common fixed point results for a new class of pair of contraction mappings having functions as contractive parameters, and satisfying minimal noncommutative operators property

Diffused metal-insulator transition in NdNiO3 film grown on BaTiO3: Likely evidence of electronic Griffiths phase

This paper reports diffused metal-insulator transitions (MITs) in an oxide with disorder that undergoes a Mott transition. The investigation was carried out in the multilayer film NdNiO3/BaTiO3/SrTiO3 (NNO/BTO/STO), where a large mismatch of lattice constants of NNO with those of BTO leads to strain relaxation and creation of quenched disorder in the NNO film. NNO film in the NNO/BTO/STO multilayer structure shows a broad Mott-type MIT at a temperature T-MI = 160 K from a high-temperature bad metallic phase (1/rho DC d rho DC/dT < 0) with dT a high value of resistivity rho DC approximate to 70 m Omega cm at 300 K to a low temperature insulating phase. Using noise spectroscopy and impedance spectroscopy which can probe the dynamics of the coexisting phases near the MIT, it was observed that in addition to the MIT at T-MI = 160 K, there exists a characteristic temperature T-G %:Z, 230 K well above the T-MI, where large low-frequency correlated fluctuations appear, signifying the appearance of a phase with slow dynamics. T-G signals the onset of a temperature region T-MI < T < T-G with coexisting phases that have been corroborated by the impedance spectroscopy and AC conductivity measurements. It is suggested that the temperature T-G may signify the onset of an electronic Griffiths phase that has been theoretically proposed for Mott transitions with disorder