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. Anal. 9, no. 1 (2018), 90-104.B. E. Rhoades, Contractive definitions and continuity, Contemporary Mathematics 72 (1988), 233-245. https://doi.org/10.1090/conm/072/956495I. A. Rus, Some variants of contraction principle, generalizations and applications, Stud. Univ. Babes-Bolyai Math. 61, no. 3 (2016), 343-358.P. 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), 186-1869. https://doi.org/10.1090/S0002-9939-07-09055-7N. Tas 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.2019.2.47H. Wu and C. Shan, Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses, Appl. Math. Modelling 33, no. 6 (2017), 2564-2574. https://doi.org/10.1016/j.apm.2008.07.022D. Zheng and P. Wang, Weak -ψ and discontinuity, J. Nonlinear Sci. Appl. 10 (2017), 2318-2323. https://doi.org/10.22436/jnsa.010.05.0

Multistability of neural networks with discontinuous non-monotonic piecewise linear activation functions and time-varying delays

This paper is concerned with the problem of coexistence and dynamical behaviors of multiple equilibrium points for neural networks with discontinuous non-monotonic piecewise linear activation functions and time-varying delays. The fixed point theorem and other analytical tools are used to develop certain sufficient conditions that ensure that the n-dimensional discontinuous neural networks with time-varying delays can have at least 5n equilibrium points, 3n of which are locally stable and the others are unstable. The importance of the derived results is that it reveals that the discontinuous neural networks can have greater storage capacity than the continuous ones. Moreover, different from the existing results on multistability of neural networks with discontinuous activation functions, the 3n locally stable equilibrium points obtained in this paper are located in not only saturated regions, but also unsaturated regions, due to the non-monotonic structure of discontinuous activation functions. A numerical simulation study is conducted to illustrate and support the derived theoretical results