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On the geometry of fixed points and discontinuity

Year 2024, Volume: 53 Issue: 1, 155 - 170, 29.02.2024

Rajendra Prasad Pant , Nihal Özgür , Bharti Joshı , Mangey Ram

https://doi.org/10.15672/hujms.1149843

Abstract

Recently, there has been a considerable effort to obtain new solutions to the Rhoades' open problem on the existence of contractive mappings that admit discontinuity at the fixed point. An extended version of this problem is also stated using a geometric approach. In this paper, we obtain new solutions to this extended version of the Rhoades' open problem. A related problem, the fixed-circle problem (resp. fixed-disc problem) is also studied. Both of these problems are related to the geometric properties of the fixed point set of a self-mapping on a metric space. Furthermore, a new result about metric completeness and a short discussion on the activation functions used in the study of neural networks are given. By providing necessary examples, we show that our obtained results are effective.

Keywords

discontinuity fixed point fixed circle fixed disc activation function

References

  • [1] H. Baghani, A new contractive condition related to Rhoades’ open question, Indian J. Pure Appl. Math. 51 (2), 565-578, 2020.
  • [2] R. K. Bisht and N. Özgür, Geometric properties of discontinuous fixed point set of $(\epsilon -\delta)$ contractions and applications to neural networks, Aequationes Math. 94 (5), 847-863, 2020.
  • [3] R. K. Bisht and R. P. Pant, A remark on discontinuity at fixed points, J. Math. Anal. Appl. 445 (2), 1239-1242, 2017.
  • [4] R. K. Bisht and R. P. Pant, Contractive definitions and discontinuity at fixed point, Appl. Gen. Topol. 18 (1), 173-182, 2017.
  • [5] R. K. Bisht and V. Rakocevic, Fixed points of convex and generalized convex contractions, Rend. Circ. Mat. Palermo (2) 69 (1), 21-28, 2020.
  • [6] O. Calin, Activation Functions, in: Deep Learning Architectures. Springer Series in the Data Sciences. Springer, Cham. 2020.
  • [7] U. Çelik and N. Özgür, A new solution to the discontinuity problem on metric spaces, Turkish J. Math. 44 (4), 1115-1126, 2020.
  • [8] X. 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. R. Soc. A 473 (2204), 20170322, 21 pp, 2017.
  • [9] Y. Du, Y. Li and R. Xu, Multistability and multiperiodicity for a general class of delayed Cohen-Grossberg neural networks with discontinuous activation functions, Discrete Dyn. Nat. Soc. 2013, 917835, 11 pp, 2013.
  • [10] M. Forti and P. Nistri, Global convergence of neural networks with discontinuous neuron activations, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50 (11), 1421-1435, 2003.
  • [11] Y. Huang, X. Yuan, H. Long, X. Fan and T. Cai, Multistability of fractional-order recurrent neural networks with discontinuous and nonmonotonic activation functions, IEEE Access 7, 116430-116437, 2019.
  • [12] Y. Huang, H. Zhang and Z. Wang, Multistability and multiperiodicity of delayed bidirectional associative memory neural networks with discontinuous activation functions, Appl. Math. Comput. 219 (3), 899-910, 2012.
  • [13] A. Hussain, H. Al-Sulami, N. Hussain, and H. Farooq, Newly fixed disc results using advanced contractions on $F$-metric space, J. Appl. Anal. Comput. 10 (6), 2313-2322, 2020.
  • [14] E. Karapınar, Recent advances on the results for nonunique fixed in various spaces, Axioms, 8 (2), 72, 2019.
  • [15] Q. Liu and J. Wang, A one-layer recurrent neural network with a discontinuous hardlimiting activation function for quadratic programming, IEEE Transactions on Neural Networks 19 (4), 558-570, 2008.
  • [16] N. Mlaiki, U. Çelik, N. Tas, N. Y. Özgür and A. Mukheimer, Wardowski type contractions and the fixed-circle problem on S-metric spaces, J. Math. 2018, 9127486, 9 pp, 2018.
  • [17] X. Nie, J. Liang and J. Cao, Multistability analysis of competitive neural networks with Gaussian-wavelet-type activation functions and unbounded time-varying delays, Appl. Math. Comput. 356, 449-468, 2019.
  • [18] X. Nie and W. X. Zheng, Multistability of neural networks with discontinuous nonmonotonic piecewise linear activation functions and time-varying delays, Neural Networks 65, 65-79, 2015.
  • [19] X. Nie and W. X. Zheng, Multistability and instability of neural networks with discontinuous nonmonotonic piecewise linear activation functions, IEEE Trans. Neural Netw. Learn. Syst. 26 (11), 2901-2913, 2015.
  • [20] X. 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 (3), 679-693, 2015.
  • [21] X. Nie and W. X. Zheng, On multistability of competitive neural networks with discontinuous activation functions, in: 4th Australian Control Conference (AUCC), IEEE, 245-250, 2014.
  • [22] M. Nour, Z. Cömert and K. Polat, A novel medical diagnosis model for COVID-19 infection detection based on deep features and Bayesian optimization, Applied Soft Computing, 97, 106580, 2020.
  • [23] N. Özgür, Fixed-disc results via simulation functions, Turkish J. Math. 43 (6), 2794- 2805, 2019.
  • [24] N. Özgür and N. Tas, New discontinuity results at fixed point on metric spaces, J. Fixed Point Theory Appl. 23 (2), 28, 14 pp, 2021.
  • [25] N. Y. Özgür and N. Tas, Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc. 42 (4), 1433-1449, 2019.
  • [26] N. Y. Özgür and N. Tas, Fixed-circle problem on S-metric spaces with a geometric viewpoint, Facta Univ. Ser. Math. Inform. 34 (3), 459-472, 2019.
  • [27] N. Y. Özgür and N. Tas, Some fixed-circle theorems and discontinuity at fixed circle, AIP Conf. Proc. 1926 (1), 020048, 2018.
  • [28] N. Y. Özgür and N. Tas, Generalizations of metric spaces: from the fixed-point theory to the fixed-circle theory, in: Rassias T. (eds) Applications of Nonlinear Analysis. Springer Optim. Appl. 134, Springer, Cham 2018.
  • [29] R. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl. 240 (1), 284-289, 1999.
  • [30] R. P. Pant, N. Y. Özgür and N. Tas, On discontinuity problem at fixed point, Bull. Malays. Math. Sci. Soc. 43 (1), 499-517, 2020.
  • [31] R. P. Pant, N. Y. Özgür and N. Tas, Discontinuity at fixed points with applications, Bull. Belg. Math. Soc.-Simon Stevin 26 (4), 571-589, 2019.
  • [32] 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. 22 (2), 39, 14 pp, 2020.
  • [33] A. Pant and R. P. Pant, Fixed points and continuity of contractive maps, Filomat 31 (11), 3501-3506, 2017.
  • [34] A. Pant, R. P. Pant, V. Rakocevic and R. K. Bisht, Generalized Meir-Keeler Type Contractions and Discontinuity at Fixed Point II, Math. Slovaca 69 (6), 1501-1507, 2019.
  • [35] S. Pourbahrami, L. M. Khanli, and S. Azimpour, An Automatic Clustering of Data Points with Alpha and Beta Angles on Apollonius and Subtended Arc Circle based on Computational Geometry, in: 28th Iranian Conference on Electrical Engineering (ICEE), IEEE 1-6, 2020.
  • [36] D. Reem and S. Reich, Fixed points of polarity type operators, J. Math. Anal. Appl. 467 (2), 1208-1232, 2018.
  • [37] B. E. Rhoades, Contractive definitions and continuity, Contemp. Math. 72, 233-245, 1988.
  • [38] H. N. Saleh, S. Sessa, W. M. Alfaqih, M. Imdad and N. Mlaiki, Fixed circle and fixed disc results for new types of $\Theta c$-contractive mappings in metric spaces, Symmetry 12 (11), 1825, 2020.
  • [39] S. Sharma, S. Sharma and A. Athaiya, Activation functions in neural networks, Int. J. Adv. Eng. Sci. Appl. Math. 4 (12), 310-316, 2020.
  • [40] K. K. Singh, M. Siddhartha and A. Singh, Diagnosis of coronavirus disease (COVID- 19) from chest X-ray images using modified XceptionNet, Romanian J. Inf. Sci. Technol. 23 (657), 91-105, 2020.
  • [41] R. G. Singh and A. P. Singh, Multiple complex extreme learning machine using holomorphic mapping for prediction of wind power generation system, Int. J. Comput. Appl. 123 (18), 24-33, 2015.
  • [42] P. V. Subrahmanyam, Completeness and fixed-points, Monatsh. Math. 80 (4), 325- 330, 1975.
  • [43] N. Tas and N. Özgür, New fixed-figure results on metric spaces, in: Debnath, P., Srivastava, H.M., Kumam, P., Hazarika, B. (eds) Fixed Point Theory and Fractional Calculus, Forum for Interdisciplinary Mathematics, Springer, Singapore, 2022.
  • [44] N. Tas and N. Y. Özgür, A new contribution to discontinuity at fixed point, Fixed Point Theory 20 (2), 715-728, 2019.
  • [45] N. Tas, N. Y. Özgür and N. Mlaiki, New types of Fc-contractions and the fixed-circle problem, Mathematics 6, 188, 2018.
  • [46] A. Tomar, M. Joshi and S. K. Padaliya, Fixed point to fixed circle and activation function in partial metric space, J. Appl. Anal. 28 (1), 57-66, 2022.
  • [47] H. Wu and C. Shan, Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses, Appl. Math. Modelling 33 (6), 2564-2574, 2017.
  • [48] L. Zhang, Implementation of fixed-point neuron models with threshold, ramp and sigmoid activation functions, in: IOP Conference Series: Materials Science and Engineering 224 (19), 012054, IOP Publishing, 2017.
  • [49] H. Zhang, Z. Wang and D. Liu, A comprehensive review of stability analysis of continuous-time recurrent neural networks, IEEE Trans. Neural Netw. Learn. Syst. 25 (7), 1229-1262, 2014.
  • [50] D. Zheng and P. Wang, Weak $\theta $-$\phi $-contractions and discontinuity, J. Nonlinear Sci. Appl. 10, 2318-2323, 2017.

Year 2024, Volume: 53 Issue: 1, 155 - 170, 29.02.2024

Rajendra Prasad Pant , Nihal Özgür , Bharti Joshı , Mangey Ram

https://doi.org/10.15672/hujms.1149843

Abstract

References

  • [1] H. Baghani, A new contractive condition related to Rhoades’ open question, Indian J. Pure Appl. Math. 51 (2), 565-578, 2020.
  • [2] R. K. Bisht and N. Özgür, Geometric properties of discontinuous fixed point set of $(\epsilon -\delta)$ contractions and applications to neural networks, Aequationes Math. 94 (5), 847-863, 2020.
  • [3] R. K. Bisht and R. P. Pant, A remark on discontinuity at fixed points, J. Math. Anal. Appl. 445 (2), 1239-1242, 2017.
  • [4] R. K. Bisht and R. P. Pant, Contractive definitions and discontinuity at fixed point, Appl. Gen. Topol. 18 (1), 173-182, 2017.
  • [5] R. K. Bisht and V. Rakocevic, Fixed points of convex and generalized convex contractions, Rend. Circ. Mat. Palermo (2) 69 (1), 21-28, 2020.
  • [6] O. Calin, Activation Functions, in: Deep Learning Architectures. Springer Series in the Data Sciences. Springer, Cham. 2020.
  • [7] U. Çelik and N. Özgür, A new solution to the discontinuity problem on metric spaces, Turkish J. Math. 44 (4), 1115-1126, 2020.
  • [8] X. 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. R. Soc. A 473 (2204), 20170322, 21 pp, 2017.
  • [9] Y. Du, Y. Li and R. Xu, Multistability and multiperiodicity for a general class of delayed Cohen-Grossberg neural networks with discontinuous activation functions, Discrete Dyn. Nat. Soc. 2013, 917835, 11 pp, 2013.
  • [10] M. Forti and P. Nistri, Global convergence of neural networks with discontinuous neuron activations, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50 (11), 1421-1435, 2003.
  • [11] Y. Huang, X. Yuan, H. Long, X. Fan and T. Cai, Multistability of fractional-order recurrent neural networks with discontinuous and nonmonotonic activation functions, IEEE Access 7, 116430-116437, 2019.
  • [12] Y. Huang, H. Zhang and Z. Wang, Multistability and multiperiodicity of delayed bidirectional associative memory neural networks with discontinuous activation functions, Appl. Math. Comput. 219 (3), 899-910, 2012.
  • [13] A. Hussain, H. Al-Sulami, N. Hussain, and H. Farooq, Newly fixed disc results using advanced contractions on $F$-metric space, J. Appl. Anal. Comput. 10 (6), 2313-2322, 2020.
  • [14] E. Karapınar, Recent advances on the results for nonunique fixed in various spaces, Axioms, 8 (2), 72, 2019.
  • [15] Q. Liu and J. Wang, A one-layer recurrent neural network with a discontinuous hardlimiting activation function for quadratic programming, IEEE Transactions on Neural Networks 19 (4), 558-570, 2008.
  • [16] N. Mlaiki, U. Çelik, N. Tas, N. Y. Özgür and A. Mukheimer, Wardowski type contractions and the fixed-circle problem on S-metric spaces, J. Math. 2018, 9127486, 9 pp, 2018.
  • [17] X. Nie, J. Liang and J. Cao, Multistability analysis of competitive neural networks with Gaussian-wavelet-type activation functions and unbounded time-varying delays, Appl. Math. Comput. 356, 449-468, 2019.
  • [18] X. Nie and W. X. Zheng, Multistability of neural networks with discontinuous nonmonotonic piecewise linear activation functions and time-varying delays, Neural Networks 65, 65-79, 2015.
  • [19] X. Nie and W. X. Zheng, Multistability and instability of neural networks with discontinuous nonmonotonic piecewise linear activation functions, IEEE Trans. Neural Netw. Learn. Syst. 26 (11), 2901-2913, 2015.
  • [20] X. 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 (3), 679-693, 2015.
  • [21] X. Nie and W. X. Zheng, On multistability of competitive neural networks with discontinuous activation functions, in: 4th Australian Control Conference (AUCC), IEEE, 245-250, 2014.
  • [22] M. Nour, Z. Cömert and K. Polat, A novel medical diagnosis model for COVID-19 infection detection based on deep features and Bayesian optimization, Applied Soft Computing, 97, 106580, 2020.
  • [23] N. Özgür, Fixed-disc results via simulation functions, Turkish J. Math. 43 (6), 2794- 2805, 2019.
  • [24] N. Özgür and N. Tas, New discontinuity results at fixed point on metric spaces, J. Fixed Point Theory Appl. 23 (2), 28, 14 pp, 2021.
  • [25] N. Y. Özgür and N. Tas, Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc. 42 (4), 1433-1449, 2019.
  • [26] N. Y. Özgür and N. Tas, Fixed-circle problem on S-metric spaces with a geometric viewpoint, Facta Univ. Ser. Math. Inform. 34 (3), 459-472, 2019.
  • [27] N. Y. Özgür and N. Tas, Some fixed-circle theorems and discontinuity at fixed circle, AIP Conf. Proc. 1926 (1), 020048, 2018.
  • [28] N. Y. Özgür and N. Tas, Generalizations of metric spaces: from the fixed-point theory to the fixed-circle theory, in: Rassias T. (eds) Applications of Nonlinear Analysis. Springer Optim. Appl. 134, Springer, Cham 2018.
  • [29] R. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl. 240 (1), 284-289, 1999.
  • [30] R. P. Pant, N. Y. Özgür and N. Tas, On discontinuity problem at fixed point, Bull. Malays. Math. Sci. Soc. 43 (1), 499-517, 2020.
  • [31] R. P. Pant, N. Y. Özgür and N. Tas, Discontinuity at fixed points with applications, Bull. Belg. Math. Soc.-Simon Stevin 26 (4), 571-589, 2019.
  • [32] 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. 22 (2), 39, 14 pp, 2020.
  • [33] A. Pant and R. P. Pant, Fixed points and continuity of contractive maps, Filomat 31 (11), 3501-3506, 2017.
  • [34] A. Pant, R. P. Pant, V. Rakocevic and R. K. Bisht, Generalized Meir-Keeler Type Contractions and Discontinuity at Fixed Point II, Math. Slovaca 69 (6), 1501-1507, 2019.
  • [35] S. Pourbahrami, L. M. Khanli, and S. Azimpour, An Automatic Clustering of Data Points with Alpha and Beta Angles on Apollonius and Subtended Arc Circle based on Computational Geometry, in: 28th Iranian Conference on Electrical Engineering (ICEE), IEEE 1-6, 2020.
  • [36] D. Reem and S. Reich, Fixed points of polarity type operators, J. Math. Anal. Appl. 467 (2), 1208-1232, 2018.
  • [37] B. E. Rhoades, Contractive definitions and continuity, Contemp. Math. 72, 233-245, 1988.
  • [38] H. N. Saleh, S. Sessa, W. M. Alfaqih, M. Imdad and N. Mlaiki, Fixed circle and fixed disc results for new types of $\Theta c$-contractive mappings in metric spaces, Symmetry 12 (11), 1825, 2020.
  • [39] S. Sharma, S. Sharma and A. Athaiya, Activation functions in neural networks, Int. J. Adv. Eng. Sci. Appl. Math. 4 (12), 310-316, 2020.
  • [40] K. K. Singh, M. Siddhartha and A. Singh, Diagnosis of coronavirus disease (COVID- 19) from chest X-ray images using modified XceptionNet, Romanian J. Inf. Sci. Technol. 23 (657), 91-105, 2020.
  • [41] R. G. Singh and A. P. Singh, Multiple complex extreme learning machine using holomorphic mapping for prediction of wind power generation system, Int. J. Comput. Appl. 123 (18), 24-33, 2015.
  • [42] P. V. Subrahmanyam, Completeness and fixed-points, Monatsh. Math. 80 (4), 325- 330, 1975.
  • [43] N. Tas and N. Özgür, New fixed-figure results on metric spaces, in: Debnath, P., Srivastava, H.M., Kumam, P., Hazarika, B. (eds) Fixed Point Theory and Fractional Calculus, Forum for Interdisciplinary Mathematics, Springer, Singapore, 2022.
  • [44] N. Tas and N. Y. Özgür, A new contribution to discontinuity at fixed point, Fixed Point Theory 20 (2), 715-728, 2019.
  • [45] N. Tas, N. Y. Özgür and N. Mlaiki, New types of Fc-contractions and the fixed-circle problem, Mathematics 6, 188, 2018.
  • [46] A. Tomar, M. Joshi and S. K. Padaliya, Fixed point to fixed circle and activation function in partial metric space, J. Appl. Anal. 28 (1), 57-66, 2022.
  • [47] H. Wu and C. Shan, Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses, Appl. Math. Modelling 33 (6), 2564-2574, 2017.
  • [48] L. Zhang, Implementation of fixed-point neuron models with threshold, ramp and sigmoid activation functions, in: IOP Conference Series: Materials Science and Engineering 224 (19), 012054, IOP Publishing, 2017.
  • [49] H. Zhang, Z. Wang and D. Liu, A comprehensive review of stability analysis of continuous-time recurrent neural networks, IEEE Trans. Neural Netw. Learn. Syst. 25 (7), 1229-1262, 2014.
  • [50] D. Zheng and P. Wang, Weak $\theta $-$\phi $-contractions and discontinuity, J. Nonlinear Sci. Appl. 10, 2318-2323, 2017.
There are 50 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Rajendra Prasad Pant 0000-0002-2044-881X

Nihal Özgür 0000-0002-8152-1830

Bharti Joshı 0000-0002-6209-0107

Mangey Ram 0000-0002-8221-092X

Early Pub Date August 15, 2023
Publication Date February 29, 2024
Published in Issue Year 2024 Volume: 53 Issue: 1

Cite

APA Pant, R. P., Özgür, N., Joshı, B., Ram, M. (2024). On the geometry of fixed points and discontinuity. Hacettepe Journal of Mathematics and Statistics, 53(1), 155-170. https://doi.org/10.15672/hujms.1149843
AMA Pant RP, Özgür N, Joshı B, Ram M. On the geometry of fixed points and discontinuity. Hacettepe Journal of Mathematics and Statistics. February 2024;53(1):155-170. doi:10.15672/hujms.1149843
Chicago Pant, Rajendra Prasad, Nihal Özgür, Bharti Joshı, and Mangey Ram. “On the Geometry of Fixed Points and Discontinuity”. Hacettepe Journal of Mathematics and Statistics 53, no. 1 (February 2024): 155-70. https://doi.org/10.15672/hujms.1149843.
EndNote Pant RP, Özgür N, Joshı B, Ram M (February 1, 2024) On the geometry of fixed points and discontinuity. Hacettepe Journal of Mathematics and Statistics 53 1 155–170.
IEEE R. P. Pant, N. Özgür, B. Joshı, and M. Ram, “On the geometry of fixed points and discontinuity”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 1, pp. 155–170, 2024, doi: 10.15672/hujms.1149843.
ISNAD Pant, Rajendra Prasad et al. “On the Geometry of Fixed Points and Discontinuity”. Hacettepe Journal of Mathematics and Statistics 53/1 (February 2024), 155-170. https://doi.org/10.15672/hujms.1149843.
JAMA Pant RP, Özgür N, Joshı B, Ram M. On the geometry of fixed points and discontinuity. Hacettepe Journal of Mathematics and Statistics. 2024;53:155–170.
MLA Pant, Rajendra Prasad et al. “On the Geometry of Fixed Points and Discontinuity”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 1, 2024, pp. 155-70, doi:10.15672/hujms.1149843.
Vancouver Pant RP, Özgür N, Joshı B, Ram M. On the geometry of fixed points and discontinuity. Hacettepe Journal of Mathematics and Statistics. 2024;53(1):155-70.

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