Abstract
In order to unify the study of Besicovitch almost periodic solutions of continuous time and discrete-time stochastic differential equations, we first propose concepts of Besicovitch almost periodic stochastic processes in p-th mean and of Besicovitch almost periodic stochastic processes in distribution on time scales, and reveal the relationship between the two random processes. Then, taking a class of stochastic Clifford-valued neural networks with time-varying delays on time scales as an example of stochastic dynamic equations with delays, we establish the existence and stability of Besicovitch almost periodic solutions in distribution for this class of networks by using Banach’s fixed point theorem, time scale calculus theory and inequality techniques.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Not applicable.
References
Andres, J., Bersani, A.M., Grande, R.F.: Hierarchy of almost periodic function spaces. Rend. Mat. Ser. VII 26, 121–188 (2006)
Besicovitch, A.S.: Almost Periodic Functions. Dover, New York (1954)
Corduneanu, C.: Almost Periodic Oscillations and Waves. Springer, New York (2009)
Li, Y., Wang, X., Huo, N.: Weyl almost automorphic solutions in distribution sense of Clifford-valued stochastic neural networks with time-varying delays. Proc. R. Soc. A 478, 20210719 (2022)
Li, Y., Huang, M., Li, B.: Besicovitch almost periodic solutions for fractional-order quaternion-valued neural networks with discrete and distributed delays, Math. Meth. Appl. Sci. (2021). https://doi.org/10.1002/mma.8070 (in press)
Kostić, M.: Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations. Banach J. Math. Anal. 13(1), 64–90 (2019)
Hilger, S.: Analysis on measure chains-a unified approach to continuous and discrete calculus. Result Math. 18(1–2), 18–56 (1990)
Buchholz, S., Sommer, G.: On Clifford neurons and Clifford multi-layer perceptrons. Neural Netw. 21(7), 925–935 (2008)
Hitzer, E., Nitta, T., Kuroe, Y.: Applications of Clifford’s geometric algebra. Adv. Appl. Clifford Algebras 23(2), 377–404 (2013)
Liu, Y., Xu, P., Lu, J., Liang, J.: Global stability of Clifford-valued recurrent neural networks with time delays. Nonlinear Dyn. 84(2), 767–777 (2016)
Breuils, S., Tachibana, K., Hitzer, E.: New applications of Clifford’s geometric algebra. Adv. Appl. Clifford Algebras 32, 17 (2022)
Xia, Z., Liu, Y., Kou, K.I., Wang, J.: Clifford-valued distributed optimization based on recurrent neural networks, IEEE Trans. Neural Netw. Learn. Syst. (2022). https://doi.org/10.1109/TNNLS.2021.3139865. (in press)
Luo, D., Jiang, Q., Wang, Q.: Anti-periodic solutions on Clifford-valued high-order Hopfield neural networks with multi-proportional delays. Neurocomputing 472, 1–11 (2022)
Li, Y., Li, B.: Pseudo compact almost automorphy of neutral type Clifford-valued neural networks with mixed delays, Discrete Contin. Dyn. Syst.-B. (2021). https://doi.org/10.3934/dcdsb.2021248. (in press)
Lv, W., Li, B.: Existence and global attractivity of pseudo almost periodic solutions for Clifford-valued fuzzy neural networks with proportional delays. Mathematics 9(24), 3306 (2021)
Huang, S., Qiao, Y.Y., Wen, G.C.: Real and Complex Clifford Analysis. Springer, New York (2006)
Bohner, M., Peterson, A.: Dynamic equations on time scales. An Introduction with Applications, Birkhäuser, Boston (2001)
Bohner, M., Georgiev, S.: Multivariable dynamic calculus on time scales. Springer, Switzerland (2016)
Li, Y., Shen, S.: Compact almost automorphic function on time scales and its application. Qual. Theory Dyn. Syst. 20, 86 (2021)
Bohner, M., Guseinov, G.: Double integral calculus of variations on time scales. Comput. Math. Appl. 54(1), 45–57 (2007)
Bohner, M., Sanyal, S.: The stochastic dynamic exponential and geometric Brownian motion on isolated time scales. Commun. Math. Anal. 8(3), 120–135 (2010)
Bohner, M., Stanzhytskyi, O.M., Bratochkina, A.O.: Stochastic dynamic equations on general time scales. Electron. J. Diff. Equ. 2013(57), 1–15 (2013)
Klenke, A.: Probability Theory: A Comprehensive Course. Springer, Berlin (2013)
Liu, Z., Sun, K.: Almost automorphic solutions for stochastic differential equations driven by Lévy noise. J. Funct. Anal. 266(3), 1115–1149 (2014)
Li, Y., Wang, C.: Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales. Abstr. Appl. Anal. 2011, 341520 (2011)
Li, Y., Wang, X.: Almost periodic solutions in distribution of Clifford-valued stochastic recurrent neural networks with time-varying delays. Chaos Solitons Fractals 153, 111536 (2021)
Acknowledgements
The authors would like to thank the Editor and the anonymous referees for their helpful comments and valuable suggestions regarding this article.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by the National Natural Science Foundation of China under Grant No. 11861072.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Li, Y., Huang, X. Besicovitch Almost Periodic Solutions to Stochastic Dynamic Equations with Delays. Qual. Theory Dyn. Syst. 21, 74 (2022). https://doi.org/10.1007/s12346-022-00606-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-022-00606-w
Keywords
- Besicovitch almost periodic stochastic processes on time scales
- Clifford-valued neural networks
- Besicovitch almost periodic solutions in distribution
- Global exponential stability
- Time scales