najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2022-13-5-491-497najo-260Research ArticleСтатьиOn existence conditions for periodic solutions to a differential equation with constant argumentMuminovM. I.mmuminov@mail.ruRadjabovT. A.radjabovtirkash@yandex.comSamarkand State University; V. I. Romanovskiy Institute of MathematicsRussian FederationSamarkand State UniversityRussian Federation202206062025135491497Copyright © Muminov M.I., Radjabov T.A., 20252025Muminov M.I., Radjabov T.A.Muminov M.I., Radjabov T.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/260

We deal with a linear differential equation with piecewise constant argument. The considered equation with the initial condition has the unique solution. We obtain the sufficient conditions for existence of n -periodic solution for the considered problem and describe the positivity conditions for the solution.

partial differential equationpiecewise constant argumentsperiodic solutionasymptotically stableReferencesCooke K.L., Wiener J. An equation alternately of retarded and advanced type. Proc. Amer. Math. Soc., 1987, 99, P. 726-732.Cooke K.L., Wiener J. An equation alternately of retarded and advanced type. Proc. Amer. Math. Soc., 1987, 99, P. 726-732.Cooke K.L., Wiener J. Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl., 1984, 99, P. 265-297.Cooke K.L., Wiener J. Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl., 1984, 99, P. 265-297.Hale J.K., Lunel V.Introduction to Functional Differential Equations. Springer, New York, 1993.Hale J.K., Lunel V.Introduction to Functional Differential Equations. Springer, New York, 1993.Hino Y., Naito T., Minh N., Shin V. Almost Periodic Solutions of Differential Equations in Banach Spaces. Stability and Control: Theory, Method and Applications, 15, Taylor & Francis, London, 2002.Hino Y., Naito T., Minh N., Shin V. Almost Periodic Solutions of Differential Equations in Banach Spaces. Stability and Control: Theory, Method and Applications, 15, Taylor & Francis, London, 2002.Busenberg S., Cooke K.L. Models of vertically transmitted diseases with sequentialcontinuous dynamics. Nonlinear Phenomena in Mathematical Sciences, V. Lakshmikantham, ed., Academic Press, New York, 1982, P. 179-187.Busenberg S., Cooke K.L. Models of vertically transmitted diseases with sequentialcontinuous dynamics. Nonlinear Phenomena in Mathematical Sciences, V. Lakshmikantham, ed., Academic Press, New York, 1982, P. 179-187.Cooke K.L., Wiener J. A survey of differential equations with piecewise continuous arguments. Delay Differential Equations and Dynamical Systems, 1991, 1475, P. 1-15.Cooke K.L., Wiener J. A survey of differential equations with piecewise continuous arguments. Delay Differential Equations and Dynamical Systems, 1991, 1475, P. 1-15.Wiener J. Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993.Wiener J. Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993.Pinto M., Robledo G. Controllability and observability for a linear time varying system with piecewise constant delay. Acta Appl. Math., 2014, P. 1-24.Pinto M., Robledo G. Controllability and observability for a linear time varying system with piecewise constant delay. Acta Appl. Math., 2014, P. 1-24.Veloz T., Pinto M. Existence, computability and stability for solutions of the diffusion equation with general piecewise constant argument. J. Math. Anal. Appl., 2015, 426 (1), P. 330-339.Veloz T., Pinto M. Existence, computability and stability for solutions of the diffusion equation with general piecewise constant argument. J. Math. Anal. Appl., 2015, 426 (1), P. 330-339.Wiener J., Heller W. Oscillatory and periodic solutions to a diffusion equation of neutral type.Int. J. Math. Math. Sci., 1999, 22 (2), P. 313-348.Wiener J., Heller W. Oscillatory and periodic solutions to a diffusion equation of neutral type.Int. J. Math. Math. Sci., 1999, 22 (2), P. 313-348.Wiener J., Debnath L. Boundary value problems for the diffusion equation with piecewise continuous time delay.Int. J. Math. and Math. Sci., 1997, 20 (1), P. 187-195.Wiener J., Debnath L. Boundary value problems for the diffusion equation with piecewise continuous time delay.Int. J. Math. and Math. Sci., 1997, 20 (1), P. 187-195.Wiener J. Boundary-value problems for partial differential equations with piecewise constant delay.Int. J. Math. and Math. Sci., 1991, 14, P. 301-321.Wiener J. Boundary-value problems for partial differential equations with piecewise constant delay.Int. J. Math. and Math. Sci., 1991, 14, P. 301-321.Muminov M.I. On the method of finding periodic solutions of second-order neutral differential equations with piecewise constant arguments. Advances in Difference Equations, 2017, 2017, 336.Muminov M.I. On the method of finding periodic solutions of second-order neutral differential equations with piecewise constant arguments. Advances in Difference Equations, 2017, 2017, 336.Muminov M.I., Ali H.M. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments. Open Mathematics, 2020, 18 (1), P. 93-105.Muminov M.I., Ali H.M. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments. Open Mathematics, 2020, 18 (1), P. 93-105.Chiu K.S., Pinto M. Periodic solutions of differential equations with a general piecewise constant argument and applications. E.J. Qualitative Theory of Diff. Equ., 2010, 46, P. 1-19.Chiu K.S., Pinto M. Periodic solutions of differential equations with a general piecewise constant argument and applications. E.J. Qualitative Theory of Diff. Equ., 2010, 46, P. 1-19.Aftabizadeh A.R., Wiener J., Jian-Ming Xu. Oscillatory and periodic solutions of delay differential equations with piecewise constant argument. Proc. Amer. Math. Sot., 1987, 99, P. 673-679.Aftabizadeh A.R., Wiener J., Jian-Ming Xu. Oscillatory and periodic solutions of delay differential equations with piecewise constant argument. Proc. Amer. Math. Sot., 1987, 99, P. 673-679.Bula I., Radin M.A., Wilkins N. Neuron model with a period three internal decay rate. Electronic J. of Qualitative Theory of Differential Equations, 2017, 46, P. 1-19.Bula I., Radin M.A., Wilkins N. Neuron model with a period three internal decay rate. Electronic J. of Qualitative Theory of Differential Equations, 2017, 46, P. 1-19.Inese Bula, Radin M.A. Eventually periodic solutions of single neuron model. Nonlinear Analysis: Modelling and Control, 2020, 25 (6), P. 903-918.Inese Bula, Radin M.A. Eventually periodic solutions of single neuron model. Nonlinear Analysis: Modelling and Control, 2020, 25 (6), P. 903-918.Wheeler D., Schieve W. Stability and chaos in an inertial two-neuron system. Phys. D: Nonlinear Phenom., 1997, 105, P. 267-284.Wheeler D., Schieve W. Stability and chaos in an inertial two-neuron system. Phys. D: Nonlinear Phenom., 1997, 105, P. 267-284.Koch C. Cable theory in neurons with active linearized membrane. Biol. Cybern., 1984, 50, P. 15-33.Koch C. Cable theory in neurons with active linearized membrane. Biol. Cybern., 1984, 50, P. 15-33.Babcock K., Westervelt R. Stability and dynamics of simple electronic neural networks with added inertia. Phys. D: Nonlinear Phenom., 1986, 23, P. 464-469.Babcock K., Westervelt R. Stability and dynamics of simple electronic neural networks with added inertia. Phys. D: Nonlinear Phenom., 1986, 23, P. 464-469.Xu M., Du B. Periodic solution for neutral-type inertial neural networks with time-varying delays. Adv. Difference Equ., 2020, 607, P. 1-21.Xu M., Du B. Periodic solution for neutral-type inertial neural networks with time-varying delays. Adv. Difference Equ., 2020, 607, P. 1-21.Liu B., Huang L. Existence and exponential stability of periodic solutions for a class of Cohen-Grossberg neural networks with time-varying delays. Chaos Solitons Fractals, 2007, 32, P. 617-627.Liu B., Huang L. Existence and exponential stability of periodic solutions for a class of Cohen-Grossberg neural networks with time-varying delays. Chaos Solitons Fractals, 2007, 32, P. 617-627.Du B. Anti-periodic solutions problem for inertial competitive neutral-type neural networks via Wirtinger inequality. J. Inequal. Appl., 2019, 187.Du B. Anti-periodic solutions problem for inertial competitive neutral-type neural networks via Wirtinger inequality. J. Inequal. Appl., 2019, 187.

The authors declare that there are no conflicts of interest present.