On a $(\omega ,c)-$periodic solution for an impulsive system of second-order differential equation with product of two nonlinear functions and mixed maxima
On a $(\omega ,c)-$periodic solution for an impulsive system of second-order differential equation
DOI:
https://doi.org/10.56143/ujmcs.v1i1.7Ключевые слова:
impulsive system of differential equations; a nonlinear function under the sign of the second-order differential; product of two nonlinear functions; $(\omega ,c)-$periodic solution, mixed maxima; contracted mapping, existence and uniqueness.Аннотация
Existence and uniqueness of $(\omega ,c)-$periodic solution of boundary value problem for an impulsive system of ordinary differential equations with a nonlinear function under the sign of the second-order differential, product of two nonlinear functions and mixed maxima are investigated. This problem is reduced to the investigation of $(\omega ,c)-$periodic solvability of the system of nonlinear functional-integral equations. The method of contracted mapping is used in the proof of unique solvability of nonlinear functional-integral equations in the space $BD\left([0,\omega ],\mathbb{R}^{n} \right)$.
Библиографические ссылки
[1] Bainov, D. D., Simeonov, P. S.: Impulsive differential equations: Periodic solutions and applications. New York, NY, USA, Wiley. (1993)
[2] Benchohra, M., Henderson, J., Ntouyas, S. K.: Impulsive differential equations and inclusions. Contemporary mathematics and its application, New York, Hindawi Publishing Corporation. (2006).
[3] Halanay, A., Veksler, D.: Qualitative theory of impulsive systems. Moscow, Mir. (1971) (Russian).
[4] Lakshmikantham, V., Bainov, D. D., Simeonov, P. S.: Theory of impulsive differential equations. Singapore, World Scientific. (1989).
[5] Perestyk, N. A., Plotnikov, V. A., Samoilenko, A. M., Skripnik, N. V.: Differential equations with impulse effect: multivalued right-hand sides with discontinuities. DeGruyter Stud. 40, Mathematics, Berlin, Walter de Gruter Co. (2011).
[6] Samoilenko, A. M., Perestyk, N. A.: Impulsive differential equations. Singapore, World Sci. (1995).
[7] Stamova, I., Stamov, G.: Impulsive biological models. In: Applied impulsive mathematical models, CMS Books in Mathematics, Springer, Cham. (2016).
[8] Catlla, J., Schaeffer, D. G., Witelski, Th. P., Monson, E. E., Lin, A. L.: On spiking models for synaptic activity and impulsive differential equations. SIAM Review. 50 (3), 553–569 (2008).
[9] Choi, S. K., Koo, N.: A note on linear impulsive fractional differential equations, J. Chungcheong Math. Soc. 28, 583–590 (2015).
[10] Fedorov, E. G., Popov, I. Yu.: Analysis of the limiting behavior of a biological neurons system with delay. J. Phys. Conf. Ser. 2086 (012109), (2021).
[11] Fedorov, E. G., Popov, I. Yu.: Hopf bifurcations in a network of Fitzhigh-Nagumo biological neurons. International J. of Nonlinear Sciences and Numerical Simulation (2021). https://doi.org/10.1515/ijnsns-2021-0188.
[12] Fedorov, E. G.: Properties of an oriented ring of neurons with the FitzHugh-Nagumo model. Nanosystems: Phys. Chem. Math. 12 (5), 553–562 (2021).
[13] Anguraj, A., Arjunan, M. M.: Existence and uniqueness of mild and classical solutions of impulsive evolution equations. Elect. J. Differ. Equat. 2005 (111), 1–8 (2005).
[14] Ashyralyev, A., Sharifov, Ya. A.: Existence and uniqueness of solutions for nonlinear impulsive differential equations with two–point and integral boundary conditions. Adv. in Difference Equat. 2013 (173) (2013).
[15] Ashyralyev, A., Sharifov, Ya. A.: Optimal control problems for impulsive systems with integral boundary conditions. Elect. J. Differ. Equat. 2013 (80), 1–11 (2013).
[16] Bai, Ch., Yang, D.: Existence of solutions for second-order nonlinear impulsive differential equations with periodic boundary value conditions. Boundary Value Problems (Hindawi Publishing Corporation) 2007 (41589), 1–13 (2007).
[17] Bin, L., Xinzhi, L., Xiaoxin, L.: Robust global exponential stability of uncertain impulsive systems. Acta Mathematika Scientia. 25 (1), 161–169 (2005).
[18] Dhayal, R., Malik, M., Abbas, S.: Solvability and optimal controls of non-instantaneous impulsive stochastic fractional differential equation of order. Stochastics. 93 780–802 (2020).
[19] Dhayal, R., Malik, M., Abbas, S., Debbouche, A.: Optimal controls for second-order stochastic differential equations driven by mixed-fractional Brownian motion with impulses. Math. Methods Appl. Sci. 43, 4107–4124 (2020).
[20] Fayziyev, A. K., Abdullozhonova, A. N., Yuldashev, T. K.: Inverse problem for Whitham type multi-dimensional differential equation with impulse effects. Lobachevskii J. Math. 44 (2), 570–579 (2023).
[21] Feckan, M., Wang, J. R.: Periodic impulsive fractional differential equations. Adv. Nonlinear Anal. 8, 482–496 (2019).
[22] Guechi, S., Dhayal, R., Debbouche, A., Malik, M.: Analysis and optimal control of phi-Hilfer fractional semilinear equations involving nonlocal impulsive conditions. Symmetry. 13 (2084), (2021).
[23] Ji, Sh., Wen, Sh.: Nonlocal cauchy problem for impulsive differential equations in Banach spaces. Intern. J. Nonlinear Science. 10 (1), 88–95 (2010).
[24] Kao, Y., Li, H.: Asymptotic multistability and local S-asymptotic omega-periodicity for the nonautonomous fractionalorder neural networks with impulses. Sci. China Inform. Sci. 64, 1–13 (2021).
[25] Li, M., Han, M.: Existence for neutral impulsive functional differential equations with nonlocal conditions, Indagationes Mathematcae 20 (3), 435–451 (2009).
[26] Mardanov, M. J., Sharifov, Ya. A., Molaei Habib, H.: Existence and uniqueness of solutions for first-order nonlinear differential equations with two-point and integral boundary conditions. Elect. J. Differen. Equat. 2014 (259), 1–8 (2014).
[27] Sharifov, Ya. A.: Optimal control problem for systems with impulsive actions under nonlocal boundary conditions. Vestnik Sam. Gos. Tekhn. Universiteta. Seria: Fiz.-Mat. Nauki. 33 (4), 34–45 (2013) (in Russian)
[28] Sharifov, Ya. A.: Optimal control for systems with impulsive actions under nonlocal boundary conditions, Russian Math. (Izv. VUZ). 57 (2), 65–72 (2013).
[29] Sharifov, Ya. A., Mammadova, N. B.: Optimal control problem described by impulsive differential equations with nonlocal boundary conditions. Differen. equat. 50 (3), 403–411 (2014).
[30] Sharifov, Ya. A.: Conditions optimality in problems control with systems impulsive differential equations with nonlocal boundary conditions. Ukrainain Math. Journ. 64 (6), 836–847 (2012).
[31] Yuldashev, T. K., Fayziyev, A. K.: On a nonlinear impulsive system of integro-differential equations with degenerate kernel and maxima. Nanosystems: Phys. Chem. Math. 13 (1), 36–44 (2022).
[32] Yuldashev, T. K., Fayziyev, A. K.: Integral condition with nonlinear kernel for an impulsive system of differential equations with maxima and redefinition vector. Lobachevskii J. Math. 43 (8), 2332–2340 (2022).
[33] Yuldashev, T. K., Fayziyev, A. K.: Inverse problem for a second order impulsive system of integro-differential equations with two redefinition vectors and mixed maxima. Nanosystems: Phys. Chem. Math. 14 (1), 13–21 (2023).
[34] Wang, J. R., Feckan, M., Zhou, Y.: A survey on impulsive fractional differential equations. Frac. Calc. Appl. Anal. 19, 806–831 (2016).
[35] Yuldashev, T. K.: Periodic solutions for an impulsive system of nonlinear differential equations with maxima. Nanosystems: Phys. Chem. Math. 13 (2), 135–141 (2022).
[36] Yuldashev, T. K.: Periodic solutions for an impulsive system of integro-differential equations with maxima. Vestnik Sam. Gos. Tekhn. Universiteta. Ser. Fiz.-Mat. Nauki. 26 (2), 368–379 (2022).
[37] Yuldashev, T. K., Sulaimonov, F. U.: Periodic solutions of second order impulsive system for an integro-differential equations with maxima. Lobachevskii J. Math. 43 (12), 3674–3685 (2022).
[38] Alvarez, E., Castillo, S., Pinto, M.: (omega ,c)-pseudo periodic functions, first order Cauchy problem and Lasota Wazewska model with ergodic and unbounded oscillating production of red cells. Bound. Value Probl. 2019, 1–20 (2019).
[39] Alvarez, E., Gomez, A., Pinto, M.: (omega ,c)-periodic functions and mild solutions to abstract fractional integrodifferential equations. Electron. J. Qual. Theory Differ. Equat. 16, 1–8 (2018).
[40] Alvarez, E., Diaz, S., Lizama, C.: On the existence and uniqueness of (N,lambda)-periodic solutions to a class of Volterra difference equations. Adv. iffer. Equ. 2019, 1–12 (2019).
[41] Agaoglou, M., Feckan, M., Panagiotidou, A. P.: Existence and uniqueness of (omega ,c)-periodic solutions of semilinear evolution equations. Int. J. Dyn. Sys. Diff. Equ. no. 10, 149–166 (2020).
[42] Khalladi, M. T., Rahmani, A.: (omega ,c)-pseudo almost periodic distributions. Nonauton. Dyn. Syst. no. 7, 237–248 (2020).
[43] Khalladi, M. T., Kostic, M., Pinto, M., Rahmani, A., Velinov, D.: On semi-c-periodic functions. J. Math. 2021, 1–5 (2021).
[44] Yuldashev, T. K.: Nonlinear optimal control of thermal processes in a nonlinear Inverse problem. Lobachevskii J. Math. 41 (1), 124–136 (2020).
[45] Yuldashev, T. K., Mamedov, Kh. R., Abduvahobov, T. A.: On a periodic solution for an impulsive system of differential equations with
Gerasimov–Caputo fractional operator and maxima. Journal of Contemporary Applied Mathematics. 13 (1), 111–122 (2023).
[46] Yuldashev, T. K., Mamedov, Kh. R., Abduvahobov, T. A.: (omega ,c)-periodic solution for an impulsive system of differential equations with the quadrate of Gerasimov–Caputo fractional operator and maxima. Journal of Contemporary Applied Mathematics. 13 (2), 65–79 (2023).
[47] Yuldashev, T. K., Abduvahobov, T. A.: Periodic solutions for an impulsive system of fractional order integro-differential equations with maxima. Lobachevskii J. Math. 44 (10), 4401–4409 (2023).
