We consider a nonlocal boundary-value problem for a system of impulsive hyperbolic equations. Conditions for the existence of a unique solution of the problem are established by the method of functional parameters. An algorithm for finding this solution is proposed.
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B. I. Ptashnik, Ill-Posed Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1984).
Yu. A. Mitropol’skii, G. P. Khoma, and M. I. Gromyak, Asymptotic Methods for the Investigation of Quasiwave Hyperbolic Equations [in Russian], Naukova Dumka, Kiev (1991).
T. Kiguradze, “Some boundary value problems for systems of linear partial differential equations of hyperbolic type,” Mem. Different. Equat. Math. Phys., 1, 1–144 (1994).
L. Cesari, “Periodic solutions of hyperbolic partial differential equations,” in: Proc. of the Internat. Symp. on Nonlinear Vibrations (Kiev, 1961), 2, Akad. Nauk Ukr. SSR, Kiev (1963), pp. 440–457.
O. Vejvoda, L. Herrmann, V. Lovicar, et al., Partial Differential Equations: Time-Periodic Solutions, Martinus Nijhoff Publishers, Prague, etc. (1982).
A. M. Samoilenko and B. P. Tkach, Numerical-Analytic Methods in the Theory of Periodic Solutions of Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1992).
A.Yu. Kolesov, E. F. Mishchenko, and N. Kh. Rozov, Asymptotic Methods for the Investigation of Periodic Solutions of Nonlinear Hyperbolic Equations [in Russian], Nauka, Moscow (1998).
A. T. Asanova and D. S. Dzhumabaev, “Periodic solutions of the systems of hyperbolic equations bounded on a plane,” Ukr. Mat. Zh., 56, No. 4, 562–572 (2004); English translation: Ukr. Math. J., 56, No. 4, 682–694 (2004).
A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations [in Russian], Vyshcha Shkola, Kiev (1987).
D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications, Halsted Press, New York (1989).
V. H. S. Lakshmikantham, “Periodic boundary value problems for second order impulsive differential systems,” Nonlin. Anal., 13, No. 1, 75–85 (1989).
V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore (1989).
S. P. Rogovchenko, Periodic Solutions for Hyperbolic Impulsive Systems [in Russian], Preprint No. 88.3, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1988).
N. A. Perestyuk and A. B. Tkach, “Periodic solutions for weakly nonlinear partial system with pulse influence,” Ukr. Math. J., 49, No. 4, 601–605 (1997).
D. D. Bainov, E. Minchev, and A. Myshkis, “Periodic boundary value problems for impulsive hyperbolic systems,” Comm. Appl. Anal., 1, No. 4, 1–14 (1997).
A. B. Tkach, “Numerical-analytic method of finding periodic solutions for systems of partial differential equations with pulse influence,” Nonlin. Oscillat., 4, No. 2, 278–288 (2001).
A. M. Samoilenko, “Numerical-analytic method for the investigation of periodic systems of ordinary differential equations. I,” Ukr. Mat. Zh., 17, No. 4, 16–23 (1965).
A. M. Samoilenko, “Numerical-analytic method for the investigation of periodic systems of ordinary differential equations. II,” Ukr. Mat. Zh., 18, No. 2, 9–18 (1966).
A. T. Asanova and D. S. Dzhumabaev, “Unique solvability of a nonlocal boundary-value problem for systems of hyperbolic equations,” Differents. Uravn., 39, No. 10, 1343–1354 (2003).
A. T. Asanova and D. S. Dzhumabaev, “On the correct solvability of a nonlocal boundary-value problem for systems of hyperbolic equations,” Dokl. Ros. Akad. Nauk, 391, No. 3, 295–297 (2003).
A. T. Asanova and D. S. Dzhumabaev, “Correct solvability of nonlocal boundary-value problems for systems of hyperbolic equations,” Differents. Uravn., 41, No. 3, 337–446 (2005).
D. S. Dzhumabaev and A. T. Asanova, “Criteria of correct solvability for linear nonlocal boundary-value problems for systems of hyperbolic equations,” Dop. Nats. Akad. Nauk Ukr., No. 4, 7–11 (2010).
A. T. Asanova, “On a boundary-value problem with data on non-characteristic intersecting lines for a system of hyperbolic equations with mixed derivative,” Nonlin. Oscillat., 15, No. 1, 3–12 (2012).
D. S. Dzhumabaev, “Method of parametrization of the solutions of boundary-value problems for ordinary differential equations,” Vestn. Akad. Nauk Kaz. SSR, No. 1, 48–52 (1988).
D. S. Dzhumabaev, “Criteria of unique solvability for linear boundary-value problems for ordinary differential equations,” Zh. Vychisl. Mat. Mat. Fiz., 29, No. 1, 50–66 (1989).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 3, pp. 315–328, March, 2013.
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Asanova, A.T. On a Nonlocal Boundary-Value Problem for Systems of Impulsive Hyperbolic Equations. Ukr Math J 65, 349–365 (2013). https://doi.org/10.1007/s11253-013-0782-x
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DOI: https://doi.org/10.1007/s11253-013-0782-x