Abstract
It is established that the linear problemu u −a 2 u xx =g(x,t),u(0,t) =u(x, t + T) =u(x,t) is always solvable in the function spaceA = {g:g(x,t) =g(x,t+T) =g(Π −x,t) = −g(−x,t)} provided thataTq = (2p − 1) Π and (2p − 1,q) = 1, wherepandq are integer numbers. To prove this statement, an exact solution is constructed in the form of an integral operator, which is used to prove the existence of a solution of a periodic boundary-value problem for a nonlinear second-order wave equation. The results obtained can be used when studying the solutions to nonlinear boundary-value problems by asymptotic methods.
Similar content being viewed by others
References
Yu. A. Mitropol'skii and G. P. Khoma, “On the periodic solutions of the second-order wave equations. I,”Ukr. Mat. Zh.,38, No.5, 593–600(1986).
Yu. A. Mitropol'skii and G. P. Khoma, “On the periodic solutions of the second-order wave equations. II,”Ukr. Mat. Zh.,38, No. 6, 733–739 (1986).
Yu. A. Mitropol'skii and G. P. Khoma, “On the periodic solutions of the second-order wave equations. III,”Ukr. Mat. Zh.,39, No. 3, 347–353 (1987).
Yu. A. Mitropol'skii and G. P. Khoma, “On the periodic solutions of the second-order wave equations. IV,”Ukr. Mat. Zh.,40, No. 6, 757–763(1988).
Yu. A. Mitropol'skii, G. P. Khoma, and M. I. Gromyak,Asymptotic Methods for the Investigation of the Quasiwave Equations of Hyperbolic Type [in Russian], Naukova Dumka, Kiev (1991).
N. A. Artem'ev, “Periodic solutions for a class of partial differential equations,”Izv. Akad. Nauk SSSR, Ser. Mat., No. 1, 15–50 (1937).
Yu. A. Mitropol'skii and G. P. Khoma,Mathematical Justifcation of the Asymptotic Methods in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1983).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1115–1121, August, 1993.
Rights and permissions
About this article
Cite this article
Mitropol'skii, Y.A., Khoma, G.P. On the periodic solutions of the second-order wave equations. V. Ukr Math J 45, 1244–1251 (1993). https://doi.org/10.1007/BF01070972
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01070972