Problem with nonlocal conditions for weakly nonlinear hyperbolic equations
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For weakly nonlinear hyperbolic equations of order n, n≥3, with constant coefficients in the linear part of the operator, we study a problem with nonlocal two-point conditions in time and periodic conditions in the space variable. Generally speaking, the solvability of this problem is connected with the problem of small denominators whose estimation from below is based on the application of the metric approach. For almost all (with respect to the Lebesgue measure) coefficients of the equation and almost all parameters of the domain, we establish conditions for the existence of a unique classical solution of the problem.
KeywordsPeriodic Solution Lebesgue Measure Nonlinear Integral Equation Nonlocal Condition Nonlocal Problem
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- 1.N. A. Artem’ev, “Periodic solutions of one class of partial differential equations,” Izv. Akad. Nauk SSSR, Ser. Mat., No. 1, 15–50 (1937).Google Scholar
- 2.O. Vejvoda, L. Harrmann, V. Lovicar, et al., Partial Differential Equations: Time-Periodic Solutions, Noordhoff, Alphen an den Rijn-Sijthoff (1981).Google Scholar
- 3.B. I. Ptashnik, Ill-Posed Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1984).Google Scholar
- 13.I. Ya. Kmit’, “On a problem with nonlocal (in time) conditions for hyperbolic systems,” Mat. Met. Fiz.-Mekh. Polya, Issue 37, 21–25 (1994).Google Scholar
- 14.T. I. Kiguradze, “One boundary-value problem for hyperbolic systems,” Dokl. RAN, Mat., 328, No. 2, 135–138 (1993).Google Scholar
- 17.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 (1990).Google Scholar
- 21.L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).Google Scholar