Abstract
We establish conditions for the unique solvability of a multipoint (with respect to the time coordinate) problem with multiple nodes for linear hyperbolic equations with constant coefficients in the class of functions periodic in the space variable. We prove metric statements concerning lower bounds of small denominators that appear in the course of construction of a solution of the problem.
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References
Yu. M. Berezanskii, “On the Dirichlet problem for the equation of string oscillation,”Ukr. Mat. Zh.,12, No. 4, 363–372 (1960).
Yu. M. Berezanskii,Expansion in Eigenvalues of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).
B. I. Ptashnyk, “Problems of the de la Valleé Poussin type for hyperbolic equations with constant coefficients,”Dopov. Akad. Nauk Ukr. SSR, No. 10, 1254–1257 (1966).
B. I. Ptashnyk, “A problem of the de la Valleé Poussin type for linear hyperbolic equations with variable coefficients,”Dopov. Akad. Nauk Ukr. SSR, No. 2, 127–130 (1967).
B. I. Ptashnyk, “An analog of ann-point problem for a system of hyperbolic equations with constant coefficients,”Dopov. Akad. Nauk Ukr. SSR, Ser. A,13, No. 4, 1254–1257 (1974).
V. I. Bernik, B. I Ptashnyk, and B. O. Salyga, “An analog of a multipoint problem for a hyperbolic equation with constant coefficients,”Differents. Uravn.,13 No. 4, 637–645 (1977).
B. I. Ptashnyk,Ill-Posed Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1984).
B. I. Ptashnyk and L. I. Komarnyts’ka, “A multipoint problem for differential equations unresolved with respect to the higher time derivative,”Dopov. Akad. Nauk Ukrainy, No. 10, 20–23 (1995).
B. I. Ptashnyk and L. P. Sylyuga, “A multipoint problem for typeless factorized differential equations with constant coefficients,”Ukr. Mat. Zh.,48, No. 1, 66–79 (1966).
B. I. Ptashnyk and L. P. Sylyuga, “A multipoint problem for typeless systems of differential equations with constant coefficients,”Ukr. Mat. Zh.,49, No. 9, 1236–1249 (1997).
P. B. Vasylyshyn, I. S. Klyus, and B. I. Ptashnyk, “A multipoint problem for hyperbolic equations with variable coefficients,”Ukr. Mat. Zh.,48, No. 11, 1468–1476 (1996).
Yu. N. Valistkii, “Well-posedness of a multipoint problem for an equation with operator coefficients,”Sib. Mat. Zh.,29, No. 4, 44–53 (1988).
Yu. N. Valistkii, “Well-posedness of a problem for a differential equation with given values of the function and its derivatives at several points,”Sib. Mat. Zh.,37, No. 2, 251–258 (1996).
V. I. Gorbachuk and M. L. Gorbachuk,Boundary-Value Problems for Differential-Operator Equations [in Russian], Naukova Dumka, Kiev (1984).
V. G. Sprindzhuk,Metric Theory of Diophantine Approximations [in Russian], Nauka, Moscow (1977).
B. I. Ptashnyk, V. V. Figol’, and P. I. Shtabalyuk, “Solvability, stability, and regularity of a multipoint problem for hyperbolic equations,”Mat. Studii, Pr. L’viv. Mat. Tov., Issue 1, 16–32 (1991).
B. I. Ptashnyk and P. I. Shtabalyuk, “A multipoint problem for hyperbolic equations in the class of functions almost periodic in the space variables,”Mat. Met. Fiz-Mekh. Polya, Issue 35, 210–215 (1992).
A. Sard, “The measure of the critical values of differential maps,”Bull. Amer. Math. Soc.,48, 883–890 (1942).
V. A. Il’in, V. A. Sadovnichii, and B. H. Sendov,Mathematical Analysis [in Russian], Vol. 2, Moscow University, Moscow (1987).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1311–1316, October, 1999.
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Bernik, V.I., Beresnevich, V.V., Vasylyshyn, P.B. et al. A multipoint problem with multiple nodes for linear hyperbolic equations. Ukr Math J 51, 1476–1483 (1999). https://doi.org/10.1007/BF02981680
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DOI: https://doi.org/10.1007/BF02981680