In a domain obtained as a Cartesian product of an interval [0,T] and the space ℝp , p ∈ ℕ, for a system of equations (with constant coefficients) unsolved with respect to the highest time derivative, we study the problem with integral conditions in the time variable for a class of functions almost periodic in the space variables. A criterion of uniqueness and sufficient conditions for the existence of solution of this problem in different functional spaces are established. We use the metric approach to solve the problem of small denominators encountered in the construction of the solution.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
M. Abramowitz and I. A. Stegun (editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [Russian translation], Nauka, Moscow (1979).
G. I. Barenblatt, Yu. P. Zheltov, and I. N. Kochina, “On the main representations of the theory of filtration in cracked media,” Prikl. Mekh. Mat., 24, No. 5, 58–73 (1960).
N. I. Bilusyak, L. I. Komarnyts’ka, and B. I. Ptashnyk, “Dirichlet-type problem for systems of partial differential equations unsolved with respect to the highest time derivative,” Ukr. Mat. Zh., 54, No. 12, 1592–1602 (2002); English translation: Ukr. Math. J., 54, No. 12, 1930–1942 (2002).
O. D. Vlasii and V. I. Ptashnyk, “Problem with nonlocal conditions for systems of partial differential equations unsolved with respect the higher time derivative,” Ukr. Mat. Visn., 1, No. 4, 501–517 (2004).
S. A. Gabov and A. G. Sveshnikov, Problems of the Dynamics of Stratified Liquid [in Russian], Nauka, Moscow (1986).
I. M. Gelfand and G. E. Shilov, Generalized Functions. Some Problems of the Theory of Differential Equations [in Russian], Fizmatgiz, Moscow (1958).
O. M. Medvid’ and M. M. Symotyuk, “Diophantine approximations of the characteristic determinant of an integral problem for a linear partial differential equation,” Nauk. Visn. Cherniv. Nats. Univ., Ser. Mat., Issue 228 74–85 (2004).
I. S. Klyus and B. I. Ptashnyk, “A multipoint problem for partial differential equations unsolved with respect to the higher time derivative,” Ukr. Mat. Zh., 51, No. 12, 1604–1613 (1999); English translation: Ukr. Math. J., 51, No. 12, 1813–1823 (1999).
A. M. Kuz’ and B. I. Ptashnyk, “Problem with integral conditions for equations unsolved with respect to the higher time derivative,” in: Proceedings of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], 11, No 2, Kyiv (2014), pp. 200–224.
L. I. Komarnyts’ka and B. I. Ptashnyk, “Boundary-value problems for differential equations unsolved with respect to the higher time derivative,” Ukr. Mat. Zh., 47, No. 9, 1197–1208 (1995); English translation: Ukr. Math. J., 47, No. 9, 1364–1377 (1995).
P. Lankaster, Theory of Matrices [Russian translation], Nauka, Moscow (1982).
Ya. T. Megraliev, “Inverse boundary-value problem for the Boussinesq–Love equation with additional integral conditions,” Sib. Zh. Industr. Mat., 16, No. 1, 75–83 (2013).
G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis. Ester Band. Reihen. Integralrechnung. Funktionentheorie [Russian translation], Nauka, Moscow (1978).
S. L. Sobolev, “One a new problem of mathematical physics,” Izv. Akad. Nauk SSSR, Ser. Mat., 18, No. 1, 3–50 (1954).
V. G. Sprindzhuk, Metric Theory of Diophantine Approximations [in Russian], Nauka, Moscow (1977).
M. A. Shubin, “Almost periodic functions and differential operators with partial derivatives,” Usp. Mat. Nauk, 33, No. 2, 3–47 (1978).
D. K. Faddeev and I. S. Somins’kyi, A Collection of Problems in Higher Algebra [in Ukrainian], Vyshcha Shkola, Kyiv (1971).
Ya. D. Tamarkin, On Some General Problems in the Theory of Ordinary Differential Equations and on the Expansions of Arbitrary Functions in Series [in Russian], Petrograd (1917).
A. S. Besicovitch, Almost Periodic Functions, Dover Publications, Cambridge (1954).
A. Bouziani and N. Merazga, “Solution to a semilinear pseudoparabolic problem with integral conditions,” Electron. J. Different. Equat., No. 115, 1–18 (2006).
A. M. Kuz and B. Yo. Ptashnyk, “Problem for hyperbolic system of equations having constant coefficients with integral conditions with respect to the time variable,” Carpath. Math. Publ., 6, No. 2, 282–299 (2014).
A. P. Oskolkov, “Nonlocal problems for the equations of Kelvin–Voight fluids and their e-approximations,” J. Math. Sci., 87, No. 2, 3393–3408 (1997).
G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev-Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht, etc. (2003).
Sh. A. Dubey, “Numerical solution for nonlocal Sobolev-type differential equations,” Electron. J. Different. Equat., 19, 75–83 (2010).
B. I. Ptashnyk is deceased.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 4, pp. 530–549, April, 2017.
About this article
Cite this article
Kuz’, A.M., Ptashnyk, B.I. Problem with Integral Conditions in the Time Variable for a Sobolev-Type System of Equations with Constant Coefficients. Ukr Math J 69, 621–645 (2017). https://doi.org/10.1007/s11253-017-1385-8