In this paper, we study the well-posedness of a problem with two multiple nodes with respect to a distinguished variable $t$ and periodicity conditions with respect to the remaining coordinates $x_1,\ldots, x_p$ for linear pseudodifferential equations. Conditions for the existence and uniqueness of a solution to the problem under consideration in spaces of exponential type on the torus are established. By means of a metric approach, theorems providing lower bounds for small denominators arising in the construction of the solution are proved.

[1] Arnol’d V.I. Small denominators. I. Mapping the circle onto itself. Izv. Akad. Nauk SSSR. Ser. Mat. 1961, 25, (1), 21–86. (Russian)
[2] Bernik V.I., Beresnevich V.V., Vasylyshyn P.B., Ptashnyk B.Yo. Multipoint problem with multiple nodes for linear hyperbolic equations. Ukr. math. journ. 1999, 51 (10), 1311–1316. (Ukrainian)
[3] Bernik V.I., Melnychuk Yu.V. Diophantine approximations and the Hausdorff dimension. Nauka i Tekhnika, Minsk, 1988. (Russian)
[4] Bobyk I.O., Ptashnyk B.Yo. Boundary value problems for hyperbolic equations with constant coefficients. Ukr. math. journ. 1994, 46 (7), 795–802. (Ukrainian)
[5] Gorbachuk V.I., Gorbachuk M.L. Boundary value problems for differential-operator equations. Nauk. Dumka, Kyiv, 1984, 284 p. (Russian)
[6] Hartman P. Ordinary Differential Equations. Mir, Moscow: Mir, 1970, 720 p. (Russian)
[7] Kalenyuk P.I., Nytrebych Z.M. Generalized scheme of separation of variables. Differential-symbol method. Publishing House of Lviv Polytechnic National University, Lviv, 2002. (Ukrainian)
[8] Kamke E. Handbook of Ordinary Differential Equations. Nauka, Moscow, 1971, 576 p. (Russian)
[9] Kartashev A.P., Rozhdestvenskii B.L. Ordinary Differential Equations and Fundamentals of the Calculus of Variations. Nauka, Moscow, 1980. – 288 p. (Russian)
[10] Nytrebych Z.M., Ilkiv V.S., Pukach P.Ya. Homogeneous problem with two-point conditions in time for some equations of mathematical physics. Azerb. Journal of Mathematics 2017, 7 (2), 180–196.
[11] Nytrebych Z.M., Malanchuk O.M., Ilkiv V.S., Pukach P.Ya. On the solvability of two-point in time problem for PDE. Italian Journal of Pure and Applied Mathematics 2017, (38), 715–726.
[12] Polya G., Szegö G. Problems and Theorems in Analysis: in 2 vols. Nauka, Moscow, 1978. – Vol. 1, 391 p. – Vol. 2, 432 p. (Russian)
[13] Ptashnik B.I. Ill-posed boundary value problems for partial differential equations. Nauk. Dumka, Kyiv, 1984. (Russian)
[14] Ptashnik B.I., Shtabalyuk P.I. A boundary value problem for hyperbolic equations in a class of functions that are almost periodic with respect to space variables. Differentsial’nye Uravneniya 1986, 22 (4), 669–678. (Russian)
[15] Sprindzhuk V. G. Achievements and problems in Diophantine approximation theory. Russian Mathematical Surveys, 1980, Volume 35, Issue 4, 1–80.
[16] Symotyuk M.M. Two-point problem for linear partial differential equations with constants coefficients. Sci. Bull. Uzhhorod Nat. University, 2002, 7, 96–107. (Ukrainian)
[17] Symotyuk M.M. Diophantine approximations of characteristic determinant of the two-point problem for partial differential equations. Mathem. Bulletin of Shevchenko Scientific Society 2004, 2, 199–212. (Ukrainian)
[18] Symotyuk M.M., Tymkiv I.R. Problem with two-point conditions for parabolic equation of second order on time. Carpathian Mathematical Publication 2014, 6 (2), 351–359. doi: 10.15330/cmp.6.2.351–359.
[19] Sprindzhuk V.G. Metric Theory of Diophantine Approximations. Nauka, Moscow, 1977, 144 p. (Russian)
[20] Symotyuk M.M. A multipoint problem for pseudodifferential equations with partial derivatives. Math. Methods and Phys.-Mech. Fields, 2003, Vol. 46, No. 2, P. 26–41.
[21] Valitskii Yu.N. Well-posedness of a problem for a differential equation with given values for a function and its derivatives at several points. Sibirsk. Mat. Zh. 1996, 37 (2), 251–258. (Russian)