Geodesic
The nonlocal A.~A.~Desin's problem for an equation of mixed elliptic-hyperbolic type
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 1, pp. 22-32.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper for the equation of mixed elliptic-hyperbolic type in rectangular area the task with the conditions of periodicity and the nonlocal problem of A. A. Desin was studied, the uniqueness criterion was set. The solution of the problem was constructed as a sum of orthogonal series in eigenfunctions of the corresponding one-dimensional spectral problem. The problem of small denominators arises in justifying the convergence of the series. Therefore the evaluation on the separation from zero of small denominators with the corresponding asymptotics was established. This assessment allowed under certain conditions relative to the set objectives and functions to prove convergence of the constructed series in the class of regular solutions and the stability of the solution.
Keywords: equation of mixed type nonlocal problem, uniqueness criterion, the existence, series, small denominators, stability. 
@article{VSGTU_2016_20_1_a1,
     author = {V. A. Gushchina},
     title = {The nonlocal {A.~A.~Desin's} problem for an equation of mixed elliptic-hyperbolic type},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {22--32},
     publisher = {mathdoc},
     volume = {20},
     number = {1},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2016_20_1_a1/}
}
TY  - JOUR
AU  - V. A. Gushchina
TI  - The nonlocal A.~A.~Desin's problem for an equation of mixed elliptic-hyperbolic type
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2016
SP  - 22
EP  - 32
VL  - 20
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2016_20_1_a1/
LA  - ru
ID  - VSGTU_2016_20_1_a1
ER  - 
%0 Journal Article
%A V. A. Gushchina
%T The nonlocal A.~A.~Desin's problem for an equation of mixed elliptic-hyperbolic type
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2016
%P 22-32
%V 20
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2016_20_1_a1/
%G ru
%F VSGTU_2016_20_1_a1
V. A. Gushchina. The nonlocal A.~A.~Desin's problem for an equation of mixed elliptic-hyperbolic type. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 1, pp. 22-32. http://geodesic.mathdoc.fr/item/VSGTU_2016_20_1_a1/

[1] Dezin A. A., “On the solvable extensions of partial differential operators”, Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), Acad. Sci. USSR Siberian Branch, Moscow, 1963, 65–66 | MR

[2] Dezin A. A., “Operators involving a first derivative with respect to time and nonlocal boundary conditions”, Math. USSR-Izv., 1:1 (1967), 57–79 | DOI | MR | Zbl

[3] Nakhusheva Z. A., “On a nonlocal problem of A. A. Dezin for the Lavrent'ev–Bitsadze equation”, Differ. Equ., 45:8 (2009), 1223–1228 | DOI | MR | Zbl

[4] Nakhusheva Z. A., “Nelokal'nye kraevye zadachi dlia osnovnykh i smeshannykh tipov differentsial'nykh uravnenii [Nonlocal boundary value problems for the main and mixed types of differential equations]”, Nal'chik, 2011, 196 pp. (In Russian)

[5] Sabitov K. B., Novikova V. A., “Nonlocal A. A. Dezin's problem for Lavrent'ev–Bitsadze equation”, Russian Mathematics (Izvestiya VUZ. Matematika), 60:6 (2016)

[6] Frankl' F. I., “Subsonic flow about a profile with a supersonic zone”, Prikl. Mat. Meh., 20:2 (1956) (In Russian) | MR | Zbl

[7] Zhegalov V. I., “A boundary-value problem for an equation of mixed type with boundary conditions on both characteristics and with discontinuities on the transition curve”, Boundary value problems in the theory of analytic functions, Uchenye Zapiski Kazanskogo Universiteta, 122, no. 3, Kazan University, Kazan, 1962, 3–16 (In Russian) | MR | Zbl

[8] Nakhushev A. M., “On Some Boundary Value Problems for Hyperbolic Equations and Equations of Mixed Type”, Differ. Uravn., 5:1 (1969), 44–59 (In Russian) | Zbl

[9] Sabitov K. B., “Dirichlet problem for mixed-type equations in a rectangular domain”, Dokl. Math., 75:2 (2007), 193–196 | DOI | MR | Zbl

[10] Sabitov K. B., Sidorenko O. G., “Problem with periodicity conditions for a degenerating equation of mixed type”, Differ. Equ., 46:1 (2010), 108–116 | DOI | MR | Zbl

[11] Arnol'd V. I., “Small Denominators And Problems Of Stability Of Motion In Classical And Celestial Mechanics”, Russian Math. Surveys, 18:6 (1963), 85–191 | DOI | MR | Zbl