Existence of solutions of a~system of two ordinary differential equations with a~modular--cubic type nonlinearity
Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 2, pp. 318-335
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We use asymptotic analysis to study the existence of solutions of a one-dimensional nonlinear system of ordinary differential equations with different powers of a small parameter at higher derivatives. A specific feature of the problem is the presence of a discontinuity of the first kind in the right-hand side of the equation $\varepsilon^4u''=f(u,v,x,\varepsilon)$ in the unknown variable $u$ at the level $u=0$, while the right-hand side of the second equation $\varepsilon^2v''=g(u,v,x,\varepsilon)$ is assumed to be smooth in all variables. We define a generalized solution of the problem is in terms of differential inclusions. Conditions under which generalized solutions turn into strong ones are proposed, and the possibility that the $u$-component of the solution intersects zero only once is studied. The existence theorems are proved by using the asymptotic method of differential inequalities.
Keywords:
system of nonlinear equations, small parameter, internal layer, upper and lower solutions, solution asymptotics, strong solutions, discontinuity of the first kind.
@article{TMF_2023_215_2_a11,
author = {B. V. Tischenko},
title = {Existence of solutions of a~system of two ordinary differential equations with a~modular--cubic type nonlinearity},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {318--335},
publisher = {mathdoc},
volume = {215},
number = {2},
year = {2023},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2023_215_2_a11/}
}
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%0 Journal Article %A B. V. Tischenko %T Existence of solutions of a~system of two ordinary differential equations with a~modular--cubic type nonlinearity %J Teoretičeskaâ i matematičeskaâ fizika %D 2023 %P 318-335 %V 215 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2023_215_2_a11/ %G ru %F TMF_2023_215_2_a11
B. V. Tischenko. Existence of solutions of a~system of two ordinary differential equations with a~modular--cubic type nonlinearity. Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 2, pp. 318-335. http://geodesic.mathdoc.fr/item/TMF_2023_215_2_a11/