Existence of solutions of a system of two ordinary differential equations with a modular–cubic type nonlinearity

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.