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Discontinuous solutions of equations not solved with respect to the unknown function. (Ukrainian. English summary) Zbl 1075.39502
This article focuses on the scalar equation $g(t,x)=0,\quad x(s_n+0)-x(s_n-0)=I_n(x).\tag{1}$ Here the function $$g:D \mapsto \mathbb R$$ is continuously differentiable almost everywhere on the direct product $$D:=\mathcal{T}\times \mathcal{X}$$ of the intervals $$\mathcal{T}$$ and $$\mathcal{X}$$, $$I_n(x)\in C^1(\mathbb R \mapsto \mathbb R)$$, $$n\in \mathcal{N}\subset \mathbb N$$, and $$\{s_n\}_{n\in \mathcal{N}}\subset\mathcal{T}$$. One can regard (1) as a generating problem for the singularly perturbed differential equation $$\epsilon \dot x=g(t,x)$$ with impulses at the set of points $$\{s_n\}$$.
A nontrivial case is considered when $$\overline D$$ contains critical points. A point $$(t_0,x_0)\in \mathcal{L}:=\{(t,x)\in D:g(t,x)=0\}$$ is said to be critical if $$V(t_0,x_0)\setminus \{(t_0,x_0)\}\cap \mathcal{L}\neq\emptyset$$ for any neighborhood $$V(t_0,x_0)$$, and either there does not exist a neighborhood $$U(t_0,x_0)$$ such that $$g\in C^1(U(t_0,x_0) \mapsto \mathbb R)$$, or $$g\in C^1(U(t_0,x_0) \to \mathbb R)$$ but $$g'_x(t_0,x_0)=0$$ for some neighborhood $$U(t_0,x_0)$$.
The authors announce a number of existence theorems for bounded and unbounded solutions of the problem (1). Some necessary as well as sufficient conditions for the existence of periodic solutions are established.
##### MSC:
 39B22 Functional equations for real functions