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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.
39B22 Functional equations for real functions