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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
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