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On boundary value problems for first order impulse functional differential equations. (English) Zbl 0842.34063
Henderson, Johnny (ed.), Boundary value problems for functional differential equations. Singapore: World Scientific. 107-117 (1995).
Scalar impulse functional differential equations of the form \[ (1) \quad (Lx) (t) = x'(t) + \sum^{m + 1}_{i = 1} \int^{t_i}_{t_{i -1}} x(s) d_s r_i(t,s) = f(t), \qquad (2) \quad x(t_j) = \beta_j x(t_j - 0) \] are considered. Here \(\beta_j > 0\), \(t \in [0,b]\), the functions \(r_i (t,s) : [0,b] \times [t_{i - 1}, t_i) \to \mathbb{R}\) are measurable in \(t\), bounded and nondecreasing in \(s\). It should be noted that (1) includes the equation with delayed argument \[ x'(t) + p(t)x \bigl( t - \tau (t) \bigr) = f(t), \quad x(\xi) = 0 \quad \text{for} \quad \xi < 0 \] as a particular case. The authors establish the equivalence between such properties of (1), (2) as nonoscillation of solutions of the homogeneous equation, positivity of Cauchy’s [or Green’s] function to (1), (2) [respectively to (1), (2) with \(x(0) = x(b)]\), existence of \(v \geq 0\), \(Lv \leq 0\) satisfying (2). Moreover, an analogous result for impulse differential equations of neutral type (IDENT) is proved. Using a concrete form of \(v\), the authors obtain an effective sufficient condition for nonoscillation of nontrivial solutions to the (IDENT).
For the entire collection see [Zbl 0834.00035].

34K10 Boundary value problems for functional-differential equations
34A37 Ordinary differential equations with impulses