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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].

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