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Nonoscillation of first order impulse differential equations with delay. (English) Zbl 0870.34010
Summary: Oscillation properties of impulsive functional-differential equations are studied for equations of the type \[ \dot x(t)= \sum^m_{i=1} p_i(t)x \bigl(t- \tau_i(t) \bigr)=0, \quad t \in[a,b], \]
\[ x(\xi)=0, \quad \xi \notin [a,b], \]
\[ x(t_j)= \beta_j x(t_j-0), \quad j=1, \dots,k, \]
\[ a<t_1 <t_2 <\cdots <t_k <b. \] The proven test for oscillation generalizes the known ones and allows consideration of the solvability of boundary value problems for the corresponding nonhomogeneous impulse equations. In particular, for the scalar impulse equation \[ \dot x(t)+ p(t)x \bigl(t- \tau (t) \bigr) =0, \quad t\in [0,\infty), \]
\[ x(\xi)= 0 \quad \text{for } \xi<0, \]
\[ x (t_j)= \beta_j (t_j-0), \quad \beta_j >0,\;j=1,2, \dots, \] let be \(B(t)= \prod_{j \in D_t} \beta_j\), where \(D_t= \{i:t_i \in[t-\tau (t),t]\}\), \(p_+(t)= \max \{p(t), 0\}\).
Proposition. Let \[ {1+ \ln B(t) \over e} \geq \int^t_{r(t)} p_+(s)ds \quad \text{where } r(t)= \max \bigl\{t- \tau (t),0 \bigr\},\;t>0. \] Then the nontrivial solution of this equation has no zeros on \([0,\infty)\).

34A37 Ordinary differential equations with impulses
34K11 Oscillation theory of functional-differential equations
Full Text: DOI
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