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**On the oscillation properties of first-order impulsive differential equations with a deviating argument.**
*(English)*
Zbl 0880.34070

For first-order linear impulsive differential equations with a deviating argument oscillation Sturm-like theorems are proved. Various criteria for oscillation or nonoscillation of the solutions of these equations are found. The oscillatory properties of some concrete equations of the considered type are investigated. The following Sturm-like oscillation theorem from the paper gives a better idea of the considered equations and the obtained results. A brief explanation follows the theorem.

Theorem. Let the intervals \(J_n=(\xi_n,\eta_n)\) with \(\lim_{n\to\infty}\xi_n=+\infty\) be regular positive hemicycles of equation \((2)\) and on each of them let conditions \((a)\) and \((b)\) hold. Then all solutions of equation \((1)\) are oscillating.

The considered impulsive equation with deviating argument is of the type \[ L[x]:=x'(t)+\sum_{i=1}^m a_i(t)x[r_i(t)]=0\quad(t\neq t_j),\quad x(t_j^+)=\alpha_j(t_j^-).\tag{1} \] The solutions of (1) are compared with the solutions of \[ \tilde L[y]:=-y'(t)+\sum_{i=1}^m q_i'(t)b_i[q_i(t)]y[q_i(t)]=0\quad(t\neq t_j),\quad y(t_j^+)=\beta_j(t_j^-).\tag{2} \] The finite interval \(J=(\xi,\eta)\) is said to be a regular positive hemicycle of equation (2) if \(q_i(\eta)>\xi,\) \(q_i(\xi)<\eta\) \((i=1,2,\dots,m)\) and there exists a solution \(y(t)\) in the interval \(J\) such that \(y(\xi^+)=y(\eta^-)=0\), \(y(t)>0\) \((t\in J)\) and \(y(t)\leq 0\) \((t\in E\text{ ext}(J):=\left(\cup_{j=1}^m q_i(J)\right)\setminus J ).\) Conditions \((a)\) and \((b)\) are the following.

\((a)\) \(b_i(t)\geq0\) \((t\in q_i(J)\setminus J\), \(i=1,2,\dots m)\), \(a_i(t)\geq0\) \((t\in J\setminus q_i(J)\), \(i=1,2,\dots m)\), \(a_i(t)\geq b_i(t)\) \((t\in J\cap q_i(J)\), \(i=1,2,\dots m)\), \(1-\alpha_j\beta_j\geq0\) \((t_j\in J)\).

\((b)\) At least one of the inequalities \(1-\alpha_j\beta_j\geq0\) is strict or at least one of the remaining inequalities in condition \((a)\) is strict in some subinterval of the respective sets where they are supposed to be valid.

Theorem. Let the intervals \(J_n=(\xi_n,\eta_n)\) with \(\lim_{n\to\infty}\xi_n=+\infty\) be regular positive hemicycles of equation \((2)\) and on each of them let conditions \((a)\) and \((b)\) hold. Then all solutions of equation \((1)\) are oscillating.

The considered impulsive equation with deviating argument is of the type \[ L[x]:=x'(t)+\sum_{i=1}^m a_i(t)x[r_i(t)]=0\quad(t\neq t_j),\quad x(t_j^+)=\alpha_j(t_j^-).\tag{1} \] The solutions of (1) are compared with the solutions of \[ \tilde L[y]:=-y'(t)+\sum_{i=1}^m q_i'(t)b_i[q_i(t)]y[q_i(t)]=0\quad(t\neq t_j),\quad y(t_j^+)=\beta_j(t_j^-).\tag{2} \] The finite interval \(J=(\xi,\eta)\) is said to be a regular positive hemicycle of equation (2) if \(q_i(\eta)>\xi,\) \(q_i(\xi)<\eta\) \((i=1,2,\dots,m)\) and there exists a solution \(y(t)\) in the interval \(J\) such that \(y(\xi^+)=y(\eta^-)=0\), \(y(t)>0\) \((t\in J)\) and \(y(t)\leq 0\) \((t\in E\text{ ext}(J):=\left(\cup_{j=1}^m q_i(J)\right)\setminus J ).\) Conditions \((a)\) and \((b)\) are the following.

\((a)\) \(b_i(t)\geq0\) \((t\in q_i(J)\setminus J\), \(i=1,2,\dots m)\), \(a_i(t)\geq0\) \((t\in J\setminus q_i(J)\), \(i=1,2,\dots m)\), \(a_i(t)\geq b_i(t)\) \((t\in J\cap q_i(J)\), \(i=1,2,\dots m)\), \(1-\alpha_j\beta_j\geq0\) \((t_j\in J)\).

\((b)\) At least one of the inequalities \(1-\alpha_j\beta_j\geq0\) is strict or at least one of the remaining inequalities in condition \((a)\) is strict in some subinterval of the respective sets where they are supposed to be valid.

Reviewer: I.Ginchev (Varna)

### MSC:

34K11 | Oscillation theory of functional-differential equations |

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

34A37 | Ordinary differential equations with impulses |

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