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Interval oscillation criteria of second order mixed nonlinear impulsive differential equations with delay. (English) Zbl 1245.34070

Summary: We study the following second order mixed nonlinear impulsive differential equations with delay
\[ (r(t) \Phi_\alpha (x'(t)))' + p_0(t) \Phi_\alpha (x(t)) + \sum^n_{i=1} p_i(t) \Phi_{\beta_i} (x(t - \sigma)) = e(t), t \geq t_0, t \neq \tau_k, \]
\[ x(\tau^+_k) = a_k x(\tau_k), x'(\tau^+_k) = b_k x'(\tau_k), k = 1, 2, \dots, \]
\[ \displaystyle \] where \(\Phi_\ast(u) = |u|^{\ast - 1}u, \sigma\) is a nonnegative constant, \(\{\tau_k\}\) denotes the impulsive moments sequence, and \(\tau_{k+1} - \tau_k > \sigma\). Some sufficient conditions for the interval oscillation criteria of the equations are obtained. The results obtained generalize and improve earlier ones. Two examples are considered to illustrate the main results.

MSC:

34K11 Oscillation theory of functional-differential equations
34K45 Functional-differential equations with impulses
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