Oscillation of solutions of second order mixed nonlinear differential equations under impulsive perturbations. (English) Zbl 1217.34055

Özbekler, A.; Zafer, A.

Comput. Math. Appl. 61, No. 4, 933-940 (2011).

Summary: New oscillation criteria are obtained for second order forced mixed nonlinear impulsive differential equations of the form \[ (r(t)\Phi_{\alpha}(x'))'q(t)\Phi_{\alpha}(x)+\sum_{k=1}^nq_k(t)\Phi_{\beta_k(x)}=e(t),~t\neq\theta_i \]
\[ x(\theta_i^+)=a_ix(\theta),~x'(\theta_i^+)=b_ix'(\theta_i) \] where \(\Phi _{\gamma }(s):=|s|^{\gamma - 1}s\) and \(\beta_{1}>\beta_{2}>\dots>\beta_{m}>\alpha >\beta_{m+1}>\dots>\beta_{n}>0\). If \(\alpha=1\) and the impulses are dropped, then the results obtained by Y. G. Sun and J. S. W. Wong [J. Math. Anal. Appl. 334, No. 1, 549–560 (2007; Zbl 1125.34024)] are recovered. Examples are given to illustrate the results.

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A37	Ordinary differential equations with impulses

Keywords:

forced equation; mixed nonlinear; interval criteria; oscillation; impulse

Citations:

Zbl 1125.34024

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