engleski

New interval oscillation criteria for forced second- order differential equations with nonlinear damping

We introduce and prove some new interval oscillation criteria for a general class of forced second-order differential equations (E):\ \ \big(r(t)k_1(x, x')\big)'+p(t)k_2(x, x')x'+q(t)f(x)= e(t), \ t\geq t_0, where the functions $k_1(u, v)$, $k_2(u, v)$ and $f(u)$ satisfy some general conditions. Our interval oscillation criteria and their proofs are different than previously published ones, and it is based on (i): a construction of a global supersolution of the generalized Riccati differential equation $(R)$: $\omega'=A_1(t)|\omega(t)|^\beta+A_2(t)|\omega(t)| ^{;\delta};+B(t)$, $t\geq T$, by using the classic Riccati transformation of a nonoscillatory solution of the main equation $(E)$, on (ii): a construction of a pair of local subsolutions of equation $(R)$ under a new oscillatory condition on the coefficients of the main equation $(E)$ and on (iii): a pointwise comparison principle between all sub- and supersolutions of equation $(R)$.

interval criteria; oscillation; second order; forced term; nonlinear damping; Riccati differential equation; pointwise comparison principle; sub and supersolutions; blow-up argument

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