Integral averages and oscillation of second order sublinear differential equations

Manojlović, Jelena V.

Czechoslovak Mathematical Journal, Tome 55 (2005) no. 1, pp. 41-60.

Résumé

New oscillation criteria are given for the second order sublinear differential equation \[ [a(t)\psi (x(t))x^{\prime }(t)]^{\prime }+q(t)f(x(t))=0, \quad t\ge t_0>0, \] where $a\in C^1([t_0,\infty ))$ is a nonnegative function, $\psi , f\in C({\mathbb R})$ with $\psi (x)\ne 0$, $xf(x)/\psi (x)>0$ for $x\ne 0$, $\psi $, $f$ have continuous derivative on ${\mathbb R}\setminus \lbrace 0\rbrace $ with $[f(x)/\psi (x)]^{\prime }\ge 0$ for $x\ne 0$ and $q\in C([t_0,\infty ))$ has no restriction on its sign. This oscillation criteria involve integral averages of the coefficients $q$ and $a$ and extend known oscillation criteria for the equation $x^{\prime \prime }(t)+q(t)x(t)=0$.

Détail

Classification : 34C10, 34C15, 34C29
Keywords: oscillation; sublinear differential equation; integral averages