Monotonicity conditions in oscillation to superlinear differential equations

Abstract

We consider the second order differential equation \[ \bigl(a(t)|x^{\prime}|^{\alpha}\operatorname{sgn\,}x^{\prime}\bigr)^{\prime }+b(t)|x|^{\beta}\operatorname{sgn\,}x=0 \] in the super-linear case $\alpha<\beta$. We prove the existence of the so-called intermediate solutions and we discuss their coexistence with other types of nonoscillatory and oscillatory solutions. Our results are new even for the Emden-Fowler equation ($\alpha=1$).

Keywords

• second order nonlinear differential equation
• nonoscillatory solution
• oscillatory solution
• intermediate solution