Oscillation of solutions of third order nonlinear neutral differential equations

Abstract

Abstract The main objective of this article is to improve and complement some of the oscillation criteria published recently in the literature for third order differential equation of the form ( r ( t ) ( z ″ ( t ) ) α ) ′ + q ( t ) f ( x ( σ ( t ) ) ) = 0 , t ≥ t 0 > 0 , $$ \bigl( r(t) \bigl( z^{\prime \prime }(t) \bigr) ^{\alpha } \bigr) ^{\prime }+q(t)f \bigl(x \bigl(\sigma (t) \bigr) \bigr)=0,\quad t\geq t_{0}>0, $$ where z ( t ) = x ( t ) + p ( t ) x ( τ ( t ) ) $z(t)=x(t)+p(t)x(\tau (t))$ and α is a ratio of odd positive integers in the two cases ∫ t 0 ∞ r − 1 α ( s ) d s < ∞ $\int _{t_{0}}^{\infty }r^{\frac{-1}{\alpha } }(s)\,\mathrm {d}s<\infty $ and ∫ t 0 ∞ r − 1 α ( s ) d s = ∞ $\int _{t_{0}}^{\infty }r^{\frac{-1}{\alpha } }(s)\,\mathrm {d}s=\infty $ . Some illustrative examples are presented.

Keywords

• Oscillation
• Third order differential equation
• Nonlinear neutral equation
• Nonoscillation