Asymptotic behavior of third order delay difference equations with a negative middle term

Abstract

Abstract In this paper, we establish some sufficient conditions which ensure that the solutions of the third order delay difference equation with a negative middle term Δ ( a n Δ ( Δ w n ) α ) − p n ( Δ w n + 1 ) α − q n h ( w n − l ) = 0 , n ≥ n 0 , $$ \Delta \bigl(a_{n}\Delta (\Delta w_{n})^{\alpha } \bigr)-p_{n}(\Delta w_{n+1})^{ \alpha }-q_{n}h(w_{n-l})=0,\quad n\geq n_{0}, $$ are oscillatory. Moreover, we study the asymptotic behavior of the nonoscillatory solutions. Two illustrative examples are included for illustration.

Keywords

• Third order delay difference equations
• Comparison method
• Oscillation
• Nonoscillation