Oscillation of higher-order differential equations with distributed delay

Abstract

Abstract This paper deals with the oscillation properties of higher-order nonlinear differential equations with distributed delay [b(ℓ)(y(n−1)(ℓ))γ]′+∫cdq(ℓ,ξ)yγ(g(ℓ,ξ))d(ξ)=0,ℓ≥ℓ0, $$ \bigl[ b ( \ell ) \bigl( y^{ ( n-1 ) } ( \ell ) \bigr) ^{\gamma} \bigr] ^{\prime}+ \int_{c}^{d}q ( \ell, \xi ) y^{\gamma} \bigl( g ( \ell, \xi ) \bigr) \,d ( \xi ) =0, \quad \ell\geq\ell_{0}, $$ under the condition ∫ℓ0∞1b1γ(ℓ)dℓ<∞. $$ \int_{\ell_{0}}^{\infty}\frac{1}{b^{\frac{1}{\gamma}} ( \ell ) }\,d\ell< \infty. $$ We obtain new oscillation criteria by employing a refinement of the generalized Riccati transformations and new comparison principles. We provide some examples to illustrate the main results.

Keywords

• Higher-order
• Distributed delay differential equations
• Oscillatory solutions
• Nonoscillatory solutions