Abstract
In this work, we present new oscillation conditions for the oscillation of the higher-order differential equations with the middle term. We obtain some oscillation criteria by a comparison method with first-order equations. The obtained results extend and simplify known conditions in the literature. Furthermore, examining the validity of the proposed criteria is demonstrated via particular examples.
Highlights
Neutral equations contribute to many applications in physics, engineering, biology, non-Newtonian fluid theory, and the turbulent flow of a polytrophic gas in a porous medium
Oscillation of neutral equations contribute to many applications of problems dealing with vibrating masses attached to an elastic bar, see [1]
We investigate the oscillatory properties of solutions of the higher-order neutral differential equation
Summary
Neutral equations contribute to many applications in physics, engineering, biology, non-Newtonian fluid theory, and the turbulent flow of a polytrophic gas in a porous medium. We investigate the oscillatory properties of solutions of the higher-order neutral differential equation. Bazighifan et al [2] examined the oscillation of higher-order delay differential equations with damping of the form α ( x )Φ [v (`−1) ( x )] 0 + α ( x )Φ [ f v (`−1) ( x ) ] + j ζ ( x )Φ [ g(v ( β ( x )))] = 0,. Bazighifan and Ramos [4] considered the oscillation of the even-order nonlinear differential equation with middle term of the form α ( x ) v (`−1) ( x ) p−1 + α ( x ) v (`−1) ( x ) p−1 + ζ ( x )v ( β( x )) = 0,. V ( x ) > 0, v 0 ( x ) > 0, v 00 ( x ) > 0, v (`−1) ( x ) > 0 and v (`) ( x ) < 0, v ( x ) > 0, v ( j) ( x ) > 0, v ( j+1) ( x ) < 0 for all odd integer j ∈ {1, 2, ..., ` − 3}, v (`−1) ( x ) > 0 and v (`) ( x ) < 0, for x ≥ x1 , where x1 ≥ x0 is sufficiently large
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