Oscillation and nonoscillation of even-order nonlinear delay-differential equations

The purpose of this paper is to compare the equations \[ y ( 2 n ) ( x ) + p ( x ) g ( y ( x ) , y τ ( x ) ) = 0 {y^{\left ( {2n} \right )}}\left ( x \right ) + p\left ( x \right )g\left ( {y\left ( x \right ),{y_\tau }\left ( x \right )} \right ) = 0 \] (1) \[ y ( 2 n ) ( x ) + p ( x ) g ( y ( x ) , y τ ( x ) ) = f ( x ) {y^{\left ( {2n} \right )}}\left ( x \right ) + p\left ( x \right )g\left ( {y\left ( x \right ),{y_\tau }\left ( x \right )} \right ) = f\left ( x \right ) \] (2) for their oscillatory and nonoscillatory nature. In Eqs. (1) and (2) y ( i ) ( x ) ≡ ( d i / d x i ) y ( x ) , i = 1 , 2 , . . . , 2 n {y^{(i)}}(x) \equiv \\ \left ( {{d^i}/d{x^i}} \right )y(x),i = 1,2,...,2n ; t 2 > t 1 {t_2} > {t_1} ; y τ ( x ) ≡ y ( x − τ ( x ) ) {y_\tau }(x) \equiv y\left ( {x - \tau \left ( x \right )} \right ) ; d y / d x dy/dx and d 2 y / d x 2 {d^2}y/d{x^2} will also be denoted by y ′ y’ and y y respectively. Throughout this paper it will be assumed that p ( x ) p\left ( x \right ) , f ( x ) f\left ( x \right ) , τ ( x ) \tau \left ( x \right ) are continuous real-valued functions on the real line ( − ∞ , ∞ ) \left ( { - \infty ,\infty } \right ) ; f ( x ) f\left ( x \right ) , p ( x ) p\left ( x \right ) and τ ( x ) \tau \left ( x \right ) , in addition, are nonnegative, τ ( x ) \tau \left ( x \right ) is bounded and f ( x ) f\left ( x \right ) , p ( x ) p\left ( x \right ) eventually become positive to the right of the origin. In regard to the function g g we assume the following: (i) g : R 2 → R g:{R^2} \to R is continuous, R R being the real line, (ii) g ( λ x , λ y ) = λ 2 q + 1 g ( x , y ) g\left ( {\lambda x,\lambda y} \right ) = {\lambda ^{2q + 1}}g\left ( {x,y} \right ) for all real λ ≠ 0 \lambda \ne 0 and some integer q ≥ 1 q \ge 1 , (iii) sgn ⁡ g ( x , y ) = sgn ⁡ x \operatorname {sgn} g\left ( {x,y} \right ) = \operatorname {sgn} x , (iv) g ( x , y ) → ∞ g\left ( {x,y} \right ) \to \infty as x , y → ∞ x,y \to \infty ; g g is increasing in both arguments monotonically. Eq. (1) is called oscillatory if every nontrivial solution y ( t ) ∈ [ t 0 , ∞ ) y\left ( t \right ) \in \left [ {{t_0},\infty } \right ) has arbitrarily large zeros; i.e., for every such solution y ( t ) y\left ( t \right ) , if y ( t 1 ) = 0 y\left ( {{t_1}} \right ) = 0 then there exists t 2 > t 1 {t_2} > {t_1} such that y ( t 2 ) = 0 y\left ( {{t_2}} \right ) = 0 . Eq. (1) is called nonoscillatory if it has a solution with a last zero or no zero in [ t 0 , ∞ ) , t 0 ≥ a > 0 \left [ {{t_0},\infty } \right ), {t_0} \ge a > 0 . A similar definition holds for eq. (2). All solutions of (1) and (2) considered henceforth are continuous and nontrivial, existing on some halfline [ t 0 , ∞ ) \left [ {{t_0},\infty } \right ) .