Even Order Half-Linear Differential Equations with Regularly Varying Coefficients

We establish nonoscillation criterion for the even order half-linear differential equation ( − 1 ) n f n ( t ) Φ x ( n ) ( n ) + ∑ l = 1 n ( − 1 ) n − l β n − l f n − l ( t ) Φ x ( n − l ) ( n − l ) = 0 , where β 0 , β 1 , … , β n − 1 are real numbers, n ∈ N , Φ ( s ) = s p − 1 sgn s for s ∈ R , p ∈ ( 1 , ∞ ) and f n − l is a regularly varying (at infinity) function of the index α − l p for l = 0 , 1 , … , n and α ∈ R . This equation can be understood as a generalization of the even order Euler type half-linear differential equation. We obtain this Euler type equation by rewriting the equation above as follows: the terms f n ( t ) and f n − l ( t ) are replaced by the t α and t α − l p , respectively. Unlike in other texts dealing with the Euler type equation, in this article an approach based on the theory of regularly varying functions is used. We establish a nonoscillation criterion by utilizing the variational technique.