Oscillation for Third-Order Nonlinear Differential Equations with Deviating Argument

and Applied Analysis 3 If any solution x of 1.1 is either oscillatory, or satisfies the condition 1.7 , or admits the asymptotic representation x i c 1 sin t − α i εi t , i 0, 1, 2, 3 , 1.8 where c / 0 and α are constants, the continuous functions εi i 0, 1, 2, 3 vanish at infinity and ε0 satisfies the inequality cε0 t > 0 for large t, then we say that 1.1 has weak property A. For n 3, the results in 16 deal with the equation x′′′ t x′ t r t f x t 0, 1.9 and read as follows. Theorem 1.1 see 16, Theorem 1.5 . Let f be a nondecreasing function satisfying ∫∞ 1 du f u < ∞, ∫−1 −∞ du f u < ∞. 1.10 Then the condition ∫∞ 0 r t dt ∞ 1.11 is necessary and sufficient in order that 1.9 has weak property A. Theorem 1.2 see 16, Corollary 1.5 . Let for some K > 0 and a > 0 r t ∣f u ∣ ≥ Kt−1|u| for u ∈ R, t ≥ a. 1.12 Then 1.9 has property A. In our previous paper 1 we have investigated 1.1 without deviating argument i.e., φ t t , especially when 1.3 is nonoscillatory. More precisely, the nonexistence of possible types of nonoscillatory solutions is examined, independently on the oscillation of 1.3 . Motivated by 1, 16 , here we continue such a study, by giving necessary and sufficient conditions in order that all solutions of 1.1 are either oscillatory or satisfy lim inft→∞x t 0. The property A for 1.1 is also considered and an extension to 1.1 of Theorem 1.1 is presented. The role of the deviating argument φ and some phenomena for 1.1 , which do not occur when 1.3 is nonoscillatory, are presented. Our results depend on a a priori classification of nonoscillatory solutions which is based on the concept of phase function 17 and on a suitable energy function. A fixed point method is also employed and sharp upper and lower estimates for bounded nonoscillatory solutions of 1.1 are established by 4 Abstract and Applied Analysis means of a suitable “cut” function. This approach enables us to assume r ∈ L1 0,∞ instead of R t ∈ L1 0,∞ , where