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Research Article Oscillation of Even-Order Neutral Delay Differential Equations Tongxing Li,1 Zhenlai Han,1, 2 Ping Zhao,3 and Shurong Sun1, 4 1

School of Science, University of Jinan, Jinan, Shandong 250022, China School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China 3 School of Control Science and Engineering, University of Jinan, Jinan, Shandong 250022, China 4 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA 2

Correspondence should be addressed to Zhenlai Han, [email protected] Received 28 November 2009; Revised 31 January 2010; Accepted 29 March 2010 Academic Editor: Josef Diblik Copyright q 2010 Tongxing Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using Riccati transformation technique, we will establish some new oscillation criteria for the even order neutral delay differential equations ztn  qtfxσt  0, t ≥ t0 , where n is even, zt  xt  ptxτt, 0 ≤ pt ≤ p0 < ∞, and qt ≥ 0. These oscillation criteria, at least in some sense, complement and improve those of Zafer 1998 and Zhang et al. 2010. An example is considered to illustrate the main results.

1. Introduction This paper is concerned with the oscillatory behavior of the even-order neutral delay differential equations ztn  qtfxσt  0,

t ≥ t0 ,

1.1

where n is even zt  xt  ptxτt. In what follows we assume that I1  p, q ∈ Ct0 , ∞, R, 0 ≤ pt ≤ p0 < ∞, qt ≥ 0, I2  τ ∈ C1 t0 , ∞, R, σ ∈ Ct0 , ∞, R, τt ≤ t, τ  t  τ0 > 0, limt → ∞ σt  ∞, τ ◦ σ  σ ◦ τ, where τ0 is a constant,  0, α is a constant. I3  f ∈ CR, R and fy/y ≥ α > 0, for y /

2

Advances in Difference Equations

Neutral differential equations find numerous applications in natural science and technology. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines; see Hale 1. In the last decades, there are many studies that have been made on the oscillatory behavior of solutions of differential equations 2–6 and neutral delay differential equations 7–23. For instance, Grammatikopoulos et al. 10 examined the oscillation of second-order neutral delay differential equations   xt  ptxt − τ  qtxt − σ  0,

1.2

t ≥ t0 ,

where 0 ≤ pt < 1. Liu and Bai 13 investigated the second-order neutral differential equations  α−1   α−1 rtZ t Z t  qtyσt yσt  0,



t ≥ t0 ,

1.3

where Zt  xt  ptxτt, 0 ≤ pt < 1. Meng and Xu 14 studied the oscillation of even-order neutral differential equations 

 n−1 α−1  n−1    qtfxσt  0, rt xt  ptxt − τ xt  ptxt − τ 

t ≥ t0 , 1.4

where 0 ≤ pt < 1. Ye and Xu 21 considered the second-order quasilinear neutral delay differential equa