Oscillation Criteria for Second-Order Delay , Difference , and Functional Equations

Consider the second-order linear delay differential equation x′′ t p t x τ t 0, t ≥ t0, where p ∈ C t0,∞ ,R , τ ∈ C t0,∞ ,R , τ t is nondecreasing, τ t ≤ t for t ≥ t0 and limt→∞τ t ∞, the discrete analogue second-order difference equation Δ2x n p n x τ n 0, where Δx n x n 1 − x n , Δ2 Δ ◦ Δ, p : N → R , τ : N → N, τ n ≤ n − 1, and limn→∞τ n ∞, and the second-order functional equation x g t P t x t Q t x g2 t , t ≥ t0, where the functions P , Q ∈ C t0,∞ ,R , g ∈ C t0,∞ ,R , g t /≡ t for t ≥ t0, limt→∞g t ∞, and g2 denotes the 2th iterate of the function g, that is, g0 t t, g2 t g g t , t ≥ t0. The most interesting oscillation criteria for the second-order linear delay differential equation, the second-order difference equation and the second-order functional equation, especially in the case where lim inft→∞ ∫ t τ t τ s p s ds ≤ 1/e and lim supt→∞ ∫ t τ t τ s p s ds < 1 for the second-order linear delay differential equation, and 0 < lim inft→∞{Q t P g t } ≤ 1/4 and lim supt→∞{Q t P g t } < 1, for the second-order functional equation, are presented.


Introduction
The problem of establishing sufficient conditions for the oscillation of all solutions to the second-order delay differential equation x t p t x τ t 0, t ≥ t 0 , 1 where p ∈ C t 0, ∞ , R here R 0, ∞ , τ ∈ C t 0 , ∞ , R , τ t is nondecreasing, τ t ≤ t for t ≥ t 0 , and lim t → ∞ τ t ∞, has been the subject of many investigations; see, for example, 1-21 and the references cited therein.

International Journal of Differential Equations
By a solution of 1 we understand a continuously differentiable function defined on τ T 0 , ∞ for some T 0 ≥ t 0 and such that 1 is satisfied for t ≥ T 0 .Such a solution is called oscillatory if it has arbitrarily large zeros, and otherwise it is called nonoscillatory.
The oscillation theory of the discrete analogue second-order difference equation ∞, has also attracted growing attention in the recent few years.The reader is referred to 22-26 and the references cited therein.
By a solution of 1 we mean a sequence x n which is defined for n ≥ min {τ n : n ≥ 0} and which satisfies 1 for all n ≥ 0. A solution x of 1 is said to be oscillatory if the terms x of the solution are neither eventually positive nor eventually negative.Otherwise the solution is called nonoscillatory.
The oscillation theory of second-order functional equations of the form and g 2 denotes the 2nd iterate of the function g, that is, has also been developed during the last decade.We refer to 27-35 and the references cited therein.By a solution of 1 we mean a real-valued function x : t 0 , ∞ → R such that sup{|x s | : s ≥ t * } > 0 for any t * ≥ t 0 and x satisfies 1 on t 0 , ∞ .
In this paper our purpose is to present the state of the art on the oscillation of all solutions to 1 , 1 , and 1 , especially in the case where for 1 , and International Journal of Differential Equations 3

Oscillation Criteria for 1
In this section we study the second-order delay equation 1 .For the case of ordinary differential equations, that is, when τ t ≡ t, the history of the problem began as early as in 1836 by the work of Sturm 16 and was continued in 1893 by Kneser 8 .Essential contribution to the subject was made by E. respectively, for the first-order delay equation x t p t x τ t 0.

2.6
The essential difference between 2.1 , 2.2 , and C 1 , C 2 is that the first two can guarantee oscillation for ordinary differential equations as well, while the last two work only for delay equations.Unlike first-order differential equations, where the oscillatory character is due to the delay only see 36 , equation 1 can be oscillatory without any delay at all, that is, in the case τ t ≡ t.Figuratively speaking, two factors contribute to the oscillatory character of 1 : the presence of the delay and the second-order nature of the equation.The conditions 2.1 , 2.2 , and C 1 , C 2 illustrate the role of these factors taken separately.
In 1999 Koplatadze et
The following corollaries being more convenient for applications can be deduced from this theorem.
Then 1 is oscillatory.
Then 1 is oscillatory.
In the case of ordinary differential equations τ t ≡ t the following theorem was given in 11 .
Theorem 2. 4  In what follows it will be assumed that the condition is fulfilled.As it follows from Lemma 4.1 in 10 , this condition is necessary for 1 to be oscillatory.The study being devoted to the problem of oscillation of 1 , the condition 2.13 does not affect the generality.
Here oscillation results are obtained for 1 by reducing it to a first-order equation.Since for the latter the oscillation is due solely to the delay, the criteria hold for delay equations only and do not work in the ordinary case.Then 1 is oscillatory.
To apply Theorem 2.5 it suffices to note that i equation 2.13 is fulfilled since otherwise k K 0; ii since τ t → ∞ as t → ∞, the relations C 4 and C 4 imply the same relations with 0 changed by any T ≥ 0. Remark 2.9 see 12 .Theorem 2.8 improves the criteria C 1 , C 2 by Koplatadze 9 and Wei 19 mentioned above.This is directly seen from C 3 , C 4 and can be easily checked if we take τ t ≡ t − τ 0 and p t ≡ p 0 / t − τ 0 for t ≥ 2τ 0 , where the constants τ 0 > 0 and p 0 > 0 satisfy

2.20
In this case neither of C 1 , C 2 is applicable for 1 while both C 3 , C 4 give the positive conclusion about its oscillation.Note also that this is exactly the case where the oscillation is due to the delay since the corresponding equation without delay is nonoscillatory.
Remark 2.10 see 12 .The criteria C 3 , C 4 look like H 1 , H 2 but there is an essential difference between them pointed out in the introduction.The condition H 2 is close to the necessary one, since according to 9 if A ≤ 1/e, then 2.16 is nonoscillatory.On the other hand, for an oscillatory equation 1 without delay we have k K 0. Nevertheless, the constant 1/e in Theorem 2.8 is also the best possible in the sense that for any ε ∈ 0, 1/e it cannot be replaced by 1/e − ε without affecting the validity of the theorem.This is illustrated by the following.
Note that there is a gap between the conditions C 3 , C 4 when 0 ≤ k ≤ 1/e, k < K.In the case of first-order equations cf., 36-48 , using results in this direction, one can derive various sufficient conditions for the oscillation of 1 .According to Remark 2.9, neither of them can be optimal in the above sense but, nevertheless, they are of interest since they cannot be derived from other known results in the literature.We combine Theorem 2.5 with Corollary 1 40 to obtain the following theorem.Theorem 2.12 see 12 .Let K and k be defined by C 3 , C 4 , 0 ≤ k ≤ 1/e, and where λ k is the smaller root of the equation λ exp kλ .Then 1 is oscillatory.
Note that the condition C 9 is analogous to the condition C 9 in 40 .

Oscillation Criteria for 1
In this section we study the second-order difference equation 1 , where Δx n

International Journal of Differential Equations
In 1994, Wyrwi ńska 26 proved that all solutions of 1 are oscillatory if lim sup while, in 1997, Agarwal et al. 22 proved that, in the special case of the second-order difference equation with constant delay all solutions are oscillatory if

3.1
In 2001, Grzegorczyk and Werbowski 23 studied 1 c and proved that under the following conditions lim sup for some n 1 > n 0 , all solutions of 1 c are oscillatory.Observe that the last condition C 2 may be seen as the discrete analogue of the condition C 2 .
In 2001 Koplatadze 24 studied the oscillatory behaviour of all solutions to 1 with variable delay and established the following.

3.3
Then all solutions of 1 oscillate.
International Journal of Differential Equations 9 Corollary 3.2 see 24 .Let α > 0 and Then all solutions of the equation oscillate.

Corollary 3.3 see 24 .
Let n 0 be an integer and Then all solutions of the equation In Then all solutions of 1 oscillate.
In the case where α 1, the following discrete analogue of Hille's well-known oscillation theorem for ordinary differential equations see 2.2 is derived.Remark 3.6 see 25 .As in case of ordinary differential equations, the constant 1/4 in 3.10 is optimal in the sense that the strict inequality cannot be replaced by the nonstrict one.More than that, the same is true for the condition 3.9 as well.To ascertain this, denote by c the right-hand side of 3.9 and by λ 0 the point where the maximum is achieved.The sequence x n n λ 0 obviously is a nonoscillatory solution of the equation x n p n x αn 0, 3.12 where p n −Δ 2 n λ 0 / αn λ 0 and α denotes the integer part of α.It can be easily calculated that

3.13
Hence for arbitrary ε > 0, p n ≥ c − ε /n 2 for n ∈ N n 0 with n 0 ∈ N sufficiently large.Using the inequality ∞ i n i 2 ≥ n −1 and the arbitrariness of ε, we obtain p i ≥ c.

3.14
This limit cannot be greater than c by Theorem 3.4.Therefore it equals c and 3.9 is violated.

Oscillation Criteria for 1
In this section we study the functional equation 1 .
In 1993 Domshlak 27 studied the oscillatory behaviour of equations of this type.In 1994, Golda and Werbowski 28 proved that all solutions of 1 oscillate if International Journal of Differential Equations 11 In the same paper they also improved condition C 1 to lim sup where k ≥ 0 is some integer.In 1995, Nowakowska and Werbowski 29 extended condition C 1 to higher-order linear functional equations.In 1996, Shen 30 , in 1997

4.8
Thus, by Corollary 4.2 all solutions of 4.7 oscillate.However, the condition C 1 is satisfied only for q > 3/4, while the condition 4.2 is satisfied for much smaller q > 5/12.
2002 Koplatadze et al. 25 studied 1 and established the following.Theorem 3.4 see 25 .Assume that

Theorem 3 . 5 10 International
see 25 .Let n 0 be an integer and lim inf Journal of Differential Equations Then all solutions of the equation Δ 2 x n p n x n − n 0 0, n ≥ n 0 , 3.11 oscillate.
see 11 .Let τ t ≡ t and Theorem 2.5 see 12 .Let 2.13 be fulfilled and let the differential inequality Turning to applications of Theorem 2.5, we will use it together with the criteria H 1 and H 2 to get the following.
1 Zhang et al. 35 , and, in 1998, Zhang et al. 34 studied functional equations with variable coefficients and constant delay, while in 1999, Yan and Zhang 33 considered a system with constant coefficients.It should be noted that conditions C 1 and C 2 may be seen as the analogues of the oscillation conditions C 1 and C 2 for 1 and C 1 and C 2 for 1 .As far as the lower bound 1/4 in the condition C 2 is concerned, as it was pointed out in 28 , it cannot be replaced by a smaller number.Recently, in 32 , this fact was generalized by proving that It is obvious that there is a gap between the conditions C 1 and C 2 when the limit lim t → ∞ {Q t P g t } does not exist.How to fill this gap is an interesting problem.Here we should mention that condition 4.1 is an attempt in this direction.In fact, from condition 4.1we can obtain see 31 that all solutions of 1 oscillate if 0 ≤ a ≤ 1/4 and In 2002, Shen and Stavroulakis 31 proved the following.Then all solutions of 1 oscillate.Then all solutions of 1 oscillate.International Journal of Differential Equations Remark 4.3 see 31 .It is to be noted that as a → 0 the condition 4.3 reduces to the condition 4.1 and the conditions 4.4 and 4.2 reduce to the condition C 1 .However the improvement is clear as 0 < a ≤ 1/4 because − 2sin 2 t − 2sin 2 t − 2sin 2 t , 4.7 where g t t − 2sin 2 t, P t ≡ 1, Q t 1/4 qcos 2 t, and q > 0 is a constant.It is easy to see that Theorem 4.1 see 31 .Assume that 0 ≤ a ≤ 1/4 and that for some integer k ≥ 0