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Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 389109, 21 pages doi:10.1155/2010/389109 Research Article Oscillation of Second-Order Mixed-Nonlinear Delay Dynamic Equations M. ¨ Unal 1 and A. Zafer 2 1 Department of Software Engineering, Bahc¸es¸ehir University, Bes¸iktas¸, 34538 Istanbul, Turkey 2 Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey Correspondence should be addressed to M. ¨ Unal, munal@bahcesehir.edu.tr Received 19 January 2010; Accepted 20 March 2010 Academic Editor: Josef Diblik Copyright q 2010 M. ¨ Unal and A. Zafer. 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. New oscillation criteria are established for second-order mixed-nonlinear delay dynamic equations on time scales by utilizing an interval averaging technique. No restriction is imposed on the coefficient functions and the forcing term to be nonnegative. 1. Introduction In this paper we are concerned with oscillatory behavior of the second-order nonlinear delay dynamic equation of the form  rtx Δ t  Δ  p 0  t  x  τ 0  t   n  i1 p i  t  | xτ i t | α i −1 x  τ i  t   e  t  ,t≥ t 0 1.1 on an arbitrary time scale T, where α 1 >α 2 > ···>α m > 1 >α m1 > ···>α n > 0,  n>m≥ 1  ; 1.2 the functions r, p i , e: T → R are right-dense continuous with r>0 nondecreasing; the delay functions τ i : T → T are nondecreasing right-dense continuous and satisfy τ i t ≤ t for t ∈ T with τ i t →∞as t →∞. We assume that the time scale T is unbounded above, that is, sup T  ∞ and define the time scale interval t 0 , ∞ T by t 0 , ∞ T :t 0 , ∞ ∩ T. It is also assumed that the reader is already familiar with the time scale calculus. A comprehensive treatment of calculus on time scales can be found in 1–3. 2 Advances in Difference Equations By a solution of 1.1 we mean a nontrivial real valued function x : T → R such that x ∈ C 1 rd T, ∞ T and rx Δ ∈ C 1 rd T, ∞ T for all T ∈ T with T ≥ t 0 ,andthatx satisfies 1.1. A function x is called an oscillatory solution of 1.1 if x is neither eventually positive nor eventually negative, otherwise it is nonoscillatory. Equation 1.1 is said to be oscillatory if and only if every solution x of 1.1 is oscillatory. Notice that when T  R, 1.1 is reduced to the second-order nonlinear delay differential equation  r  t  x   t     p 0  t  x  τ 0  t   n  i1 p i  t  | x  τ i  t  | α i −1 x  τ i  t   e  t  ,t≥ t 0 1.3 while when T  Z, it becomes a delay difference equation Δ  r  k  Δx  k   p 0  k  x  τ 0  k   n  i1 p i  k  | xτ i k | α i −1 x  τ i  k   e  k  ,k≥ k 0 . 1.4 Another useful time scale is T  q N : {q m : m ∈ N and q>1 is a real number}, which leads to the quantum calculus. In this case, 1.1 is the q-difference equation Δ q  r  t  Δ q x  t    p 0  t  x  τ 0  t   n  i1 p i  t  | x  τ i  t  | α i −1 x  τ i  t   e  t  ,t≥ t 0 , 1.5 where Δ q ftfσt − ft/μt, σtqt,andμtq − 1t. Interval oscillation criteria are more natural in view of the Sturm comparison theory since it is stated on an interval rather than on infinite rays and hence it is necessary to establish more interval oscillation criteria for equations on arbitrary time scales as in T  R.Asfar as we know when T  R, an interval oscillation criterion for forced second-order linear differential equations was first established by El-Sayed 4. In 2003, Sun 5 demonstrated nicely how the interval criteria method can be applied to delay differential equations of the form x   t   p  t  | xτt | α−1 x  τ  t   e  t  ,  α ≥ 1  , 1.6 where the potential p and the forcing term e may oscillate. Some of these interval oscillation criteria were recently extended to second-order dynamic equations in 6–10. Further results on oscillatory and nonoscillatory behavior of the second order nonlinear dynamic equations on time scales can be found in 11–23, and the references cited therein. Therefore, motivated by Sun and Meng’s paper 24, using similar techniques introduced in 17 by Kong and an arithmetic-geometric mean inequality, we give oscillation criteria for second-order nonlinear delay dynamic equations of the form 1.1. Examples are considered to illustrate the results. Advances in Difference Equations 3 2. Main Results We need the following lemmas in proving our results. The first two lemmas can be found in 25, Lemma 1. Lemma 2.1. Let {α i }, i  1, 2, ,nbe the n-tuple satisfying α 1 >α 2 > ···>α m > 1 >α m1 > ···> α n > 0. Then, there exists an n-tuple {η 1 ,η 2 , ,η n } satisfying n  i1 α i η i  1, n  i1 η i < 1, 0 <η i < 1. 2.1 Lemma 2.2. Let {α i }, i  1, 2, ,nbe the n-tuple satisfying α 1 >α 2 > ···>α m > 1 >α m1 > ···> α n > 0. Then there exists an n-tuple {η 1 ,η 2 , ,η n } satisfying n  i1 α i η i  1, n  i1 η i  1, 0 <η i < 1. 2.2 The next two lemmas are quite elementary via differential calculus; see 23, 25. Lemma 2.3. Let u, A, and B be nonnegative real numbers. Then Au γ  B ≥ γ  γ −1  1/γ−1 A 1/γ B 1−1/γ u, γ > 1. 2.3 Lemma 2.4. Let u, A, and B be nonnegative real numbers. Then Cu − Du γ ≥  γ − 1  γ γ/1−γ C γ/γ−1 D 1/1−γ , 0 <γ<1. 2.4 The last important lemma that we need is a special case of the one given in 6. For completeness, we provide a proof. Lemma 2.5. Let τ : T → T be a nondecreasing right-dense continuous function with τt ≤ t, and a, b ∈ T with a<b.Ifx ∈ C 1 rd τa,b T is a positive function such that rtx Δ t is nonincreasing on τa,b T with r>0 nondecreasing, then x  τ  t  x σ  t  ≥ τ  t  − τ  a  σ  t  − τ  a  ,t∈  a, b  T . 2.5 Proof. By the Mean Value Theorem 2, Theorem 1.14 x  t  − x  τ  a  ≥ x Δ  η   t − τ  a  , 2.6 for some η ∈ τa,t T , for any t ∈ τa,b T . Since rtx Δ t is nonincreasing and rt is nondecreasing, we have r  t  x Δ  t  ≤ r  η  x Δ  η  ≤ r  t  x Δ  η  ,t>η 2.7 4 Advances in Difference Equations and so x Δ t ≤ x Δ η, t ≥ η.Now x  t  − x  τ  a  ≥ x Δ  t  t − τ  a  ,t∈  τa,b  T . 2.8 Define μ  s  : x  s  −  s − τ  a  x Δ  s  ,s∈  τ  t  ,σ  t  T ,t∈  a, b  T . 2.9 It follows from 2.8 that μs ≥ xτa > 0fors ∈ τt,σt T and t ∈ a, b T . Thus, we have 0 <  σt τt μ  s  x  s  x σ  s  Δs   σt τt  s − τa xs  Δ Δs  σ  t  − τ  a  x σ  t  − τ  t  − τ  a  x  τ  t  , 2.10 which completes the proof. In what follows we say that a function Ht, s : T 2 → R belongs to H T if and only if H is right-dense continuous function on {t, s ∈ T 2 : t ≥ s ≥ t 0 } having continuous Δ-partial derivatives on {t, s ∈ T 2 : t>s≥ t 0 },withHt, t0 for all t and Ht, s /  0 for all t /  s.Note that in case H R ,theΔ-partial derivatives become the usual partial derivatives of Ht, s.The partial derivatives for the cases H Z and H N will be explicitly given later. Denoting the Δ-partial derivatives H Δ t t, s and H Δ s t, s of Ht, s with respect to t and s by H 1 t, s and H 2 t, s, respectively, the theorems below extend the results obtained in 5 to nonlinear delay dynamic equation on arbitrary time scales and coincide with them when H 2 t, s is replaced by Ht, s. Indeed, if we set Ht, s  Ut, s, then it follows that H 1  t, s   U 1  t, s   U  σ  t  ,s    U  t, s  ,H 2  t, s   U 2  t, s   U  t, σ  s    U  t, s  . 2.11 When T  R, they become ∂H  t, s  ∂t  ∂U  t, s  /∂t 2  U  t, s  , ∂H  t, s  ∂s  ∂U  t, s  /∂s 2  U  t, s  2.12 as in 5. However, we prefer using H 2 t, s instead of Ut, s for simplicity. Theorem 2.6. Suppose that for any given (arbitrarily large) T ∈ T there exist subintervals a 1 ,b 1  T and a 2 ,b 2  T of T, ∞ T ,wherea 1 <b 1 and a 2 <b 2 such that p i  t  ≥ 0 for t ∈  a 1 ,b 1  T ∪  a 2 ,b 2  T ,  i  0, 1, 2, ,n  ,  −1  l e  t  > 0 for t ∈  a l ,b l  T ,  l  1, 2  , 2.13 where a l  min  τ j  a l  : j  0, 1, 2, ,n  2.14 Advances in Difference Equations 5 hold. Let {η 1 ,η 2 , η n } be an n-tuple satisfying 2.1 of Lemma 2.1. If there exist a function H ∈H T and numbers c ν ∈ a ν ,b ν  T such that 1 H 2  c ν ,a ν   c ν a ν  Q  t  H 2  σ  t  ,a ν  − r  t  H 2 1  t, a ν   Δt  1 H 2  b ν ,c ν   b ν c ν  Q  t  H 2  b ν ,σ  t  − r  t  H 2 2  b ν ,t   Δt>0 2.15 for ν  1, 2,where Q  t   p 0  t  τ 0  t  − τ 0  a ν  σ  t  − τ 0  a ν   k 0 | et | η 0 n  i1  p i t  η i  τ i t − τ i a ν  σt − τ i a ν   α i η i , k 0  n  i0 η −η i i ,η 0  1 − n  i1 η i , 2.16 then 1.1 is oscillatory. Proof. Suppose on the contrary that x is a nonoscillatory solution of 1.1. First assume that xt and xτ j t j  0, 1, 2 ,n are positive for all t ≥ t 1 for some t 1 ∈ t 0 , ∞ T . Choose a 1 sufficiently large so that τ j τ j a 1  ≥ t 1 .Lett ∈ a 1 ,b 1  T . Define w  t   −r  t  x Δ  t  x  t  ,t≥ t 1 . 2.17 Using the delta quotient rule, we have w Δ  t   −  r  t  x Δ  t   Δ x  t  − r  t   x Δ t  2 x  t  x σ  t   −  r  t  x Δ  t   Δ x σ  t   r  t   x Δ t  2 x  t  x σ  t  . 2.18 Notice that x  t  x σ  t   x  t   x  t   μ  t  x Δ  t    x 2  t   1 − μ  t  w  t  r  t    x 2  t  r  t   r  t  − μ  t  w  t   2.19 which implies r  t  − μ  t  w  t   r  t  x σ  t  x  t  > 0. 2.20 Hence, we obtain w Δ  t   −  r  t  x Δ  t   Δ x σ  t   w 2  t  r  t  − μ  t  w  t  . 2.21 6 Advances in Difference Equations Substituting 2.21 into 1.1 yields w Δ  t   p 0  t  x  τ 0  t  x σ  t   w 2  t  r  t  − μ  t  w  t   n  i1 p i  t  | xτ i t | α i −1 x  τ i  t  x σ  t  − e  t  x σ  t  . 2.22 By assumption, we can choose a 1 ,b 1 ≥ t 1 such that p i t ≥ 0 i  1, 2, 3 ,n and et ≤ 0 for all t ∈  a 1 ,b 1  T , where a 1 is defined as in 2.14. Clearly, the conditions of Lemma 2.5 are satisfied when, τ replaced with τ j for each fixed j  0, 1, 2, ,n. Therefore, from 2.5,we have x  τ j  t   x σ  t  ≥ τ j  t  − τ j  a 1  σ  t  − τ j  a 1  ,t∈  a 1 ,b 1  T 2.23 and taking into account 2.22 yields w Δ  t  ≥ p 0  t  τ 0  t  − τ 0  a 1  σ  t  − τ 0  a 1   w 2  t  r  t  − μ  t  w  t   n  i1 p k  t   τ i t − τ i a 1  σt − τ i a 1   α i  x σ t  α i −1  | e  t  | x σ  t  . 2.24 Denote Q ∗ 0  t  : p 0  t  τ 0  t  − τ 0  a 1  σ  t  − τ 0  a 1  ,Q ∗ i  t  : p i  t   τ i t − τ i a 1  σt − τ i a 1   α i . 2.25 From 2.24, we have w Δ  t  ≥ Q ∗ 0  t   w 2  t  r  t  − μ  t  w  t   n  i1 Q ∗ i  t  x σ t  α i −1  | e  t  | x σ  t  . 2.26 Now recall the well-known arithmetic-geometric mean inequality, see 26, n  i0 u i η i ≥ n  i0 u ηi i , 2.27 where η 0  1 −  n i1 η i and η i > 0, i  1, 2, ,n. Setting u 0 η 0 : | e  t  | x σ  t  ,u i η i : Q ∗ i  t  x σ t  α i −1 2.28 in 2.26 yields w Δ  t  ≥ Q ∗ 0  t   w 2  t  r  t  − μ  t  w  t   n  i1 u i η i  u 0 η 0  Q ∗ 0  t   w 2  t  r  t  − μ  t  w  t   n  i0 u i η i . 2.29 Advances in Difference Equations 7 From 2.29 and taking into account 2.27,weget w Δ  t  ≥ Q ∗ 0  t   w 2  t  r  t  − μ  t  w  t   n  i0 u η i i 2.30 and hence, w Δ  t  ≥ Q ∗ 0  t   w 2  t  r  t  − μ  t  w  t   η −η 0 0 | et | η 0  x σ t  η 0 n  i1 η −η i i  Q ∗ i t  η i   x σ t  α i −1  η i  Q ∗ 0  t   w 2  t  r  t  − μ  t  w  t   η −η 0 0 | et | η 0 n  i1 η −η i i  Q ∗ i t  η i  x σ t  −η 0   n j1 α j η j −η j   Q ∗ 0  t   w 2  t  r  t  − μ  t  w  t   η −η 0 0 | et | η 0 n  i1 η −η i i  Q ∗ i t  η i 2.31 which yields w Δ  t  ≥ Q ∗ 0  t   w 2  t  r  t  − μ  t  w  t   η −η 0 0 | et | η 0 n  i1 η −η i i  p i t  ηi  τ i t − τ i a 1  σt − τ i a 1   α i η i  Q  t   w 2  t  r  t  − μ  t  w  t  , 2.32 where Q  t   Q ∗ 0  t   η −η 0 0 | et | η 0 n  i1 η −η i i  p i t  ηi  τ i t − τ i a 1  σt − τ i a 1   α i η i . 2.33 Multiplying both sides of 2.32 by H 2 σt,a 1  and integrating both sides of the resulting inequality from a 1 to c 1 ,a 1 <c 1 <b 1 yield  c 1 a 1 w Δ  t  H 2  σ  t  ,a 1  Δt ≥  c 1 a 1 Q  t  H 2  σ  t  ,a 1  Δt   c 1 a 1 w 2  t  H 2  σ  t  ,a 1  r  t  − μ  t  w  t  Δt. 2.34 Fix s and note that  wtH 2 t, s  Δ t  H 2  σ  t  ,s  w Δ  t    H 2 t, s  Δ t w  t   H 2  σ  t  ,s  w Δ  t   H 1  t, s  H  σ  t  ,s  w  t   H  t, s  H 1  t, s  w  t  , 2.35 from which we obtain H 2  σ  t  ,s  w Δ  t    wtH 2 t, s  Δ t − H 1  t, s  H  σ  t  ,s  w  t  − H  t, s  H 1  t, s  w  t  . 2.36 8 Advances in Difference Equations Therefore,  c 1 a 1 w Δ  t  H 2  σ  t  ,a 1  Δt   c 1 a 1  wtH 2 t, a 1   Δ t Δt −  c 1 a 1  H 1  t, a 1  H  σ  t  ,a 1  w  t   H  t, a 1  H 1  t, a 1  w  t  Δt. 2.37 Notice that  c 1 a 1  wtH 2 t, a 1   Δ t Δt  w  c 1  H 2  c 1 ,a 1  − w  a 1  H 2  a 1 ,a 1   w  c 1  H 2  c 1 ,a 1  2.38 since Ha 1 ,a 1 0 and hence, we obtain from 2.34 that w  c 1  H 2  c 1 ,a 1  ≥  c 1 a 1 Q  t  H 2  σ  t  ,a 1  Δt   c 1 a 1 w 2  t  r  t  − μ  t  w  t  H 2  σ  t  ,a 1  Δt   c 1 a 1  H 1  t, a 1  H  σ  t  ,a 1  w  t   H  t, a 1  H 1  t, a 1  w  t  Δt. 2.39 On the other hand, w 2  t  H 2  σ  t  ,s  r  t  − μ  t  w  t   w  t  H  σ  t  ,s  H 1  t, s   H  t, s  H 1  t, s  w  t    wtHσt,s  rt − μtwt   rt − μtwtH 1 t, s  2 −  r  t  − μ  t  w  t   H 2 1  t, s  − w  t  H  σ  t  ,s  H 1  t, s   H  t, s  H 1  t, s  w  t  . 2.40 Taking into account that Hσt,sHt, sμtH 1 t, s, we have w 2  t  H 2  σ  t  ,a 1  r  t  − μ  t  w  t   w  t  H  σ  t  ,a 1  H 1  t, a 1   H  t, a 1  H 1  t, a 1  w  t  ≥−r  t  H 2 1  t, a 1  . 2.41 Using this inequality in 2.39, we have w  c 1  H 2  c 1 ,a 1  ≥  c 1 a 1  Q  t  H 2  σ  t  ,a 1  − r  t  H 2 1  t, a 1   Δt. 2.42 Advances in Difference Equations 9 Similarly, by following the above calculation step by step, that is, multiplying both sides of 2.32 this time by H 2 b 1 ,σs after taking into account that H 2  t, σ  s  w Δ  s    wsH 2 t, s  Δ s − H 2  t, s  H  t, σ  s  w  s  − H  t, s  H 2  t, s  w  s  , 2.43 one can easily obtain −w  c 1  H 2  b 1 ,c 1  ≥  b 1 c 1  Q  s  H 2  b 1 ,σ  s  − r  s  H 2 2  b 1 ,s   Δs. 2.44 Adding up 2.42 and 2.44,weobtain 0 ≥ 1 H 2  c 1 ,a 1   c 1 a 1  Q  t  H 2  σ  t  ,a 1  − r  t  H 2 1  t, a 1   Δt  1 H 2  b 1 ,c 1   b 1 c 1  Q  t  H 2  b 1 ,σ  t  − r  s  H 2 2  b 1 ,t   Δt. 2.45 This contradiction completes the proof when xt is eventually positive. The proof when xt is eventually negative is analogous by repeating the above arguments on the interval  a 2 ,b 2  T instead of a 1 ,b 1  T . Corollary 2.7. Suppose that for any given (arbitrarily large) T ≥ t 0 there exist subintervals a 1 ,b 1  and a 2 ,b 2  of T, ∞ such that p i  t  ≥ 0 for t ∈  a 1 ,b 1  ∪  a 2 ,b 2  ,  i  0, 1, 2, ,n  ,  −1  l e  t  ≥ 0 for t ∈  a l ,b l  ,  l  1, 2  , 2.46 where a l  min{τ j a l  : j  0, 1, 2, ,n} holds. Let {η 1 ,η 2 , ,η n } be an n-tuple satisfying 2.1 of Lemma 2.1. If there exist a function H ∈H R and numbers c ν ∈ a ν ,b ν  such that 1 H 2  c ν ,a ν   c ν a ν  Q  t  H 2  t, a ν  − r  t  H 2 1  t, a ν   dt  1 H 2  b ν ,c ν   b ν c ν  Q  t  H 2  b ν ,t  − r  t  H 2 2  b ν ,t   dt > 0 2.47 for ν  1, 2,where Q  t   p 0  t  τ 0  t  − τ 0  a ν  t − τ 0  a ν   k 0 | et | η 0 n  i1  p i t  η i  τ i t − τ i a ν  t − τ i a ν   α i η i , k 0  n  i0 η −η i i ,η 0  1 − n  i1 η i , 2.48 then 1.3 is oscillatory. 10 Advances in Difference Equations Corollary 2.8. Suppose that for any given (arbitrarily large) T ≥ t 0 there exist a 1 ,b 1 ,a 2 ,b 2 ∈ Z with T ≤ a 1 <b 1 and T ≤ a 2 <b 2 such that for each i  0, 1, 2, ,n, p i  t  ≥ 0 for t ∈ { a 1 , a 1  1, a 1  2, ,b 1 } ∪ { a 2 , a 2  1, a 2  2, ,b 2 } ,  −1  l e  t  ≥ 0 for t ∈ { a l , a l  1, a l  2, ,b l }  l  1, 2  , 2.49 where a l  min{τ j a l  : j  0, 1, 2, ,n}holds. Let {η 1 ,η 2 , ,η n } be an n-tuple satisfying 2.1 of Lemma 2.1. If there exist a function H ∈H Z and numbers c ν ∈{a ν  1,a ν  2, ,b ν −1} such that 1 H 2  c ν ,a ν  c ν −1  ta ν  Q  t  H 2  t  1,a ν  − r  t  H 2 1  t, a ν    1 H 2  b ν ,c ν  b ν −1  tc ν  Q  t  H 2  b ν ,t 1  − r  t  H 2 2  b ν ,t   > 0 2.50 for ν  1, 2,where H 1  t, a ν  : H  t  1,a ν  − H  t, a ν  ,H 2  b ν ,t  : H  b ν ,t 1  − H  b ν ,t  , Q  t   p 0  t  τ 0  t  − τ 0  a ν  t  1 − τ 0  a ν   k 0 | et | η 0 n  i1  p i t  η i  τ i t − τ i a ν  t  1 − τ i a ν   α i η i , k 0  n  i0 η −η i i ,η 0  1 − n  i1 η i , 2.51 then 1.4 is oscillatory. Corollary 2.9. Suppose that for any given (arbitrarily large) T ≥ t 0 there exist a 1 ,b 1 ,a 2 ,b 2 ∈ N with T ≤ a 1 <b 1 and T ≤ a 2 <b 2 such that for each i  0, 1, 2, ,n, p i  t  ≥ 0 for t ∈  q a 1 ,q a 1 1 , ,q b 1  ∪  q a 2 ,q a 2 1 , ,q b 2  ,  −1  l e  t  ≥ 0 for t ∈  q a l ,q a l 1 , ,q b l  ,  l  1, 2  2.52 where q a l  min{τ j q a l  : j  0, 1, 2, ,n} holds. Let {η 1 ,η 2 , ,η n } be an n-tuple satisfying 2.1  of Lemma 2.1. If there exist a function H ∈H q and numbers q c ν ∈{q a ν 1 ,q a ν 2 , ,q b ν −1 } such that 1 H 2  q c ν ,q a ν  c ν −1  ma ν q m  Q  q m  H 2  q m1 ,q a ν  − r  q m  H 2 1  q m , q a ν    1 H 2  q b ν ,q c ν  b ν −1  mc ν q m  Q  q m  H 2  q b ν ,q m1  − r  q m  H 2 2  q b ν ,q m  > 0 2.53 [...]... 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Zafer 2 1 Department of Software. Tisdell, Oscillation and nonoscillation of forced second order dynamic equations,” Pacific Journal of Mathematics, vol. 230, no. 1, pp. 59–71, 2007. 12  M. Bohner and S. H. Saker, Oscillation of. 0 is oscillatory. Proof. We will just highlight the proof since it is the same as the proof of Theorem 2.6.We should remark here that taking et ≡ 0andη 0  0 in proof of Theorem 2.6, we arrive