Abstract and Applied Analysis
Volume 2013, Issue 1 740568
Research Article
Open Access

Oscillation Criteria for Fourth-Order Nonlinear Dynamic Equations on Time Scales

Xin Wu, 

Xin Wu

College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China gxu.edu.cn

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Taixiang Sun, 

Corresponding Author

Taixiang Sun

College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China gxu.edu.cn

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Hongjian Xi, 

Hongjian Xi

Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China gxu.edu.cn

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Changhong Chen, 

Changhong Chen

College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China gxu.edu.cn

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First published: 16 July 2013
https://doi.org/10.1155/2013/740568
Citations: 3
Academic Editor: Delfim F. M. Torres

Abstract

We establish some new oscillation criteria for nonlinear dynamic equation of the form (a(t)(b(t)(c(t)xΔ(t)) Δ) Δ) Δ + q(t)f(x(σ(t))) = 0 on an arbitrary time scale T with sup T = ∞, where a(t), b(t), c(t) are positive rd-continuous functions. An example illustrating the importance of our result is included.

1. Introduction

A time scale  T  is an arbitrary nonempty closed set of real numbers  R  with the topology and ordering inherited from  R. The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D thesis [1] in order to unify continuous and discrete analysis. The cases when a time scale  T  is equal to  R  or the set of all integers  Z  represent the classical theories of differential and difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies and helps avoid proving results twice once for differential equations and once again for difference equations. The general is to prove a result for a dynamic equation where the domain of the unknown function is a time scale  T. In this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. Therefore, not only can the theory of dynamic equations unify the theories of differential equations and difference equations, but also extends these classical cases to cases “in between,” for example, to the so-called q-difference equations when  T = {1, q, q2, …, qn, …}, which has important applications in quantum theory (see [2]). In the last years there has been much research activity concerning the oscillation and asymptotic behavior of solutions of some dynamic equations on time scales, and we refer the reader to the paper [3–8] and the references cited therein.

Recently, Hassan in [9] studied the third-order dynamic equation
()
on a time scale  T, where  γ ≥ 1  is the quotient of odd positive integers,  a  and  r  are positive rd-continuous functions on  T, and the so-called delay function  τ : T → T  satisfies  τ(t) ≤ t  for  t ∈ T  and  lim t→∞τ(t) = ∞  and  f ∈ C(T × R, R)  and obtained some oscillation criteria, which improved and extended the results that have been established in [10–12].

Li et al. in [13] also discussed the oscillation of (1), where  γ > 0  is the quotient of odd positive integers,  f ∈ C(T × R, R)  is assumed to satisfy  uf(t, u) > 0  for  u ≠ 0, and there exists a positive rd-continuous function  p  on T such that  f(t, u)/uγ ≥ p(t)  for  u ≠ 0. They established some new sufficient conditions for the oscillation of (1).

Wang and Xu in [14] extended the Hille and Nehari oscillation theorems to the third-order dynamic equation
()
on a time scale  T, where  γ ≥ 1  is a ratio of odd positive integers and the functions  ri(t)  (i = 1,2), q(t)  are positive real-valued rd-continuous functions defined on  T.
Erbe et al. in [15] were concerned with the oscillation of the third-order nonlinear functional dynamic equation
()

on a time scale  T, where  γ  is the quotient of odd positive integers,  a  and  r  are positive rd-continuous functions on  T, and  g : T → T  satisfies  lim t→∞g(t) = ∞  and  f ∈ C(T × R, R). The authors obtain some new oscillation criteria and extend many known results for oscillation of third-order dynamic equations.

Qi and Yu in [16] obtained some oscillation criteria for the fourth-order nonlinear delay dynamic equation
()

on a time scale  T, where  γ  is the ratio of odd positive integers,  p  is a positive real-valued rd-continuous function defined on  T,  τ ∈ Crd (T, T),  τ(t) ≤ t, and  lim t→∞τ(t) = ∞.

Grace et al. in [17] were concerned with the oscillation of the fourth-order nonlinear dynamic equation
()

on a time scale  T, where  λ  is the ratio of odd positive integers,  q  is a positive real-valued rd-continuous function defined on  T. They reduce the problem of the oscillation of all solutions of (5) to the problem of oscillation of two second-order dynamic equations and give some conditions ensuring that all bounded solutions of (5) are oscillatory.

Grace et al. in [18] establish some new criteria for the oscillation of fourth-order nonlinear dynamic equations
()

where  a  is a positive real-valued rd-continuous function satisfying that,  f : [t0, ∞) × R → R  is continuous satisfying  sgn  f(t, x) = sgn  x  and  f(t, x) ≤ f(t, y)  for  x ≤ y  and  t ≥ t0. They also investigate the case of strongly superlinear and the case of strongly sublinear equations subject to various conditions.

Agarwal et al. in [19] were concerned with oscillatory behavior of a fourth-order half-linear delay dynamic equation with damping
()

on a time scale  T  with   sup  T = ∞, where  λ  is the ratio of odd positive integers,  r, p, q  are positive real-valued rd-continuous functions defined on  T,  r(t) − μ(t)p(t) ≠ 0,  τ ∈ Crd (T, T),  τ(t) ≤ t, and  τ(t) → ∞  as  t → ∞. They establish some new oscillation criteria of (7).

Zhang et al. in [20] were concerned with the oscillation of a fourth-order nonlinear dynamic equation
()

on an arbitrary time scale  T  with  sup  T = ∞, where  p, q ∈ Crd (T, (0, ∞))  withand there exists a positive constant  L  such that  f(y)/y ≥ L  for all  y ≠ 0; they give a new oscillation result of (8).

Motivated by the previous studies, in this paper, we will study the oscillation criteria of the following fourth-order nonlinear dynamic equation:
()
where  T  is a time scale with  sup  T = ∞  and  t0 ∈ T  is a constant and. Throughout this paper, we assume that the following conditions are satisfied:
  • (H1)

    and cΔ(t) ≥ 0.

  • (H2)

  • (H3)

    f ∈ C(T, R)  and there exists a positive constant  M  such that for any  u ≠ 0,  f(u)/u ≥ M.

By a solution of (9), we mean a nontrivial real-valued functionwith  Tx ≥ t0, which has the property thatand satisfies (9) on  [Tx, ∞) T, whereis the space of differentiable functions whose derivative is rd-continuous. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution  x(t)  of (9) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory.

2. Some Auxiliary Lemmas

We shall employ the following lemmas.

Lemma 1. Assume that  x(t)  is an eventually positive solution of (9). Then there exists  t1 ∈ [t0, ∞) T  sufficiently large, such that, for  t ∈ [t1, ∞) T, one of the following cases holds:

  • (1)

    x(t) > 0,   xΔ(t) < 0, , ,

  • (2)

    x(t) > 0,   xΔ(t) > 0, (c(t)xΔ(t)) Δ > 0, ,

  • (3)

    x(t) > 0,   xΔ(t) > 0, (c(t)xΔ(t)) Δ > 0, ,

  • (4)

    x(t) > 0,   xΔ(t) > 0, (c(t)xΔ(t)) Δ < 0, .

Proof. Let  x(t)  be an eventually positive solution of (9). Then there is a  t1 ≥ t0, sufficiently large, such that,  x(t) > 0  for  t ≥ t1. By (9) we have

()
which implies that a(t)(b(t)(c(t)xΔ(t)) Δ) Δ  is decreasing and one of the following two cases holds.
  • (a)

    (b(t)(c(t)xΔ(t)) Δ) Δ > 0  for  t ≥ t1.

  • (b)

    There is a  t2 ≥ t1  such that  (b(t)(c(t)xΔ(t)) Δ) Δ < 0  for  t ∈ [t2, ∞) T.

If case (a) holds, then  b(t)(c(t)xΔ(t)) Δ  is strictly increasing on  [t1, ∞) T  and there exist the following two subcases.

  •  

    a1 (c(t)xΔ(t)) Δ < 0  for  t ≥ t1.

  •  

    a2 There exists a  t3 ≥ t2  such that  (c(t)xΔ(t)) Δ > 0  for  t ∈ [t3, ∞) T.

If subcase  (a1)  holds, then we claim  xΔ(t) > 0. If not, there exists a  t4 ≥ t3  such that  c(t)xΔ(t) ≤ c(t4)xΔ(t4) < 0  for  t ≥ t4. Thus, we get

()

which contradicts x(t) > 0  eventually. Therefore, we obtain case (4).

If subcase  (a2)  holds, then let  b(t)(c(t)xΔ(t)) Δ ≥ b(t3)(c(t3)xΔ(t3)) Δ > 0 we get

()

Therefore, we obtain case (3).

If case (b) holds, then we claim  (c(t)xΔ(t)) Δ > 0  for  t ≥ t2. If not, there exists a  t5 ≥ t2  such that  b(t)(c(t)xΔ(t)) Δ ≤ b(t5)(c(t5)xΔ(t5)) Δ < 0  for  t ≥ t5. Integrating this inequality from  t5  to  t, we get

()
Then, there exists a t6 ≥ t5 such that  c(t)xΔ(t)≤−M < 0  for  t ≥ t6. Integrating this inequality from  t6  to  t, we get
()
which contradicts x(t) > 0  eventually. The proof is completed.

Lemma 2 (see [12].)Assume that there exists  T ∈ T  such that  U  satisfies

()
Then
()
where.

3. The Main Result

Now we state and prove our main result.

Theorem 3. Assume that one of the following conditions holds:

()
()
()
If there exist two positive functionssuch that for all sufficiently large  t1 ∈ [t0, ∞) T, and  t4 > t3 > t2 > t1, and some constant  d ∈ (0,1),
()
()
()
where
()
Then, every solution  x(t)  of (9) is oscillatory.

Proof. Assume that (9) has a nonoscillatory solution  x(t)  on  [t0, ∞) T. Then, without loss of generality, there is a  t1 ≥ t0, sufficiently large, such that  x(t) > 0  for  t ≥ t1. By Lemma 1, there exist the following four possible cases:

  • (1)

    x(t) > 0, xΔ(t) < 0, (c(t)xΔ(t)) Δ > 0, (b(t)(c(t)xΔ(t)) Δ) Δ < 0,

  • (2)

    x(t) > 0, xΔ(t) > 0, (c(t)xΔ(t)) Δ > 0, (b(t)(c(t)xΔ(t)) Δ) Δ < 0,

  • (3)

    x(t) > 0, xΔ(t) > 0, (c(t)xΔ(t)) Δ > 0, (b(t)(c(t)xΔ(t)) Δ) Δ > 0,

  • (4)

    x(t) > 0, xΔ(t) > 0, (c(t)xΔ(t)) Δ < 0, (b(t)(c(t)xΔ(t)) Δ) Δ > 0.

If case (1) holds, then

()
which implies that  a(t)(b(t)(c(t)xΔ(t)) Δ) Δ  is decreasing on  [t1, ∞) T, and so
()
Dividing the previous inequality by  a(s)  and integrating the resulting inequality from  t  to  l, we get
()
Let  l → ∞, we obtain
()
Hence, there exists a constant  m > 0  such that
()
Integrating (28) from  t0  to  t, we get
()
which implies that
()

which contradicts assumption (17).

Integrating (28) from  t  to  ∞, we get

()
Integrating the previous inequality from  t0  to  t  gives
()
which implies
()

which contradicts assumption (18).

Let  A(t) = a(t)(b(t)(c(t)xΔ(t)) Δ) Δ. Integrating (27) from  t  to  ∞  gives

()
Integrating (34) from  t  to  ∞, we get
()
Set
()
Then,  R(t) < 0  for  t ∈ [t1, ∞) T  and
()
By (34), we get
()
Combining (36) with (38) gives
()
In view of (35), we get
()
From (39), we obtain
()
Integrating (41) from  t1  to  t  gives
()
which implies
()

Which contradicts assumption (19).

If case (2) holds, then set

()
and  R(t) < 0  for  t ∈ [t1, ∞) T  and
()
On the other hand, letfor  t ∈ [t1, ∞) T, where  u(t) = c(t)xΔ(t); it is easy to check that  U(t) > 0,   UΔ(t) > 0, UΔΔ(t) > 0.  In view of
()
by  (H1), we get
()
Therefore, by Lemma 2, for any  d ∈ (0,1), there exists  td ∈ [t1, ∞) T  such that
()
Then, we see that
()
Since
()
we get
()
In view of (49), we obtain that for all  t ∈ [td, ∞) T,
()
On the other hand, there exists  t2 ≥ td  such that for any  t ∈ [t2, ∞) T,
()
It follows from (52) and (53) that
()
Combining (45) with (54) gives
()
By (27) we get
()
Multiplying both sides of (55) with  t  replaced by  s, by  Qσ(s), and integrating with respect to  s  from  t2  to  t  (t ≥ t2), one gets
()
Thus,
()
which implies that
()

which contradicts assumption (20).

If case (3) holds, then since

()
we have
()
Hence, there exists  t2 ∈ [t1, ∞) T  such that
()
which implies that
()
Hence, there exists  t3 ∈ [t2, ∞) T  such that
()
Combining (62) with (64) gives
()
Write
()
Thus,  R(t) > 0  and for any  t ∈ [t1, ∞) T,
()
By (61) and (65), we get
()
Integrating the last inequality from  t4  (t4 ∈ [t3, ∞) T)  to  t, we get
()

which contradicts assumption (21).

If case (4) holds, then

()
Integrating the previous inequality from  t  to  z, we get
()
Letting  z → ∞  in this inequality, we obtain
()
Integrating the previous inequality from  t  to  z, we get
()
Letting  z → ∞  in this inequality, we obtain
()
Now we set
()
Thus,  R(t) > 0  and for any t ∈ [t1, ∞) T,
()
Since
()
we get
()
Hence, by (74) and (78), we get
()
Integrating the previous inequality from  t1  to  t, we get
()
which contradicts assumption (22). The proof is completed.

4. Example

Finally, we give an example to illustrate our main result.

Example 1. Consider the fourth-order nonlinear dynamic equation

()
where  ϱ > 0  is a constant, and
()
So  M = 1. It is easy to calculate that
()
It is obvious that
()
Therefore, we get
()
Then, condition (18) holds. By Lemma 2, we get
()
while  t  sufficiently large. Let  α(t) = 1, β(t) = t. We have that if  ϱ ≥ 1/2d, then
()
Since
()
we get
()
Since
()
()
we obtain
()
Hence, conditions (18), (20), (21), and (22) of Theorem 3 are satisfied. By Theorem 3, we see that every solution  x(t)  of (81) is oscillatory if  ϱ ≥ 1/2d.

Acknowledgments

This project is supported by NNSF of China (11261005) and NSF of Guangxi (2011GXNSFA018135, 2012GXNSFD A276040) and SF of ED of Guangxi (2013ZD061).

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