Oscillatory Behavior of Quasilinear Neutral Delay Dynamic Equations on Time Scales

Abstract

By means of the averaging technique and the generalized Riccati transformation technique, we establish some oscillation criteria for the second-order quasilinear neutral delay dynamic equations , , where , and the time scale interval is . Our results in this paper not only extend the results given by Agarwal et al. (2005) but also unify the oscillation of the second-order neutral delay differential equations and the second-order neutral delay difference equations.

1. Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. Thesis in 1988 in order to unify continuous and discrete analysis (see Hilger [1]). Several authors have expounded on various aspects of this new theory and references cited therein. A book on the subject of time scale, by Bohner and Peterson [2], summarizes and organizes much of the time scale calculus; we refer also the last book by Bohner and Peterson [3] for advances in dynamic equations on time scales.

A time scale is an arbitrary closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist (see Bohner and Peterson [2]).

In the last few years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations on time scales which attempts to harmonize the oscillation theory for the continuous and the discrete to include them in one comprehensive theory and to eliminate obscurity form both, for instance, the papers [4–20] and the reference cited therein.

For oscillation of delay dynamic equations on time scales, see recently papers [21–32]. However, there are very few results dealing with the oscillation of the solutions of neutral delay dynamic equations on time scales; we refer the reader to [33–44].

Agarwal et al. [33] and Saker [37] consider the second-order nonlinear neutral delay dynamic equations on time scales:

(1.1)

where , is a quotient of odd positive integer, are positive constants, such that for all nonzero , and there exists a nonnegative function defined on satisfing .

Agwo [35] examines the oscillation of the second-order nonlinear neutral delay dynamic equations:

(1.2)

Li et al. [36] discuss the existence of nonoscillatory solutions to the second-order neutral delay dynamic equation of the form

(1.3)

Saker et al. [38, 39], Sahíner [40], and Wu et al. [43] consider the second-order neutral delay and mixed-type dynamic equations on time scales:

(1.4)

where , is a quotient of odd positive integer, , such that for all nonzero , and there exists a nonnegative function defined on satisfing .

Zhu and Wang [44] study existence of nonoscillatory solutions to neutral dynamic equations on time scales:

(1.5)

Recently, Tripathy [42] has established some new oscillation criteria for second-order nonlinear delay dynamic equations of the form

(1.6)

where , is a quotient of odd positive integer, are positive constants, and .

To the best of our knowledge, there are no results regarding the oscillation of the solutions of the following second-order nonlinear neutral delay dynamic equations on time scales up to now:

(1.7)

where , and the time scale interval is .

In what follows we assume the following:

• (A1) and are positive constants with ;

• (A2)

• (A3) and constant;

• (A4) for and .

To develop the qualitative theory of delay dynamic equations on time scales, in this paper, by using the averaging technique and the generalized Riccati transformation, we consider the second-order nonlinear neutral delay dynamic equation on time scales (1.7) and establish several oscillation criteria. Our results in this paper not only extend the results given but also unify the oscillation of the second-order quasilinear delay differential equation and the second-order quasilinear delay difference equation. Applications to equations to which previously known criteria for oscillation are not applicable are given.

By a solution of (1.7), we mean a nontrivial real-valued function , , which has the property and satisfying (1.7) for . Our attention is restricted to those solutions of (1.7) which exist on some half line with for any . A solution of (1.7) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory. Equation (1.7) is called oscillatory if all solutions are oscillatory.

Equation (1.7) includes many other special important equations; for example, if , (1.7) is the prototype of a wide class of nonlinear dynamic equations called Emden-Fowler neutral delay superlinear dynamic equation:

(1.8)

where .

If , (1.7) is the prototype of nonlinear dynamic equations called Emden-Fowler neutral delay sublinear dynamic equation:

(1.9)

where .

We note that if , then (1.7) becomes second-order nonlinear delay dynamic equation on time scales:

(1.10)

If , then (1.7) becomes second-order nonlinear delay dynamic equation on time scales:

(1.11)

If , then (1.7) becomes second-order nonlinear delay dynamic equation on time scales:

(1.12)

If , then (1.7) becomes second-order nonlinear delay dynamic equation on time scales:

(1.13)

It is interesting to study (1.7) because the continuous version and its special cases have several physical applications, see [1] and when is a discrete variable, and include its special cases also, are important in applications.

The paper is organized as follows: In the next section we present the basic definitions and apply a simple consequence of Keller's chain rule, Young's inequality:

(1.14)

and the inequality

(1.15)

where and are nonnegative constants, devoted to the proof of the sufficient conditions for oscillation of all solutions of (1.7). In Section 3, we present some corollaries to illustrate our main results.

2. Main Results

In this section we shall give some oscillation criteria for (1.7) under the cases when and . It will be convenient to make the following notations in the remainder of this paper. Define

(2.1)

We define the function space as follows: provided that is defined for and is rd-continuous function and nonnegative. For given function , we set

(2.2)

In order to prove our main results, we will use the formula

(2.3)

where is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller's chain rule (see Bohner and Peterson [2, Theorem ]).

Also, we assume the condition and , and there exists such that and

Lemma 2.1.

Assume . If is an eventually positive solution of (1.7), then, there exists a such that for . Moreover,

(2.4)

Lemma 2.2.

Assume ,

(2.5)

If is an eventually positive solution of (1.7), then

(2.6)

The proof of Lemmas 2.1 and 2.2 is similar to that of Saker et al. [39, Lemma ]; so it is omitted.

Lemma 2.3.

Assume that the condition holds:

(2.7)

If is an eventually positive solution of (1.7), then

(2.8)

(2.9)

or .

Proof.

Since is an eventually positive solution of (1.7), there exists a number such that , , and for all . In view of (1.7), we have

(2.10)

By , then , we get

(2.11)

Let , then (2.8) holds, and is an eventually decreasing function. It follows that

(2.12)

We prove that is eventually positive.

Otherwise, there exists a such that , then we have for , and hence which implies that

(2.13)

Therefore, there exists and such that

(2.14)

We can choose some positive integer such that for Thus, we obtain

(2.15)

The above inequality implies that for sufficiently large which contradicts the fact that eventually. Hence eventually. Consequently, there are two possible cases:

• (i) eventually;

• (ii) eventually.

If Case holds, we can get

(2.16)

Actually, by , we can easily verify that Using (2.3), we get

(2.17)

From , we have that is eventually negative.

Let since is eventually positive, so is eventually increasing. Therefore, is either eventually positive or eventually negative. If is eventually negative, then there is a such that for . So,

(2.18)

which implies that is strictly increasing for . Pick so that for . Then

(2.19)

so that for .

By (2.8), we have

(2.20)

which implies that

(2.21)

which contradicts (2.7). Hence, without loss of generality, there is a such that , that is, for . Consequently,

(2.22)

and we have that is strictly decreasing for .

If there exists a such that Case holds, then exists, and we claim that Otherwise, We can choose some positive integer such that for Thus, we obtain

(2.23)

which implies that and which contradicts

Now, we assert that is bounded. If it is not true, there exists with as such that

(2.24)

From and

(2.25)

which implies that it contradicts the existence of Therefore, we can assume that

(2.26)

By we get

(2.27)

thus and Hence, The proof is complete.

Theorem 2.4.

Assume that (2.5) holds, ,

(2.28)

Then (1.7) is oscillatory on .

Proof.

Suppose that (1.7) has a nonoscillatory solution . We may assume that is eventually positive. We shall consider only this case, since the proof when is eventually negative is similar. In view of Lemmas 2.1 and 2.2, there exists a such that , and for .

From (2.4) we have for

(2.29)

and hence

(2.30)

So,

(2.31)

So,

(2.32)

Now note that imply

(2.33)

This contradicts (2.28). The proof is complete.

Remark 2.5.

Theorem 2.4 includes results of Agarwal et al. [21, Theorem ] and Han et al. [25, Theorem ].

Theorem 2.6.

Assume that the condition and (2.7) hold:

(2.34)

Then every solution of (1.7) either oscillates or tends to zero as .

Proof.

Suppose that (1.7) has a nonoscillatory solution . We may assume that is eventually positive. In view of Lemma 2.3, either or there exists a such that and is strictly decreasing for .

Then from (2.8), we have for

(2.35)

Since the rest of the proof is similar to Theorem 2.4, so we omit the detail. The proof is complete.

Theorem 2.7.

Assume that (2.5) holds. , let ,

(2.36)

then (1.7) is oscillatory on , where .

Proof.

Suppose that (1.7) has a nonoscillatory solution . We may assume that is eventually positive. In view of Lemmas 2.1 and 2.2, there exists a such that , and for . Define the function by

(2.37)

We get

(2.38)

If , by (2.3), we get

(2.39)

So, from (2.4) we have

(2.40)

By Young's inequality (1.14), we obtain that

(2.41)

From being strictly decreasing, , by (2.37) and (2.40), we get that

(2.42)

that is,

(2.43)

So,

(2.44)

Using the inequality (1.15) we have

(2.45)

Integrating the inequality above from to we obtain

(2.46)

Therefore,

(2.47)

which contradicts (2.36).

If , proceeding as the proof of above, we have (2.37) and (2.38). By (2.3), we get that

(2.48)

So, from (2.4) we have

(2.49)

Since the rest of the proof is similar to that of above, so we omit the detail. The proof is complete.

Theorem 2.8.

Assume that the condition and (2.7) hold, let ,

(2.50)

then every solution of (1.7) either oscillates or tends to zero as , where .

Proof.

Suppose that (1.7) has a nonoscillatory solution . We may assume that is eventually positive. In view of Lemma 2.3, either or there exists a such that and for .

Since the rest of the proof is similar to Theorem 2.7, so we omit the detail. The proof is complete.

Theorem 2.9.

Assume that (2.5) holds. , let ,

(2.51)

then (1.7) is oscillatory on .

Proof.

We prove only case . The proof of case is similar. Proceeding as the proof of Theorem 2.7, we have (2.37) and (2.43). Replacing in (2.43) by , then multiplying (2.43) by , and integrating from to , , we have

(2.52)

Integrating by parts and using the fact that , we get

(2.53)

So,

(2.54)

that is,

(2.55)

Hence,

(2.56)

Using the inequality (1.15) we have

(2.57)

Set , so,

(2.58)

This contradicts (2.51) and finishes the proof.

Theorem 2.10.

Assume that the condition and (2.7) hold, let ,

(2.59)

then every solution of (1.7) either oscillates or tends to zero as .

Proof.

Suppose that (1.7) has a nonoscillatory solution . We may assume that is eventually positive. In view of Lemma 2.3, either or there exists a such that and for .

Since the rest of the proof is similar to Theorem 2.9, so we omit the detail. The proof is complete.

Following the procedure of the proof of Theorem 2.7, we can also prove the following theorem.

Theorem 2.11.

Assume that (2.5) holds. , let ,

(2.60)

(2.61)

there exists such that

(2.62)

or

(2.63)

and for any ,

(2.64)

and then (1.7) is oscillatory on , where .

Proof.

We prove only case . The proof of case is similar. Proceeding as the proof of Theorem 2.9, we have (2.56) and (2.57). So we have for all ,

(2.65)

By (2.64), we obtain

(2.66)

Define

(2.67)

Then, by (2.56) and (2.64), we have that

(2.68)

We claim that

(2.69)

Suppose, to the contrary, that

(2.70)

By (2.60), there exists a positive constant such that

(2.71)

Let be an arbitrary positive number, then it follows from (2.70) that there exists a such that

(2.72)

Hence,

(2.73)

By (2.71), there exists a such that , which implies that . Since is arbitrary, then

(2.74)

In view of (2.68), we consider a sequence with satisfying

(2.75)

Then, there exists a constant such that for all sufficiently large . Since (2.70) ensures that

(2.76)

so,

(2.77)

hold for all sufficiently large . Therefore,

(2.78)

On the other hand, from the definition of we can obtain, by Hölder's inequality,

(2.79)

and, accordingly,

(2.80)

So, because of (2.78), we get

(2.81)

which gives that

(2.82)

contradicting (2.61). Hence, (2.69) holds. In view of (2.66), from (2.69), we have

(2.83)

which contradicts (2.62). The proof is complete.

Theorem 2.12.

Assume that the condition and (2.7) hold, let , there exists such that (2.60) and (2.61), and either (2.62) or (2.63) hold, and for any ,

(2.84)

and then every solution of (1.7) either oscillates or tends to zero as .

Proof.

Suppose that (1.7) has a nonoscillatory solution . We may assume that is eventually positive. In view of Lemma 2.3, either or there exists a such that and for .

Since the rest of the proof is similar to Theorem 2.11, so we omit the detail. The proof is complete.

Following the procedure of the proof of Theorem 2.8, we can also prove the following theorem.

Theorem 2.13.

Assume that (2.5) and (2.60) hold, , let ,

(2.85)

there exists such that (2.62) or (2.63) holds, and for any ,

(2.86)

and then (1.7) is oscillatory on .

Theorem 2.14.

Assume that the condition , (2.7), and (2.60) hold, let , there exists such that (2.62) or (2.63) holds, and for any ,

(2.87)

and then every solution of (1.7) either oscillates or tends to zero as .

Remark 2.15.

As Theorem 2.7–Theorem 2.14 are rather general, it is convenient for applications to derive a number of oscillation criteria with the appropriate choice of the functions and .

3. Examples

In this section, we give some examples to illustrate our main results.

Example 3.1.

Consider the following delay dynamic equations on time scales:

(3.1)

where

(3.2)

with , for and . Then, by Theorem 2.4, we have

(3.3)

Then (3.1) is oscillatory on .

Example 3.2.

Consider the second-order delay dynamic equations on time scales:

(3.4)

where

(3.5)

with , for and . Then,

(3.6)

let , and by Theorem 2.7, we have

(3.7)

Then (3.4) is oscillatory on .