Oscillation of Second-Order Mixed-Nonlinear Delay Dynamic Equations

Abstract

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

(1.1)

on an arbitrary time scale , where

(1.2)

the functions , , : are right-dense continuous with nondecreasing; the delay functions are nondecreasing right-dense continuous and satisfy for with as .

We assume that the time scale is unbounded above, that is, and define the time scale interval by . 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].

By a solution of (1.1) we mean a nontrivial real valued function such that and for all with , and that satisfies (1.1). A function is called an oscillatory solution of (1.1) if 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 of (1.1) is oscillatory.

Notice that when , (1.1) is reduced to the second-order nonlinear delay differential equation

(1.3)

while when , it becomes a delay difference equation

(1.4)

Another useful time scale is and is a real number, which leads to the quantum calculus. In this case, (1.1) is the -difference equation

(1.5)

where , , and .

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 . As far as we know when , 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

(1.6)

where the potential and the forcing term 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.

2. Main Results

We need the following lemmas in proving our results. The first two lemmas can be found in [25, Lemma ].

Lemma 2.1.

Let , be the tuple satisfying . Then, there exists an tuple satisfying

(2.1)

Lemma 2.2.

Let , be the tuple satisfying . Then there exists an tuple satisfying

(2.2)

The next two lemmas are quite elementary via differential calculus; see [23, 25].

Lemma 2.3.

Let , and be nonnegative real numbers. Then

(2.3)

Lemma 2.4.

Let , and be nonnegative real numbers. Then

(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 be a nondecreasing right-dense continuous function with , and with . If is a positive function such that is nonincreasing on with nondecreasing, then

(2.5)

Proof.

By the Mean Value Theorem [2, Theorem ]

(2.6)

for some , for any . Since is nonincreasing and is nondecreasing, we have

(2.7)

and so , . Now

(2.8)

Define

(2.9)

It follows from (2.8) that for and . Thus, we have

(2.10)

which completes the proof.

In what follows we say that a function belongs to if and only if is right-dense continuous function on having continuous -partial derivatives on , with for all and for all . Note that in case , the -partial derivatives become the usual partial derivatives of . The partial derivatives for the cases and will be explicitly given later.

Denoting the -partial derivatives and of with respect to and by and , respectively, the theorems below extend the results obtained in [5] to nonlinear delay dynamic equation on arbitrary time scales and coincide with them when is replaced by . Indeed, if we set , then it follows that

(2.11)

When , they become

(2.12)

as in [5]. However, we prefer using instead of for simplicity.

Theorem 2.6.

Suppose that for any given (arbitrarily large) there exist subintervals and of , where and such that

(2.13)

where

(2.14)

hold. Let be an tuple satisfying (2.1) of Lemma 2.1. If there exist a function and numbers such that

(2.15)

for , where

(2.16)

then (1.1) is oscillatory.

Proof.

Suppose on the contrary that is a nonoscillatory solution of (1.1). First assume that and are positive for all for some . Choose sufficiently large so that . Let .

Define

(2.17)

Using the delta quotient rule, we have

(2.18)

Notice that

(2.19)

which implies

(2.20)

Hence, we obtain

(2.21)

Substituting (2.21) into (1.1) yields

(2.22)

By assumption, we can choose such that () and for all , where is defined as in (2.14). Clearly, the conditions of Lemma 2.5 are satisfied when, replaced with for each fixed . Therefore, from (2.5), we have

(2.23)

and taking into account (2.22) yields

(2.24)

Denote

(2.25)

From (2.24), we have

(2.26)

Now recall the well-known arithmetic-geometric mean inequality, see [26],

(2.27)

where and , . Setting

(2.28)

in (2.26) yields

(2.29)

From (2.29) and taking into account (2.27), we get

(2.30)

and hence,

(2.31)

which yields

(2.32)

where

(2.33)

Multiplying both sides of (2.32) by and integrating both sides of the resulting inequality from to yield

(2.34)

Fix and note that

(2.35)

from which we obtain

(2.36)

Therefore,

(2.37)

Notice that

(2.38)

since and hence, we obtain from (2.34) that

(2.39)

On the other hand,

(2.40)

Taking into account that , we have

(2.41)

Using this inequality in (2.39), we have

(2.42)

Similarly, by following the above calculation step by step, that is, multiplying both sides of (2.32) this time by after taking into account that

(2.43)

one can easily obtain

(2.44)

Adding up (2.42) and (2.44), we obtain

(2.45)

This contradiction completes the proof when is eventually positive. The proof when is eventually negative is analogous by repeating the above arguments on the interval instead of .

Corollary 2.7.

Suppose that for any given (arbitrarily large) there exist subintervals and of such that

(2.46)

where holds. Let be an tuple satisfying (2.1) of Lemma 2.1. If there exist a function and numbers such that

(2.47)

for , where

(2.48)

then (1.3) is oscillatory.

Corollary 2.8.

Suppose that for any given (arbitrarily large) there exist with and such that for each ,

(2.49)

where holds. Let be an tuple satisfying (2.1) of Lemma 2.1. If there exist a function and numbers such that

(2.50)

for , where

(2.51)

then (1.4) is oscillatory.

Corollary 2.9.

Suppose that for any given (arbitrarily large) there exist with and such that for each ,

(2.52)

where holds. Let be an tuple satisfying (2.1) of Lemma 2.1. If there exist a function and numbers such that

(2.53)

for , where

(2.54)

then (1.5) is oscillatory.

Notice that Theorem 2.6 does not apply if there is no forcing term, that is, . In this case we have the following theorem.

Theorem 2.10.

Suppose that for any given (arbitrarily large) there exists a subinterval of , where such that

(2.55)

where holds. Let be an tuple satisfying (2.2) in Lemma 2.2. If there exist a function and a number such that

(2.56)

where

(2.57)

then (1.1) with 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 and in proof of Theorem 2.6, we arrive at

(2.58)

The arithmetic-geometric mean inequality we now need is

(2.59)

where and , are as in Lemma 2.2.

Corollary 2.11.

Suppose that for any given (arbitrarily large) there exists a subinterval of , where with such that

(2.60)

where holds. Let be an tuple satisfying (2.2) in Lemma 2.2. If there exist a function and a number such that

(2.61)

where

(2.62)

then (1.3) with is oscillatory.

Corollary 2.12.

Suppose that for any given (arbitrarily large) there exists with such that

(2.63)

where holds. Let be an tuple satisfying (2.2) in Lemma 2.2. If there exist a function and a number such that

(2.64)

where

(2.65)

then (1.4) with is oscillatory.

Corollary 2.13.

Suppose that for any given (arbitrarily large) there exist with such that

(2.66)

where holds. Let be an tuple satisfying (2.2) in Lemma 2.2. If there exist a function and a number such that

(2.67)

where

(2.68)

then (1.5) with is oscillatory.

It is obvious that Theorem 2.6 is not applicable if the functions are nonpositive for . In this case the theorem below is valid.

Theorem 2.14.

Suppose that for any given (arbitrarily large) there exist subintervals and of , where and such that

(2.69)

where holds. If there exist a function , positive numbers and satisfying

(2.70)

and numbers such that

(2.71)

for where

(2.72)

with

(2.73)

then (1.1) is oscillatory.

Proof.

Suppose that (1.1) has a nonoscillatory solution. Without losss of generality, we may assume that and are eventually positive on when is sufficiently large. If is eventually negative, one may repeat the same proof step by step on the interval

Rewriting (1.1) for as

(2.74)

and applying Lemma 2.3 to each term in the first sum, we obtain

(2.75)

where for . Setting

(2.76)

yields

(2.77)

Substituting the above last equality into (2.75), we have

(2.78)

It follows from (2.5) that

(2.79)

(2.80)

(2.81)

Notice that the second sum in (2.78) can be written as

(2.82)

and hence applying the Lemma 2.4 yields

(2.83)

where and for Using (2.79), (2.80), and (2.78) into (2.78), we obtain

(2.84)

Setting

(2.85)

we have

(2.86)

The rest of the proof is the same as that of Theorem 2.6 and hence it is omitted.

Corollary 2.15.

Suppose that for any given (arbitrarily large) there exist subintervals and of , where and such that

(2.87)

where holds. If there exist a function , positive numbers and satisfying

(2.88)

and numbers such that

(2.89)

for where

(2.90)

with

(2.91)

then (1.3) is oscillatory.

Corollary 2.16.

Suppose that for any given (arbitrarily large) there exist with and such that for each ,

(2.92)

where holds. If there exist a function , positive numbers and satisfying

(2.93)

and numbers such that

(2.94)

for , where

(2.95)

with

(2.96)

then (1.4) is oscillatory.

Corollary 2.17.

Suppose that for any given (arbitrarily large) there exist with and such that for each ,

(2.97)

where holds. If there exist a function , positive numbers and satisfying

(2.98)

and numbers such that

(2.99)

for , where

(2.100)

with

(2.101)

then (1.5) is oscillatory.

3. Examples

In this section we give three examples when , and , in (1.1). That is, we consider

(3.1)

For simplicity we take , thus . Note that and by Lemma 2.2.

Example 3.1.

Let and be constants. Consider the differential equation

(3.2)

Let , , and , .

We calculate

(3.3)

and see that (2.61) holds if

(3.4)

Since all conditions of Corollary 2.11 are satisfied, we conclude that (3.2) is oscillatory when (3.4) holds.

Example 3.2.

Let and be constants. Define , , and for , , ; otherwise, the functions are defined arbitrarily. Consider the difference equation

(3.5)

Let , , and . We derive

(3.6)

and see that positivity in (2.64) satisfies if

(3.7)

Since all conditions of Corollary 2.12 are satisfied, we conclude that (3.5) is oscillatory if (3.7) holds.

Example 3.3.

Let and be constants. Define , and for , , ; otherwise, the functions are defined arbitrarily. Consider the -difference equation, (),

(3.8)

Let , , and . We have

(3.9)

We see that (2.67) holds for all and . Since all conditions of Corollary 2.12 are satisfied, we conclude that (3.8) is oscillatory if and are positive.