Research Article | Open Access

# Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

**Academic Editor:**Shengqiang Liu

#### Abstract

We establish several oscillation criteria for a class of second-order neutral delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals. Our results extend and unify a number of other existing results and handle the cases which are not covered by known criteria. The new results we obtain are of significance because the equations we study allow an infinite number of nonlinear terms and even a continuum of nonlinearities.

#### 1. Introduction

In this paper, we consider the oscillatory behavior of solutions of the following second-order neutral delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals: where , , is a time scale which is unbounded above, , is a constant, and the following conditions are satisfied:, is another time scale, denotes the collection of all functions which are right-dense continuous on ;, , , , is a strictly increasing and satisfying , ;, , for , , , , for , , , for any ; is right-dense continuous on , and for all , where is the forward jump operator on ; is a continuous function such that , for all and there exists a positive right-dense continuous function defined on such that for all and for all ; is strictly increasing; denotes the Riemann-Stieltjes integral of the function on with respect to .

By a solution of (1), we mean a function such that and , and satisfying (1) for all , where denotes the set of right-dense continuously -differentiable functions on . In the sequel, we will restrict our attention to those solutions of (1) which exist on the half-line and satisfy for any . A nontrivial solution of (1) is called oscillatory if it has arbitrary large zeros; otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of neutral functional equations on time scales, and we refer the reader to the papers [1–10] and the references cited therein. For an introduction to time scale calculus and dynamic equations, we refer the reader to the landmark paper of Hilger [11] and the seminal book by Bohner and Peterson [12] for a comprehensive treatment of the subject.

Recently, Saker and O’Regan [13] studied the the quasi-linear equation of the form where , is an odd positive integer.

Wu et al. [14] obtained several oscillation criteria for the equation with , is a quotient of odd positive integers.

Chen [15] investigated the following second-order Emden-Fowler neutral delay dynamic equation with , , is a constant.

It is obvious that (2)–(4) are special cases of (1). In the present paper, we will establish several oscillation criteria for the more general (1), which is of significance because (1) allows an infinite number of nonlinear terms and even a continuum of nonlinearities determined by the function . Our results extend and unify a number of other existing results and handle the cases which are not covered by known criteria. Finally, two examples are demonstrated to illustrate the efficiency of our work.

#### 2. Preliminaries

In the sequel, we denote by the set of Riemann-Stieltjes integrable functions on with respect to , and we use the convention that , .

Lemma 1 (see [15]). *Suppose that () holds. Let . If exists for all sufficiently large , then for all sufficiently large .*

Lemma 2 (see [12]). *Assume that is -differentiable and eventually positive or eventually negative, then
*

Lemma 3 (see [16]). *Suppose that and are nonnegative, then
**
where equality holds if and only if .*

Lemma 4 (see [17]). *Let and satisfy , on , and
**
Then,
*

Lemma 5 (see [17]). *Let satisfying , and . Assume such that is nonincreasing on , where , is a constant. Then,
*

#### 3. Main Results

Theorem 6. *Assume that hold. If there exist a function and a function such that on ,
**
where**then (1) is oscillatory.*

*Proof. *Suppose that (1) has a nonoscillatory solution , then there exists such that for all . Without loss of generality, we assume that , , , and for , , because a similar analysis holds for , , , and . Then, from (1), , and , we get
Therefore, is a nonincreasing function, and is eventually of one sign.

We claim that
Otherwise, if there exists a such that for , then, from (14), there exists some positive constant such that
that is,
and integrating the above inequality from to , we have
Letting , from (), we get , which contradicts (14). Thus, we have proved (15).

We choose some such that for . Therefore, from (14), (15), and the fact , we have that
which follows that
On the other hand, from (1), , and (15), we obtain
Notice (15) and the fact , we get
where .

Define
Obviously, . From (22), (23), it follows that
Now, we consider the following two cases.

In the first case, . By (15), , and Lemmas 1 and 2, we have
From , (20), (23)–(25), and the fact that is nondecreasing, we obtain

In the second case, . By (15), , and Lemmas 1 and 2, we get
From , (20), (23)-(24), (27), and the fact that is nondecreasing, we haveTherefore, for , from (26) and (28), we get
On the other hand, it is obvious that the conditions in Lemma 5 are satisfied with , , replaced by , , and , respectively. So, we have
in the view of , we get
From (29) and (31), we obtain
By (10) and (11), we have
Therefore, by Lemma 4 and (33), we have that for Substituting (34) into (32), we obtain
where and are defined by (13).

Taking
by Lemma 3 and (35), we obtain
Integrating above inequality (37) from to , we have
Since for , we have
which contradicts (12). This completes the proof of Theorem 6.

*Remark 7. *If we take and use the convention that , , then Theorems 6 reduces to [15, Theorems 3.1]. If furthermore is a quotient of odd positive integer, then Theorem 6 reduces to [14, Theorem 3.1].

*Remark 8. *The function satisfying (10) and (11) in Theorem 6 can be constructed explicitly for any nondecreasing function . In fact, if we assume that , and let ,
It is easy to see that and

Moreover,
Let
Then, we obtain that
By the continuous dependence of on , there exists such that satisfies

*Remark 9. *Set , , for , and ;
, satisfying ; , ; , ;Then, (1) reduces to
So, if we take for some peculiar cases in Theorem 6, we can obtain various results. For example, if we take , , , and in (46), then Theorem 6 generalizes the results by [18, Theorem 2].

Next, we use the general weighted functions from the class which will be extensively used in the sequel.

Let , we say that a continuous function belongs to the class if(i) for and for where ;(ii) has a nonpositive right-dense continuous -partial derivative with respect to the second variable.

Theorem 10. *Assume that hold. If there exist functions , , such that on , (10) and (11) hold, and
**
where
** and are defined as in Theorem 6, then (1) is oscillatory.*

*Proof. *We proceed as in the proof of Theorem 6 to have (35). From (35), we obtain
Multiplying (50) (with replaced by ) by , integrating it with respect to from to for , and using integration by parts and (i)-(ii), we get
where is defined as in (49).

Taking
by Lemma 3 and (51), we obtain
where is defined as in (48).

Then, it follows that
Thus, from (54), we get

Therefore,
which contradicts (47). This completes the proof of Theorem 10.

*Remark 11. *In the literature, there are so many results for second-order nonlinear neutral functional dynamic equation; however, to the best of our knowledge, there is no work done attempting to study the neutral functional dynamic equation with an infinite number of nonlinear terms. Hence, our paper seems to be the first one dealing with this untouched problem. Our results not only unify the existing results in the literature, but also extend the existing results to a wider class of dynamic equations.

#### 4. Examples

*Example 1. *Consider the following dynamic equation:
In (57), , , , , , , , , and is a constant.

If and , , , and , where is an arbitrary positive integer, then . We can choose , . Then, it is easy to get that , and therefore,
Hence, by Theorem 6, (57) is oscillatory.

*Example 2. *Consider on the following differential equation:
In (59), , , , , , are constants, , , , , , , , and .

We can choose , . Then, it is easy to get that , , and therefore, from Theorem 6,
Hence, by Theorem 6, (59) is oscillatory if