Research Article | Open Access
Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals
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.
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  and the seminal book by Bohner and Peterson  for a comprehensive treatment of the subject.
Recently, Saker and O’Regan  studied the the quasi-linear equation of the form where , is an odd positive integer.
Wu et al.  obtained several oscillation criteria for the equation with , is a quotient of odd positive integers.
Chen  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.
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 ). Suppose that () holds. Let . If exists for all sufficiently large , then for all sufficiently large .
Lemma 2 (see ). Assume that is -differentiable and eventually positive or eventually negative, then
Lemma 3 (see ). Suppose that and are nonnegative, then where equality holds if and only if .
Lemma 4 (see ). Let and satisfy , on , and Then,
Lemma 5 (see ). 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 , wherethen (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 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.
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.
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