Abstract
By using a generalized arithmetic-geometric mean inequality on time scales, we study the forced oscillation of second-order dynamic equations with nonlinearities given by Riemann-Stieltjes integrals of the form , where , is a time scale which is unbounded from above; ; is a strictly increasing right-dense continuous function; , , , and are right-dense continuous functions; is strictly increasing. Some interval oscillation criteria are established in both the cases of delayed and advanced arguments. As a special case, the work in this paper unifies and improves many existing results in the literature for equations with a finite number of nonlinear terms.
1. Introduction
Following Hilger’s landmark paper [1], there have been plenty of references focused on the theory of time scales in order to unify continuous and discrete analysis, where a time scale is an arbitrary nonempty 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. The oscillation theory has been developed very rapidly since the discovery of time scale calculus with this understanding. Throughout this paper, a knowledge and understanding of time scale calculus is assumed. For an introduction to time scale calculus and dynamic equations, we refer to the seminal books by Bohner and Peterson [2, 3].
In this paper, we consider the following second-order dynamic equation with the nonlinearity given by a Riemann-Stieltjes integral of the form where , , is a time scale (a closed nonempty subset of real numbers) which is unbounded from above; ; , , is another time scale; is a strictly increasing right-dense continuous function satisfying , , are right-dense continuous with ; is right-dense continuous; , are right-dense continuous functions satisfying ; is strictly increasing. Here denotes the Riemann-Stieltjes integral of the function on with respect to , is the forward jump operator.
We restrict our attention to those solutions of (1.1) which exist on the time scale half-line , where may depend on the particular solution, a nontrivial function in any neighborhood of infinity. As usual, such a solution of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Equation (1.1) is said to be oscillatory if every proper solution is oscillatory.
Recently, people have been interested in the combined effects of linear, superlinear, sublinear terms, and a forced term in oscillation. For instance, Sun and Wong [4] investigated the following forced differential equation with mixed nonlinearities where , and . The authors obtained interval oscillation criteria for (1.2) by using an arithmetic-geometric inequality and employing arguments developed earlier in [5–9]. Sun and Meng [10] studied (1.2) again by making use of some of the arguments developed by Kong [11]. In [12], Agarwal and Zafer extended the results in [4] to dynamic equations on time scales of the form where ,
Very recently, Agarwal et al. [13] further to extend the results in [12] to the case of several delays of the form where and are right-dense continuous functions satisfying for . Sun and Kong [14] studied the oscillation of the second-order forced differential equation with the nonlinearity given by a Riemann-Stieltjes integral of the form where is a strictly increasing function satisfying . Some interval oscillation criteria of the El-Sayed type and the Kong type are established which unify many existing results in the literature.
It is obvious that (1.2), (1.3), (1.5), and (1.6) are special cases of (1.1). Some other particular cased of (1.1) can be found in [15–20]. In this paper, we will establish interval oscillation criteria for the more general (1.1). Clearly, our work is of significance because (1.1) allows an infinite number of nonlinear terms and even a continuum of nonlinearities determined by the function . Moreover, even for the special cases of (1.2), (1.3), (1.5), and (1.6), our results generalize many existing oscillation criteria in the literature.
This paper is organized as follows. We present some lemmas in Section 2 which play a key role in the proof of the main results. The main results are given in Section 3. Two examples are given to illustrate the main results in Section 4.
2. Preliminaries
We here present four lemmas which play a key role in the proof of the main results in the next section. In the sequel, we denote by the set of Riemann-Stieltjes integrable functions on with respect to . Assume that , . Let .
We first present the following two Lemmas 2.1, and 2.2, which generalize Lemma 2.1 and Lemma 3.1 in [14].
Lemma 2.1. Let Then for any , there exists such that on ,
Proof. By the choice of and the definitions of and , we have that . Set It is easy to see that and Moreover, Let Then we have that By the continuous dependence of on , there exists such that satisfies . Note that on and . This completes the proof of Lemma 2.1.
The next lemma is a generalized arithmetic-geometric mean inequality on time scales.
Lemma 2.2. Assume that is right-dense continuous, , , on and . Then
Proof. Define an operator as follows: It is obvious that is a linear operator satisfying that and . To derive inequality (2.9) it suffices to show that Note that for . Thus, for any we have which follows that Taking the operator on both sides of (2.13), we get which implies (2.11). This completes the proof.
The following two lemmas generalize Lemma 2.4 and Lemma 6.1 in [13].
Lemma 2.3. Let be a right-dense continuous function satisfying , with , and . Assume is a positive right-dense continuous function such that is nonincreasing on . Then where .
Proof. Set . It is not difficult to verify that is nonincreasing on since is nonincreasing on . Then we have Next, for , and we define Then (2.16) yields that for and . Consequently, for , we have This implies (2.15). The proof of Lemma 2.3 is complete.
Similar to the proof of Lemma 2.3, we can get the following result.
Lemma 2.4. Let be a right-dense continuous function satisfying , with , and . If is a positive right-dense continuous function for which is nonincreasing on , then where is defined as in Lemma 2.3.
3. Main Results
We note from the definition of and that . In the following we will use the values of in the interval to establish interval criteria for oscillation of (1.1). For with , we define the function class , where denotes the set of right-dense continuously —differentiable functions on . In the following, let and be defined as in Lemmas 2.3 and 2.4. Set
Theorem 3.1. Assume that for and . Suppose also that for any , there exist subintervals of , , such that , , and where . For each , let be defined as in Lemma 2.1. If there exists for such that where Here we use the convention that , , and and for due to the fact that . Then (1.1) is oscillatory.
Proof. We prove this result by the contradiction method. Assume the contrary. Then (1.1) has an extendible solution which is eventually positive or negative. Without loss of generality, we may assume that for all . When is eventually negative, the proof is in the same way except that the interval , instead of , is used. Define
It follows that
It is obvious that the conditions in Lemma 2.3 are satisfied with replaced by . By (2.15) we have for
By (3.2), (3.6), and (3.7), and the fact that is increasing, we get
We first consider the case where the supremum in (3.3) is assumed at . From (3.2) and (3.8) we have that for
Let be defined as in Lemma 2.1 with . Then satisfies (2.2) and (2.3) with . This follows that
Therefore, by Lemma 2.2 we have that for
Substituting (3.11) into (3.9) we obtain
where is defined by (3.4) with and . Multiplying both sides of the above inequality by and proceeding as in the proof of Theorem 3.1 in [13], we can get a contradiction with (3.3).
Now we consider the case where the supremum in (3.3) is assumed at . Let . Then from (2.2) and (2.3), we get
Hence for
where
On the other hand, by the basic arithmetic-geometric mean inequality, we have that
Substituting (3.16) into (3.14), using Lemma 2.2 and similar to the computation in (I), for we can get
which also implies (3.12) for . The rest of the proof is similar to Part (I) and hence is omitted. This completes the proof of Theorem 3.1.
For the case when for and , using Lemma 2.4 and following the proof of Theorem 3.1, we have the following oscillation result for (1.1) immediately.
Theorem 3.2. Assume that for and . Suppose also that for any , there exist subintervals of , , such that , , and (3.2) holds for and , where . For each , let be defined as in Lemma 2.1. If there exists for such that (3.4) holds, where Then (1.1) is oscillatory.
Remark 3.3. We see from the proof of Lemma 2.1 in Section 2 that for each , the function can be constructed explicitly for any nondecreasing function , and hence the functions in Theorems 3.1 and 3.2 are explicitly given.
Remark 3.4. We observe that in Theorems 3.1 and 3.2, if the supremum in (3.3) is assumed at , the effect of is neglected in some extent. This implies that the magnitude of in cannot be large. For otherwise, the supremum would have been taken at some .
Now, we interpret the results for (1.1) to the special case of (1.5). Set , , for , and
Then (1.1) reduces to (1.5). By a straightforward computation, we have that
Then Lemma 2.1 can be restated as the following: for any , there exists a positive -tuple satisfying
Therefore, by Theorems 3.1 and 3.2, we obtain the following oscillation results for (1.5) which generalize the results in [13].
Corollary 3.5. Assume that for and . Suppose also that for any , there exist subintervals of , , such that , , and where , and . For each , let be defined by (3.21). We further assume that there exists a function such that (3.4) holds, where Then (1.5) is oscillatory.
Corollary 3.6. Assume that for and . Suppose also that for any , there exist subintervals of , , such that , , and where , and . For each , let be defined by (3.21). We further assume there exists a function such that (3.4) holds, where Then (1.5) is oscillatory.
Remark 3.7. Corollaries 3.5 and 3.6 generalize those results in [13] since the sufficient condition for oscillation of (1.5) is given here in the form of .
4. Examples
We will give two examples to illustrate Theorems 3.1 and 3.2 in the case when , , and .
Example 4.1. Consider on the following differential equation
where , are constants, is an arbitrary nonnegative function. Here we have , , , , , , , , and . For any , we choose large enough so that and let , , and . Then it is easy to verify that (3.2) holds, and
A straightforward computation yields that and . By Lemma 2.1, for any , there exists a positive Riemann integrable function on such that (2.2) and (2.3) hold. Particularly, we can choose and hence . If we choose for , then we have
If there exist positive constants and such that
then condition (3.3) holds. By Theorem 3.1, we have that (4.1) is oscillatory.
Example 4.2. Consider on the following difference equation where , for and are positive constants, () are arbitrary real-valued functions with and . For any , we can choose large enough so that . Let , , , and . Then (3.2) is valid. Choose and for . Then (2.2) and (2.3) hold with . By the straightforward computation, we get By Theorem 3.2, (4.5) is oscillatory if .
Acknowledgments
The author thanks the reviewer for his/her valuable comments on this paper. This paper was supported by the national natural science foundations of China (60704039, 61174217) and the natural science foundations of Shandong Province (ZR2010AL002, JQ201119).