Journal of Function Spaces

Journal of Function Spaces / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 5597947 | https://doi.org/10.1155/2021/5597947

Dandan Yang, Chuanzhi Bai, "On the Oscillation Criteria for Fourth-Order -Laplacian Differential Equations with Middle Term", Journal of Function Spaces, vol. 2021, Article ID 5597947, 10 pages, 2021. https://doi.org/10.1155/2021/5597947

On the Oscillation Criteria for Fourth-Order -Laplacian Differential Equations with Middle Term

Academic Editor: Maria Alessandra Ragusa
Received18 Feb 2021
Revised26 Feb 2021
Accepted05 Apr 2021
Published20 Apr 2021

Abstract

In this paper, we study the oscillatory properties of the solutions of a class of fourth-order -Laplacian differential equations with middle term. The new oscillation criteria obtained by using the theory of comparison with first- and second-order differential equations and a refinement of the Riccati transformations. The results in this paper improve and generalize the corresponding results in the literatures. Three examples are provided to illustrate our results.

1. Introduction

In this paper, we are concerned with the oscillation behavior of solutions of the fourth-order -Laplacian differential equations with middle term where , , are real numbers, , , , , , , , , , , , and . Furthermore, we investigate (1) under the condition.

A function , , is called a solution of equation (1), if , and satisfies (1) on .

Since the fourth-order delay differential equations described many real-life applications, such as models related to physical, chemical, and biological phenomena, in the last decades, a lot of research has been done on the oscillatory behavior of fourth-order delay differential equations, see [114] and the references cited therein. On the other hand, the -Laplacian differential equations there are some important applications in continuum mechanics and elasticity theory [1523], the oscillatory behavior of the solutions of fourth-order differential equations with -Laplacian like operator have been investigated in recent years by using different methods and various techniques, for example, [2426].

In this paper, motivated by [2628], we will give some new sufficient conditions for that oscillatory behavior of (1). In Section 2, we will provide some lemmas that will help us to prove our main results. In Section 3, based on the comparison with first- and second-order differential equations and a refinement of the Riccati transformations, we establish some new oscillation criteria of (1).

2. Preliminaries

First, we give the following lemmas that can discuss our main results.

Lemma 1 ([1], Lemma 2.21). Let be a positive and -times differentiable function on an with its th derivative nonpositive on and not identically zero on any interval of the form , . Then, there exists an integer , , with odd such that for some large ,

Lemma 2 ([29]). Let the function satisfies , , and eventually. Then,

Lemma 3 ([30], Lemma 2.2.3). Let and . Assume that is of a fixed sign, on , not identically zero and that there exists a such that, for all , . If we have , then there exists such that for every and .

Lemma 4 ([31], Lemma 2.3). Assume that is a quotient of odd positive integers; and are constants. Then

The following lemma will be used in the proof of our main results in the next section.

Lemma 5. Let be an eventually positive solution of (1); if () holds, then and .

Proof. Since is an eventually positive solution of (1), then , , and for . Thus, , . From (1), we obtain Multiplying by on both sides of the above equation, we get Thus is decreasing, and hence, is eventually of one sign. Hence, we assert that for any . Otherwise, if for any , we get by (8) that where . From (10), one has that is, Consequently, So is an eventually negative function which contradicts . Thus, we have . From (1), we get from which it follows that By (15) and Lemma 1 (set and ), one has that , . The proof is completed.

Lemma 6 ([32]). Let be a quotient of two positive integers. Assume that is a positive continuous function on . If then the first-order delay differential equation is oscillatory.

3. Main Results

In the following theorem, we then by using a comparison strategy involving first-order differential equations to provide an oscillation criterion for equation (1).

For convenience, let

Theorem 7. Assume that () and hold. If the differential equation is oscillatory for some , then the differential equation (1) is oscillatory.

Proof. Assume that (1) has a nonoscillatory solution in . Without loss of generality, we may let be an eventually positive solution of (1). Then, there exists a such that , , and for . Let which having in mind (1) gives From the definition of , one has By repeating the same process, we have Set in Lemma 2, we obtain , which implies that is nonincreasing. Moreover, by the fact that gives Combining (24) and (25), which yields Between equations (1) and (26), we obtain Since is positive and increasing (by Lemma 5), we have . So, from Lemma 3, one has for some . It follows between (27) and (28) that, for all , is a positive solution of the first-order delay differential inequality It is well known (see [33] and Theorem 7) that the corresponding equation (20) also has a positive solution, which is a contradiction. The theorem is proved.

Corollary 8. Assume that () and () hold, and . If for some , then the differential equation (1) is oscillatory.

Proof. From Lemma 6, we know that (30) implies the oscillatory of (20).

Theorem 9. Assume that () and () , hold. If the differential equation is oscillatory, then every solution of equation (1) is oscillatory.

Proof. Assume the contrary that is an eventually positive solution of equation (1). Thus, we may suppose that , , and are positive for all that are sufficiently large. From and (by Lemma 5), we obtain by and that Combining (21) and (32), one has By using (28) and (33), we get that From (21), (34) implies that It is well known (see [33] and Theorem 7) that the corresponding equation (31) also has a positive solution, which is a contradiction. The theorem is proved.

Corollary 10. Assume that () and () hold, and . If for some , then the differential equation (1) is oscillatory.

Proof. From Lemma 6, we know that (36) implies the oscillatory of (31).

Lemma 11. Assume that () holds, is an eventually positive solution of (1), and for some constants . Then, we have .

Proof. Our proof by reduction to the absurd. Assume that . From Lemma 2, we obtain Integrating the above equality from to , one find that Let in Lemma 3, then for all and every sufficiently large . Now, we define a function by By differentiating (41) and using the inequalities (39) and (40), we get Since , there exist a and a constant such that , for all . Without loss of generality, we may let . By using Lemma 4 with we obtain This implies that which contradicts (37). The proof is completed.

For convenience, let

Theorem 12. Assume that () and (37) hold for some and . If is oscillatory, then (1) is also oscillatory.

Proof. We use the reduction to the absurd arguments. Assume that (1) has a nonoscillatory solution in . Without loss of generality, we only need to be concerned with positive solutions of equation (1). Then, there exists a such that , , and for . From Lemmas 4 and 11, one has that for , where is sufficiently large. Integrating (8) from to , we obtain By using Lemma 3 in [26] together with (48), we have for all . This coupled with (49); we can arrive at Since , then (51) reduced to the following inequality Taking in (52), one has from which we readily obtain Integrating the above inequality from to , we obtain which implies that Now, if we define , then for , and By using (56) and the definition of , we see that Since , there exists a constant such that , for all , where is sufficiently large. Without loss of generality, we may let . Then, by (56), one has It is well known (see [34]) that the differential equation (47) is oscillatory if and only if there exists a such that (59) holds, which is a contradiction. The theorem is proved.

For convenience, let

Theorem 13. Assume that () and (37) hold for some , , and . If is oscillatory, then (1) is also oscillatory.

Proof. As in the proof of Theorem 12, one has (49). Hence, it follows between and that By (62), similar to (56), we obtain We now define by then for , and By using (63) and the definition of , similar to (59), we see that It is well known (see [34]) that the differential equation in (61) is nonoscillatory if and only if there exists a such that (66) holds, which is a contradiction. So, the theorem is proved.

In the following, we employ the integral averaging technique to establish a Philos-type oscillation criterion for (1).

Let

Corollary 14. Assume that () and (37) hold for some and . Let there exists a continuous function such that for ,, , and has a nonpositive continuous partial derivative with respect to the second variable in . Suppose that there exists function such that If or then (1) is oscillatory.

Example 15. Consider the equation where , , , , and . Let , , , , , , and . It is easy to check that () and () are satisfied. Moreover, we have If satisfies inequality (30), that is then equation (71) is oscillatory by Corollary 8.

Example 16. Consider the equation where , , and . Let , , , , , , </