Set of Oscillation Criteria for Second Order Nonlinear Forced Differential Equations with Damping
By employing a generalized Riccati technique and an integral averaging technique, some new oscillation criteria are established for the second order nonlinear forced differential equation with damping. These results extend, improve, and unify some known oscillation criteria in the existing literature.
The oscillatory problem for second order nonlinear forced differential equation with damping is concerned, where and and H is a continuous function on .
Throughout this paper we will also suppose that there are positive constants , and satisfying the following:), (), (), (), (A5) for all ,() is continuous function such that and ,().
We will consider only nontrivial solutions of (1) which are defined for all large . A solution of (1) is said to be oscillatory if it has a sequence of zeros clustering at and nonoscillatory otherwise. Equation (1) is said to be oscillatory if all its solutions are oscillatory.
In the late 19th century, some scholars focus on sufficient conditions for the oscillation theorems of different classes of differential equations with damping. We refer to the new published papers [1–11]. The oscillatory theory of second order nonlinear differential equations has been widely applied in research of lossless high-speed computer network and physical sciences.
Recently, the oscillatory behavior for various particular cases of (1), such as the nonlinear differential equations has been studied extensively by numerous authors with different methods; see, for example, [9–11] and the references quoted therein.
In this paper, by using a generalized Riccati and integral averaging technique, several new oscillation criteria for (1) are established.
A significant drawback of many oscillation results for differential equations with damping reported in the literature is a necessity to impose a variety of additional restrictions on the sign of the damping term . We emphasize that our theorems are free of particular restrictions on .
2. Main Results
For convenience, we introduce the class of the function . Let . A function is said to belong to the class , if(1) for and for ,(2) has continuous and nonpositive partial derivatives on with respect to the second variable,(3)there exists a function such that .
In this section, several oscillation conditions for (1) are established under the assumptions .
Theorem 1. Let assumptions be fulfilled and . If there exist functions and such that and where then (1) is oscillatory.
Proof. Let be a nonoscillatory solution of (1). Then there exists a such that for all . Without loss of generality, we may assume that on interval . A similar argument holds also for the case when is eventually negative. Defining a generalized Riccati transformation by for all , then differentiating Equation (5), and using (1) and (A1)–(A6), it follows for all with defined as above. Then we obtain Multiplying both sides of (7) by , integrating it with respect to from to , and using the properties of the function , we get, for all , Therefore, for all , Applying inequality (9), for , yields It follows that which contradicts assumption (3), so (1) is oscillatory.
Example 3. Consider the nonlinear damped differential equation where and . Since , , and the assumptions (A1)–(A6) hold. If we take and , then and . A direct computation yields that the conditions of Theorem 1 are satisfied; Example 3 is oscillatory.
Theorem 4. Let assumptions (A1)–(A6) be fulfilled and . Suppose that If there exist functions and such that and and for any where and are the same as defined in Theorem 1, and then (1) is oscillatory.
Proof. Let be a nonoscillatory solution of (1). Then there exists a such that for all . Without loss of generality, we may assume that on interval . A similar argument holds also for the case when is eventually negative.
Define the function as in (5). Similar to the proof of Theorem 1, we obtain inequality (9). Further, it follows for and therefore Thus, by (18), we get for all . This implies that Define for all . Then In order to show that suppose that By (15), there exists a positive constant such that On the other hand, according to (27) for any positive constant there exists a such that For , By (28) we can easily see that Then there exists such that for all . Therefore, by (30), for all , and since is an arbitrary constant, we can make a conclusion that Next, let us consider a sequence in with and such that Now, by (25), there exists a constant such that and hence (32) leads to By taking into account (32), from (34), we derive where . Thus The above inequality and (35) imply that Further, by Schwarz inequality, we have, for any positive integer , and therefore It follows from (38) that Consequently, but the latter contradicts assumption (16). Hence, (27) fails to hold. Finally, by (23), we obtain This contradicts the assumption (12). Therefore, (1) is oscillatory.
Theorem 5. Let assumptions (A1)–(A6) be fulfilled and . Suppose that (15) holds. If there exist functions and , such that , and (17) holds, and and for every where and are defined as in Theorem 1, , then (1) is oscillatory.
Proof. Without loss of generality, we assume that there exists a solution of (1) such that on for some . The function is defined as in (5). Then, following the proof of Theorem 4, we have (20). Now it follows that for every . By (45), we know that (23) holds and Then, where and are defined as in the proof of Theorem 4 It follows from (44) and (45) that Then there exists a sequence in such that and Now, suppose that (27) holds. With the same argument as in Theorem 4, we conclude that (32) is satisfied. By (48), there exists a constant such that Then, similar to the proof of Theorem 4, we obtain (41) which contradicts (50), and hence (27) fails. From (23) and (26) we have which contradicts assumption (17).
Theorem 6. Let assumptions (A1)–(A6) be fulfilled and . Suppose that (15) holds. If there exist functions and , such that , and (17) and (45) hold, and where and are defined as in Theorem 1 and , then (1) is oscillatory.
Example 7. Consider the nonlinear damped differential equation
Obviously, for all and , is a constant. Since , , and , the assumptions (A1)–(A6) hold. If and , then and , for all .
A direct computation yields We conclude by Theorem 6 that all solutions of this equation are oscillatory.
Remark 8. If (5) is replaced by and (A5) by for , we can obtain similar oscillation results that are derived in the present paper.
Remark 9. If we take , , it is easy to see that Theorems 1–6 reduce to Theorems 1−4 of Wang . If we take , , and for , then (1) reduces to , and by taking , Theorems 1–5 reduce to Theorems 1–3 of Rogovchenko and Tuncay .
The following lemma will significantly simplify the proofs of next theorems. First recall class functions defined on . A function is said to belong to the class if(i) for and when ,(ii) has partial derivatives on such that for some .
Lemma 11. Let with , . If there exist and such that
for all , where
The proof of this lemma is similar to that of  and hence will be omitted.
In the next theorems we define the following functions that will be used in the proofs. Let
Theorem 12. Suppose that (A1)–(A7) hold. Assume that If there exists a continuously differentiable function such that is nonnegative and decreasing function, we have There exists an interval , and that there exists , and for any constant , such that where Then (1) is oscillatory.
Proof. Without loss of generality, we may assume that there exists a nonoscillatory solution of (1) such that on for some . The similar argument holds also for . Define the function as
Differentiating (70), using (1) and (A1)–(A7), we get
Integrating (71) from to we get that
Since , then by Bonnet’s Theorem, there exist for every such that
where is a constant. Then, we have, for ,
Three cases of the oscillatory solutions are discussed below.
Case 1. Assume that is oscillatory; then there exists a sequence such that and , on . From (74) we get
Using (67) we obtain
Then there exists a constant , such that
Using Schwarz inequality, (A7), and (77) we have
where is a positive constant.
Let ; applying (78) we have