Research Article | Open Access

# Fite-Wintner-Leighton-Type Oscillation Criteria for Second-Order Differential Equations with Nonlinear Damping

**Academic Editor:**Yuriy Rogovchenko

#### Abstract

Some new oscillation criteria for a general class of second-order differential equations with nonlinear damping are shown. Except some general structural assumptions on the coefficients and nonlinear terms, we additionally assume only one sufficient condition (of Fite-Wintner-Leighton type). It is different compared to many early published papers which use rather complex sufficient conditions. Our method contains three items: classic Riccati transformations, a pointwise comparison principle, and a blow-up principle for sub- and supersolutions of a class of the generalized Riccati differential equations associated to any nonoscillatory solution of the main equation.

#### 1. Introduction

In the paper, we develop some new oscillation criteria for the following class of second-order differential equations with nonlinear damping: where the coefficients , and the functions , are continuous in all their variables, and solution , . A function is said to be oscillatory if there is a sequence such that and as .

In Section 2, we present some basic structural assumptions on the coefficients: , , and and on the nonlinear functions: and , which are slightly more general than those of the previously published results such as in Zhao et al. [1, Theorem 2.1] (see also Theorem A, Section 2), [1, Theorems 2.2–2.8], [2, Theorem 2], [3, Theorem 2.1]. In Section 3, we study some new oscillation criteria for (1) based on an additional sufficient condition of Fite-Wintner-Leighton type, which is rather simpler than Kamenev-type conditions or related complex ones. Equation (1) in various different forms has been considered in many several published papers, see, for instance, [4–12] and references therein. In Section 4, we state and prove a pointwise comparison principle between all sub- and supersolutions of the corresponding generalized Riccati differential equation associated with every nonoscillatory solution of (1). Furthermore, under the main assumption of Fite-Wintner-Leighton type, we construct a subsolution of the Riccati differential equation which blows up in time. It together with classic Riccati transformation gives the proof of the main result.

#### 2. Main Assumptions and Remarks

In particular, for , in [1] authors firstly supposed the next five basic conditions on the coefficients , and the functions , , and : Such a set of assumptions, with slightly different (6) and , was introduced for the first time in [2], see also [3]. Just the same as in [1], besides (6) we also consider a similar assumption: And, in the case when and may change the sign, instead of (2)–(4) and (6), one considers also:

Here, assumptions (5) and (6) are generalized in the following sense, see Theorem 5—(ii) and (iii), respectively, which are weaker than (5) and (6), respectively. One of the reasons for that is presented in the next remarks.

*Remark 1. *(1)The most simple second-order differential operator which satisfies assumption (5) for is linear in variable ; that is,
where , , and is an arbitrary function satisfying . It is because for all and , see also Corollary 6. However, it is easy to check that the differential operator from (7) does not satisfy assumption (5) for every .(2)Next, we consider the corresponding second-order quasilinear differential operator:
where , , and is an arbitrary function satisfying and in order to ensure that , we take since . Unfortunately, the differential operator from (8) does not satisfy assumption (5) for every , . It is because , which is different from (5).(3)Unlike (5), the differential operator from (8) satisfies assumption , and hence, (8) is also included in our study of the oscillation of (1), see Corollary 11. (4)Although both differential operators from (7) and (8) do not satisfy assumption (5) for every , the so-called generalized prescribed mean curvature-like differential operator:
satisfies assumption (5) for every , where , , , and is an arbitrary function satisfying , see Corollary 9. (5)The simple case is involved in unlike (6), and hence, the nonlinear equation can be considered as a special case of (1).

We pay attention to the recently published paper [13] in which authors show that any generalization of the assumptions (2)–(6) should be done very carefully.

Now, we can recall [1, Theorems 2.5].

Theorem A. *Let (2)–(6) hold. Assume that there exist , , , and some such that for all **
where and are defined, respectively, by
**
and satisfies
**
Then, (1) is oscillatory.*

In Theorem A, the set . And the assumption means that , is continuous on , for all and for all . It is easy to see that the coefficients: , , and are involved in the assumptions (10)–(12), often called the general Kamenev-type conditions, about the Kamenev-type conditions and their several generalization we refer the reader, for instance, to [14–18]. The main purpose of supposing the existence of the functions: , , and satisfying the corresponding assumptions (10)–(12) is to ensure the nonexistence of continuous function which satisfies the corresponding Riccati differential inequality: where , , and depend on , , , and , and .

Instead of Kamenev-type conditions (10)–(12), we consider the next one (which can be called the Fite-Wintner-Leighton-type condition by a reason given in Remark 2): for the explicitly given two functions and which depend on the data , , , , , , and , (see Theorems 5 and 15), let there be a function and a point such that Combining a pointwise comparison principle and a blow-up argument, which is a different method than that in the case of Kamenev-type conditions, we are able to prove the nonexistence of any continuous function which satisfies the corresponding Riccati differential inequality: where , , and are arbitrary functions. On the various aspects of the comparison principles, we refer the reader, for instance, to [19, 20]—the comparison principles for Volterra integral operators, [21, 22]—the pointwise comparison principle for ODEs and [23]—the abstract form of comparison principles.

*Remark 2. * It is simple to check that in particular for , , and , the conditions (3), (5) with and still hold where the inequality “” is replaced by “.” Then (1) becomes the linear second-order differential equation : . Hence, the inequality in (14) for can be replaced by the corresponding equality, where and (see the case (iii) of Theorem 5), and so, we conclude that in this case, (14) is equivalent to:
which presents the classic Fite-Wintner-Leighton oscillation criterion for linear second-order differential equation , where “” appears instead of “.” In Fite [24], Wintner [25], and Leighton [26] equation was considered, respectively, with and , and may change sign, and arbitrary and may change sign. Nonlinear version of such a class of oscillation criteria was due to Wong [27], and th-order extension for linear equations can be found in Travis [28].

In order to simplify notation, we firstly introduce the following definition for the pointwise comparison principle of the corresponding Riccati differential equation: where , , and are three arbitrary functions, and .

*Definition 3. *Let and be two arbitrary real numbers, . Two functions, and , are said to be, respectively, subsolution and supersolution of the Riccati differential equation (17) provided that
Moreover, if the statement:
is fulfilled for all sub- and supersolutions of (17), then we say that comparison principle (19) holds for (17) with arbitrary and , .

*Remark 4. *The possibilty that (19) holds for all sub- and supersolutions and with arbitrary and , plays an essential role in some concrete situations. According to it, when the comparison principle (19) holds for the Riccati differential equation (17) with arbitrary and , , then we can choose some concrete sub- and supersolutions as well as and with some suitable properties.

Our method contains the next three steps: (i)at the first step, we give a sufficient condition on such that comparison principle (19) holds for the Riccati differential equation (17) with arbitrary , and , ; (ii)at the second step, for a supersolution of (17), where , , and are three arbitrary functions, and under assumption (14), we find two real numbers and , , and construct a subsolution , of (17) such that the following initial and blow-up arguments are satisfied: (iii)at the third step, under conditions (2)–(6) or related ones such as and , we show that if the main equation (1) allows a nonoscillatory solution , then the function: is well defined for some , , and is a supersolution of (17) with some concrete , , and ; in the case when and change the sign, instead of (21), we consider the function:

In conclusion, combining (19) and (20), we obtain the nonexistence of any continuous supersolution of the Riccati differential equation (17), and hence, the function given by (21) or (22) is not possible. Therefore, (1) does not allow any nonoscillatory solution.

#### 3. Main Results and Examples

As usual, we recognize two main different cases: the first one is when and are positive and the second one is when they may change the sign. Moreover, in the first case, depending on the combination of assumptions (5), (6), , , and , we consider five subcases such as is done in our first oscillation criterion for (1).

Theorem 5 (positive coefficients). *Let assumptions (2)–(4) be fulfilled. Then, (1) is oscillatory if one of the next five cases is met.*(i)*Let and (5), (6) hold. One supposes (14) with respect to provided that or , otherwise,
*(ii)*Let and , (6) hold. One supposes (14) with respect to and given by
*(iii)*Let and (5), hold. One supposes (14) with respect to and given by
*(iv)*Let and (5), hold. One supposes (14) with respect to and given by
*(v)*Let , , and (5), hold. One supposes (14) with respect to and given by
*

For each of the cases (i)–(v) of Theorem 5, we derive some consequences and examples, which show the importance of our oscillation criterion.

The case (i) of Theorem 5 for allows us to consider the following class of equations: where the functions , , and satisfy for some and . Under assumption (29), it is easy to see that the functions and satisfy both required assumptions (5) and (6) with . Hence, as an easy consequence of Theorem 5, we obtain the following result.

Corollary 6. *Let (2)–(4) and (14) hold with respect to and given in case (i) of Theorem 5 with . If and satisfy (29), then (28) is oscillatory. *

*Example 7. *Let , or , and . Then, the equation:
is oscillatory. Indeed, it is enough to check that the coefficients , , and and the functions , , and satisfy all the assumptions of Corollary 6 with respect to and for some .

*Example 8. *Let , or , and . Then, the equation:
is oscillatory. In fact, it is easy to check that the coefficients , , and and the functions , , and satisfy all the assumptions of Corollary 6 with respect to and for some .

The case (i) of Theorem 5 for proposes the following class of differential equations: where , , and the functions , , and satisfy

As a consequence of Theorem 5, we derive the next interesting corollary.

Corollary 9. *Let (2)–(4) and (14) hold with respect to and given in case (i) of Theorem 5. If and satisfy (33), then (32) is oscillatory. *

*Example 10. *Let , , , or , and . Then, according to Corollary 9, we conclude that the equation:
is oscillatory.

We have pointed out in Remark 1 that assumption unlike (5) allows to consider the oscillation of the following quasilinear differential equation: where and the functions , , and satisfy for some . It is clear that (28) is a particular case of (35) for . Under assumption (36), the functions satisfy both required assumptions and (6) with . Therefore, we can derive the following easy consequence of the case (ii) of Theorem 5.

Corollary 11. *Let (2)–(4) and (14) hold with respect to and given in case (ii) of Theorem 5 with . If and satisfy (36), then (35) is oscillatory. *

*Example 12. *Let , , and . Then the equation:
is oscillatory. In fact, it is enough to check that the coefficients , , and the functions , and satisfy all assumptions of Corollary 11 with respect to for some .

The case (iii) of Theorem 5 allows us to consider the following class of equations: where the functions , , and satisfy: for some . Under this assumption, it is easy to see that the functions and satisfy both required assumptions (5) and . Hence, as an easy consequence of case (iii) of Theorem 5, we obtain the following result.

Corollary 13. *Let (2), (3), and (14) hold with respect to given in case (iii) of Theorem 5. If , , and satisfy (40), then (39) is oscillatory. *

*Example 14. *Let , , , , and . Then, the equation:
is oscillatory. In order to show that, it is enough to check that the coefficients: , and the functions: , , , and satisfy all the assumptions of Corollary 13 with respect to and for some .

Next, we consider the oscillation of (1) in the case when the coefficients and may change the sign.

Theorem 15 (coefficients may change the sign). *Let and assumptions , (5), and hold. Then, (1) is oscillatory provided that one of the following two cases is met. (vi) One assumes (14) with respect to and given by
**
(vii) Let . One assumes (14) with respect to and given by
*

The case (vi) of Theorem 15 allows us to consider the following class of equations: where , and the function satisfies Under (45), one can easily check that the functions and satisfy both required assumptions (5) and . Hence, as an easy consequence of case (vi) of Theorem 5, we conclude the next consequence.

Corollary 16. *Let and (14) hold with respect to and given in case (vi) of Theorem 5. If satisfies (45), then (44) is oscillatory. *

*Example 17. *Let and . Then, the equations:
are oscillatory. In order to show that, it is enough to check that the coefficients: , , and and the functions: , , and satisfy all the assumptions of Corollary 16 with respect to and for some , .

#### 4. Proofs of the Main Results

In this section, we study the oscillation of (1) in the view of a pointwise comparison principle presented below, which will be shown for the corresponding Riccati differential equation.

*Definition 18. *A function is said to be locally Lipschitz in the second variable if for any bounded interval and there is a constant depending on such that

Now, we state and use the following general comparison principle, which will be proved at the end of this section.

Lemma 19. *Let and be two arbitrary real numbers such that . Let and , , be two functions satisfying:
**
where is a locally Lipschitz function in the second variable. Then, we have
*

*Definition 20. *A function is said to be locally bounded on , if for any bounded interval there is a constant depending on such that for all .

According to Lemma 19, we are able to give a sufficient condition on the functions: such that the Riccati differential equation (17) satisfies the comparison principle (19).

Lemma 21. *If and are two locally bounded functions on , then comparison principle (19) holds for the Riccati differential equation (17) with arbitrary , , and , where . *

* Proof. *Let and , be, respectively, sub- and supersolution of (17); that is, they satisfy (18). It is not difficult to check that is a locally Lipschitz function in the second variable. Indeed, for any bounded interval , , for all and , we have
where . Hence, Lemma 19 can be applied to and . If we set , and , then statement (48) is fulfilled because of assumption (18), and therefore, the desired conclusion (19) immediately follows from (49).

Corollary 22. *If and are two continuous functions on , then comparison principle (19) holds for the Riccati differential equation (17) with arbitrary , , and , where . *

*Proof. *Since and are two continuous functions on , they are also locally bounded functions on , and hence, this corollary immediately follows from Lemma 21.

Next, we present an essential lemma in which we construct a subsolution of (17) which has a blow-up desired property.

Lemma 23. *Let , , and be three arbitrary functions, and let assumption (14) hold, where if and otherwise. Let be a supersolution of the Riccati differential equation (17). Then, there are two real numbers and , , and a subsolution of (17) satisfying
*

*Proof. *In particular from (14), we obtain a sequence as such that
where (determined in (14)) can be chosen so that . From the previous statement, we conclude that there is a such that
Since is a continuous function in the variable , there is a such that
Consequently, we derive that
which together with (53) and (54) shows

Next, let be such that , where is from (54)-(56). Such exists since the tangent function is a bijection from to . Let
Because of (56), we have , , and . Since is a continuous function, it implies the existence of a such that
As a consequence, the function
is well defined and obviously satisfies
Also, since is continuous on , we have . Now, by taking the derivative of for every , we obtain
According to (14), we observe that(1)if , then
(2)if , then
(3)if , then
Thus, in all three cases of , we have
Putting the previous inequality into (61) and taking into account of (60), we conclude that
It proves that is a subsolution of the Riccati differential equation (17) which satisfies the statement (51).

Next, we are concerned with the following technical but crucial lemma.

Lemma 24. *Let the assumptions of Theorem 5 in the cases (i)–(iii) hold. If the main equation (1) allows a nonoscillatory solution , then the function given by (21) is well-defined with respect to such an and some , , and is a supersolution of the Riccati differential equation (17). *

*Proof. *If the main equation (1) allows a nonoscillatory solution , then there is a such that for all . Hence, the function given by (21) is well defined for such an . Next, making the derivative of , using that satisfies (1) and taking common assumptions of Theorem 5 for the functions , , , , and , we obtain
Depending on each of the three cases (i)–(iii) of Theorem 5, from the previous equality, we obtainNext from (21), we also have
Now, from (68) and (69), we immediately obtain: , . According to the definition of a supersolution, the previous inequality shows this lemma.

Lemma 25. *Let the assumptions of Theorem 5 in the cases (iv)-(v) hold. If the main equation (1) allows a nonoscillatory solution , then the function given by
**
is well defined with respect to such an and some such that , and is a supersolution of the Riccati differential equation (17).*

The proof of Lemma 25 is omitted because it is very similar to the proof of the following lemma.

Lemma 26. *Let assumptions of Theorem 15 hold. If the main equation (1) allows a nonoscillatory solution , then the function given by
**
is well defined with respect to such an and some , , and is a supersolution of the Riccati differential equation (17), where and , are given in the case (vi) of Theorem 5. *

* Proof. *Let be a nonoscillatory solution of (1), and thus, we can take a such that on . Let be a function defined by
From the assumptions of Theorem 15 and from equalities (1) and (72), we can easily make the following computation:
that is,
Now, if the middle term on the right-hand side of (74) is moved into the left-hand side, and multiplying such equality by , we conclude that the function
satisfies the Riccati differential equation (17) with respect to and given in the case (vi) of Theorem 15, which proves the first statement of this lemma.

However, if we group the first two terms on the right-hand side of (74) by the purpose of getting the complete square, then from (74) we easily conclude that the function:
satisfies the Riccati differential equation (17) with respect to and given in the case (vii) of Theorem 15, which proves the second statement of this lemma.

Now, we are able to present a common proof of the main results of the paper.

*Proof of Theorems 5 and 15. *At first, it is worth pointing out that the functions: , , , and , which are appearing at the same time in the main assumption (14) and the Riccati differential equation (17), only depend on the appropriate combination of basic assumptions on the coefficients: , and and the functions: and , which are formulated in one of the five cases of Theorem 5 and one of the two cases of Theorem 15.

Now, if we assume the contrary to the main assertion of the theorem; that is, if (1) is not oscillatory, then there is a nonoscillatory solution of (1) and a point and , where is appearing in (14), such that for all . Then by Lemmas 24, 25 and 26, the function given by (21) or (70), and (71) is well defined with respect to such an , smooth enough on , and it is a supersolution of the Riccati differential equation (17). Taking into account the main results of Lemma 23, we obtain the two numbers and , , and a subsolution of (17) such that the blow-up argument (51) is satisfied. By Corollary 22, we can apply the comparison principle (19) to (17) with arbitrary and , where . Hence, combining (19) and (51), we get as , which contradicts the fact that . Thus, is not possible, and therefore, (1) does not allow any nonoscillatory solution.

*Proof of Lemma 19. *Let and ; that is,
If statement (49) does not hold, then there is a point such that ; that is,
Moreover, since from (77) and (78), we obtain a such that
Since , we may use (47) in particular for
Hence, from (47), (48), and (79), we get