Abstract and Applied Analysis

Volume 2013 (2013), Article ID 735360, 12 pages

http://dx.doi.org/10.1155/2013/735360

Research Article

New Oscillation Criteria for Second-Order Forced Quasilinear Functional Differential Equations

Mervan Pašić

Department of Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, 10000 Zagreb, Croatia

Received 29 April 2013; Accepted 17 July 2013

Academic Editor: Milan Tvrdy

Copyright © 2013 Mervan Pašić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We establish some new interval oscillation criteria for a general class of second-order forced quasilinear functional differential equations with ϕ-Laplacian operator and mixed nonlinearities. It especially includes the linear, the one-dimensional p-Laplacian, and the prescribed mean curvature quasilinear differential operators. It continues some recently published results on the oscillations of the second-order functional differential equations including functional arguments of delay, advanced, or delay-advanced types. The nonlinear terms are of superlinear or supersublinear (mixed) types. Consequences and examples are shown to illustrate the novelty and simplicity of our oscillation criteria.

1. Introduction

We study the oscillation of the following three kinds of second-order forced quasilinear functional differential equations of delay, advanced, and delay-advanced types:

where and , , , and are continuous functions on , and , , is a classic solution of (1). A continuous function is said to be nonoscillatory if there is a such that on . Otherwise, is said to be oscillatory. Equation (1) is called oscillatory if all of its classic solutions are oscillatory.

In the delay case ,, and is nondecreasing; in the advanced case , , and is nonincreasing; in the delay-advanced case , , ,, and . The exponents satisfy a superlinear or a supersublinear (mixed) condition. On the function which appears in the first term of (1), we impose such conditions that the following three main classes of second-order differential operators are especially included: the linear , the one-dimensional-Laplacian , and the prescribed mean curvature operator . The function satisfies a usual growth condition and the coefficients , and are positive only on some intervals where changes the sign.

Recently, Bai and Liu [1] have studied the oscillation of second-order delay differential equation:

where . In Murugadass et al. [2], authors have studied the oscillation of the second-order quasilinear delay differential equation:

where and are ratio of odd positive integers. Later, in Hassan et al. [3], authors consider the oscillation of the half-linear functional differential equation:

where the function is positive continuous functions with . Moreover, in [1–3] some doubts concerning the proof of the main result of [4] are resolved. The well-known variational technique that uses the generalized Philos’ results based on the so-called -function has been used in [1, Theorem 2.2], [2, Theorems 2.3, and 2.6], and [3, Theorems 2.5, 2.6, and 2.7] (see also [5–11] and references therein). Since the second-order quasilinear differential operator usually causes some difficulties in many problems, the application of previous method on (1) is stated here as an open problem. In contrast to the preceding, we use a combination of the Riccati classic transformation, a blow-up argument, and a comparison pointwise principle recently established in [12, 13] but for differential equations without functional arguments. It seems that our criteria are slightly simpler to be verified, which is discussed on some examples given in the next section.

Among some recently published papers on the oscillation of quasilinear functional differential equations with both delay and advanced terms, we point out an oscillation criterion by Zafer [5] obtained for equation

where ,, and . On an interesting example in [5], the author established the oscillations provided at least one of constants appearing in the coefficients is sufficiently large (see also [6] for ), which is presented here in the next section as a particular case of our main results.

On the properties of some classes of the one-dimensional quasilinear differential equations with -Laplacian operators we refer reader to [14–20] and the references therein. About the applications of second-order functional differential equations in the mathematical description of certain phenomena in physics, technics, and biology (oscillation in a vacuum tube; interaction of an oscillator with an energy source; coupled oscillators in electronics, chemistry, and ecology; relativistic motion of a mass in a central field; ship course stabilization; moving of the tip of a growing plant; etc.), we suggest reading Kolmanovskii and Myshkis book [21].

2. Main Results and Examples

Let be a function which appears in the first term of (1) such that

These two assumptions are fulfilled, respectively, in the linear case , where and ; in the one-dimensional-Laplacian quasilinear case , where and ; and in the mean prescribed curvature quasilinear case

where and . The importance of the prescribed mean curvature quasilinear differential operators lies in the capillarity type problems in fluid mechanics, flux-limited diffusion phenomena, and prescribed mean curvature problems; see for instance [14, 22, 23].

The function which appears in the second term of (1) satisfies

where is from assumption (7).

In the first two theorems below, the exponents satisfy

where is from assumption (7). For instance if ,, , and , then . On assumption (10) see for instance [9, Lemma 1].

Unlike recently published oscillation criteria for the linear and half-linear second-order forced functional differential equations, our oscillation criterion is only based on the following elementary integral inequality:

where,is from (7),,, and functions, and are explicitly expressed by the coefficients of (1) just like it is done in the next three main results.

Theorem 1 (delay equation). One assumes (6), (7), (9), and (10). Let be a nondecreasing positive function on . Let on and ,. Let for every there exist , such that

where . Equation (1) is oscillatory provided there are two real parameters such that (11) is fulfilled, where

for , , and positive constants,,appearing, respectively, in (7), (9), and (10).

As we can see in (13), the forcing term can be an oscillatory function. The main property of any interval oscillation criterion (see [8]) is that the coefficients of the considered equation do not satisfy some conditions on the whole than only on intervals . The main consequence of Theorem 1 is the following oscillation criterion for (1) with and , .

Corollary 2. One assumes (6), (7), (10), (12), and (13). Then equation

is oscillatory provided

where is defined in (15).

In particular for , previous corollary takes the following form.

Corollary 3. Let (10) hold with , on , and , . Let for every there exist , such that assumptions (12) and (13) hold with . Equation

is oscillatory provided

where is defined in (15).

In the next examples and remarks, we discuss the application of oscillation criteria from Corollary 3 and [1, Theorem 2.2] on the following equation:

where , , and ,  , and are positive constants.

Example 4. As a consequence of Corollary 3, we show in this example that (20) is oscillatory provided the constants ,  , and satisfy the following simple inequality:

wheresatisfy (10), , and are two real numbers defined by

Indeed, let , , ,, , , , and . Firstly, it is elementary to check that the required inequalities (12) and (13) are satisfied. Next, we prove that required inequality (19) immediately follows from assumption (21). On the first hand, we estimate from above the term on the right hand side in (19) using that :

that is

On the other hand, we calculate the integral on the left hand side in (19):

Now, from previous two equalities, and inequalities (21) and (24), we conclude

Thus, assumption (21) proves desired inequality (19), and thus, Corollary 2 verifies that (20) is oscillatory provided (21) holds.

Remark 5. One way to obtain the inequality (21) is to suppose, for instance, that at least one of the constants ,  , andis large enough.

Remark 6. We can consider the slightly more general equation than (20):

where and satisfy (6) and (7), and. In a similar way as in Example 4, one can show that (27) is oscillatory provided at least one of the constants,  , andis large enough.

Example 7. In this example we present an application of [1, Theorem 2.2] for getting another condition on the constants,  , andfor oscillation of (20). Let,

, , and . It is clear thaton,,on, andon,. Therefore we may apply [1, Theorem 2.2] on (20) provided the following inequality holds:

Now, if we putandfor, then previous inequality takes the following concrete form:

Similarly as in Example 4, we can calculate previous integrals and conclude by [1, Theorem 2.2] that (20) is oscillatory provided the following inequalities hold:

where

and .

We leave to the reader to compare and verify which one of the two inequalities (21) and (31) is simpler to be verified. Also, similar to the previous example, it is possible to get corresponding inequalities analogously to (31) for other criteria published in the papers cited in the references.

Remark 8. In [1, Section 3] the authors give an application of [1, Theorem 2.2] to (20) where,,, and. However, the following choice (see [1, Section 3])

is not correct since the desired conditionson,are not fulfilled. Hence, we suggest reader to use the intervals proposed in the previous example.

Open Question 9.  The well-known variational technique based on the generalized Philos’ -function has been used in [1–11] to obtain some oscillation criteria for (1), where the second-order quasilinear differential operator is linear or half-linear. Is it possible to use this technique in the case of any function satisfying general conditions (6) and (7)?

Theorem 10 (advanced equation). Under assumptions (6), (7), (9), and (10), letbe a nonincreasing positive function on and on ,. Let for everythere exist numbers , such that and

where. Equation (1) is oscillatory provided there are two real parameterssuch that (11) is fulfilled, where

for,, and positive constants,, andappearing, respectively, in (7), (9), and (10).

Now, analogously with (16), we consider the oscillation of the advanced equation:

As a consequence of Theorem 10, we have the following criterion.

Corollary 11. One assumes (6), (7), (10), and (34). Then (37) is oscillatory provided condition (17) is satisfied, whereis defined in (36).

As the third case, we consider second-order functional differential equations with both delay and advanced arguments.

Theorem 12 (delay-advanced equation). One assumes (6), (7), and (9),for, and,. Leton,for,  andforon. Let,, andsatisfy (12)-(13) forand (34) for. Equation (1) is oscillatory provided there are two real parameterssuch that (11) is fulfilled, where

for , , and positive constants , appearing, respectively, in (7) and (9).

Analogously with (16) and (37), we consider the oscillation of delay-advanced equation:

As a consequence of Theorem 12 we derive the following oscillation criterion for (40).

Corollary 13. Let all assumptions of Theorem 12 hold with respect to and . Equation (40) is oscillatory provided condition (17) is satisfied, where is defined in (39).

In Zafer [5] (see also [6]) the author has illustrated its main oscillation result on the following example of the second-order functional differential equations with delay and advanced arguments. Since , it can be rewritten in the form

where,  , and are positive constants; ; ; and . It has been proved that previous equation is oscillatory provided at least one of the constants , and is large enough. Here according to Corollary 13 we can repeat this interesting conclusion for slightly complicated equation but with .

Example 14. We consider the following -Laplacian functional differential equation with delay and advanced arguments:

where satisfies (6), (7), ,, and . We claim that the previous equation is oscillatory provided at least one of the constants ,  , and is large enough. Indeed, it is enough to show that condition (17) is fulfilled by using Corollary 13. It can be done in a similar way as in Example 4 with (20) and Corollary 3. We leave it to the reader.

3. Auxiliary Results: Qualitative Properties of Concave-Like Functions

Let be two arbitrary real numbers. It is known that if is a concave and smooth enough function on , that is, if for all , then for all , . Moreover, if for all , then from previous inequality we obtain

However, often we do not have any information about the sign of the second-order linear differential operator in (1) except only in a particular case when and . Hence the main goal of this section is to find some sufficient conditions on and such that the assumption “ for all ” implies the desired inequality (43). It is done in the following results.

Lemma 15. Let satisfy (6). Let for all , . For any function such that for all , the following statement holds:

Proof. From assumption for all , it follows that

To the end of this proof, let be fixed. Thanks to (6), if , then too, becauseis odd and increasing. In this case, since and on, it implies that

Thus, it remains to show (44) for the case of . From for all ,, we obtain which together with and (45) gives

Acting on (47) with and using that is an increasing function because of (6), we obtain

Next, since on and , we have . From the mean-value theorem for on , we get a depending on and such that , and since , from (48) we obtain

which proves the desired inequality in (44).

In the advanced case of (1), that is, when , we have the analogous result to Lemma 15.

Lemma 16. Let satisfy (6). Let for all , . For any function such that for all , the following statement holds:

Proof. From assumption for all , we have

To the end of this proof, let be fixed. If , then because is odd and increasing, and since by assumption for all , we have that

which proves (50) in this case. Let now . It implies that and because of (51) and for all . Hence, from for all ,, from (51), and , we especially conclude that

Acting on (53) with we obtain

By the Lagrange’s mean-value theorem, there exists a depending on such that . Since for all and , from (54) we obtain

which proves the desired inequality.

Statement (44) will be frequently used in the following form.

Corollary 17. Let satisfy (6), , and for all , . Let be two arbitrary real numbers such that . Then for any function such that for all , the following statement holds:

Proof. By assumptions of this corollary, we have , , and for all . In particular from statement (44) applied on , we get

Integrating this inequality over the interval for all , we obtain

which proves the desired inequality in (56).

Statement (50) will appear in the following form.

Corollary 18. Let be two arbitrary real numbers. Let satisfy (6), , , and for all , . Then for any function such that for all , the following statement holds:

Proof. Analogously with the proof of Corollary 17, we just need to use the second inequality in (50) on and integrate it over the interval for all .

Next, by Corollary 17 and the arithmetic-geometric mean inequality

we can prove the following proposition.

Proposition 19 (with delay arguments). Let for all such that and let satisfy (6). Let exponents and the -tuple satisfy (10). Let on , , and . Let and on, where . Then for any function such that and for all , we have

Proof. By inequality (60) in particular for and , together with (56), we obtain

which proves this proposition.

Proposition 20 (with advanced arguments). Let for all such that and let satisfy (6). Let exponents and the -tuple satisfy (10). Let on , , and . Let and on , where . Then for any function such that and for all , we have

Proof. It is very similar to the proof of Proposition 19 but instead of Corollary 17 we need to use Corollary 18.

According to Corollary 17, we are able to prove the next useful proposition in which we also use the well-known Young inequality :

Proposition 21 (with delay-advanced arguments). Let on , real numbers for all and ,. Let satisfy (6), ,and , on , and , . Let and on , where . Then for any function such that and on , we have

for all .

Proof. First of all, by inequality (64) in particular for and

together with (56), we obtain

In the same way, one can show that

The previous two inequalities prove this proposition.

4. Proof of Main Results

Proof of Theorem 1. If the assertion of this theorem does not hold, then there is a nonoscillatory solution of (1) such that for all and some . Moreover, it is enough to work only with the case and on , since the second case and on can be transformed into the first one. Indeed, multiplying equation (1) by and using assumptions that and are odd functions, we have that (1) is equivalent to

The proof is outlined in the following three steps.

Step  1. Next, for any the following function is well defined:

and . Since and on , from the previous equality we get

Using (1) and assumptions (7) and (9), from the second equality of (71), we get

where . Next, using assumptions (12) and (13) in (1) for all we easily conclude that satisfies all assumptions of Corollary 17 and Proposition 19. Hence from (56), (61), and (72) we observe

Step  2. In this step, we need the next elementary proposition.

Proposition 22. Let and let be a positive number defined by

Then there is an increasing odd function , , such that

Moreover, for we have and we can take , .

Proof. Let be a function defined by

It is not difficult to see that (see [24]) and is a bijection from on interval and that the function determined by the formula , where is the inverse function of , satisfies all properties given in (75).

Let now be such that (such an exists since is a bijection from to ),, and let be a function defined by

Because of (11), we have . Hence,

Next, let and be two functions defined by

Since for all , we have and . Since is continuous, it gives the existence of such that . Hence, the function , , satisfies

and because of (78), we obtain

that is

Step  3. We claim that

In order to prove inequality (83), we need the following proposition.

Proposition 23. Let and be two functions satisfying on , respectively,

Then the following statement holds:

Proof. If we denote by

then (84) can be rewritten in the form andon. Also, it is not difficult to check thatis locally Lipschitz function in the second variable and bounded in the first variable on bounded interval. Hence, we may apply the pointwise comparison principle from [12, Lemma 19] and, consequently, we conclude that statement (85) holds, which proves this proposition.

Next, we proceed with the proof of inequality (83). By (80) we know thatwhich, together with (73) and (82), ensures that we may apply Proposition 23. Now, the desired inequality (83) immediately follows from statement (85).

Next, from the second equality of (80) and inequality (83) we conclude that . It contradicts the fact that . Hence, can not be oscillatory as it is supposed at the beginning of this proof and we conclude that (1) is oscillatory.

Proof of Corollary 2. From (16), we see that and. Next, to the end of this proof, let be defined by

Now, we can rewrite (17) in the form

Since , it is clear that

and so

Together with assumption (88) we show that

It proves that desired condition (11) is fulfilled. Thus, this corollary follows from Theorem 1.

Proofs of Theorems 10 and 12. It follows by the same line of arguments as in the proof of Theorem 1, but instead of Proposition 19 we use Propositions 20 and 21, respectively.

Acknowledgment

This work is supported by the Ministry of Science of the Republic of Croatia under Grant no. 036-0361621-1291. The author has dedicated this work to his professors: Asim Musaefendić—Gymnasium Maglaj; Svetozar Kurepa, Nikola Sarapa, Branko Najman and Darko Žubrinić—graduate and postgraduate studies on Faculty of Sciences University of Zagreb; Jean-Pierre Puel and Jacques-Louis Lions—postdoctoral study in Paris 1996/97.

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