Abstract and Applied Analysis

Volume 2014 (2014), Article ID 818732, 11 pages

http://dx.doi.org/10.1155/2014/818732

Research Article

On Eventually Positive Solutions of Quasilinear Second-Order Neutral Differential Equations

Simona Fišnarová and Robert Mařík

Department of Mathematics, Mendel University in Brno, Zemědělská 1, 613 00 Brno, Czech Republic

Received 17 February 2014; Accepted 16 April 2014; Published 13 May 2014

Academic Editor: Samir Saker

Copyright © 2014 Simona Fišnarová and Robert Mařík. 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 study the second-order neutral delay differential equation , where , and . Based on the conversion into a certain first-order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral . A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case.

1. Introduction

In the paper we study the equation

where , , is the power type nonlinearity. The coefficients and are subject to the usual conditions , and the coefficient is positive, .

We assume that ,

and there exist numbers and such that and .

Under the solution of (1) we understand any differentiable function which does not identically equal zero eventually, such that is differentiable and (1) holds for large .

Following the widely accepted terminology, the solution of (1) is said to be oscillatory if it has infinitely many zeros tending to infinity. Equation (1) is said to be oscillatory if all its solutions are oscillatory. In the opposite case, that is, if there exists an eventually positive solution of (1), (1) is said to be nonoscillatory.

In the paper we study nonoscillatory solutions of (1). Since is a solution of (1) if and only if is a solution of (1), we can focus our attention on positive solutions.

The paper is organized as follows. In the remaining part of the current section we summarize selected important facts related to (1) and trends in the oscillation theory of this equation. In Section 2 we summarize tools like inequalities and oscillation criteria used in the proofs of main results. The main results are presented in the next three sections. Results on eventually positive solutions are separated into Sections 3 and 4 according to different asymptotic behavior: in Section 3 and in Section 4. In both cases we provide an efficient condition which ensures that solutions of this type do not exist. Note that under some additional conditions (namely, divergence of integral (3) below) the results from Section 3 immediately yield also oscillation criteria. If (3) fails, we can formulate oscillation criteria using a suitable combination of results from Sections 3 and 4, as shown in Section 5. The results of the paper improve several recently published results even in the linear case. We discuss these improvements in detail in remarks and examples accompanying the main theorems.

Neutral differential equation (1) as well as other related equations have been studied frequently in the literature. There are two main methods in the oscillation theory of (1). One of them is based on a modification of the classical Riccati substitution which is known to be a powerful tool in theory of second-order linear differential equations. Following this method, neutral equation (1) is in some sense considered as a perturbation of some second-order ordinary differential equation. An alternative approach, used for example, in a series of papers by Baculíková et al. [1–4] and Li [5] is based on the fact that it is possible to derive neutral first-order differential inequality for quasiderivative from (1) and the resulting inequality can be studied in the scope of theory elaborated for first-order delay differential inequalities. In this paper we use the later approach. The resulting theorems are sometimes referred to as comparison theorems for neutral differential equations.

Two main approaches are used to put the shift in the differential term under the control. If , then (1) can be “majorized” (in the sense of the classical Sturm comparison theory, which however has no extension to delay equations) by a delay equation of the form (1) with . Oscillation criteria for second-order delay differential equations can be then used to conclude results for neutral equation (1) (see, e.g., [6–8]). An alternative approach deals with a suitable combination of (1) and the same equation with independent variable shifted from to . This approach, which is used also in our paper, does not require but yields other restrictions, such as commutativity of the composition of delays (2).

Neglecting which method is used to study the oscillation of (1), it turns out that it is necessary to distinguish two cases: either

or

The absolute majority of oscillation results in the literature concerns case (3), since in this case the positive solutions of (1) exhibit simpler behavior than in case (4); see Lemma 5 below. Case (4) has been studied, for example, in [9–16]. Note that for this case it is typical that the oscillation criterion consists of two relatively independent conditions. One of them is used to eliminate positive solutions with ; the other one to eliminate positive solutions with . There are also results which treat both cases and in one unified approach. However, following this approach a typical conclusion is weaker: the equation is proved to be almost oscillatory (all nonoscillatory solutions, if exist any, tend to zero). Note also that the paper [16] does not satisfy these rules (makes use of unified approach to both cases but concludes oscillation), but there are several inaccuracies in this paper; see [10, 17] for corrected version of [16].

In this paper we essentially use the method from [1, 2] with a modification for case (4) presented in [12]. However, to keep the influence of each condition as transparent as possible we used different organization of the paper, as we explained above. The main improvement with respect to these papers is that we replace inequalities and estimates used in these papers by suitable parametrized versions depending on parameters and (see below). This yields criteria with some degree of freedom and optimization with respect to the parameters which yields sharper results, as we carefully explain on examples of equations with proportional delay. A similar method where we use parameters and to refine the widely used inequalities has been used in the recent paper [18].

Finally, note that [12] in fact deals with linear equations and the extension to nonlinear equations is suggested in Remark 11 at the end of the paper [12]. However here we use an advanced technique rather than the method suggested in [12].

2. Preliminary Results

In the paper we derive results related to the existence or nonexistence of certain equations and inequalities in terms of several parameters. The following two lemmas allow to find the values of the parameters, which yield sharpest results.

The function introduced in the following lemma plays a role in a formulation of oscillation criteria in the case .

Lemma 1. Let . The function

satisfies

for every and .

Proof. It follows from the fact that

and as the function of on attains its minimal value at the point and

The following functions appear in the examples and allow to find the optimal values of the parameters which yield the sharpest result.

Lemma 2. Let , be positive numbers.

(i)The function is increasing on , decreasing on , and satisfies

on with the equality, if and only if .(ii)The function is increasing on , decreasing on , and satisfies

on with the equality, if and only if .

Proof. By a direct computation

Hence has a local maximum at the point and the value of this local maximum is . Similarly,

and hence has a local maximum at the point and the maximal value is .

Lemma 3. Let , , , , . Then

Proof. From the fact that the function is a convex function for we have

for nonnegative and . From here we obtain the desired inequality for and .

Lemma 4. Let . The inequality

holds for positive mutually conjugate numbers , and every which satisfies and .

Proof. It follows from Lemma 3, from the definition of and from condition (2).

The following lemma is well known in theory of neutral differential equations. It states (among others) that if is an eventually positive solution, then is eventually of one sign and the negative sign of is excluded if (3) holds.

Lemma 5. Let be an eventually positive solution of (1). The corresponding function satisfies

eventually if (3) holds and either (16) or

eventually if (4) holds.

Proof. It follows from [7, Lemma 10] and from the proof of that lemma.

In the following lemma we summarize effective oscillation criteria for delay and advanced first-order equation which appear in the analysis of (1). Note that (iii) is sharper version of the related condition from [2, Lemma 4].

Lemma 6. Let .

(i)If and

then

has no eventually positive solution.(ii)If and

then

has no eventually positive solution.(iii)Let , . If

then

has no eventually positive solution.(iv)Let , . If

then

has no eventually positive solution.

Proof. See [9, Lemmas 2.1–2.4] and [19, Lemma 2.2.9]. Note that the original proof of condition (i) is due to [20] and the proofs of conditions (iii) and (iv) for equations are due to [21].

3. Positive Solutions with Eventually

In this section we give sufficient conditions which exclude the possibility that the equation possesses an eventually positive solution such that the corresponding function is eventually increasing. Note that Lemma 5 excludes other types of eventually positive solutions if (3) holds. Hence if (3) holds as well, then the criteria from this section guarantee oscillation of (1).

Denote

The following theorem allows us to relate positive solutions of (1) with a certain first-order neutral equation. This neutral equation can be further compared with a certain nonneutral differential equation. The form of this nonneutral differential equation depends on the fact whether the deviating argument in the differential term is delay or advanced argument. If , , and , then Theorem 7 reduces to [2, Theorems 4, 5 and 6].

Theorem 7. Let , , and a function which satisfies and . Suppose that there exists a number and a solution of (1) which satisfy

Let be such that

for every and let be such that for . Then the following statements are true.

(i)The inequality

has a positive decreasing solution on .(ii)If , then

has a positive solution on .(iii)If , then

has a positive solution on .

Proof. Let be a solution of (1) which satisfies and for . Inequalities (15), , and imply

for .

We shift (1) from to and get

Substituting from this inequality and from (1) to (33) and using (26) we obtain

for . Denoting and using the obvious fact that is positive and decreasing on we have

for . Thus is an eventually positive and eventually decreasing solution of (30) and claim (i) is proved.

Denote

Since is a positive decreasing function, we have for and for . Hence if we have

where the function is defined by (5), and, if , then similarly

Hence we have

if , and

if . This and claim (i) prove claims (ii) and (iii) since in each case we have found an eventually positive solution of the corresponding inequality.

Remark 8. Note that in the proof of Theorem 7 we constructed the solutions of the inequalities (30), (31), and (32).

In the following corollary we give an efficient condition for nonexistence of the solutions mentioned in the points (ii) and (iii) of Theorem 7. According to Lemma 6 we distinguish the cases and .

Corollary 9. Let . Equation (1) has no solution which satisfies

if there exists number and a function satisfying and such that one of the following conditions holds.

(i) and for every there exists such that

if , and

if .(ii) and for every there exists such that

if , and (44) holds if .

Proof. It follows from Theorem 7 and Lemmas 6 and 1.

In the following example we compare our results with the results of [1]. Note that in this example (3) holds and hence the results of this section ensure oscillation of the equation.

Example 10 (linear equation). Baculíková and Džurina studied in [1] differential equation

where , , , and obtained that the equation is oscillatory if either

or

We have , , , , , , and

and hence (3) holds. Using Corollary 9 with , and we have and consequently,

Hence, (46) is oscillatory if either

or

Even in case , if we view the right hand sides as linear functions of , we easily see from the slope and -intercept of these lines that these conditions are sharper than those from [1]. Note that the fact that the equation is linear causes that the parameter does not have any influence on the sharpenes of these conditions, since the function from Lemma 1 is a constant function with respect to the first variable for . Hence, the improvement with respect to the results from [1] is purely in the presence of the parameter in the definition of the function . Figure 1 reveals also different asymptotic behavior of the right hand sides of (51) and (52) with respect to the corresponding constants from (47) and (48). Based on this fact we see that the improvement is significant especially if is sufficiently far from .

Figure 1: The graph of right hand sides (RHS) of (47) and (48) for compared to (51) and (52) as functions of .

When looking for optimal conditions for oscillation of (46) it is easy to ensure that the case is not optimal for every . Really, if we replace inequality signs in (51) and (52) by equality signs and view the resulting equality as a formula which defines as a function of , we get U-shaped function with one local minimum (see Figure 2). Since can be any positive number smaller than , it turns out that the optimal choice for is on the decreasing part and on the increasing part, where is the point where the function attains its global maximum. Using Lemma 2, we have and . Hence we get that the equation is oscillatory if

From the graphical point of view these conditions arise from (51) and (52) by isolating and replacing the increasing part of the resulting curve by a constant function; see Figure 2 for more details and for comparison with the oscillation constant resulting from (48).

Figure 2: A comparison of the lower bounds for the coefficient which guarantee oscillation of (46) for different values of . Parameters used for the graphs are and .

Example 11 (half-linear equation). Consider the differential equation

We have , , , , , , , and,

and hence condition (3) holds. Using results of Baculíková and Džurina [2, Corollaries 3 and 4] we obtain that (54) is oscillatory if

or

Using Corollary 9 with , and we have and consequently,

Hence,

and (54) is oscillatory if

or

If , the comparison of our lower bound and the upper bound if and if is on Figure 3. Note that in contrast to the linear case, both curves do not intersect at .

Figure 3: The graph of right hand sides (RHS) of (56) and (57) for compared to (60) and (61) as functions of .

The function is positive on the interval and it follows from Lemma 1 that the maximal value of is at the point . Hence, (54) is oscillatory if

Figure 4 shows how the critical constant which ensures oscillation of (54) is improved with respect to the results of [2] (dashed line) for various values of the delay . For readers convenience we graphed also a dotted curve which is only partial improvement of [2]: the values of and are chosen as in [2] and the value of (which plays role in ) is choosen to equal to .

Figure 4: A comparison of the lower bounds for the coefficient which guarantee oscillation of (54) for different values of . Parameters are and .

The following theorem and corollary are variants of Theorem 7 and Corollary 9 for sublinear case .

Theorem 12. Let , , and a function which satisfies and . Suppose that there exists a number and a solution of (1) which satisfy

Let be such that

for every and let be such that for . Then the following statements are true.

(i)The inequality

has a positive decreasing solution on .(ii)If , then

has a positive solution on .(iii)If , then

has a positive solution on .

Proof. The proof is the same as the proof of Theorem 7 where we formally put and use

instead of Lemma 3.

Corollary 13. Let , . Equation (1) has no solution which satisfies

if there exists and a function satisfying and such that one of the following conditions holds.

(i) and for every there exists such that

if , and (44) holds if .(ii) and for every there exists such that

if , and (44) holds if .

Proof. It follows from Theorem 12 and Lemmas 6 and 1.

Remark 14. If and , then Theorem 12 reduces to [2, Theorems 1, 2, and 3] and Corollary 13 reduces to [2, Corollaries 1 and 2].

4. Positive Solutions with Eventually

In this section we modify the methods from previous section for positive solutions which satisfy eventually. Troughout this section we will suppose that (4) holds, since if it fails, then evetually positive solutions with eventually do not exist.

The function defined by the relation

and the following Theorem 15 are the corresponding modifications of the function and Theorem 7.

Theorem 15. Let , , , and let be a function which satisfies . Suppose that there exists a number and a solution of (1) which satisfy

Let be such that

for every . Then the following statements are true.

(i)The inequality

has a positive increasing solution on .(ii)If , then

has a positive solution on .(iii)If , then

has a positive solution on .

Proof. Let be a solution of (1) which satisfies and for . Inequalities (15), , and imply

Combining this inequality with (1) and (1) shifted from to , similarly as in the proof of Theorem 7, we obtain

for . The function defined by is negative and decreasing. Hence for we have

and hence

Since we have

which implies

Combining this inequality with (79) and multiplying by we find that is a positive and increasing solution of (75). Claim (i) is proved. Denote

Since is positive and increasing, we have for and for . Hence, if , we have

which implies

Analogously, if ,

which implies

Claims (ii) and (iii) then follow from (i) and positivity of .

Corollary 16. Let and . Equation (1) has no solution which satisfies

if there exists and a function satisfying such that one of the following conditions holds.

(i) and

if , and

if .(ii) and

if , and (91) holds if .

Proof. It follows from Theorem 15 and Lemmas 6 and 1.

In a similar way as in Theorem 12 and Corollary 13, if we suppose in Theorem 15 and Corollary 16 and use inequality (68) instead of inequality from Lemma 3, we get the following statements.

Theorem 17. Let , , and a function which satisfies . Suppose that there exists a number and a solution of (1) which satisfy

Let be such that

for every . Then the following statements are true.

(i)The inequality

has a positive increasing solution on .(ii)If , then

has a positive solution on .(iii)If , then

has a positive solution on .

Corollary 18. Let , . Equation (1) has no solution which satisfies

if there exists and a function satisfying such that one of the following conditions holds.

(i) and

if , and (91) holds if .(ii) and

if , and (91) holds if .

5. Oscillation Criteria If (4) Holds

As we explained before, it follows from Lemma 5 that if (3) holds, then the criteria from Section 3 are in fact oscillation criteria. If (3) fails and (4) holds, then the set of all possible eventually positive solutions is more comprehensive and may contain also solution which satisfy eventually. Hence to ensure oscillation of (1) in the case (4) we have to eliminate both cases; criteria from both Sections 3 and 4 apply. For example, in the half-linear case , (1) is oscillatory if either conditions

or

see the example below.

Example 19. Consider the equation

We have , , , , , . We will apply Corollaries 9 and 16 with , , and . We have . Since

condition (4) holds. Next we have

Consequently, if , then (43) and (92) give

If , then (45) and (90) give

Hence, (103) is oscillatory if

or

The function is positive on the interval and it follows from Lemma 1 that the maximal value of is at the point . Hence, (103) is oscillatory if

6. Conclusion

In the paper we derived asymptotic results for neutral quasilinear equation (1). Note that this equation covers several types of second-order differential equations studied in the literature, namely, the linear and half-linear second-order differential equations.

Using the comparison method we derived sufficient conditions for nonexistence of eventually positive solutions with various asymptotic behaviors. Additional assumptions (such as (3)) or suitable combinations of the results yield oscillation criteria for this equation. The novelty of the presented results is in the fact that we used parametrized versions of inequalities used typically in comparison theory of neutral differential equations. Despite the fact that we introduced three parameters (, , and ), the results remain simple and effective. We have shown on several examples that effective oscillation criteria can be formulated for particular equations by establishing the optimal values for these parameters.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for valuable comments to the paper. Research supported by the Grant P201/10/1032 of the Czech Science Foundation.

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