Abstract and Applied Analysis

Volume 2012 (2012), Article ID 979138, 19 pages

doi:10.1155/2012/979138

Research Article

Limit-Point/Limit-Circle Results for Equations with Damping

M. Bartušek¹ and John R. Graef²

¹Faculty of Science, Masaryk University Brno, Kotlářská 2, 611 37 Brno, Czech Republic

²Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA

Received 27 September 2012; Accepted 12 December 2012

Academic Editor: Elena Braverman

Copyright © 2012 M. Bartušek and John R. Graef. 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.

The authors study the nonlinear limit-point and limit-circle properties for the second order nonlinear damped differential equation , where , , and . Some examples are given to illustrate the main results.

1. Introduction

In this paper, we study the nonlinear equation

and its special case

We set and and assume throughout that , , , , , and .

We will only consider solutions defined on their maximal interval of existence to the right.

Remark 1.1. The functions , , and are smooth enough so that all nontrivial solutions of (1.1) defined on are nontrivial in any neighborhood of (see Theorem 13(i) in [1]). Moreover, if either or on , then all nontrivial solutions of (1.1) are defined on .

We can write (1.1) as the equivalent system

where the relationship between a solution of (1.1) and a solution of the system (1.3) is given by

We are interested in what is known as the nonlinear limit-point and nonlinear limit-circle properties of solutions as given in the following definition (see the monograph [2] as well as the papers [3–14]).

Definition 1.2. A solution of (1.1) defined on is said to be of the nonlinear limit-circle type if

and it is said to be of the nonlinear limit-point type otherwise, that is, if

Equation (1.1) will be said to be of the nonlinear limit-circle type if every solution of (1.1) defined on satisfies and to be of the nonlinear limit-point type if there is at least one solution for which holds.

The properties defined above are nonlinear generalizations of the well-known linear limit-point/limit-circle properties introduced by Weyl [15] more than 100 years ago. For the history and a survey of what is known about the linear and nonlinear problems as well as their relationships to other properties of solutions such as boundedness, oscillation, and convergence to zero, we refer the reader to the monograph by Bartušek et al. [2] as well as the recent papers of Bartušek and Graef [4, 6, 9–11].

Here, we are also interested in what we call the strong nonlinear limit-point and strong nonlinear limit-circle properties of solutions of (1.1) as given in the following definitions. These notions were first introduced in [7] and [8], respectively, and further studied, for example, in [4, 6]. We define the function by

and let constant be given by

Definition 1.3. A solution of (1.1) defined on is said to be of the strong nonlinear limit-point type if

Equation (1.1) is said to be of the strong nonlinear limit-point type if every nontrivial solution defined on is of the strong nonlinear limit-point type and there is at least one nontrivial solution defined on .

Definition 1.4. A solution of (1.1) defined on is said to be of the strong nonlinear limit-circle type if

Equation (1.1) is said to be of the strong nonlinear limit-circle type if every solution defined on is of the strong nonlinear limit-circle type.

From the above definitions we see that for an equation to be of the nonlinear limit-circle type, every solution must satisfy ; whereas for an equation to be of the nonlinear limit-point type, there needs to be only one solution satisfying . For an equation to be of the strong nonlinear limit-point type, every solution defined on must satisfy and there must be at least one such nontrivial solution.

If , (1.1) becomes

and moreover, if , then (1.7) reduces to the well-known half-linear equation,

a general discussion of which can be found in the monograph by Došlý and Řehák [16]. Using terminology introduced by the authors in [5–7], if , we say that (1.1) is of the super-half-linear type, and if , we will say that it is of the sub-half-linear type. Since in this paper we are assuming that , we are in the half-linear and sub-half-linear cases.

The limit-point/limit-circle problem for the damped equation

with was considered in the papers [17, 18], where is the ratio of odd positive integers and is an odd integer, respectively. The results in both of these papers tend to be modifications of results in [12–14] to accommodate the damping term.

It will be convenient to define the following constants:

Notice that ,  ,  , and . We define the function by

and in the reminder of this paper we will make use of the assumption that

If holds, we define the constants

For any solution of (1.1), we let

Note that on for every solution of (1.1).

For any continuous function , we let and so that .

2. Lemmas

In this section we present a number of lemmas that will facilitate proving our main results.

Lemma 2.1. For every nontrivial solution of (1.1) defined on , for .

Proof. Suppose, to the contrary, that (1.1) has a nontrivial solution such that for a number . Then (1.13) implies and so (1.1) has the solution defined by

But this contradicts Remark 1.1 and proves the lemma.

Lemma 2.2. Let be a solution of (1.1). Then:

(i)for , we have

(ii)for , we have

Proof. Let be a solution of (1.1). Then it is a solution of the equation

with . Then (2.2) and (2.3) follow from Lemma  1.2 in [5] applied to (2.5). Relation (2.4) follows from (2.2).

The following two lemmas give us sufficient conditions for the boundedness of from above and from bellow by positive constants.

Lemma 2.3. Let hold and assume that

Then for any nontrivial solution of (1.1) defined on , the function is bounded from bellow on by a positive constant depending on .

Proof. Suppose, to the contrary, that there is a nontrivial solution of (1.1) such that

By Lemma 2.1, on . Let be such that

the existence of such a follows from and (2.6). Then, for any such that , there exist and such that and

for . Then (2.2) implies

on . From this, (2.3) (with and ), (2.4), (2.9), and the fact that , we have

Hence, using (2.8) and the facts that ,  , and , we obtain

This contradiction to proves the lemma.

Lemma 2.4. Assume that for large , holds,

and either (i) , or (ii) and

Then for every solution of (1.1) the function is bounded on .

Proof. Let be a nontrivial solution of (1.1). Then according to Remark 1.1 and Lemma 2.1, is defined on and on . In view of and (2.13), we can choose such that

Suppose that is not bounded, that is,

Then, for any with , there exist and such that ,  , and

Since is of bounded variation and , we see that

Setting and in (2.2)–(2.4), we have (2.10) and

From this, (2.8), (2.15), and (2.18), we obtain

where .

If , then and (2.20) gives us a contradiction.

Now let and (2.14) hold. Then and (2.20) implies

Hence,

for all such that , where . At the same time, (2.20) implies for these values of . Thus, (2.22) holds for all . On the other hand, if , then (1.13) implies

for . So,

From this and (2.22),

which contradicts (2.14). Hence, is bounded from above on . Since on , the conclusion follows.

Lemma 2.5. Let and (2.6) hold. Then there exists a solution of (1.1) defined on , a constant , and such that

Moreover, can be chosen arbitrary small.

Proof. Condition implies that is bounded, so we can choose ,  , and such that

Consider a solution of (1.1) such that . First, we will show that

Suppose (2.28) does not hold. Then there exist such that

for . Lemma 2.2 (with and ), and the facts that ,  , and imply

Hence, which contradicts the choice of , and so (2.28) holds.

Now, Lemma 2.2 (with ,   ) similarly implies

and the statement of the lemma is proved.

Lemma 2.6. Suppose that and (2.6) hold and

In addition, assume that either

or

holds. If is a solution of (1.1) with

on for some positive constants and , then

Moreover, if does not tend to zero as , then

Proof. Let be a nontrivial solution of (1.1) satisfying (2.35). Then in view of (1.13), (2.32), and (2.35)

as . Now, (2.2) and (2.35) imply

for so there exists such that

for . It follows from (1.1) that

where and

Hence, from (2.39),

with for . Moreover, (1.13), (2.35), and (2.38) imply

for with and .

If (2.33) holds, then (2.43) implies is bounded on , and in view of (2.41), we have

for with and . Note, that is bounded on .

Now let (2.34) hold. Then, using (2.2), (2.35), and (2.40) and setting , we have

for , where by (2.34). From (2.46) together with (2.39), (2.41), and (2.42), inequality (2.45) holds with and . Thus, (2.45) holds with if either (2.33) or (2.34) holds, and is bounded for . Moreover,

for . Adding the left hand inequalities in (2.44) and (2.47) gives

as , and subtracting gives

as .

If is oscillatory, let be a sequence of zeros of . Then letting in (2.48) and (2.49), it is clear that the conclusion of the lemma holds.

Let be nonoscillatory. Then either

or

We first prove (2.36). It clearly holds if (2.50) does. So suppose (2.51) holds. Then for large and (2.48) gives us the contradiction. Hence, (2.36) holds.

Finally, we prove (2.37). From (2.49), (2.37) holds if (2.50) does. Let (2.51) hold and assume that (2.37) does not. Then (2.49) implies

In view of (2.39) and (2.51),  , so . This contradicts the assumptions of the lemma and completes the proof.

3. LP/LC Problem for (1.1)

In this section we present our main results for (1.1) and give some examples to illustrate them.

Theorem 3.1. Let for large and assume that and (2.13) hold. In addition, if , assume that (2.14) also holds. Then (1.1) is of the strong nonlinear limit-circle type if and only if

Proof. Let be a nontrivial solution of (1.1). By Remark 1.1, is defined on . The hypotheses of Lemmas 2.3 and 2.4 are satisfied, so there are constants and such that

on . Hence, from this and (1.13),

The conclusion of the theorem then follows from (3.1).

In case , the results in this paper reduce to previously known results by the present authors except that the necessary part of Theorem 3.1 is new.

Theorem 3.2. Let , (2.6), and either (2.33) or (2.34) hold. If

then (1.1) is of the nonlinear limit-point type.

Proof. The hypotheses of Lemmas 2.5 and 2.6 are satisfied, so if is a solution given by Lemma 2.5, then (2.36) holds, and the conclusion follows.

Theorem 3.3. Let for large and let conditions , (2.13), and either (2.33) or (2.34) hold. In addition, if , assume that (2.14) holds. If

then every nontrivial solution of (1.1) is of the nonlinear limit-point type. If, moreover, does not tend to zero as , then (1.1) is of the strong nonlinear limit-point type.

Proof. Note that the hypotheses of Lemmas 2.3, 2.4, and 2.6 are satisfied. Let be a nontrivial solution of (1.1). Then Remark 1.1 implies is defined on , and by Lemmas 2.3 and 2.4, there are positive constants and such that

Thus, by Lemma 2.6, (2.36) holds, and if does not tend to zero as , then (2.37) holds. This proves the theorem.

Remark 3.4. Note that Lemmas 2.1, 2.2, and 2.6 are valid without the assumption that .

4. LP/LC Problem for (1.2)

One of the main assumptions in Section 3 is (2.6), which takes the form

for (1.2). It is possible to remove this condition when studying (1.2). The technique to accomplish this is contained in the following lemma; a direct computation proves the lemma.

Lemma 4.1. Equation (1.2) and the equation

are equivalent where

That is, every solution of (1.2) is a solution of (4.2) and vice versa.

Based on this lemma, results for (1.2) can be obtained by combining Lemma 4.1 and known results for (4.2), such as those that can be found, for example, in [3, 5, 7, 9, 10]. Here we only state a sample of the many such possible results.

Define

Theorem 4.2. Assume that

and either (i) or (ii) and

Then (1.2) is of the strong nonlinear limit-circle type if and only if

Proof. The conclusion follows from Theorem  2.11 in [5] applied to (4.2) and Lemma 4.1.

Our next result follows from Theorem 3.2 being applied to (4.2) and Lemma 4.1.

Theorem 4.3. Let (4.5) and either

hold. If

then (1.2) is of the nonlinear limit-point type.

Our final theorem is a strong nonlinear limit-point result for (1.2).

Theorem 4.4. Assume that (4.5) holds and

If , then (1.1) is of the nonlinear limit-point type. If, in addition, does not tend to zero as , then (1.2) is of the strong nonlinear limit-point type.

Proof. This result follows from Theorem  2.16 in [5] and Lemma 4.1 above. Note that Theorem  2.16 in [5] is proved for for , but it is easy to see from (2.34) in [5] and the end of its proof that Theorem 2.16 holds as long as does not tend to zero as .

We conclude the paper with some examples to illustrate our main results.

Example 4.5. Consider the equation

Assume that and . If satisfies either

or

then the conditions of Theorem 3.2 are satisfied, so (4.11) is of the nonlinear limit-point type.

Example 4.6. Consider the special case of (4.11) with , namely,

Then we have , so if

and either

or

holds, (4.14) is of the nonlinear limit-point type.

Example 4.7. Consider the equation

with . Note that here . Assume that

Then by Theorem 3.1, (4.18) is of the strong nonlinear limit-circle type if and only if . By Theorem 3.3, (4.18) is of the strong nonlinear limit-point type if . It is worth noting that this agrees with the well-known limit-circle criteria

of Dunford and Schwartz [19, page 1414] (also see the discussion in [2]).

For our next example we consider the case where . It may be convenient to refer to this case as the fully half-linear equation.

Example 4.8. Consider the equation

where . If , then (4.21) is of the strong nonlinear limit-circle type by Theorem 3.1. On the other hand, if and , then (4.21) is of the strong nonlinear limit-point type by Theorem 3.3.

Our final example will illustrate several of our theorems as well allow us to compare our results to those in [17, 18].

Example 4.9. Consider the equation

with , , and . Calculations show the following.

(i)Equation (4.22) is of the nonlinear limit-circle type if

(a) and (by Theorem 4.2); (b) and (by Theorem 4.2); (c) and (by Theorem 4.2); (d), , and (by Theorem 3.1); (e), , and (by Theorem 3.1).

(ii)Equation (4.22) is of the nonlinear limit-point type if

(f) and (by Theorem 4.3); (g) and (by Theorem 3.2).

Now by [17, Corollary 2.3], (4.22) is of the nonlinear limit-circle type if and . The nonlinear limit-point result [17, Theorem 2.6] does not apply to (4.22). This shows that our results substantially extend the ones in [17] in the case of nonlinear limit-circle type results and are new in the case of nonlinear limit-point results. The results in [17] follow from ours if and for and . There are errors in the proofs of the results in [18].

More specifically, for (4.22) with , that is,

the results in [17, 18] show that (4.23) is of the nonlinear limit-circle type if and . and by results in the present paper (4.23) is of the nonlinear limit-point type if and only if

(h) and ;(i) and ;(j) and is arbitrary.

Hence, the results in [17, 18] follow from ours, and our results are substantially better; note that we obtain necessary and sufficient condition for (4.23) to be of the nonlinear limit-circle type.

Acknowledgment

This research is supported by Grant no. 201/11/0768 of the Grant Agency of the Czech Republic.

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