#### Abstract

The authors study the nonlinear limit-point and limit-circle properties for second-order nonlinear damped differential equations of the form where , , and . Examples to illustrate the main results are included.

#### 1. Introduction

In this paper, we continue the study of the nonlinear limit-point and limit-circle properties for the second-order damped equation where , , , , , , , and . We also consider the special case of (1) with , namely, Previous results of this type for damped equations can be found in the papers of Shao and Song [1], Xing et al. [2], and the present authors [3]. Later in this paper we will compare the results here with those previously known.

The limit-point/limit-circle problem has its origins in the work of Weyl [4] over 100 years ago. Weyl considered the second-order linear eigenvalue problem and classified this equation to be of the limit-circle type if every solution belongs to , that is, and to be of the limit-point type if at least one solution does not belong to , that is, The limit-point/limit-circle problem then becomes that of determining conditions on the coefficient function that allows us to distinguish between these two cases. Weyl also proved that the linear equation always has at least one square integrable solution provided . The problem then reduces to whether has one (limit-point case) or two (limit-circle case) square integrable solutions; this has come to be known as the Weyl Alternative. Weyl also showed that if is limit-circle for some , then it is limit-circle for all . In particular, this is true for , so that if we can show that the equation is limit-circle, then is limit-circle for all values of , and if is not limit-circle, then is not limit-circle for any value of . However, for we are not guaranteed that there is at least one square integrable solution.

In the years since Weyl’s original work there has been a great deal of interest in this problem due to its relationship with the solution of certain boundary value problems. By comparison, the analogous problem for nonlinear equations is relatively new and has not been as extensively studied as the linear case.

In what follows, we will only consider solutions defined on their maximal interval of existence to the right. We next define what we mean by a proper solution.

Definition 1. A solution of (1) is said to be proper if it is defined on and is nontrivial in any neighborhood of .

Remark 2. Under the covering assumptions here, the functions , , and are smooth enough so that all solutions of (1) are defined for large (see [5, Theorem 2(i)]). Moreover, all nontrivial solutions of (1) are proper if either on or (see [5, Theorem 4]).

The nonlinear limit-point/limit-circle problem originated in the work of Graef [6, 7] and Graef and Spikes [8]. The history and a survey of what is known about the linear and nonlinear problems as well as their relationships with other properties of solutions such as boundedness, oscillation, and convergence to zero, can be found in the monograph by Bartušek et al. [9] as well as the recent papers of Bartušek and Graef [1014]. The nonlinear limit-point and limit-circle properties of solutions are defined as follows (see [9] and the papers [3, 6, 7, 1018]).

Definition 3. A solution of (1) 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) will be said to be of the nonlinear limit-circle type if every solution of (1) satisfies and to be of the nonlinear limit-point type if there is at least one solution for which holds.
We can write (1) as the equivalent system where the relationship between a solution of (1) and a solution of the system (5) is given by
Also of interest here is what we call the strong nonlinear limit-point and strong nonlinear limit-circle properties of solutions of (1) as can be found in the following definitions. These notions were first introduced in [17, 18], respectively, and further studied, for example, in [10, 11]. We define the function by and the constant by

Definition 4. A solution of (1) is said to be of the strong nonlinear limit-point type if Equation (1) is said to be of the strong nonlinear limit-point type if every proper solution is of the strong nonlinear limit-point type and there is at least one proper solution.

Definition 5. A solution of (1) is said to be of the strong nonlinear limit-circle type if Equation (1) is said to be of the strong nonlinear limit-circle type if every solution is of the strong nonlinear limit-circle type.

Notice that if , (1) reduces to and moreover, if , then (11) takes the form This is the well-known half-linear equation, a general discussion of which can be found in the monograph by Došlý and Rehák [19]. Using the terminology introduced by the authors in [3, 11, 16], if , we say that (1) or (11) 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 super-half-linear cases. In [3], we considered the sub-half-linear case of (1) and (11).

The limit-point/limit-circle problem for the damped equation with was studied in [1] with being the ratio of odd positive integers and in [2] with being an odd integer. The results in both of these papers tend to be modifications of results in [68] to accommodate the damping term. We will say more about the relationship between the results in [1, 2] and the present paper in Section 5.

It will be convenient to define the following constants: Notice that , , and . We define the function by and sometimes we will make use of the assumption that If (16) holds, we define the constants For any solution of (1), we let Note that on for every solution of (1).

For any continuous function , we let and so that .

In Section 2, we give some preliminary lemmas. Section 3 contains our main results on (1), and in Section 4 we study (2). Examples to illustrate our results and to compare our results to previously known ones are given in Section 5.

#### 2. Lemmas

In this section we establish some lemmas that will be needed to prove the main results in this paper.

Lemma 1. Let either or on . Then for every nontrivial solution of (1) one has for large .

Proof. Let be a nontrivial solution of (1). Then (18) implies the existence of such that . Suppose, to the contrary, that for some . Then (18) implies and so (1) has the solution defined by But this contradicts Remark 2 and proves the lemma.

Lemma 2. Let be a solution of (1). Then(i)for , one has (ii)for , one has

Proof. Let be a solution of (1). Then it is a solution of the equation with . The expressions (20) and (21) follow from Lemma   in [16] applied to (23). Inequality (22) follows from (20).

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

Lemma 3. In addition to (16), assume that and one of the following conditions holds:(i)   ;(ii)    for large and or(iii)   and (25) holds.
Then for any nontrivial solution of (1) defined on , the function is bounded from below for large by a positive constant depending on .

Proof. Suppose, to the contrary, that there is a nontrivial solution of (1) such that By Lemma 1, for large . Let be such that for and the existence of such a follows from (16) and (24). Then, for any such that , there exist and such that and for . Then (20) implies on . From (21) (with and ), inequalities (22) and (29), and the fact that , we have Since , (27) and (30) imply It is easy to see that if and only if .
In this case (31) and give us a contradiction and the statement of the lemma holds in case (i).
Suppose that does not hold; this implies . Since is of bounded variation and , we see that From this, (28), and (30), we obtain or with . Hence, for all such that . At the same time, (31) implies for these values of . Thus, (35) holds for all and so .
Now in cases (ii) and (iii) we can actually estimate a bound from below on . Let Then (18) and (36) imply Define
Suppose case (ii) holds. Then (36) and (37) imply
Suppose case (iii) holds. Then (36) and (37) imply
Hence, (39) holds in both cases (ii) and (iii) for . So and in view of (35) and the fact that , which contradicts (25). Notice that with in case (ii) and in case (iii).

Lemma 4. Let (16) and (24) hold. Then for every solution of (1), the function is bounded.

Proof. Let be a solution of (1) and let be such that (27) holds. Suppose that is not bounded, that is, Then for any with , there exist and such that , , and
Setting and in (20)–(22), we have (29) and Since , (27) and (45) imply This contradiction proves that is bounded.

Lemma 5. Let (16) and (24) hold. Then there exists a solution of (1) and positive constants and and such that

Proof. Assumption (16) implies that is bounded, so we can choose such that
Consider a solution of (1) such that . First, we will show that Suppose (50) does not hold. Then there exist such that for . Lemma 2 (with and ), together with the facts that and , implies This contradiction shows that (50) holds.
Now, Lemma 2 (with , ) similarly implies for . From this and from (50) we have
Next, we prove that is defined on . Suppose to the contrary that is defined on with and cannot be extended to . Then . The change of variables and transforms (1) into with the noncontinuable solution defined on and . This contradicts Remark 2 applied to (55) and proves that is defined on . The conclusion of the lemma then follows from (54).

Lemma 6. Suppose that (16) and (24) hold and In addition, assume that either or holds. If is a solution of (1) with for large and some positive constants and , then Moreover, if then That is, is of the strong nonlinear limit-point type.

Proof. Let be a nontrivial solution of (1) satisfying (59) on . Then, (20) and (59) imply the existence of such that for .
Let be such that holds for . It follows from (5) that where and From (63) and (66) Moreover, (18) and (59) imply for with .
Suppose (57) holds. By l’Hôspital’s Rule, there exists such that for . From this, (65) and (67), we have for .
Now let (58) hold. Applying l’Hôspital’s Rule, there exists such that for . Then using (64), (71), and the fact that , we have for . This together with (65) and (67) implies that inequality (70) again holds. Thus, (70) holds if either (57) or (58) does. Moreover, for . Adding (68) and the left-hand inequality in (73) gives as , while adding (68) and the right-hand inequality in (73) gives as .
If is oscillatory, let be a sequence of zeros of . Then letting in (74) and (75), we see that is a strong nonlinear limit-point type solution of (1).
If is nonoscillatory, then either or for large . First we show (60) holds. Clearly (60) holds if (76) does, so suppose (77) holds. Then for large and (60) follows from (74).
Finally, assume (61) holds. From (61) and (63), it follows that there is a such that for ; hence, (62) follows from this and from (75).

Remark 6. Lemma 6 actually holds for all positive , , and regardless of their relative size.

#### 3. LP/LC Problem for (1)

In this section we present our main results for (1).

Theorem 7. Assume that (16) and (24) hold. Then (1) is of the strong nonlinear limit-circle type if and only if

Proof. Suppose (79) holds and let be any nontrivial solution of (1) defined on . Then, by Lemma 4, there is a positive constant such that . Hence, from this and (18), so is of the strong nonlinear limit-circle type. Thus, (1) is of the strong nonlinear limit-circle type.
Now suppose that (79) does not hold, that is, Let be a solution of (1) given by Lemma 5. Then there is such that on . Hence, from this and (18), Thus, either and so and (1) are not of the strong nonlinear limit-circle type.

Theorem 8. Let (16), (24), and either (57) or (58) hold. If then (1) is of the nonlinear limit-point type.

Proof. The hypotheses of Lemmas 5 and 6 are satisfied; so if is a solution given by Lemma 5, then (60) holds and the conclusion follows.

Theorem 9. Let conditions (16), (24), and either (57) or (58) hold. Assume, in addition, that either (i) ; (ii) for large and (25) holds; or (iii) and (25) holds. If (84) holds, then every nontrivial solution of (1) is of the nonlinear limit-point type. If, moreover, (61) holds, then (1) is of the strong nonlinear limit-point type.

Proof. Note that the hypotheses of Lemmas 36 are satisfied. Let be a nontrivial solution of (1) defined on . Then by Lemmas 3 and 4, there are positive constants and such that Thus, by Lemma 6, (60) holds; moreover, if (61) holds, then so does (62). The existence of a nontrivial solution of (1) is given by Lemma 5. This proves the theorem.

Remark 10. If is nondecreasing for large , then condition (25) becomes If is nonincreasing for large , then (25) becomes

#### 4. LP/LC Problem for (2)

One of the main assumptions is Section 3 is condition (24), which takes the form for (2). It is possible to remove this condition when studying (2). The technique to accomplish this is contained in the following lemma; a direct computation proves it (or see [3, Lemma 7]).

Lemma 11. Equation (2) and the equation are equivalent where That is, every solution of (2) is a solution of (89) and vice versa.

Based on this lemma, we can obtain results for (2) by combining Lemma 11 with known results for (89), such as those in [1014, 16, 18]. Here we only present a sample of the many possibilities.

Set

Theorem 12. (i) Let and Then (2) is of the nonlinear limit-circle type.
(ii) Assume that Then (2) is of the strong nonlinear limit-circle type if and only if

Proof. If is not identically a constant on the result follows from Lemma 11 and from [13, Theorem 1] applied to (89). If on , then the statement follows from Theorem 7 applied to (89) and Lemma 11. Part (ii) follows from Lemma 11 and Theorem 7 applied to (89).

The next result follows from Lemma 11 and Theorem 8 applied to (89).

Theorem 13. Let (93) hold and assume that either or If then (2) is of the nonlinear limit-point type.

Our final theorem is a strong nonlinear limit-point result for (2).

Theorem 14. Assume that (93), (97), and either (95) or (96) hold. In addition, assume that either (i)   ; (ii)    on and or (iii)    and (98) holds. Then every nontrivial solution of (2) is of the nonlinear limit-point type. If, moreover, then (2) is of the strong nonlinear limit-point type.

Proof. This follows from Lemma 11 and Theorem 9 applied to (89).

#### 5. Examples

In this section we present some examples to illustrate our results.

Example 1. Consider the equation with and . We have the following results.(i)If and , then (100) is of the strong nonlinear limit-circle type by Theorem 7.(ii)If and , then (100) is of the strong nonlinear limit-point type by Theorem 8.(iii)If , , and , then (100) is of the strong nonlinear limit-point type by Theorem 9(i).(iv)If , , and , then (100) is of the strong nonlinear limit-point type by Theorem 9(iii).

The following example will allow us to compare our results to those in [2].

Example 2. Again consider (100) with and . The following results hold.(i)Equation (100) is of the nonlinear limit-circle type if(a) and (by Theorem 12(ii));(b) and (by Theorem 12(i) if and by Theorem 12(ii) if );(c), , and (by Theorem 12(i));(d), and (by Theorem 12(ii)).(ii)Equation (100) is of the nonlinear limit-point type if(e) and (by Theorem 13);(f) and (by Theorem 13).By [2, Corollary 2.1], (100) is of the nonlinear limit-circle type if either (i) , , and , or (ii) , , and . This shows that, in the case of nonlinear limit-circle type results, the results in [2] are a special case of those in this paper.

Remark 15. Theorem  2.3 in [2] appears to show that (100) is of the nonlinear limit-point type if , , and . We have a contradiction to our case (b) in Example 2 above if , , and . The proof of Theorem  2.3 in [2] is incorrect; in their expression for the term “” is missing.

In our next example we have that in (1) (or (2)) is negative.

Example 3. Consider the equation with , . Calculations show the following.(i)Equation (101) is of the nonlinear limit-circle type if(a) and (by Theorem 7); (b) and (by Theorem 12(ii)). (ii)Equation (101) is of the nonlinear limit-point type if(c) and (by Theorem 8); (d) and (by Theorem 13); (e) and (by Theorem 13);(f), , and (by Theorem 13).

#### Acknowledgment

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