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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 803137, 12 pages
http://dx.doi.org/10.1155/2011/803137
Research Article

Limit Circle/Limit Point Criteria for Second-Order Sublinear Differential Equations with Damping Term

1Department of Mathematics, Jining University, Jining 273155, China
2The Thirteen Middle School of Jining, Jining 22100, China

Received 6 September 2011; Accepted 16 October 2011

Academic Editor: Zhenya Yan

Copyright © 2011 Jing Shao and Wei Song. 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 purpose of the present paper is to establish some new criteria for the classification of the sublinear differential equation as of the nonlinear limit circle type or of the nonlinear limit point type. The criteria presented here generalize some known results in the literature.

1. Introduction

In 1910, Weyl [1] published his now classical paper on eigenvalue problems for second-order linear differential equations of the form

He classified this equation to be of the limit circle type if each solution is square integrable (denoted by ), that is,

and to be of the limit point type if at least one solution does not belong to , that is,

Weyl showed that the linear equation (1.1) always has at least one square integrable solution if Im . Thus, for second-order linear equations with Im , the problem reduces to whether (1.1) has one (limit point type) or two (limit circle type) linearly independent square integrable solutions. This is known as the Weyl Alternative. Weyl also proved that if (1.1) is of the limit circle type for some , then it is of the limit circle type for all . In particular, this is true for , that is, if we can show the following equation

is of limit circle type, then (1.1) is of the limit circle type for all values of . There has been considerable interest in this problem over the years (see [110] and references cited therein). The analogous problem for nonlinear equations is relatively new and not as extensively studied as the linear cases. For a survey of known results on 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 recent monograph [10]. In this paper, we will discuss the equation with damping term

where and are continuous, , , , , and , , say , and are positive integers, we can write , where . When , then (1.5) turns into the following equation

which is widely researched by many authors (see [10] and references cited therein).

Definition 1.1 (see [2]). A nontrivial solution of (1.5) is said to be of the nonlinear limit circle type if and it is of the nonlinear limit point type otherwise, that is, there exists a nontrivial solution satisfying Equation (1.5) is said to be of the nonlinear limit circle type if all its solutions satisfy (1.7), and it is said to be of the nonlinear limit point type if there is at least one solution satisfying (1.8).
In this paper, we will give sufficient and necessary conditions to guarantee the nonlinear limit circle type or nonlinear limit point type for (1.5).

2. Main Results

To simplify notations, let and . We define

Then (1.5) becomes

where

Note that in the transformation (2.1), is no longer an integer here. In fact, .

We begin with a boundedness result.

Theorem 2.1. If the condition holds, then each solution of (1.5) is bounded.

Proof. We rewrite (1.5) as the system and we define then we have Gronwall’ inequality and condition (2.4) imply that is bounded, so is bounded.

To prove our main limit-circle result, we write (2.2) as the system

Theorem 2.2. Assume that condition (2.4) holds and and condition is satisfied. If then (1.5) is of the nonlinear limit circle type, that is, each solution of (1.5) satisfies

Proof. Define By direct calculation, we have Since condition (2.4) and Theorem 2.1 are satisfied, the solution of (1.5) is bounded, that is, there exists a constant such that . So Now Let denote the inverse function of , we obtain that is convergent by condition (2.9). Hence, integrating , applying Gronwall’s inequality, and using condition (2.9), we obtain that is bounded, so for some constant . Condition (2.11) then implies that is of the nonlinear limit circle type.
When , the (1.5) becomes In this case, , . We get the following corollary.

Corollary 2.3. Assume condition (2.4) and and condition is satisfied. If then (2.19) is of the nonlinear limit circle type, that is, each solution of (2.19) satisfies

Example 2.4. Consider the following second-order nonlinear differential equation here , , . We can easily verify that all the conditions in Corollary 2.3 are fulfilled, so each solution of (2.24) is of the nonlinear limit circle type. We note further that the type of (2.24) cannot be determined since .

Next, we give a necessary condition for the sublinear (1.5) to be of the nonlinear limit circle type.

Theorem 2.5. Suppose condition (2.4), hold. If is a nonlinear limit circle type solution of (1.5), then

Proof. Let be a nonlinear limit circle type solution of (1.5), then is bounded by Theorem 2.1. Multiplying (1.5) by , noting that and integrating by parts, we obtain Using (1.5), we have Denote by Schwartz inequality, the boundedness of , and condition (2.14), we obtain for some constant . By Schwartz inequality, the boundedness of , and condition (2.25), for some constant .
If is not eventually monotonic, let be an increasing sequence of zeros of . Then, by (2.28), there exists some constant , such that This implies for all and some , so (2.26) holds.
If is eventually monotonic, then for all ( large enough). Using (2.28), we can repeat the type of argument used above to obtain that (2.26) holds. This completes the proof of Theorem 2.5.

The following theorem gives sufficient conditions to ensure that (1.5) being of the nonlinear limit point type.

Theorem 2.6. Suppose condition (2.4), (2.9), (2.10), (2.25) hold. If then (1.5) is of the nonlinear limit point type.

Proof. As in the proof of Theorem 2.2, we define we differentiate it to obtain Let be any nontrivial solution of (1.5) with . Theorem 2.1 implies that is bounded, so for some constant . Hence Let , then we can write . So Integrating the above inequality, we obtain Condition (2.9) implies and since , we have Rewriting in terms of and and dividing (2.42) by , we obtain If is a limit circle type solution, then is bounded, , and by condition (2.4) and (2.25), we get according to Theorem 2.5. By condition (2.34), we get By the Schwartz inequality, Noticing condition (2.35) and being bounded, we get so, by the Schwartz inequality, we have Consequently, integrating both sides of (2.44), we see that the integrand of the left side of (2.44) is bounded, but the integrand of the right side of (2.44) tends to infinity according to condition (2.36). This leads to a contradiction, so is a limit point case solution of (1.5), and (1.5) is of the nonlinear limit point type.

Similarly, when , (1.5) becomes (2.19) and we get the following corollary.

Corollary 2.7. Suppose condition (2.4), (2.10), (2.20) hold. If then (2.19) is of the nonlinear limit point type.

Acknowledgments

The authors thank the referee for his helpful suggestions on this paper which improved some of our results. This research was partially supported by the NSF of China (Grants 10801089 and 11171178).

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