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The authors study the nonlinear limit-point and limit-circle properties for second-order nonlinear damped differential equations of the form

In this paper, we continue the study of the nonlinear limit-point and limit-circle properties for the second-order damped equation

The limit-point/limit-circle problem has its origins in the work of Weyl [

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.

A solution of (

Under the covering assumptions here, the functions

The nonlinear limit-point/limit-circle problem originated in the work of Graef [

A solution

We can write (

Also of interest here is what we call the strong nonlinear limit-point and strong nonlinear limit-circle properties of solutions of (

A solution

A solution

Notice that if

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

It will be convenient to define the following constants:

For any continuous function

In Section

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

Let either

Let

Let

for

for

Let

The next two lemmas give us sufficient conditions for the boundedness of

In addition to (

or

Then for any nontrivial solution

Suppose, to the contrary, that there is a nontrivial solution of (

In this case (

Suppose that

Now in cases (ii) and (iii) we can actually estimate a bound from below on

Suppose case (ii) holds. Then (

Suppose case (iii) holds. Then (

Hence, (

Let (

Let

Setting

Let (

Assumption (

Consider a solution

Now, Lemma

Next, we prove that

Suppose that (

Let

Let

Suppose (

Now let (

If

If

Finally, assume (

Lemma

In this section we present our main results for (

Assume that (

Suppose (

Now suppose that (

Let (

The hypotheses of Lemmas

Let conditions (

Note that the hypotheses of Lemmas

If

One of the main assumptions is Section

Equation (

Based on this lemma, we can obtain results for (

Set

(i) Let

(ii) Assume that

The next result follows from Lemma

Let (

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

Assume that (

This follows from Lemma

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

Consider the equation

If

If

If

If

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

Again consider (

Equation (

Equation (

Theorem 2.3 in [

In our next example we have that

Consider the equation

Equation (

Equation (

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