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

In this paper, we study the nonlinear equation

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

The functions

We can write (

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 [

A solution

The properties defined above are nonlinear generalizations of the well-known linear limit-point/limit-circle properties introduced by Weyl [

Here, we are also interested in what we call the strong nonlinear limit-point and strong nonlinear limit-circle properties of solutions of (

A solution

A solution

From the above definitions we see that for an equation to be of the nonlinear limit-circle type, every solution must satisfy

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

For every nontrivial solution

Suppose, to the contrary, that (

Let

for

for

Let

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

Let

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

This contradiction to

Assume that

Let

If

Now let

Let

Condition

Consider a solution

Now, Lemma

Suppose that

Let

If (

Now let (

If

Let

Finally, we prove (

In this section we present our main results for (

Let

Let

In case

Let

The hypotheses of Lemmas

Let

Note that the hypotheses of Lemmas

Note that Lemmas

One of the main assumptions in Section

Equation (

Based on this lemma, results for (

Define

Assume that

The conclusion follows from Theorem 2.11 in [

Our next result follows from Theorem

Let (

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

Assume that (

This result follows from Theorem 2.16 in [

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

Consider the equation

Consider the special case of (

Consider the equation

For our next example we consider the case where

Consider the equation

Our final example will illustrate several of our theorems as well allow us to compare our results to those in [

Consider the equation

Equation (

Equation (

Now by [

More specifically, for (

Hence, the results in [

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