We state the following lemma, which is an extension of [3, Lemma 2] and improvement of [10, Lemma 2].

Lemma 2.1.

Let be a nonoscillatory solution of (1.9). If

then

where

Proof.

Since (1.9) is linear, we may assume that is an eventually positive solution. Then, is eventually nonincreasing. Let for all , where . In view of (2.1), there exists and an increasing divergent sequence such that

Now, consider the function defined by

We see that and for all . Therefore, there exists such that and for all . Clearly, is a nondecreasing divergent sequence. Then, for all , we have

and

Thus, for all , we can calculate

and using (2.3),

Letting tend to infinity, we see that (2.2) holds.

For the statement of our main results, we introduce

for , where .

Lemma 2.2.

Let be a nonoscillatory solution of (1.9). If there exists such that

then

where is defined in (2.3).

Proof.

Since (1.9) is linear, we may assume that is an eventually positive solution. Then, is eventually nonincreasing. There exists such that for all . Thus, for all . We rewrite (1.9) in the form

for . Integrating (2.13) from to , where , we get

which implies . From (2.13), we see that

and thus

where . Note for . Now define

By the definition (2.17), we have for all and all , which yields for all . Then, we see that

holds for all (see also [13, Corollary 2.11]). Therefore, from (2.13), we have

for . Integrating (2.19) from to , where , we get

which implies that . Thus, for all , where , and we see that

for all . By induction, there exists with and

for all . To prove now (2.12), we assume on the contrary that . Taking on both sides of (2.22), we get

which implies that , contradicting (2.11). Therefore, (2.12) holds.

Theorem 2.3.

Assume (2.1). If there exists such that (2.11) holds, then every solution of (1.9) oscillates on .

Proof.

The proof is an immediate consequence of Lemmas 2.1 and 2.2.

We need the following lemmas in the sequel.

Lemma 2.4 (see [7, Lemma 2]).

For nonnegative with , one has

Now, we introduce

for and , where .

Lemma 2.5.

If there exists such that

holds, then (2.1) is true.

Proof.

There exists such that (see the proof of Lemma 2.2). Then, Lemma 2.4 implies

which yields

In view of (2.26), taking on both sides of the above inequality, we see that (2.1) holds. Hence, the proof is done.

Theorem 2.6.

Assume that there exists such that (2.26) and (2.11) hold. Then, every solution of (1.9) is oscillatory on .

Proof.

The proof follows from Lemmas 2.1, 2.2, and 2.5.

Remark 2.7.

We obtain the main results of [7, 8] by letting in Theorem 2.6. In this case, we have for all . Note that (2.1) and (2.26), respectively, reduce to

which indicates that (2.26) is implied by (2.1).