Lemma 1. Let (5) hold and $$ be a real valued function with $$ for large ϱ. If $$, then one of these (8) to (9) holds and if $$, then one of these (9) to (13) holds, where

Lemma 2. Let $$ and Lemma 1hold. Then, $$ for $$, where

Theorem 1. Let $$. If (5) and

hold, then (4) is oscillatory.

Proof. Let $$ be a nonoscillatory solution of (4) and $$ for ϱ ≥ ϱ₀ > μ.

If we set

then (4) becomes

Hence, we find ϱ₁ ≥ ϱ₀ such that ℒ₁z(ϱ),  1 = 1,2,3 are eventually of one sign on [ϱ₁, ∞).

Consider the potential cases 2.3 to 2.6 of Lemma 1.

Case 2.5. For $$, it follows from the Taylor series that

Also,

which implies

That is,

Consequently,

For $$, the above disparity can be composed as

Using (4), it is not difficult to check that

Expected to (16), (17) and (31) give

That is,

As a result,

which is logical inconsistency to (21) due to (15).

Case 2.6. For $$, it follows from (25) that

Since

we have

That is,

Consequently,

For $$, the above disparity can be composed as

Applying (16) and (17) in (31) yields that

Expected to (40), the above equation becomes

Therefore,

which is logical inconsistency to (22) due to (15).

Cases 2.3 and 2.4 can be managed also as the above cases.

Finally, $$ for ϱ ≥ ϱ₀. Using (16) and $$ in (4), we obtain x(ϱ) > 0 and

Proceeding as above, we see that every solution of (44) oscillates.

This completes the proof of theorem.

Theorem 2. Let $$ and ς ≥ 2ξ. If (5) to (16) and

hold, then every solution of (4) oscillates.

Proof. Let the opposition that $$ is a nonoscillatory solution of (4) and $$ for ϱ ≥ ϱ₀ > μ.

The case $$ for ϱ ≥ ϱ₀ > μ can be dealt with similarly.

Setting z(ϱ) by (23), we get (24).

Consequently, we find ϱ₁ > ϱ₀ such that z(ϱ) and ℒ_iz(ϱ),  i = 1,2,3 are eventually of one sign on [ϱ₁, ∞).

Let z(ϱ) > 0. Then, $$ for ϱ ≥ ϱ₂ > ϱ₁ and (24) becomes

Applying Lemma 1 to (52) and Theorem 1, we get contradictions to (45) to (48) expected to ς > 2ξ > ξ.

Next, z(ϱ) < 0 for ϱ ≥ ϱ₁ and $$ for ϱ ≥ ϱ₂≥ implies $$.

By Lemma 1, the cases of 2.4 to 2.8 hold for.

Case 2.4. Since

it follows that

For k ≥ ϵ + 2 > ϱ₁, summing the above inequality from ϵ = ε to $$, we have

That is,

which implies for k ≥ ϵ + 2 > ϱ₂.

Therefore, for $$,

Since (4) can be viewed as

using (58), (59), and (16) yields

Summing the last inequality form $$ to $$, we get

Hence,

which gives the contradiction to (49).

Case 2.5. From (25), we have

for $$ and which implies that

That is,

Consequently,

Also hence,

holds for $$, using (16) and (68) in (59) and summing from ϵ + ξ − ς to ϵ + ξ − ς + 1, we obtain

Therefore,

which is logical inconsistency to (50).

Case 2.6. We use (25) and it follows that

for $$. Since we have which implies that

Hence, for $$, it follows that

Consequently, (59) becomes

As a result,

which is logical inconsistency to (51).

In both cases 2.7 and 2.8, we see that lim_n⟶∞z(ϱ) = −∞.

On the other hand, z(ϱ) < 0 for ϱ ≥ ϱ₁ implies $$ for ϱ ≥ ϱ₁.

That is,

Hence, $$ is bounded and ⇒z(ϱ) is bounded, which is a contradiction.

This completes the proof of the theorem.

Example 1. Consider the NFDE,

Here,

Clearly, all conditions of Theorem 1 are satisfied.

Hence, (79) is oscillatory and $$ is one of the oscillatory solutions of (79).