Theorem 3.1 Suppose that (1.2) and (1.3) hold with . If
(3.1)
and
(3.2)
for every sufficiently large T, where , then every solution of (1.1) is oscillatory.
Proof Let x be a nonoscillatory solution of (1.1). Firstly, we suppose that x is an eventually positive solution of (1.1). Then there exists such that for . Let and take , , and , then from Part (i) of Lemma 2.1 we conclude
(3.3)
where H is defined as in Theorem 3.1. From (1.4), (1.2), (1.3) and (3.3), we obtain
(3.4)
where
(3.5)
and
(3.6)
Multiplying (3.4) by , we have, for ,
(3.7)
Take . Next, we consider the cases and , respectively.
Case (i). Let . Then we get , ,
(3.8)
and
(3.9)
It follows from (3.7)-(3.9) that for . Therefore, we find , which contradicts (3.1).
Case (ii). Let . Then we have ,
(3.10)
and
(3.11)
From (3.7), (3.10) and (3.11), we conclude for . Hence, we obtain , which contradicts (3.1).
Finally, we assume that x is an eventually negative solution of (1.1). Then a similar argument leads to a contradiction with (3.2). The proof is complete. □
Remark 3.1 In [20], the plus sign ‘+’ in (2.9) in Theorem 2.2, (2.13) in Theorem 2.3, (2.17) in Theorem 2.4, (3.6) in Theorem 3.2, (3.8) in Theorem 3.3 and (3.10) in Theorem 3.4 should be the minus sign ‘−’.
Remark 3.2 Theorems 2.2 and 2.3 in [20] are the special cases of our Theorem 3.1 with and , respectively. Our Theorem 3.1 improves and extends the results of Theorems 2.2-2.4 in [20] since these theorems did not include the cases and for (1.1).
The following example shows that the condition (3.1) cannot be dropped.
Example 3.1 Consider the Riemann-Liouville fractional differential equation
(3.12)
where .
In (3.12), , , , , and . Taking , and , we find that the conditions (1.2) and (1.3) are satisfied. But the condition (3.1) is not satisfied since for every sufficiently large and all , we have and
where H is defined as in Theorem 3.1. Taking , by Definition 2.1 we get
Integrating by parts twice, we obtain
(3.13)
Therefore, by Definition 2.2 we conclude
which implies that satisfies the first equality in (3.12). From (3.13) we get , which yields that satisfies the second equality in (3.12). Hence, is a nonoscillatory solution of (3.12).
Next, we consider the case when (1.5) holds, which was not considered in [20].
Theorem 3.2 Let and suppose that (1.2) and (1.5) hold with . If
(3.14)
and
(3.15)
for every sufficiently large T, where H is defined as in Theorem 3.1, then every bounded solution of (1.1) is oscillatory.
Proof Let x be a bounded nonoscillatory solution of (1.1). Then there exist constants and such that
(3.16)
Firstly, we suppose that x is a bounded eventually positive solution of (1.1). Then there exists such that for . Similar to the proof of (3.3), by Part (ii) of Lemma 2.1 we find
(3.17)
where H is defined as in Theorem 3.1. Define Φ and Ψ as in (3.5) and (3.6), respectively. Similar to the proof of (3.7), from (1.4), (1.2), (1.5) and (3.17), we get, for ,
(3.18)
Take . Next, we consider the cases and , respectively.
Case (i). Let . Then (3.8) and (3.9) are still true. From (3.16), (3.8), (3.9) and (3.18), we conclude for . Thus, we see , which contradicts (3.14).
Case (ii). Let . Then (3.10) and (3.11) are still valid. From (3.16), (3.10), (3.11) and (3.18), we conclude for . Since , we obtain , which contradicts (3.14).
Finally, we suppose that x is a bounded eventually negative solution of (1.1). Then a similar argument leads to a contradiction with (3.15). The proof is complete. □