In this section, we obtain sufficient conditions for the oscillation of all solutions of Equation (E±). First we consider the Equation (E-), viz,
To prove our main results we need the following lemma, which can be found in [9].
Lemma 2.1. Let A ≥ 0, B ≥ 0 and γ ≥ 1. Then
(2.1)
and
(2.2)
Theorem 2.2. Let σ1 > τ1, σ2 > τ2 and {q
n
} and {p
n
} are positive real nonincreasing sequences. Assume that the difference inequalities
i)
(2.3)
has no eventually positive increasing solution,
ii)
(2.4)
has no eventually positive decreasing solution,
iii)
(2.5)
has no eventually positive solution.
Then every solution of Equation (E-) is oscillatory.
Proof. Let {x
n
} be a nonoscillatory solution of Equation (E-). Without loss of generality, we may assume that there exists n1 ∈ ℕ(n0) such that xn-θ> 0 for all n ≥ n1. Set .
Then
which implies that {z
n
} and {Δz
n
} are of one sign for all n2 ≥ n1. We claim that z
n
> 0 eventually. To prove it assume that z
n
< 0. Then we let
Thus
From Equation (E-), we get
Hence {u
n
} is a positive solution of inequality (2.5), a contradiction.
Therefore z
n
≥ 0. We define
(2.6)
Then, we have
(2.7)
Using the inequality (2.1) in (2.7), we obtain
Now using the inequality (2.2), we obtain
(2.8)
Consequently {y
n
} and {Δy
n
} are of one sign, eventually. Now we shall prove that y
n
> 0. If not, then let
Hence
and (2.8) implies
We obtain that {v
n
} is a positive solution of inequality (2.5), a contradiction. Next we consider the following two cases:
Case 1: Let Δz
n
< 0 for n ≥ n3 ≥ n2. We claim that Δy
n
< 0 for n ≥ n3. If not, then we have y
n
> 0, Δy
n
> 0 and Δ2y
n
≥ 0 which implies that . On the other hand, z
n
> 0, Δz
n
< 0 implies that . Then applying limits on both sides of (2.6) we obtain a contradiction. Thus Δy
n
< 0 for n ≥ n3. Using the monotonicity of {z
n
}, we now get
This together with (2.8) implies
Thus {y
n
} is a positive decreasing solution of inequality (2.4), a contradiction. Case 2: Let Δz
n
> 0 for n ≥ n3. Now we consider the following two cases. Case (i): Assume that Δy
n
< 0 for n ≥ n3. Proceeding similarly as above and using the monotonicity of {z
n
} we obtain
Then using this in (2.8) we obtain
and again {y
n
} is a positive decreasing solution of inequality (2.4), a contradiction.
Case (ii): Assume that Δy
n
> 0 for n ≥ n3. Then which in view of (2.8) implies
that is, (2.3) has a positive increasing solution, a contradiction. The proof is complete.
Remark 2.1. Theorem 2.2 permits us to obtain various oscillation criteria for Equation (E-). Moreover we are able to study the asymptotic properties of solutions of Equation (E-) even if not all assumptions of Theorem 2.2 are satisfied. If the difference inequality (2.3) has an eventually positive increasing solution then the conclusion of Theorem 2.2 is replaced by "Every solution of Equation (E-) is either oscillatory or |x
n
| → ∞ as n → ∞".
Remark 2.2. In [2, Theorem 7.6.26], the author considered the Equation (E-) with α = β = 1, p
n
≡ p, and q
n
≡ q and obtain oscillation results with (1 + a - b) > 0. Hence Theorem 2.2 generalize and improve the results of [2, Theorem 7.6.26].
Remark 2.3. Applying existing conditions sufficient for the inequalities (2.3), (2.4), and (2.5) to have no above mentioned solutions, we immediately obtain various oscillation criteria for Equation (E-).
Theorem 2.3. Let σ1 > τ1, σ2 ≥ 2, and β = α. Assume that
(2.9)
and
(2.10)
and that the difference inequality (2.5) has no eventually positive solution. Then every solution of Equation (E-) is oscillatory.
Proof. Conditions (2.9) and (2.10) are sufficient for the inequality (2.3) to have no increasing positive solution and for (2.4) to have no decreasing positive solution, respectively (see e.g., [2, Lemma 7.6.15]). The proof then follows from Theorem 2.2.
Remark 2.4. Taking into account the result of [2], we see that the absence of positive solution of (2.5) can be replaced by the assumption that for the corresponding equation
every solution of this equation are oscillatory.
Next we consider the difference Equation (E+)
and present conditions for the oscillation of all solutions of Equation (E+).
Theorem 2.4. Assume that σ1 ≥ τ1, σ2 ≥ τ2 + 2, and . If
(2.11)
has no eventually positive increasing solution, and
(2.12)
has no eventually positive decreasing solution, then every solution of (E+) is oscillatory.
Proof. Let {x
n
} be a nonoscillatory solution of (E+). Without loss of generality, we assume that there exists an integer n1 ∈ ℕ(n0) such that xn-θ> 0 for all n ≥ n1. Setting
and
(2.13)
Then z
n
> 0, y
n
> 0 and
(2.14)
Then {Δz
n
} is of one sign, eventually. On the other hand
(2.15)
Using (2.1) in (2.15) we obtain
or
(2.16)
Next we consider the following two cases:
Case 1: Assume that Δz
n
> 0. Then Δy
n
> 0. In view of (2.16), we have
(2.17)
Applying the monotonicity of z
n
, we find
(2.18)
Combining (2.17) and (2.18) we have
(2.19)
Thus
Therefore {y
n
} is a positive increasing solution of the difference inequality (2.11), a contradiction.
Case 2: Assume that Δz
n
< 0. Then Δy
n
< 0. In view of (2.16) we see that
From the monotonicity of {z
n
} we find
Combining the last two inequalities, we obtain
(2.20)
Therefore {y
n
} is a positive decreasing solution of the difference inequality (2.12), a contradiction. This completes the proof.
Theorem 2.5. Assume that σ1 ≥ τ1, σ2 ≥ τ2+2, β = α, and . If
(2.21)
has no eventually positive increasing solution, and
(2.22)
has no eventually positive decreasing solution, then every solution of (E+) is oscillatory.
Proof. Conditions (2.21) and (2.22) are sufficient for the inequality (2.11) to have no increasing positive solution and for (2.12) to have no decreasing positive solution, respectively (see e.g., [2, Lemma 7.6.15]). The proof then follows from Theorem 2.4.
Remark 2.5. When a = β = 1, Theorem 2.5 involves result of Theorem 7.6.6 of [2].
Theorem 2.6. Let , and . Suppose that there exist two positive real sequence {ϕ
n
} and {ψ
n
} with Δϕ
n
≥ 0 and Δψ
n
≤ 0, such that
(2.23)
and
(2.24)
where , are as in Theorem 2.4. If the difference inequality
(2.25)
has no eventually negative solution, and
(2.26)
has no eventually positive solution, then every solution of Equation (E+) is oscillatory.
Proof. Let {x
n
} be a nonoscillatory solution of Equation (E+). Without loss of generality, we assume that there exists an integer n1 ∈ ℕ(n0) such that xn-θ> 0 for all n ≥ n1. Define z
n
and y
n
as in Theorem 2.4. Proceeding as in the proof of Theorem 2.4, we obtain (2.16). Next we consider the following two cases.
Case 1: Assume Δz
n
> 0. Clearly Δy
n
> 0. Then as in case 1 of Theorem 2.4, we find that {y
n
} is a positive increasing solution of inequality (2.19). Let . Then B
n
> 0. Using (2.24), we have
Define B
n
= ψ
n
v
n
. Then {v
n
} is a positive solution of (2.26), a contradiction. Case 2: Assume that Δz
n
< 0. Clearly Δy
n
< 0. Then as in case 2 of Theorem 2.4, we find that {y
n
} is a positive decreasing solution of inequality (2.20). Let . Then A
n
< 0. Using (2.23), we have
Define A
n
= ϕ
n
v
n
. Then {v
n
} is a negative solution of inequality (2.25), a contradiction. This completes the proof.
From Theorem 2.6 and the results given in [7] we have the following oscillation criteria for Equation (E+).
Corollary 2.7. Let , and . Suppose that there exist two positive real sequence {ϕ
n
} and {ψ
n
} with Δϕ
n
≥ 0 and Δψ
n
≤ 0 such that (2.23) and (2.24) holds. If
(2.27)
and
(2.28)
then every solution of Equation (E+) is oscillatory.
Proof. It is known (see [7]) that condition (2.27) is sufficient for inequality (2.25) to have no eventually negative solution. On the other hand, condition (2.28) is sufficient for inequality (2.26) to have no eventually positive solution.
Remark 2.6. From the results presented in this section, we observe that when the coefficient p
n
= 0 or the condition on {p
n
} is violated the conclusion of the theorem may be replaced by "Every solution {x
n
} of equation (E±) is oscillatory or x
n
→ ∞ as n → ∞".
Once again from the proofs, we see that if q
n
= 0 or condition on {q
n
} is violated then the conclusion of the theorems may be replaced by "Every solution {x
n
} of Equation (E±) is oscillatory or x
n
→ 0 as n → ∞".