Oscillation and Asymptotic Properties of Second Order Half-Linear Differential Equations with Mixed Deviating Arguments

: In this paper, we study oscillation and asymptotic properties for half-linear second order differential equations with mixed argument of the form ( r ( t )( y (cid:48) ( t )) α ) (cid:48) = p ( t ) y α ( τ ( t )) . Such differential equation may possesses two types of nonoscillatory solutions either from the class N 0 (positive decreasing solutions) or N 2 (positive increasing solutions). We establish new criteria for N 0 = ∅ and N 2 = ∅ provided that delayed and advanced parts of deviating argument are large enough. As a consequence of these results, we provide new oscillatory criteria. The presented results essentially improve existing ones even for a linear case of considered equations.

It is known (see, e.g., [7]) that if y(t) is a nonoscillatory solution of (1), then eventually either: y(t)r(t)(y (t)) α < 0 and y(t) r(t)(y (t)) α > 0 and we say that y(t) is of degree 0, and we denote the set of such solutions by N 0 or y(t)r(t)(y (t)) α > 0 and y(t) r(t)(y (t)) α > 0, and we say that y(t) is of degree two, and we denote the corresponding set by N 2 . Consequently, the set N of all nonoscillatory solutions of (1) has the following decomposition. N = N 0 ∪ N 2 .
In this paper, we establish the desired criteria when deviating argument τ(t) is of a mixed type, which means that its delayed part: are both unbounded subsets of (t 0 , ∞).
The second aim of this paper is to join the criteria obtained for N 0 = ∅ and N 2 = ∅ to establish the oscillation of (1).
Our basic results will be formulated for general Equation (1), i.e., without additional conditions imposed on function r(t). Then, we provide significant improvements for two partial cases, namely when (1) is in either canonical form, that is, when it has the following form.
When this situation occurs we employ the following function: or in noncanonical form (opposite case) when the following is the case.
In this case, we shall use the auxiliary function of the following form.

Materials and Methods
We have used the methods of mathematical analysis.

Basic Results
Our first result is applicable to both canonical and noncanonical equations. In all our results, we employ two sequences {t k } and {s k } such that the following is the case: and we have the following.
Theorem 1. Assume that there exist two sequences {t k } and {s k } satisfying (4) and (5), respectively. If the following is the case: and lim sup k→∞ τ(s k ) then, (1) is oscillatory.
Proof. Assume on the contrary, that (1) possesses an eventually positive solution y(t).
Extracting the α root and integrating from τ(t k ) to t k , we are led to the following: Now, we assume that y(t) ∈ N 2 . Then, y(t) is increasing. It is useful to notice that since τ(t) is increasing, it follows from s k ∈ A τ that (s k , τ(s k )) ⊂ A τ . Let u ∈ (s k , τ(s k )) for some k ∈ {1, 2, . . . }. By integrating (1) from s k to u, one obtains the following.
An integration of the last inequality from s k to τ(s k ) provides the following:  (7) and so N 2 = ∅, and the proof is complete.
To illustrate the above mentioned criteria, we provide the following couple of examples.

Example 1.
We consider the second order linear functional differential equation in the canonical form.

Example 2.
We consider the second order linear functional differential equation in the noncanonical form.
which means that for a > 3.25892, the class N 0 = ∅.
To eliminate class N 2 , we set s k = e π/2+2kπ , k = 1, 2, . . . . Condition (11) simplifies to the following: which ensures that N 2 = ∅ provided that a > 13.86352. Therefore, the following condition guarantees oscillation of (13). a > 13.86352 In the next two sections, we essentially improve conditions (6)-(11) for eliminations of classes N 0 and N 2 . To achieve our goals, it is necessary to study canonical and nocanonical equations separately.

Canonical Equations
We establish new monotonic properties of possible nonoscillatory solutions and then apply them to improve the above mentioned criteria. The progress will be presented via Equation (12). In the first part, we focus our considerations to eliminate class N 0 . (2) hold. Assume that there exist a sequence {t k } satisfying (4) and a positive constant γ c such that for k ∈ {1, 2, . . . }, we have the following.

Lemma 1. Let
If y(s) is a positive solution of (1) such that y(s) ∈ N 0 , then the following is the case.
It is easy to see that the last inequality, in view of (14), implies the following.
Consequently, we have the following: , t k ) and k = 1, 2, . . . , and the proof is complete.
Assume that there exists a sequence {t k } satisfying (4) and a positive constant γ c such that (14) holds. If the following is the case: then, the class N 0 = ∅ for (1) is the case.
Extracting the α root and integrating from τ(t k ) to t k , one obtains the following.
This is a contradiction, and the proof is now complete.

Corollary 3.
Let α = 1 and (2) hold. Assume that there exists a sequence {t k } satisfying (4) and a positive constant γ c such that (14) holds. If the following is the case: then the class N 0 = ∅ for (1). Now, we turn our attention to the class N 2 .

Lemma 2.
Assume that there exists a sequence {s k } satisfying (5) and a positive constant δ c such that for k ∈ {1, 2, . . . }, the following is the case.
If y(u) is a positive solution of (1) such that y(u) ∈ N 2 , then the following is the case.
By extracting the α root and integrating from s k to τ(s k ), we observe that the following is the case.
This is a contradiction, and the proof is complete now.
By picking up the above results, we are prepared to formulate the improvement of Theorem 1 provided that (1) is a canonical form. Note that if γ c = δ c = 0, Theorem 4 reduces to Theorem 1. In the opposite case, the progress that Theorem 4 yields will be demonstrated by means of Equation (12).
2 a 1 2 We used Matlab for evaluating (with γ c = 2.9483028) the following.
Finally, we conclude that the following is the case: which by Corollary 3 guarantees that N 0 = ∅. We obtain essentially better results for value of a than it has been presented in Example 1. We claim that N 2 = ∅ for a ≥ 7.364929976. To verify this, we let a = 7.364929976 and s k = e π/2+2kπ and k = 1, 2, . . . . Then, τ(s k )) = 1.5 e π/2+2kπ . Equation (18) implies the following. 1 + 0.5 sin(ln t) − 1 ≥ δ c on each (s k , τ(s k )).
Consequently, we have the following.
Condition (20) reduces to the following.
To simplify the last integral, we use the substitution t = e π/2+2kπ x, and we obtain the following. x (−3+δ c )/2 (1 + 0.5 cos ln x) δ c /2 √ 1.5 − √ x dx > 1 By employing Matlab, we find out that for δ c = 6.1300057, the following is the case.
Therefore, the following is the case: which by Corollary 4 implies that N 2 = ∅. Again we obtain better results than in Example 1. By combining both criteria, we bserve that condition a ≥ 7.364929976 implies oscillation of (12), while Theorem 1 requires a > 13.8635.

Noncanonical Equations
Now, we turn our attention to noncanonical equation. Similarly as in the previous section, we introduce new monotonic properties of nonoscillatory solutions and then apply them to improve criteria concerning noncanonical equations. The progress will be demonstrated via Equation (13). (3) hold. Assume that there exists a sequence {t k } satisfying (4) and a positive constant γ n such that for k ∈ {1, 2, . . . }, we have the following.

Lemma 3. Let
If y(s) is a positive solution of (1) such that y(s) ∈ N 0 , then the following is the case.
Assume that there exists a sequence {t k } satisfying (4) and a positive constant γ n such that (21) holds. If the following is the case: then the class N 0 = ∅ for (1).
By extracting the α root and integrating τ(t k ) to t k , we obtain the following.
This is a contradiction, and the proof is complete now.
Corollary 5. Let α = 1 and (3) hold. Assume that there exists a sequence {t k } satisfying (4) and a positive constant γ n such that (21) holds. If the following is the case: then the class N 0 = ∅ for (1). Now, we turn our attention to the class N 2 . Since the proofs of the following results are very similar to those presented for canonical equations, they will be omitted.
If y(u) is a positive solution of (1) such that y(u) ∈ N 2 , then the following is the case.

Corollary 6.
Let α = 1 and (3) hold. Assume that there exists a sequence {s k } satisfying (5) and a positive constant δ n such that (24) holds. If the following is the case: then the class N 2 = ∅ for (1).
Picking up the above results, we immediately obtain the following improvement of Theorem 1 for noncanonical (1). The progress that Theorem 7 yields will be demonstrated via equation (13).
Taking the monotonicity of the above function into account, we see that the following is the case.
Criterion (17) in terms of coefficients of (13) yields the following. By substituting s = t e (3π/2)+2kπ , one can observe that the above inequality transforms into the following. a 2 γ n 1 0.5 We employ Matlab for evaluating the following (with γ n = 2.4750025).
Thus, we have the following.

Discussion
In this paper, we tried to fulfill the certain gap in the oscillation theory concerning differential equations with mixed arguments. Our results are of high generality. Our basic criteria are applicable to the general equation, and the improved ones are applicable to canonical and noncanonical equations, separately. The progress is demonstrated via a set of examples.