On the oscillation of second-order half-linear functional differential equations with mixed neutral term

In this article, the authors establish new sufficient conditions for the oscillation of solutions to a class of second-order half-linear functional differential equations with mixed neutral term. The results obtained improve and complement some known results in the relevant literature. Examples illustrating the results are included.

By a solution of equation (1) we mean a function x : [t x , ∞) → R, t x ≥ t 0 , such that z ∈ C 1 ([t x , ∞), R), r(z ) α ∈ C 1 ([t x , ∞), R), and which satisfies (1) on [t x , ∞). We consider only those solutions x(t) of (1) that satisfy sup{|x(t)| : t ≥ T} > 0 for all T ≥ t x ; moreover, we tacitly assume that (1) possesses such solutions. Such a solution x(t) of (1) is said to be oscillatory if it has arbitrarily large zeros on [t x , ∞); otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.
However, oscillation results for mixed neutral differential and dynamic equations are relatively scarce in the literature; some results can be found, for example, in [20][21][22][23][24][25][26][27][28][29][30][31][32], and the references cited therein. We would like to point out that the results obtained in [20][21][22][23][24][25][26][27][28][29][30][31][32] require both of p 1 and p 2 to be constants or bounded functions, and hence, the results established in these papers cannot be applied to the cases where lim t→∞ p 1 (t) = ∞ and /or lim t→∞ p 2 (t) = ∞. In view of the observations above, we wish to develop new sufficient conditions which can be applied to the cases where lim t→∞ p 1 (t) = ∞ and /or lim t→∞ p 2 (t) = ∞. In this connection, the results obtained in the present paper are new, improve and complement some existing results in the relevant literature. Furthermore, the results in this paper can easily be extended to more general second-order mixed neutral differential and dynamic equations to obtain more general oscillation results. It is therefore hoped that the present paper will contribute significantly to the study of oscillation of solutions of second-order mixed neutral differential equations.

Some preliminary lemmas
In this section, we present some lemmas that will play an important role in establishing our main results. For notational purposes, we let, for any continuous function d, and throughout this paper, we define , ϕ (t) := 1 for all sufficiently large t, where g −1 1 and g −1 2 denote the inverse functions of g 1 and g 2 , respectively, and m is a function to be specified later.
where equality holds if and only if D = E.

Lemma 2.2:
Assume that conditions (C1)-(C3 ξ ) (or (C1), (C2), and (C3 ϕ )) hold, and let x(t) be an eventually positive solution of (1). Then there exists t 1 ≥ t 0 such that, for t ≥ t 1 , Proof: The proof is standard and so we omit the details of its proof.

Lemma 2.4: Let conditions (C1)-(C3 ξ ) hold and ξ(t) > 0. If x(t) is an eventually positive solution of (1) such that (2) holds, then z(t) satisfies the inequality
Proof: Let x(t) be an eventually positive solution of (1) such that (2) for t ≥ t 1 and for some t 1 ∈ [t 0 , ∞). From the definition of z(t), (see also (8.6) in [1]), we obtain . ( 8 ) Using the fact that the functions z, g 1 and g 2 are strictly increasing, and noting that g 1 (t) < t < g 2 (t), we get and Using (9) and (10) in (8) gives Thus, from (11) we have Substituting (12) into (1) gives (7) and completes the proof.  (2) for t ≥ t 1 and for some t 1 ∈ [t 0 , ∞). Following a similar argument as in the proof of Lemma 2.4, we obtain Using the fact that g 1 and g 2 are strictly increasing, and noting that g 1 (t) < t < g 2 (t), we get and Since (3) (15) and (16) that and respectively.

Main results
In this section, we present some sufficient conditions for the oscillation of all solutions of equation (1). We begin with the following result. where then every solution of equation (1) is oscillatory.
(The proof if x(t) is eventually negative is similar, so we omit the details of that case here as well as in the remaining proofs in this paper). Proceeding as in the proofs of Lemmas 2.3 and 2.4, we see that (4), (5) and (7) hold on [t 2 , ∞) ⊆ [t 1 , ∞). Define the Riccati substitution Clearly ω(t) > 0 for t ≥ t 1 , and from (7) we obtain for t ≥ t 2 . (23) From g 2 (t) ≥ h(t) and the fact that g 2 is strictly increasing, we see that t ≥ g −1 2 (h(t)). Hence, by (4) we get Substituting (24) into (23) gives for t ≥ t 2 . (25) In view of (3) and (5), it is easy to see that From (26), z(t) > 0, and z (t) > 0, (25) yields for t ≥ t 2 . An integration of (27) from t 2 to t gives t t 2 which contradicts condition (21). This completes the proof.
Proof: Let x(t) be a nonoscillatory solution of (1). Without loss of generality, we may assume that there exists Proceeding as in the proof of Theorem 3.1, we again arrive at (25) for t ≥ t 2 . From (22) and the definition of χ 1 (t), inequality (25) can be written as Applying Lemma 2.1 with λ := (α + 1)/α, Substituting (30) into (29) gives Integrating the last inequality from t 2 to t yields t t 2 which is a contradiction to our assumption (28). This proves the theorem.  ([t 0 , ∞), R) such that (3) holds. If there exists a positive function η ∈ C 1 ([t 0 , ∞), R) such that, for all sufficiently large t 1 ∈ [t 0 , ∞) and for some T ∈ (t 1 , ∞), Proof: Let x(t) be a nonoscillatory solution of (1). Without loss of generality, we may assume that there exists Proceeding as in the proof of Theorem 3.2, we again arrive at (29) which can be written as From (22) and (5), we obtain Using (33) in (32), we conclude that Completing the square with respect to ω, it follows from (34) that Integrating (35) from t 2 to t gives t t 2 which contradicts condition (31). This completes the proof of the theorem.  ([t 0 , ∞), R) such that, for all sufficiently large t 1 ∈ [t 0 , ∞) and for some T ∈ (t 1 , ∞), where then every solution of equation (1) is oscillatory.
Proof: Let x(t) be a nonoscillatory solution of (1). Without loss of generality, we may assume that there exists Proceeding exactly as in the proof of Theorem 3.1, we again arrive at (23) for t ≥ t 2 . From g 2 (t) ≤ h(t) and the fact that g 2 is strictly increasing, we see that and so Using (38) in (23) yields Taking into account that (5) holds, and using the fact that z (t) > 0, inequality (39) takes the form The remainder of the proof is similar to that of Theorem 3.1, and so the details are omitted.

Disclosure statement
No potential conflict of interest was reported by the authors.