Asymptotic Behavior of Higher-Order Quasilinear Neutral Differential Equations

and Applied Analysis 3 Proof. Assume that (1) has a nonoscillatory solution x(t) which is eventually positive and such that lim t→∞ x (t) ̸ = 0. (15) Then z satisfies z (σ (t)) = x (σ (t)) + p (σ (t)) x (τ (σ (t))) ≤ x (σ (t)) + p0x (τ (σ (t))) . (16) In view of (1), we have 0 = p0 β τ󸀠 (t)(r (τ (t)) (z (n−1) (τ (t)))α) 󸀠 + p0βq (τ (t)) x (σ (τ (t))) ≥ p0 β τ∗ (r (τ (t)) (z (τ (t)))α) 󸀠 + p0βq (τ (t)) x (σ (τ (t))) . (17) Using (16) and [9, Lemma 2], we obtain q (t) x (σ (t)) + p0βq (τ (t)) x (σ (τ (t))) = q (t) x (σ (t)) + p0βq (τ (t)) x (τ (σ (t))) ≥ Q (t) z (σ (t)) . (18) It follows from (1), (17), and (18) that (r (t) (z (t))α + p0 β τ∗ r (τ (t)) (z (τ (t)))α) 󸀠 + Q (t) z (σ (t)) ≤ 0. (19) As in the proof of [25, Theorem 2.1], we conclude that, by virtue of (1) and Lemma 1, there are two possibilities, either z (t) > 0, z (t) > 0, z (t) > 0, z (t) ≤ 0, (r (t) (z (t))α) 󸀠 ≤ 0, (20) or z (t) > 0, z (t) > 0, z (t) < 0, (r (t) (z (t))α) 󸀠 ≤ 0, (21) for all t ≥ t1, where t1 ≥ t0 is large enough. Case I. Suppose first that conditions (20) hold. Using inequality (19) and assumption η1(t) ≤ σ(t), we conclude that (r (t) (z (t))α + p0 β τ∗ r (τ (t)) (z (τ (t)))α) 󸀠 + Q (t) z (η1 (t)) ≤ 0. (22) Furthermore, by the monotonicity of z(t), there exists a constant M > 0 such that z (η1 (t)) = z (η1 (t)) z (η1 (t)) ≥ Mz (η1 (t)) . (23) Combining (22) and (23), we have (r (t) (z (t))α + p0 β τ∗ r (τ (t)) (z (τ (t)))α) 󸀠 + M1Q (t) z (η1 (t)) ≤ 0, (24) where M1 = M. An application of conditions (20) allows us to deduce that the function w (t) := r (t) (z (t))α (25) is positive and nonincreasing. By Lemma 2, we have z (t) ≥ λt n−1 (n − 1)!r1/α (t) r 1/α (t) z (t) = λt n−1 (n − 1)!r1/α (t)w 1/α (t) , (26) for every λ ∈ (0, 1) and for all sufficiently large t. Using (26) in (24), we conclude that w(t) is a positive solution of a delay differential inequality (w (t) + p0 β τ∗ w (τ (t))) 󸀠 + M1( λ (n − 1)!) γ Qγ (t) w (η1 (t)) ≤ 0. (27) Define now a function y(t) by y (t) := w (t) + p0 β τ∗ w (τ (t)) . (28) Then, by the monotonicity of w(t), y (t) ≤ w (t) (1 + p0 β τ∗ ) . (29) Substituting (29) into (27), we observe that y(t) is a positive solution of a delay differential inequality y (t)+M1( λ (n − 1)!) γ ( τ∗ τ∗ + p0 ) γ/α Qγ (t) y (η1 (t)) ≤ 0. (30) Then, by virtue of [21, Theorem 1], the associated delay differential equation y (t)+M1( λ (n − 1)!) γ ( τ∗ τ∗ + p0 ) γ/α Qγ (t) y (η1 (t)) = 0 (31) 4 Abstract and Applied Analysis also has a positive solution. However, the result by Kitamura and Kusano [15, Theorem 2] implies that, under assumption (12), (31) is oscillatory. Therefore, (1) cannot have positive solutions. Case II. Assume now that conditions (21) hold. By virtue of (15), we have that lim t→∞ z (t) ̸ = 0. (32) An application of Lemma 2 yields z (t) ≥ λ (n − 2)! t z (t) , (33) for any λ ∈ (0, 1) and for all sufficiently large t. Hence, by (19) and (33), we obtain (r(t)(z (t))α + p0 β τ∗ r (τ (t)) (z (τ (t)))α) 󸀠 + ( λ (n − 2)!) β Qβ (t) (z (σ (t))) β ≤ 0. (34) Using conditions z(t) < 0, σ(t) ≤ η2(t), and inequality (34), we have (r(t)(z (t))α + p0 β τ∗ r (τ (t)) (z (τ (t)))α) 󸀠 + ( λ (n − 2)!) β Qβ (t) (z (η2 (t))) β ≤ 0. (35) Furthermore, by the monotonicity of z(t), there exists a constant N > 0 such that (z (η1 (t))) β = (z (η1 (t))) β−λ(z(n−2) (η1 (t))) λ ≥ N(z (η2 (t))) λ. (36) Combining (35) and (36), we arrive at (r (t) (z (t))α + p0 β τ∗ r (τ (t)) (z (τ (t)))α) 󸀠 + N1( λ (n − 2)!) β Qβ (t) (z (η2 (t))) λ ≤ 0, (37) where N1 = N. Using the monotonicity of w(t), for s ≥ t ≥ t1, we conclude that r (s) z (s) ≤ r (t) z (t) . (38) Dividing (38) by r(s) and integrating the resulting inequality from t to l, we obtain z (l) ≤ z (t) + r (t) z (t) ∫ l t r (s) ds. (39) Passing to the limit as l → ∞, we deduce that 0 ≤ z (t) + r (t) z (t) A (t) , (40)

Our principal goal is to analyze the asymptotic behavior of solutions to (1) in the case where condition (3) holds.We provide sufficient conditions which ensure that solutions to (1) are either oscillatory or approach zero at infinity.In some cases, we reveal oscillatory nature of (1).However, we do not discuss in this paper nonoscillation results referring to the recent monograph by Agarwal et al. [3] for an excellent analysis of recent advances in this direction.
As usual, all functional inequalities are supposed to hold for all  large enough.Without loss of generality, we deal only with positive solutions of (1) since, under our assumptions, if () is a solution, then −() is a solution of this equation too.
In the sequel, we denote by  −1 the function which is inverse to .We also adopt the following notation for a compact presentation of our results: where the meaning of , ,  1 , and  3 will be explained later.

Asymptotic Behavior of Solutions to Even-Order Equations
In what follows, () can be both a delayed or an advanced argument.Throughout this section, in addition to the basic assumptions listed in the introduction, it is also supposed that (3) holds along with We need the following auxiliary results.
Case I. Suppose first that conditions (20) hold.Using inequality (19) and assumption  1 () ≤ (), we conclude that Furthermore, by the monotonicity of (), there exists a constant  > 0 such that Combining ( 22) and ( 23), we have where  1 =  − .An application of conditions (20) allows us to deduce that the function is positive and nonincreasing.By Lemma 2, we have for every  ∈ (0, 1) and for all sufficiently large .Using (26) in ( 24), we conclude that () is a positive solution of a delay differential inequality Define now a function () by Then, by the monotonicity of (), Substituting ( 29) into (27), we observe that () is a positive solution of a delay differential inequality Then, by virtue of [21, Theorem 1], the associated delay differential equation also has a positive solution.However, the result by Kitamura and Kusano [15,Theorem 2] implies that, under assumption ( 12), (31) is oscillatory.Therefore, (1) cannot have positive solutions.
Case II.Assume now that conditions (21) hold.By virtue of (15), we have that lim An application of Lemma 2 yields for any  ∈ (0, 1) and for all sufficiently large .Hence, by ( 19) and (33), we obtain Using conditions  (−1) () < 0, () ≤  2 (), and inequality (34), we have Furthermore, by the monotonicity of  (−2) (), there exists a constant  > 0 such that Combining ( 35) and (36), we arrive at where  1 =  − .Using the monotonicity of (), for  ≥  ≥  1 , we conclude that Dividing (38) by  1/ () and integrating the resulting inequality from  to , we obtain (39) Passing to the limit as  → ∞, we deduce that which yields Combining ( 37) and (41), we have Using again monotonicity of (), we conclude that Substituting ( 43) into (42), we observe that () is a negative solution of an advanced differential inequality which implies that () := −() is a positive solution of an advanced differential inequality Consequently, by [7, Lemma 2.3], the associated advanced differential equation also has a positive solution.However, it follows from [15, Theorem 1] that if condition (13) holds, (46) is oscillatory.Therefore, (1) cannot have positive solutions.This contradiction with our initial assumption completes the proof.
Proof.Assume that () is an eventually positive solution of (1) that satisfies (15).Proceeding as in the proof of Theorem 3, one comes to the conclusion that, for every  ∈ (0, 1), a delay differential equation and an advanced differential equation both have positive solutions.On the other hand, condition (47) and [9, Lemma 4] imply that (49) is oscillatory, a contradiction.Likewise, by virtue of [6, Theorem 2.4.1],condition (48) yields that (50) has no positive solutions.This contradiction completes the proof.
Theorem 5. Let  ≥ 2 be even and 0 <  ≤ 1. Assume that conditions ( 1 ) and ( 2 ) are satisfied, and there exist two numbers ,  ∈ R as in Theorem 3 and two functions If conditions (12) and (13) hold, the conclusion of Theorem 3 remains intact.
Proof.As above, let () be an eventually positive solution of (1) that satisfies (15).As in the proof of Theorem 3, we split the argument into two parts.
Case I. Assume first that (20) is satisfied.It has been established in the proof of Theorem 3 that the function () defined by ( 25) is positive, nonincreasing, and satisfies inequality (27).Introducing again () by (28) and using the monotonicity of (), we conclude that Substitution of (52) into (27) implies that, for sufficiently large , () is a positive solution of a delay differential inequality Then, by virtue of [21, Theorem 1], the associated delay differential equation  25) is negative, nonincreasing, and satisfies the inequality (42).Introducing again () by ( 28) and using the monotonicity of (), we conclude that Substituting (55) into (42), we observe that () is a negative solution of an advanced differential inequality That is, () := −() is a positive solution of an advanced differential inequality Then, by [ (60) Then the conclusion of Theorem 3 remains intact.
Proof.Assuming that () is an eventually positive solution of (1) that satisfies (15) and proceeding as in the proof of Theorem 5, one concludes that, for every  ∈ (0, 1), a delay differential equation and an advanced differential equation have positive solutions.On the other hand, application of condition (59) along with [9, Lemma 4] implies that (61) is oscillatory, a contradiction.Likewise, by virtue of [6, Theorem 2.4.1],condition (60) yields that (62) has no positive solutions.This contradiction completes the proof.
Note that Theorems 3-6 ensure that every solution () of ( 1) is either oscillatory or tends to zero as  → ∞ and, unfortunately, cannot distinguish solutions with different behaviors.In the remaining part of this section, we establish several results which guarantee that all solutions of (1) are oscillatory.
Therefore, we have to analyze the only remaining case, and we assume now that all the conditions in (66) are satisfied.Then, inequality (41) holds.Integrating (41) from  to ∞  − 2 times, we obtain where () is defined by (25).Taking into account that   () < 0, () ≤  3 (), and using (19), we have By virtue of monotonicity of (), there exists a constant  2 > 0 such that Combining ( 68) and (69), we obtain Using (67) in (70), we conclude that in this case, the function () defined by ( 25) is negative, nonincreasing, and satisfies the inequality Introducing again () by ( 28) and using the monotonicity of (), we arrive at (43).Substitution of (43) into (71) leads to the conclusion that () is a negative solution of an advanced differential inequality in which case the function () := −() is a positive solution of an advanced differential inequality Then, by [7, Lemma 2.3], the associated advanced differential equation also has a positive solution.However, [15, Theorem 1] implies that (74) is oscillatory under assumption (64).Therefore, (1) cannot have positive solutions.This contradiction with our initial assumption completes the proof.
Proof.Let () be a nonoscillatory solution of (1) which is eventually positive.As in the proof of Theorem 7, one can have either (20) or (65), or (66).However, conditions (47) and (48) exclude cases (20) and (65).Thus, all the inequalities in (66) should be satisfied.Along the same lines as in the proof of Theorem 7, one comes to the conclusion that an advanced differential equation has positive solutions.On the other hand, if condition (75) holds, a well-known result [6, Theorem 2.4.1]implies that (76) has no positive solutions.This contradiction completes the proof.
Proof.Let () be an eventually positive nonoscillatory solution of (1).The same argument as in the proof of Theorem 7 yields that (66) holds.Define the function () by (25).From the proof of Theorem 7, we already know that () is negative, nonincreasing, and satisfies the inequality (71).Introducing then the function () by ( 28) and using the monotonicity of (), we arrive at (55).Substituting (55) into (71), we observe that () is a negative solution of an advanced differential inequality while () := −() is a positive solution of an advanced differential inequality In this case, the result due to Baculíková [7, Lemma 2.3] allows one to deduce that the associated advanced differential equation

Asymptotic Behavior of Solutions to Odd-Order Equations
In this section, in addition to conditions ( 1 ), ( 2 ), and (3), we also assume that The validity of the following four propositions can be established in the same manner as it has been done for Theorems 3-6.Therefore, to avoid unnecessary repetition, we only formulate counterparts of Theorems 3-6 for the case of odd-order equations.
Then the conclusion of Theorem 3 remains intact.
Theorem 13.Let  ≥ 3 be odd and let 0 <  ≤ 1. Assume that conditions ( the conclusion of Theorem 3 remains intact. Note that Theorems 11-14 apply only if  is a delayed argument, () < .Hence, it is important to complement such results with the following theorems that can be applied in the case where  is an advanced argument, () ≥ .
For the case  > 1, one arrives at the contradiction with the assumptions of the theorem by using another auxiliary result obtained by Baculíková [7,Lemma 2.1].Thus, we conclude that   () > 0 eventually.The rest of the proof follows the same lines as in Theorem 3 and is omitted.
Combining the ideas exploited in the proofs of Theorems 4-6 and 15, one can derive the following results.