Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 81, pp. 1–13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu OSCILLATION OF THIRD-ORDER NEUTRAL DAMPED DIFFERENTIAL EQUATIONS MIROSLAV BARTUŠEK Abstract. We study a third-order damped neutral sublinear differential equa- tion whose differential operator is non-oscillatory. Specifically, we obtain suf- ficient conditions for all solutions to be oscillatory. 1. Introduction Consider the third-order differential equation z′′′ + q(t)z′ + r(t)f ( x(σ(t) )) = 0 , t ≥ 0, (1.1) z(t) = x(t) + a(t)x ( τ(t) ) . (1.2) In this article we impose er the following assumptions: (H1) q ∈ C(R+), q(t) ≥ 0 for large t, r ∈ C(R+), r(t) > 0 for large t, R+ = [0,∞); (H2) σ ∈ C(R), R = (−∞,∞), σ(t) ≤ t for t ∈ R, limt→∞ σ(t) =∞, there exists a constant σ1 such that 0 < σ′(t) ≤ σ1 for all t ∈ R; (H3) τ ∈ C3(R), σ(t) ≤ τ(t) ≤ t for all t ∈ R, limt→∞ τ(t) =∞, and there exists a τ0 exists such that 0 < τ0 ≤ τ ′(t) for all t ∈ R; (H4) a ∈ C3(R+), there exists a number a1 such that 0 ≤ a(t) ≤ a1 for all t ∈ R+; (H5) f ∈ C(R), f(u)u > 0 for u 6= 0 and there exists a λ ∈ (0, 1] such that |f(u)| ≥ |u|λ ∀u ∈ R ; (H6) The associated second-order linear equation h′′ + q(t)h = 0 , t ≥ 0 (1.3) has a solution h(t) > 0 for all t large enough. Definition 1.1. Let T ∈ R+ and T0 = σ(T ). A function x is said to be a solution of (1.1) on [T,∞) if x is defined and continuous on [T0,∞), z ∈ C3[T,∞), and x satisfies (1.1) on [T,∞). A solution is said to be non-oscillatory if x(t) 6= 0 for all large t, otherwise it is said to be oscillatory. Equation (1.1) is oscillatory if all its solutions are oscillatory. 2010 Mathematics Subject Classification. 34C10, 34K11. Key words and phrases. Functional differential equations; neutral equation; oscillation; nonlinear equation. c©2021. This work is licensed under a CC BY 4.0 license. Submitted July 9, 2021. Published September 21, 2021. 1 2 M. BARTUŠEK EJDE-2021/81 In recent years, a great attention has been paid to qualitative theory of third- order neutral differential equations. Such equations have applications in mathe- matical modeling in biology and physics, see for example [10, 11, 12, 15]. A great effort has been devoted to oscillation theory of the damped equations of the forms x′′′ + q(t)x′ + r(t)f ( x(σ(t)) ) = 0, (1.4)( r2(t)(r1(t)x′)′ )′ + q(t)x′(t) + r(t)f ( x(σ(t)) ) = 0 (1.5) with ri ∈ C(R+), ri(t) > 0 for t ∈ R+ and i = 1, 2. An equation is said to have Property A if every solution is either oscillatory or x(t)x′(t) < 0 for all large t. Sufficient (and or necessary) conditions have been studied under which equation either (1.4) or (1.5) has Property A. Equation (1.4) has been studied in [8] (where there is a nice review of the results.), in [2], and the references therein. For studies of (1.5), see for example [1, 3, 14]. Property A has been generalized for the neutral differential equation z′′′ + r(t)f ( x(σ(t)) ) = 0 (1.6) in [13], and for the equation( r2(t)(r1(t)z′)′ )′ +R(t)x ( σ(t) ) = 0 (1.7) in [5, 6], where ri ∈ C(R+), R ∈ C(R+), ri > 0 for i = 1, 2, R > 0, and z is given by (1.2). An interesting question was solved in [6] for (1.7) in the canonical case, i.e. when ∫ ∞ 0 1 ri(t) dt =∞ for i = 1, 2. (1.8) Reference [5] shows sufficient conditions for (1.5) (with q ≡ 0) no having a solution x such that z(t)z′(t) < 0 for large t. Since (1.3) is non-oscillatory and q ≥ 0, every eventually positive solution of (1.3) is nondecreasing for large t, and the following holds, see [9]. Lemma 1.2. Equation (1.3) has a solution h which is positive and nondecreasing for t ≥ t0 ≥ 0 and ∫ ∞ t0 dt h2(t) =∞ , ∫ ∞ t0 h(t) dt =∞ . (1.9) If ∫∞ 0 tq(t) dt < ∞ then limt→∞ h(t) ∈ (0,∞). Also if ∫∞ 0 tq(t) dt = ∞, then limt→∞ h(t) =∞. Note that if a solution h satisfies (1.9), then a positive constant times h also satisfies (1.9). This solution is called a principal solution. Definition 1.3. Let h be a principal solution of (1.3) such that h(t) > 0 on [t∗,∞) ⊂ R+. In the case ∫∞ 0 tq(t) dt <∞, h is chosen such that limt→∞ h(t) = 1. It is easy to see that for for t ≥ t∗, (1.1) can be rewritten as( h2(t) ( z′ h(t) )′)′ + h(t)r(t)f ( x(σ(t)) ) = 0 . (1.10) For t ≥ t∗, we denote the quasiderivatives of z as follows: z[1](t) = z′(t) h(t) , z[2](t) = h2(t) ( z[1](t) )′ z[3](t) = ( z[2](t) )′ . (1.11) EJDE-2021/81 THIRD-ORDER NEUTRAL DAMPED DIFFERENTIAL EQUATIONS 3 Then we rewrite (1.1) as (1.10) and using (1.11), z[3](t) + h(t)r(t)f ( x(σ(t)) ) = 0 . (1.12) Note, that For t ≥ t∗, (1.10) is a special case of the equation( r2(t) ( r1(t)z′ )′)′ +R(t)f ( x(σ(t)) ) = 0, (1.13) where r1(t) = 1 h(t) , r2(t) = h2(t) , R(t) = h(t)r(t) . (1.14) Because of (1.9), equation (1.13) is in canonical form, i.e. (1.8) holds. Our goal is to find sufficient conditions for (1.1) to be oscillatory. A crucial problem is to prove nonexistence of non-oscillatory solutions such that z(t)z′(t) < 0 for large t. So, if f(u) = u on R it is possible to use results from [6] for equation (1.7) with (1.14). However, a very restrictive assumption τ(σ(t)) ≡ σ(τ(t)) is used in [6]. We give sufficient conditions for the nonexistence of such solutions without this assumption and without the assumption f(u) ≡ u. Note, that our assumption 0 < σ′(t) ≤ σ1 is not assumed in [6]. Let N be the set of all non-oscillatory solutions of (1.1) which are defined on subintervals of R+ and which are positive for large t. We shall study only the set N . Non-oscillatory solutions which are negative for large t can be study by a similar way. It is known (see, e.g., [6, Lemma ]) that N can be divided into two subsets N = N0 ∪N1 where z is given by (1.2) and N0 = { x ∈ N : z(t) > 0, z[1](t) < 0, z[2(t) > 0, z[3](t) < 0 for large t } , N1 = { x ∈ N : z[i](t) > 0, i = 0, 1, 2, z[3](t) < 0 for large t } ] . In this article, τ−1 and σ−1 denote the inverse functions of τ and σ, respectively. Also we define N00 = { x ∈ N0 : lim t→∞ z(t) = 0 } , N01 = { x ∈ N0 : lim t→∞ z(t) ∈ (0,∞) } . For simplicity, for t ≥ 0, we define r∗(t) = min { r(σ−1(t)), r(σ−1(τ(t))) } . (1.15) Note, that by (H3), σ−1 ( τ(t) ) ≥ t , (1.16) where e denotes the Euler number. 2. Preliminaries Here we state some auxiliary results which will be needed later. Lemma 2.1. Let x ∈ N be defined on [T,∞) and T0 = σ(T ). Let A ∈ C[T,∞) be positive and ∫ ∞ T A(t) |x ( σ(t) ) |λ dt <∞ . (2.1) Then ∫ ∞ T A∗(t) |z(t)|λ dt <∞ (2.2) 4 M. BARTUŠEK EJDE-2021/81 where A∗(t) = min ( A(σ−1(t)), A(σ−1(τ(t))) ) . Proof. Let x ∈ N and t1 ≥ T be such that x(t) > 0 for t ≥ σ(t1). The substitution s = σ(t) and (2.1) yield 1 σ1 ∫ ∞ σ(t1) A ( σ−1(s) ) xλ(s) ds ≤ ∫ ∞ σ(t1) A ( σ−1(s) ) xλ(s) ds σ′(σ−1(s)) = ∫ ∞ t1 A(t)xλ ( σ(t ) dt <∞ . (2.3) From this, applying substitution s = τ(t), for t0 = τ−1(σ(t1)), we obtain τ0 σ1 ∫ ∞ t0 A ( σ−1(τ(t)) ) xλ ( τ(t) ) dt ≤ 1 σ1 ∫ ∞ t0 A ( σ−1(τ(t)) ) xλ ( τ(t) ) τ ′(t) dt = 1 σ1 ∫ ∞ σ(t1) A ( σ−1(s) ) xλ(s) ds <∞ . (2.4) We have zλ(t) ≤ ( x(t) + a1x ( τ(t) ))λ ≤M(xλ(t) + xλ ( τ(t) ) (2.5) with M = 2λ(1 + aλ1 ). As τ is increasing and σ(t1) ≤ t0, (2.3), (2.4), (2.5) imply min { 1 σ1 , τ0 σ1 }∫ ∞ t0 A∗(t)zλ(t) dt ≤M { 1 σ1 ∫ ∞ σ(t1) A ( σ−1(t) ) xλ(t) dt } + τ0 σ1 ∫ ∞ t0 A ( σ−1(τ(t)) ) xλ ( τ(t) ) dt <∞ . Hence, (2.2) is valid. � Lemma 2.2. There exist k0 ≥ k > 0 such that k0t ≥ h(t) ≥ k exp {∫ t 0 sq(s) ds } for t ≥ t∗ (2.6) where t∗ and h are given by Definition 1.3. Moreover, if ε > 0, then h(u) h(v) ≤ (1 + ε) u v for u ≥ v ≥ 1 + ε ε t∗ > t∗ . (2.7) Proof. As for (2.6), see [4, Lemma 2] and 1.3. Now we prove (2.7). We have ε 1+εv ≥ t ∗ which is equivalent to v − t∗ ≥ v 1+ε . From this we have u− t∗ v − t∗ ≤ u v − t∗ ≤ (1 + ε) u v for u ≥ v ≥ 1 + ε ε t∗ . (2.8) As h′(t) > 0 and h′ is non-increasing for t ≥ t∗, we obtain h(t) = h(t∗) + ∫ t t∗ h′(s) ds ≥ h′(t)(t− t∗) . This inequality and (2.8) imply h(u) h(v) = exp {∫ u v h′(s) h(s) ds } ≤ exp {∫ u v ds s− t∗ } = u− t∗ v − t∗ ≤ (1 + ε) u v for u ≥ v ≥ 1+ε ε t∗; hence, (2.7) holds. � Lemma 2.3. Let x ∈ N and T ≥ 0 be such that x is positive on [σ(T ),∞). EJDE-2021/81 THIRD-ORDER NEUTRAL DAMPED DIFFERENTIAL EQUATIONS 5 (i) If x ∈ N0 and ∫∞ 0 t q(t) dt <∞, then∫ ∞ T t2r∗(t)zλ(t) dt <∞ . (2.9) (ii) If x ∈ N , then∫ ∞ T exp {∫ t 0 sq(s) ds } r∗(t) zλ(t) dt <∞ . (2.10) Proof. (i) Let x ∈ N0 and t0 ≥ max(T, t∗) be such that x(t) > 0 for t ≥ σ(t0), z[i](t) 6= 0 for t ≥ t0, i = 1, 2. Then limt→∞ z(t) = C ≥ 0. It is easy to see that (1.9), (1.11) and x ∈ N0 imply limt→∞ z[i](t) = 0 for i = 1, 2. Hence (1.11) and (H5) yield z[1](t) = − ∫ ∞ t h−2(s)z[2](s) ds, z[2](t) = − ∫ ∞ t z[3](s) ds = ∫ ∞ t h(s)r(s)xλ ( σ(s) ) ds (2.11) for t ≥ t0. As t0 ≥ t∗, by Definition 1.3 there exist positive constants C1 and C2 such that C1 ≤ h(t) ≤ C2 for t ≥ t0. From this, (1.11), (2.11), and Fubini’s theorem, we have ∞ > z(t0)− C = − ∫ ∞ t0 h(s)z[1](s) ds ≥ ∫ ∞ t0 h(s) ∫ ∞ s 1 h2(v) ∫ ∞ v h(w)r(w)xλ ( σ(w) ) dw dv ds ≥ (C1 C2 )2 ∫ ∞ t0 ∫ ∞ s ∫ ∞ v r(w)xλ ( σ(w) ) dw dv ds = C3 ∫ ∞ t0 ∫ ∞ s (w − s)r(w)xλ ( σ(w) ) dw ds = 1 2 C3 ∫ ∞ t0 (w − t0)2r(w)xλ ( σ(w) ) dw ≥ C3 8 ∫ ∞ 2t0 w2r(w)xλ ( σ(w) ) dw with C3 = (C1/C2)2. From this and Lemma 2.1 (with A(t) = t2r(t), T = 2t0), I := ∫ ∞ 2t0 min {( σ−1(t) )2 r ( σ−1(t) ) , ( σ−1(τ(t)) )2 r ( σ−1(τ(t)) )} zλ(t)dt <∞ . Using (1.15) and (1.16) we obtain (2.9). (ii) Let x ∈ N be defined on [T,∞). Then there exists t0 ≥ max(T, t∗) such that x(t) > 0 for t ≥ σ(t0), z[2](t) > 0 for t ≥ t0 . From this, (1.11), (1.12), (H5), and Lemma 2.2, we have ∞ > z[2](t0) ≥ z[2](t0)− z[2](∞) = − ∫ ∞ t0 z[3](s) ds = ∫ ∞ t0 h(t)r(t)f ( x(σ(t)) ) dt 6 M. BARTUŠEK EJDE-2021/81 ≥ k ∫ ∞ t0 exp {∫ t t0 sq(s) ds } r(t)xλ(σ(t)) dt . Therefore, (2.10) follows Lemma 2.1 (with A(t) = exp { ∫ t 0 sq(s) ds } r(t)). � 3. Main results We begin with the following lemma which states sufficient conditions for N0 to be empty in case f(u) = u. Lemma 3.1. Let f(u) ≡ u on R and let one of the following assumptions hold. (i) There exists a function ξ ∈ C(R+) such that t < ξ(t) < σ−1(τ(t)) for large t and either I =∞ or 2σ1(τ0 + a1) τ0e < I <∞ (3.1) where I := lim inf t→∞ ∫ t τ−1(σ(ξ(t))) r∗(s) h(s) h(ξ(s)) ( ξ(s)− s )2 ds ; (ii) there exists a function η ∈ C(R+) such that τ−1(σ(t)) ≤ η(t) ≤ t for large t and either J =∞ or 2σ1 ( 1 + a1 τ0 ) < J <∞ (3.2) where J := lim sup t→∞ h(t) h(σ−1(τ(η(t))) ( σ−1(τ(η(t)))− t )2 ∫ t η(t) r∗(s) ds . Then N0 = ∅. Proof. Let x ∈ N0. Then there exists T ≥ t∗ (see Definition 1.3) such that for t ≥ T and i = 0, 1, 2, h(t) > 0 , x ( σ(t) ) > 0 , (−1)iz[i](t) > 0, (3.3) and t < ξ(t) < σ−1(τ(t)) (resp. τ−1(σ(t)) ≤ η(t) ≤ t) in case (i) (resp. (ii)). From this, (H2), and (H3), we obtain σ1 τ0 ( z[2](σ−1(τ(t))) )′ + h ( σ−1(τ(t)) ) r ( σ−1(τ(t)) ) x ( τ(t) ) ≤ 1 (σ−1(τ(t)))′ ( z[2](σ−1(τ(t))) )′ + h ( σ−1(τ(t)) ) r ( σ−1(τ(t)) ) x ( τ(t) ) = 0 , where ′ = d dt . Similarly, σ1 ( z[2](σ−1(t)) )′ + h ( σ−1(t) ) r ( σ−1(t) ) x(t) ≤ 1 (σ−1(t))′ ( z[2](σ−1(t)) )′ + h ( σ−1(t) ) r ( σ−1(t) ) x(t) = 0 . Hence, using (H4) for t ≥ T , we have[ σ1z [2] ( σ−1(t) ) + a1σ1 τ0 z[2] ( σ−1(τ(t)) )]′ + h ( σ−1(τ(t)) ) r∗(t)z(t) ≤ [ σ1z [2] ( σ−1(t) ) + a1σ1 τ0 z[2] ( σ−1(τ(t)) )]′ + h ( σ−1(t) ) r ( σ−1(t) ) x(t) + a1h ( σ−1(τ(t)) ) r ( σ−1(τ(t)) ) x ( τ(t) ) ≤ 0 . (3.4) EJDE-2021/81 THIRD-ORDER NEUTRAL DAMPED DIFFERENTIAL EQUATIONS 7 Furthermore, for v ≥ t ≥ T , we have −z[1](t) ≥ z[1](v)− z[1](t) = ∫ v t z[2](s) h2(s) ds ≥ z[2](v) h2(v) (v − t) and thus using (1.11), and integration from u to v, with v ≥ u, imply z(u) ≥ z[2](v) h2(v) ∫ v u h(s)(v − s) ds ≥ h(u) 2h2(v) (v − u)2z[2](v) . (3.5) Assuming Case (i), we define v(t) = σ1z [2] ( σ−1(t) ) + a1σ1 τ0 z[2] ( σ−1(τ(t)) ) (3.6) for t ≥ T . Then (3.4) and (3.5) with u = t, v = ξ(t) imply v′(t) + h(t)h(σ−1(τ(t))) 2h2(ξ(t)) ( ξ(t)− t )2 r∗(t)z[2] ( ξ(t) ) ≤ 0 . As ξ(t) < σ−1(τ(t)) and h is nondecreasing, we obtain v′(t) + h(t) 2h(ξ(t)) ( ξ(t)− t )2 r∗(t)z[2] ( ξ(t) ) ≤ 0 . (3.7) As z[2] > 0 is non-increasing, (3.6) implies v(t) ≤ [ σ1 + a1σ1 τ0 ] z[2] ( σ−1(τ(t)) ) , and, hence, z[2] ( ξ(t) ) ≥ τ0 σ1(τ0 + a1) v ( τ−1(σ(ξ(t))) ) . Substituting this into (3.7) yields v′(t) + τ0 2σ1(τ0 + a1) h(t) h(ξ(t)) ( ξ(t)− t )2 r∗(t)v ( τ−1(σ(ξ(t))) ) ≤ 0 . (3.8) Using (3.1), τ−1(σ(ξ(t))) < t, and the well-known criterion for (3.8) to be oscillatory (see [7, Theorem 2.1.1]) implies a contradiction. Now assume Case (ii). According to (3.5) for u = t, v = σ−1(τ(η(t))) ≥ u we have z(t) ≥ h(t) 2h2(σ−1(τ(η(t)))) ( σ−1(τ(η(t)))− t )2 z[2] ( σ−1(τ(η(t))) ) . (3.9) Integrating (3.4) from η(t) to t, we have σ1z [2] ( σ−1(η(t)) ) + a1σ1 τ0 z[2] ( σ−1(τ(η(t))) ) ≥ σ1z [2] ( σ−1(t) ) + a1σ1 τ0 z[2] ( σ−1(τ(t)) ) + ∫ t η(t) h ( σ−1(τ(s)) ) r∗(s)z(s) ds ≥ h ( σ−1(τ(η(t))) ) z(t) ∫ t η(t) r∗(s) ds . From this, (3.3), (3.9), and z[2] > 0 and decreasing, we have σ1 ( 1 + a1 τ0 ) z[2] ( σ−1(τ(η(t))) ) ≥ h ( σ−1(τ(η(t))) ) z(t) ∫ t η(t) r∗(s) ds 8 M. BARTUŠEK EJDE-2021/81 ≥ h(t) 2h(σ−1(τ(η(t)))) ( σ−1(τ(η(t)))− t )2 ∫ t η(t) r∗(s) dsz[2] ( σ−1(τ(η(t))) ) . This contradicts (3.2) and proves the statement. � Note, that some ideas from [6] are used in the second part of the proof of Lemma 3.1. Theorem 3.2. (i) Let either∫ ∞ 0 tq(t) dt <∞ and ∫ ∞ 0 t2r∗(t) dt =∞ (3.10) or ∫ ∞ 0 tq(t) dt =∞ and ∫ ∞ 0 exp {∫ t 0 sq(s) ds } r∗(t) dt =∞ . (3.11) Then the set N01 is empty. (ii) If ∫ ∞ 0 exp {∫ t 0 sq(s) ds } tλr∗(t) dt =∞ (3.12) then the set N1 is empty. Proof. (i) Let x ∈ N01 be such that x(t) > 0 for t ∈ [σ(T ),∞). Then limt→∞ z(t) = C ∈ (0,∞) and (3.10), (resp. (3.12)) contradicts (2.9) (resp. (2.10)). (ii) Let x ∈ N1. From this and from (1.11), positive constants T0 ≥ T and M exist such that z(t) ≥ Mt for t ≥ T0. Now, this fact and (3.12) contradict (2.10). � Now we can formulate the main results. For ξ ∈ C(R+) and η ∈ C(R+), we denote I1 = lim inf t→∞ ∫ t τ−1(σ(ξ(t))) r∗(s) ( ξ(s)− s )2 ds , (3.13) J1 = lim sup t→∞ ( σ−1(τ(η(t)))− t )2 ∫ t η(t) r∗(s) ds , (3.14) I2 = lim inf t→∞ ∫ t τ−1(σ(ξ(t))) s ξ(s) r∗(s) ( ξ(s)− s )2 ds , (3.15) J2 = lim sup t→∞ t σ−1(τ(η(t))) ( σ−1(τ(η(t)))− t )2 ∫ t η(t) r∗(s) ds . (3.16) Lemma 3.3. Suppose K > 0, C > 0, ∫∞ 0 tq(t) dt < ∞, f(u) ≥ Ku for u ∈ [0, C] and one of the following assumptions holds. (i) There exists a function ξ(t) ∈ C(R+) such that t ≤ ξ(t) < σ−1(τ(t)) for large t, and either I1 =∞ or M := 2σ1(τ0 + a1) K τ0e < I1 <∞ ; (ii) There exists a function η ∈ C(R+) such that τ−1(σ(t)) ≤ η(t)) ≤ t for large t, and either J1 =∞ or 2σ1 K ( 1 + a1 τ0 ) < J1 <∞ . EJDE-2021/81 THIRD-ORDER NEUTRAL DAMPED DIFFERENTIAL EQUATIONS 9 Then (1.1) has no solution x ∈ N0 such that z(t) ≤ C for large t. Proof. (i) Let x ∈ N0 and T ≥ 0 be such that 0 < x ( σ(t) ) ≤ C , 0 < z(t) ≤ C for t ≥ T , t ≤ ξ(t) < σ−1 ( τ(t) ) for t ≥ T , 1− ε ≤ h(t) ≤ 1 for t ≥ T , (3.17) where ε = { 1 2 − M 2I1 if I1 <∞, 1 2 if I1 =∞. (3.18) Note, that (3.17) and (3.18) imply h(t) h(ξ(t)) ≥ 1− ε = 1 2 + M 2I1 > I1 + 3M 4I1 = NM I1 (3.19) with N = I1 4M + 3 4 for t ≥ T in case I1 <∞. Then x is the solution of the equation z′′′ + q(t)z′ + r0(t)x ( σ(t) ) = 0 (3.20) for t ≥ T with r0(t) = f(x(σ(t))) x(σ(t)) r(t) ≥ Kr(t) . (3.21) Now we apply Lemma 3.1 to (3.20), considering the assumption posed in I. If I1 =∞, then using (3.19) and (3.21), I =∞. Let I1 <∞. Then (3.19) and (3.21) imply lim inf t→∞ ∫ t τ−1(σ(ξ(t))) min { r0(σ−1(s), r0(σ−1(τ(s)))} h(s) h(ξ(s)) ( ξ(s)− s )2 ds ≥ lim inf t→∞ KMN I1 ∫ t τ−1(σ(ξ(t))) r∗(s) ( ξ(s)− s )2 ds > 2σ1(τ0 + a1) τ0e ; hence, all assumptions of Lemma 3.1 applied on (3.20) are satisfied andN0 is empty. The contradiction proves the statement. Statement (ii) can be proved similarly. � Lemma 3.4. Suppose K > 0, C > 0, ∫∞ 0 tq(t) dt = ∞, f(u) ≥ Ku for u ∈ [0, C] and one of the following assumptions holds (i) There exists a function ξ(t) ∈ C(R+) such that t ≤ ξ(t) < σ−1(τ(t)) for large t, and either I2 =∞ or M = 2σ1(τo + a1) Kτ0e < I2 <∞ ; (ii) There exists a function η ∈ C(R+) such that τ−1(σ(t) ≤ η(t)) ≤ t for large t, and either J2 =∞ or 2σ1(τ0 + a1) Kτ0 < J2 <∞ . Then (1.1) has no solution x ∈ N0 such that z(t) ≤ C for large t. 10 M. BARTUŠEK EJDE-2021/81 Proof. It is similar as the one of Lemma 3.3; instead of (3.19), we apply (2.7) with 0 < ε ≤ I2−M I2+M in case (i). We obtain h(t) h(ξ(t)) ≥ (1 2 + M 2I2 ) t ξ(t) . Case (ii) is similar. � The following results solve our problem for λ = 1. Recall, that I1, J1, I2 and J2 are given by (3.13)–(3.16), respectively. Theorem 3.5. Suppose λ = 1, ∫∞ 0 tq(t) dt <∞ and one of the following assump- tions holds. (i) There exists a function ξ(t) ∈ C(R+) such that t ≤ ξ(t) < σ−1(τ(t)) for large t, and either I1 =∞ or 2σ1(τ0 + a1) τ0e < I1 <∞ ; (ii) there exists a function η ∈ C(R+) such that τ−1(σ(t)) ≤ η(t) ≤ t for large t, and either J1 =∞ or 2σ1 ( 1 + a1 τ0 ) < J1 <∞ . Then the set N0 is empty. If, moreover,∫ ∞ 0 tr∗(t) dt =∞ , then (1.1) is oscillatory. Proof. Let x ∈ N0. As λ = 1, we can put K = 1 and C = 1 + 2 limt→∞ z(t). Then a contradiction follows from Lemma 3.3. The nonexistence of x ∈ N1 follows from Theorem 3.2(ii), (3.12) and ∫∞ 0 tq(t) dt <∞. � Theorem 3.6. Suppose λ = 1, ∫∞ 0 tq(t) dt =∞, and one of the following assump- tions holds: (i) There exists a function ξ(t) ∈ C(R+) such that t ≤ ξ(t) < σ−1(τ(t)) for large t, and either I2 =∞ or 2σ1(τ0 + a1) τ0e < I2 <∞ ; (ii) there exists a function η ∈ C(R+) such that τ−1(σ(t)) ≤ η(t)) ≤ t for large t, and either J2 =∞ or 2σ1 ( 1 + a1 τ0 ) < J2 <∞ . Then the set N0 is empty. If, moreover,∫ ∞ 0 exp {∫ t 0 sq(s) ds } tr∗(t) dt =∞ then (1.1) is oscillatory. The proof of the above theorem is similar to that of Theorem 3.5. Theorem 3.7. Let λ ∈ (0, 1), ∫∞ 0 tq(t) dt < ∞ and one of the following assump- tions hold. EJDE-2021/81 THIRD-ORDER NEUTRAL DAMPED DIFFERENTIAL EQUATIONS 11 (i) Thee exists a function ξ(t) ∈ C(R+) such that t ≤ ξ(t) < σ−1(τ(t)) for large t, and either I1 =∞ or I1 > 0; (ii) there exists a function η ∈ C(R+) such that τ−1(σ(t)) ≤ η(t)) ≤ t for large t, and either J1 =∞ or J1 > 0. Then set N00 is empty. If, moreover,∫ ∞ 0 tλr∗(t) dt =∞ (3.22) then (1.1) is oscillatory. Proof. (i) Let x ∈ N00. Put K = 3σ1(τ0 + a1) τ0eI1 and C = K− 1 1−λ in case I1 <∞ and K = C = 1 if I1 =∞. Then f(u) ≥ uλ ≥ Ku on [0, C]. Hence, all assumptions of Lemma 3.3 are satisfied and Lemma 3.3 contradicts x ∈ N00. The nonexistence of x ∈ N01 ∪N1 follows from (3.22) and Theorem 3.2. Statement (ii) can be proved similarly. � Theorem 3.8. Let λ ∈ (0, 1), ∫∞ 0 tq(t) dt = ∞ and one of the following assump- tions hold. (i) There exists a function ξ(t) ∈ C(R+) such that t ≤ ξ(t) < σ−1(τ(t)) for large t, and either I2 =∞ and ∫ ∞ 0 exp {∫ t 0 sq(s) ds } tλr∗(t) dt =∞ (3.23) or 0 < I2 <∞ and ∫ ∞ 0 exp {∫ t 0 sq(s) ds } r∗(t) dt =∞ ; (3.24) (ii) there is a function η ∈ C(R+) such that τ−1(σ(t)) ≤ η(t)) ≤ t for large t, and either J2 =∞ and ∫ ∞ 0 exp {∫ t 0 sq(s) ds } tλr∗(t) dt =∞ or 0 < J2 <∞ and ∫ ∞ 0 exp {∫ t 0 sq(s) ds } r∗(t) dt =∞ . Then (1.1) is oscillatory. Proof. (i) Suppose (3.23) holds. Then Theorem 3.2(ii) implies N1 = ∅. Let x ∈ N0. Then limt→∞ z(t) = C0 ∈ [0,∞) and z(t) ≤ C = 2C0 + 1 for large t. Moreover, f(u) ≥ K u for [0, C] where K = Cλ−1. Then all assumptions of Lemma 3.4(i) are satisfied whose statement contradicts x ∈ N0. Suppose (3.24) holds. Then Theorem 3.2 impliesN01∪N1 = ∅. The nonexistence of x ∈ N00 can be proved as in Theorem 3.7(i). The proof of (ii) is similar. � 12 M. BARTUŠEK EJDE-2021/81 4. Examples Remark 4.1. (i) It follows from the assumptions of Theorems 3.5–3.8 that σ(t) ≤ τ(t) for large t as it is supposed in (H3). (ii) In Theorems 3.5–3.8, it is possible to choose e.g. ξ(t) = 1 2 (t + σ−1(τ(t))); similarly, we can choose either τ(t) ≤ t, τ(t) 6≡ t in any neighborhood of ∞ and η(t) ≡ τ(t), or τ(t) ≡ t for large t and η(t) = 1 2 (t+ σ(t)), σ(t) < t. Example 4.2. Consider the equation z′′′ + q(t)z′ + r(t)|x(C1t)|λ sgnx(C1t) = 0 (4.1) with z(t) = x(t) + a(t)x(C0t) where 0 < λ ≤ 1, 0 < C1 < C0 ≤ 1, r(t) ≥ r0 tv for large t with r0 > 0 and v ≥ 0, 0 ≤ a(t) ≤ a1 <∞ and (H6) holds. Put ξ(t) = C2t, 1 < C2 < C0 C1 . Let λ = 1. Then N0 is empty for (4.1) if either v < 3 or v = 3 and (C2 − 1)2 log C0 C1C2 > m C0 + a1 r0eC0C2 1 where m = 1 for ∫∞ 0 tq(t) dt < ∞ and m = C2 for ∫∞ 0 tq(t) dt = ∞ (see Theo- rems 3.5 and 3.6). Moreover, (4.1) is oscillatory if v ≤ 2. Let 0 < λ < 1. Equation (4.1) is oscillatory if v ≤ λ+ 1 (Theorems 3.7 and 3.8). Example 4.3. Consider the equation z′′′ + q(t)z′ + r(t)|x(t− C1)|λ sgnx(t− C1) = 0 (4.2) with z(t) = x(t) + a(t)x(t − C0) where 0 ≤ C0 < C1, 0 ≤ a(t) ≤ a1 < ∞ on R+, r(t) ≥ r0t v, v ≥ 0 and (H6) holds. Put C2 ∈ (0, C1 − C0), ξ(t) = t+ C2. If λ = 1, then Theorems 3.5 and 3.6 imply that (4.2) is oscillatory if either v > 0 or v = 0 and C2 2 [C1 − C0 − C2] > 2σ1(τ0 + a1) r0τ0e . If λ ∈ (0, 1), then Theorems 3.7 and 3.8 imply that (4.2) is oscillatory if v ≥ 0. Acknowledgements. This research was supported by the Grant GA 20-11846S from the Czech Grant Agency. References [1] M. F. Aktaş, D. Çakmak, A. Tiryaki; On the qualitative behaviors of solutions of third order nonlinear differential equations, Comput. Math. Appl., 62 (4) (2011), 2029-2036. doi: 10.1016/j.camwa.2011.06.045 [2] M. Bartušek, M. Cecchi, Z. Došlá, M. Marini; Oscillation for third-order differential equa- tion with deviating argument, Abstr. Appl. Anal., (2010), 19 pp. Art. ID 278962. doi: 10.1155/2010/278962 [3] M. Bartušek, M. 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Engin., 42 (1970), 259-305. Miroslav Bartušek Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic Email address: bartusek@math.muni.cz 1. Introduction 2. Preliminaries 3. Main results 4. Examples Acknowledgements References