Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 215, pp. 1–15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu OSCILLATION CRITERIA FOR THIRD-ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS WITH DAMPING MARTIN BOHNER, SAID R. GRACE, IRENA JADLOVSKÁ Abstract. This paper is a continuation of the recent study by Bohner et al [9] on oscillation properties of nonlinear third order functional differential equation under the assumption that the second order differential equation is nonoscillatory. We consider both the delayed and advanced case of the studied equation. The presented results correct and extend earlier ones. Several illustrative examples are included. 1. Introduction In this article, we consider nonlinear third-order functional differential equations of the form 
α

′ ′ (t) + p(t) y ′ (t) + q(t)f y(g(t)) = 0, t ≥ t0 , (1.1) r2 r1 (y ′ )α where t0 is fixed and α ≥ 1 is a quotient of odd positive integers. Throughout the whole paper, we assume that the following hypotheses hold: (i) r1 , r2 , q ∈ C(I, R+ ), where I = [t0 , ∞) and R+ = (0, ∞); (ii) p ∈ C(I, [0, ∞)); (iii) g ∈ C 1 (I, R), g ′ (t) ≥ 0, g(t) → ∞ as t → ∞; (iv) f ∈ C(R, R) such that xf (x) > 0 and f (x)/xβ ≥ k > 0 for x 6= 0, where k is a constant and β ≤ α is the ratio of odd positive integers. By a solution of equation (1.1) we mean a function y ∈ C([Ty , ∞)), Ty ∈ I, which has the property r1 y ′ , r2 (r1 (y ′ )α )′ ∈ C 1 ([Ty , ∞)) and satisfies (1.1) on [Ty , ∞). Our attention is restricted to those solutions y of (1.1) which exist on I and satisfy the condition sup{|y(t)| : t1 ≤ t < ∞} > 0 for all t1 ≥ t0 . We make the standing hypothesis that (1.1) admits such a solution. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on [Ty , ∞) and otherwise it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory. The study on asymptotic behavior of third-order differential equations was initiated in a pioneering paper of Birkhoff [7] which appeared in the early twentieth century. Since then, many authors contributed to the subject studying different 2010 Mathematics Subject Classification. 34C10, 34K11. Key words and phrases. Oscillation; delay; advance; third order; damping; functional differential equation. c 2016 Texas State University. Submitted June 23, 2016. Published August 12, 2016. 1 2 M. BOHNER, S. R. GRACE, I. JADLOVSKÁ EJDE-2016/215 classes of equations and applying various techniques. A summary of the most significant efforts on oscillation theory of third-order differential equations as well as an extensive bibliography can be found in the survey paper by Barrett [6] and monographs by Greguš [10], Swanson [13] and the recent one of Padhi and Pati [12]. The aim of this note is to complement the very recent study [9] on asymptotic and oscillatory properties of (1.1). The method and arguments used in the present paper are different than those used in [9]. We rely on the assumption that the related second-order ordinary differential equation p(t) v(t) = 0 (1.2) (r2 v ′ )′ (t) + r1 (t) is nonoscillatory. We consider both the delay and advanced case of (1.1). While oscillation of all solutions is attained in the delay case, we state in the advanced case some new sufficient conditions for all solutions to either oscillate or converge to zero. It is interesting to note how the asymptotic behavior of (1.1) changes when the middle term is inserted. As is customary, we choose a third-order Euler-type differential equation for demonstration. Example 1.1. The equation 1 1 ′ y (t) + 3 y(t) = 0 2 4t 4t admits oscillatory solutions and the nonoscillatory solution, where the roots of the characteristic equation are λ1,2 = 1.5490 ± 0.3925i and λ3 = −0.097912. But the corresponding equation without damping 1 y ′′′ (t) + 3 y(t) = 0 4t has only nonoscillatory solutions where the characteristic roots are λ1 = 1.2696, λ2 = 1.8376, λ3 = −0.10716. Clearly, the middle term generates oscillation. y ′′′ (t) + Because of the middle term p(y ′ )α , the problem of convergence to zero as t → ∞ and/or nonexistence of a nonoscillatory solution y with yy ′ < 0 seems to be especially crucial and challenging. We recall the related existing results. Lemma 1.2 (See [4, Lemma 2.4]). Assume that α = 1. Let ρ2 be a sufficiently smooth positive function and define φ := (r2 ρ′2 )′ r1 + ρ2 p. Suppose that there exists t1 ∈ I such that ρ′2 ≥ 0, Z ∞ φ ≥ 0, φ′ ≤ 0 on [t1 , ∞), (kρ2 (s)q(s) − φ′ (s)) ds = ∞, t1 where kρ2 q − φ′ ≥ 0 on [t1 , ∞) and not identically zero on any subinterval of [t1 , ∞). If (1.2) is nonoscillatory and y is a solution of (1.1) with yL1 y < 0, then limt→∞ y(t) = 0. However, since the proof of Lemma 1.2 is based on integration by parts, it cannot be generalized for α 6= 1. The proposed method will take this problem into account. On the other hand, in [9], the authors offered a partial result for (1.1) in the sense EJDE-2016/215 OSCILLATION FOR THIRD-ORDER DIFFERENTIAL EQUATIONS 3 that either (1.1) is oscillatory or r2 (r1 (y ′ )α )′ is oscillatory (see [9, Theorem 3.1]). Oscillation of (1.1) has been left as an interesting open problem. So far, very little is known when g(t) > t. Some attempts in unifying results for both delay and advanced case have been made in [3]. We also extend these results by employing Riccati type transformation and comparison with oscillatory first-order advanced differential equations. 2. Preliminary lemmas and definitions As in [9], we define L0 y = y, L1 y = r1 (y ′ )α , L2 y = r2 (L1 y)′ , L3 y = (L2 y)′ on I. With this notation, (1.1) can be rewritten as L3 y(t) + p(t) L1 y(t) + q(t)f (y(g(t))) = 0. r1 (t) (2.1) Following [9], we define the functions: Z t Z t ds ds , R1 (t, t1 ) = , R (t, t ) = 2 1 1/α r (s) 2 t1 r1 (s) t1 Z t 1/α R2 (s, t1 ) ∗ R (t, t1 ) = ds, 1/α t1 r1 (s)  ∗ 1)  R (g(t),t if g(t) < t, R∗ (t,t1 ) R(g(t), t1 ) := R (g(t),t ) 1  1 if g(t) ≥ t, R1 (t,t1 ) for t0 ≤ t1 ≤ t < ∞. Note that the above definition of R(g(t), t1 ) will allow us to consider delayed and advanced type equations simultaneously in the proof of our main results. Throughout and without further mentioning, it will be assumed that Ri (t, t0 ) → ∞ as t → ∞ for i = 1, 2. All the functional inequalities considered in the paper are assumed to hold eventually, that is, they are satisfied for all t large enough. Now, we provide several auxiliary results that are of importance in establishing our main results. Lemma 2.1. Let v be a solution of (1.2) which is positive on [t1 , ∞). Then v′ > 0 and  ′ v ≤0 R2 (·, t1 ) (2.2) (2.3) on [t1 , ∞). Proof. Let v be a solution of (1.2) with v > 0 on [t1 , ∞). Then (r2 v ′ )′ < 0 on [t1 , ∞) so that r2 v ′ is decreasing on [t1 , ∞). First assume v ′ (t2 ) < 0 for some t2 ≥ t1 . Then r2 (t)v ′ (t) ≤ r2 (t2 )v ′ (t2 ) =: c < 0 for all t ≥ t2 and thus Z t Z t ds v(t) = v(t2 ) + v ′ (s) ds ≤ v(t2 ) + c r t2 t2 2 (s) 4 M. BOHNER, S. R. GRACE, I. JADLOVSKÁ = v(t2 ) − c Z t2 t1 EJDE-2016/215 ds + cR2 (t, t1 ) → −∞ as t → ∞, r2 (s) a contradiction. Thus (2.2) holds. Now let t ≥ t1 . Then Z t 1 v(t) ≥ v(t) − v(t1 ) = r2 (s)v ′ (s) ds ≥ r2 (t)v ′ (t)R2 (t, t1 ) t1 r2 (s) and we see that  ′ r2 (t)v ′ (t)R2 (t, t1 ) − v(t) v (t) = ≤ 0. R2 (·, t1 ) r2 (t)R22 (t, t1 ) Hence v/R2 (·, t1 ) is nonincreasing on [t1 , ∞).  Lemma 2.2 (See [5, Theorem 1.1]). Assume that v is a positive solution of (1.2) on I. Then  ′ 1  2 r1 ′ α ′  ′ r2 (r1 (y ′ )α )′ (t) + p(t)(y ′ (t))α = r2 v ( (y ) ) (t), (2.4) v(t) v for t ∈ I. If (1.2) is nonoscillatory, the classical work of Hartmann [11] has termed a nontrivial solution v of (1.2) a principal solution (unique up to a constant multiple) such that Z ∞ ds = ∞. r2 (s)v 2 (s) Since every eventually positive solution of (1.2) is increasing, the principal solution of (1.2) satisfies Z ∞ Z ∞ ds v(s) 1/α ds = ∞. (2.5) = ∞, 2 r1 (s) t0 t0 r2 (s)v (s) In the proofs of our theorems, an equivalent binomial form of (1.1) will be used repeatedly. This will also allow us to take correctly into account the possible case of L2 y being oscillatory that was missing in the previous results. Lemma 2.3 (See [9, Lemma 2.2]). Suppose that (1.2) is nonoscillatory. If y is a nonoscillatory solution of (1.1) on [t1 , ∞), t1 ≥ t0 , then there exists t2 ≥ t1 such that yL1 y > 0 (2.6) or yL1 y < 0 (2.7) on [t2 , ∞). Lemma 2.4. If y is a nonoscillatory solution of (1.1) with y(t)L1 y(t) > 0 for t ≥ t1 , t1 ∈ I. Then yL2 y ≥ 0, yL3 y < 0 on [t1 , ∞). Proof. Let y be a nonoscillatory solution of (1.1), say y(t) > 0, y(g(t)) > 0, and L1 y(t) > 0 for all t ≥ t1 . By (2.1), we see that L3 y(t) < 0 for all t ≥ t1 so L2 y is strictly decreasing on [t1 , ∞). Now assume there exists t2 ≥ t1 with L2 y(t2 ) < 0. Then for t ≥ t2 , Z t Z t L2 y(s) ds L1 y(t) = L1 y(t2 ) + (L1 y)′ (s) ds = L1 y(t2 ) + t2 t2 r2 (s) EJDE-2016/215 OSCILLATION FOR THIRD-ORDER DIFFERENTIAL EQUATIONS ≤ L1 y(t2 ) + L2 y(t2 )R2 (t, t2 ) → −∞ as 5 t → ∞, a contradiction.  Lemma 2.5 (See [9, Lemma 2.3]). Let y be a nonoscillatory solution of (1.1) with y(t)L1 y(t) > 0 for t ≥ t1 , t1 ∈ I. Then L1 y(t) ≥ R2 (t, t1 )L2 y(t), 1/α y(t) ≥ R∗ (t, t1 )L2 y(t), t ≥ t1 , (2.8) t ≥ t1 . (2.9) Lemma 2.6. Let y be a solution of (1.1) with y(t)L1 y(t) > 0 for t ≥ t1 , t1 ∈ I. If Z ∞ Z ∞ p(s) 1 + kq(s)R1β (g(s), t1 )) ds du = ∞, (2.10) ( r1 (s) t1 r2 (u) u then limt→∞ L1 y(t) = ∞. Proof. Let y be a nonoscillatory solution of (1.1), say y(t) > 0, y(g(t)) > 0, and L1 y(t) > 0 for t ≥ t1 . Then by Lemma 2.4, L2 y ≥ 0 and L1 y is increasing, so L1 y(t) ≥ L1 y(t1 ) =: ℓ > 0. Obviously, y(g(t)) ≥ ℓ1/α R1 (g(t), t1 ) for t ≥ t1 . Setting both estimates into (1.1) and integrating from t to ∞, one gets Z ∞ Z ∞ p(s) β/α ds + kℓ q(s)R1β (g(s), t1 ) ds. L2 y(t) ≥ ℓ r (s) 1 t t By integrating the last inequality from t1 to ∞, we obtain (2.10).  Lemma 2.7. Assume (2.10) holds. Let y be a solution of (1.1) with y(t)L1 y(t) > 0 for t ≥ t1 , t1 ∈ I. Then there exists t2 > t1 such that y(g(t)) ≥ R(g(t), t1 )y(t), for all t ≥ t2 . (2.11) Proof. Let y be a nonoscillatory solution of (1.1), say y(t) > 0, y(g(t)) > 0, and L1 y(t) > 0 for t ≥ t1 . We first prove (2.11) if g(t) ≤ t holds for all t ∈ I. From (2.8), we have  L y ′ L2 y(t)R2 (t, t1 ) − L1 y(t) 1 (t) = ≤ 0. R2 (·, t1 ) r2 (t)R22 (t, t1 ) Thus L1 y R2 (·,t1 ) is nonincreasing on [t1 , ∞) and moreover, this fact yields Z t 1/α 1/α R2 (u, t1 )L1 y(u) y(t) = y(t1 ) + du 1/α 1/α t1 r1 (u)R2 (u, t1 ) Z t 1/α 1/α 1/α L1 y(t) L1 y(t)R∗ (t, t1 ) R2 (u, t1 ) du = ≥ 1/α 1/α 1/α R2 (t, t1 ) t1 r1 (u) R2 (t, t1 ) (2.12) for t ≥ t1 . Consequently, 1/α 1/α  ′ y L1 y(t)R∗ (t, t1 ) − y(t)R2 (t, t1 ) (t) = ≤ 0 for all t ≥ t1 , 1/α R∗ (·, t1 ) r1 (t)(R∗ (t, t1 ))2 which implies that y R∗ (·,t1 ) is nonincreasing on [t1 , ∞). Thus, if g(t) ≥ t1 , then y(g(t)) ≥ R∗ (g(t), t1 ) y(t) = R(g(t), t1 )y(t). R∗ (t, t1 ) 6 M. BOHNER, S. R. GRACE, I. JADLOVSKÁ EJDE-2016/215 1/α Now, we show that (2.11) holds in case of g(t) ≥ t for all t ∈ I. Since L1 y is increasing on [t1 , ∞), it is easy to see that, where t3 > t2 , y(t) = y(t3 ) + Z t 1/α L1 y(s) t3 1/α r1 (s) ds 1/α ≤ y(t3 ) + L1 y(t)R1 (t, t3 ) 1/α 1/α = y(t3 ) − L1 y(t)R1 (t3 , t1 ) + L1 y(t)R1 (t, t1 ), for all t ≥ t3 . On the other hand, it follows from (2.10) that 1/α lim L1 y(t) = ∞. t→∞ Therefore, there exists t2 > t3 such that 1/α y(t) ≤ L1 y(t)R1 (t, t1 ) (2.13) on [t2 , ∞). Now, one can see that  1/α ′ y L y(t)R1 (t, t1 ) − y(t) ≥ 0 for all t ≥ t2 , (t) = 1 1/α R1 (·, t1 ) r1 (t)R12 (t, t1 ) so we conclude that y R1 (·,t1 ) is nondecreasing on [t2 , ∞). Hence, if g(t) ≥ t2 , then y(g(t)) ≥ R1 (g(t), t1 ) y(t) = R(g(t), t1 )y(t). R1 (t, t1 ) The proof is complete.  Lemma 2.8. Let y be a solution of (1.1) with y(t)L1 y(t) > 0 for t ≥ t1 , t1 ∈ I. Assume that Z ∞ p(s) ( R2 (s, t1 ) + kq(s)(R∗ (g(s), t1 ))β ) ds = ∞. (2.14) r1 (s) t1 Then limt→∞ y(t)/R∗ (t, t1 ) = 0. Proof. Let y be a nonoscillatory solution of (1.1), say y(t) > 0, y(g(t)) > 0, and L1 y(t) > 0 for t ≥ t1 . By l’Hospital’s rule, it is easy to see that lim y(t) t→∞ R∗ (t, t1 ) = lim L2 y(t). t→∞ Assume to the contrary that L2 y(t) ≥ ℓ > 0 for all t ≥ t1 . Integrating (1.1) from t1 to t and using (2.8) and (2.9), we find Z t Z t p(s) q(s)f (y(g(s))) ds L1 y(s) ds + L2 y(t1 ) ≥ t1 t1 r1 (s) Z t Z t p(s) β/α q(s)(R∗ (g(s), t1 ))β ds. R2 (s, t1 ) ds + kℓ ≥ℓ t1 t1 r1 (s) Letting t → ∞, one gets a contradiction with (2.14) and so ℓ = 0.  EJDE-2016/215 OSCILLATION FOR THIRD-ORDER DIFFERENTIAL EQUATIONS 7 3. Main results Now, we are prepared to present the main results of this paper. Lemma 3.1. Let (1.2) be nonoscillatory. If R∞ Z ∞ 1/α Z 1/α q(s) ds R2 (x, t1 )  ∞ u dx = ∞, du 1/α r2 (u)R2 (u, t1 ) t1 x r (x) (3.1) 1 then any solution y of (1.1) with yL1 y < 0 converges to zero as t → ∞. Proof. Assume to the contrary that y is a nonoscillatory solution of (1.1), say y(t) > 0, y(g(t)) > 0, and L1 y(t) < 0 for t ≥ t1 , t1 ∈ I such that lim y(t) = ℓ > 0. t→∞ Using assumption (iv) on f and (2.4) in (1.1), we have  ′ r1 r2 v 2 ( (y ′ )α )′ (t) + kq(t)v(t)y β (g(t)) ≤ 0. v Then by [5, Lemma 1.6], y satisfies r1 ′ α
′ (y ) > 0, y ′ < 0, v  r2 v 2 r1 ′ α
′  ′ <0 (y ) v on [t1 , ∞). Integrating (3.2) from t to ∞ and using y(g(t)) ≥ ℓ, we obtain Z ∞ r1 kℓβ ( (y ′ )α )′ (t) ≥ q(s)v(s) ds. v r2 (t)v 2 (t) t (3.2) (3.3) (3.4) Taking (2.2) into account, (3.4) becomes ℓ1 r1 ′ α
′ (y ) (t) ≥ v r2 (t)v(t) Z ∞ q(s) ds, t where ℓ1 = kℓβ > 0. Integrating the last inequality from t to ∞ and using (2.3) from Lemma 2.1, we arrive at R∞ Z v(t) ∞ u q(s) ds ′ α du −(y (t)) ≥ ℓ1 r1 (t) t r2 (u)v(u) R∞ Z q(s) ds R2 (t, t1 ) ∞ u ≥ ℓ1 du. r1 (t) r (u)R 2 2 (u, t1 ) t Finally, by integrating the above inequality from t1 to t, we have R∞ Z Z t 1/α 1/α q(s) ds R2 (x, t1 )  ∞ 1/α u dx. du y(t1 ) ≥ ℓ1 1/α r2 (u)R2 (u, t1 ) x t1 r1 (x) Letting t → ∞, we obtain a contradiction with (3.1). Hence ℓ = 0. The proof is complete.  Theorem 3.2. Suppose that (1.2) is nonoscillatory and that (2.10) and (2.14) hold. If there exists a constant c > 0 and a function ρ ∈ C 1 (I, R+ ) such that Z t A2 (s)  ds = ∞, (3.5) lim sup kρ(s)q(s)Rβ (g(s), t1 ) − 4B(s) t→∞ t1 8 M. BOHNER, S. R. GRACE, I. JADLOVSKÁ EJDE-2016/215 where, for t ≥ t1 , p(t) ρ′ (t) − R2 (t, t1 ), ρ(t) r1 (t) R2 (t, t1 )
1/α , B(t) = βcβ−α ρ−1 (t)(R∗ (t, t1 ))β−1 r1 (t) A(t) = (3.6) then any solution y of (1.1) is either oscillatory or converges to zero as t → ∞. Proof. Let y be a nonoscillatory solution of (1.1) on [t1 , ∞), t ≥ t1 . Without loss of generality, we may assume that y(t) > 0 and y(g(t)) > 0 for t ≥ t1 , t1 ≥ t0 . From Lemma 2.3, it follows that L1 y < 0 or L1 y > 0 on [t1 , ∞). First, we assume L1 y > 0. By Lemma 2.4, L2 y(t) ≥ 0 for t ≥ t1 . Setting the estimate (2.11) into (2.1) and using the assumption (iv) on f , we obtain L3 y(t) + p(t) L1 y(t) + kRβ (g(t), t1 )q(t)y β (t) ≤ 0 r1 (t) (3.7) on [t2 , ∞) for some t2 > t1 . We define ω=ρ L2 y > 0 on [t2 , ∞). yβ (3.8) Differentiating the function ω and using (3.7) and (2.8) in the resulting equation, we have y ′ (t) ω ′ (t) ≤ −kρ(t)q(t)Rβ (g(t), t1 ) + A(t)ω(t) − β ω. (3.9) y(t) From the definition of L1 y and (2.8), we obtain  L y(t) 1/α  R (t, t ) 1/α 1 2 1 1/α y ′ (t) = ≥ L2 y(t). r1 (t) r1 (t) Thus 1/α y ′ (t)  R2 (t, t1 ) 1/α ρ1/α (t)L2 y(t) β/α−1 y (t) ≥ y(t) ρ(t)r1 (t) y β/α (t)  R (t, t ) 1/α 2 1 = w1/α (t)y β/α−1 (t), ρ(t)r1 (t) and the inequality (3.9) becomes ω ′ (t) ≤ −kρ(t)q(t)Rβ (g(t), t1 ) + A(t)ω(t)  R (t, t ) 1/α 2 1 − βω 1+1/α (t)y β/α−1 (t) . ρ(t)r1 (t) (3.10) By Lemma 2.8, it follows from (2.14) that 0< y(t) R∗ (t, t1 ) ≤ L2 y(t1 ) =: c for all t ≥ t1 . Hence y β/α−1 (t) ≥ cβ/α−1 (R∗ (t, t1 ))β/α−1 . (3.11) From the definition of ω and (2.9), we obtain ω(t) = ρ(t) L2 y(t) ≤ ρ(t)(R∗ (t, t1 ))−α y α−β (t), y β (t) t ≥ t2 . EJDE-2016/215 OSCILLATION FOR THIRD-ORDER DIFFERENTIAL EQUATIONS 9 Using (3.11) in the above inequality, we have ω(t) ≤ cα−β ρ(t)(R∗ (t, t1 ))−β , and since α ≥ 1, w1/α−1 (t) ≥ c(α−β)(1/α−1) ρ1/α−1 (t)(R∗ (t, t1 ))−β(1/α−1) . (3.12) Using (3.11) and (3.12) in (3.10), we have ω ′ (t) ≤ −kρ(t)q(t)Rβ (g(t), t1 ) + A(t)ω(t)  R (t, t ) 1/α 2 1 − βcβ−α ρ−1 (t)(R∗ (t, t2 ))β−1 w2 (t) r1 (t) = −kρ(t)q(t)Rβ (g(t), t1 ) + A(t)ω(t) − B(t)ω 2 (t) p A(t) 2 A2 (t) B(t)ω(t) − p + = −kρ(t)q(t)Rβ (g(t), t1 ) − 4B(t) 2 B(t) ≤ −kρ(t)q(t)Rβ (g(t), t1 ) + (3.13) A2 (t) 4B(t) for all t ≥ t2 , where A and B are as in (3.6). Integrating the inequality (3.13) from t2 to t, we find Z t A2 (s)  ds ≤ ω(t2 ) − ω(t) ≤ ω(t2 ), kρ(s)q(s)Rβ (g(s), t1 ) − 4B(s) t2 which contradicts condition (3.5). Assume L1 y < 0. By Lemma 3.1, condition (4.1) ensures that any solution of (1.1) tends to zero as t → ∞. The proof is complete.  For t ≥ t1 ≥ t0 , we let Z ∞ p(s) 1 P (t) = ds, r2 (t) t r1 (s) Q1 (t) = (R∗ (g(t), t1 ))β β/α r2 (t)R2 (g(t), t1 )   Z t P (s) ds . µ(t) = exp − Z ∞ kq(s) ds, t t1 Now, we present the following comparison result for the advanced case, which complements [9, Theorem 3.5]. Theorem 3.3. Assume that g(t) ≥ t holds for all t ∈ I. Let all the hypotheses of Theorem 3.2 hold, except (3.5). If every solution of the first-order advanced equation z ′ (t) − (µ(g(t)))1−β/α Q1 (t)z β/α (g(t)) = 0 (3.14) is oscillatory, then any solution y of (1.1) is either oscillatory or converges to zero as t → ∞. Proof. Let y be a nonoscillatory solution of (1.1) on [t1 , ∞), t ≥ t1 . Without loss of generality, we may assume that y(t) > 0 and y(g(t)) > 0 for t ≥ t1 for some t1 ≥ t0 . From Lemma 2.3, it follows that L1 y(t) < 0 or L1 y(t) > 0 for t ≥ t1 . 10 M. BOHNER, S. R. GRACE, I. JADLOVSKÁ EJDE-2016/215 First, we assume L1 y > 0. Then by Lemma 2.4, L2 y > 0 on [t1 , ∞). Integrating (1.1) from t to ∞ and using the assumption (iv), we obtain Z ∞ Z ∞ p(s) L1 y(s) ds + kq(s)y β (g(s)) ds L2 y(t) ≥ r1 (s) t t (3.15) Z ∞ Z ∞ p(s) ≥ L1 y(t) kq(s) ds ds + y β (g(t)) r1 (s) t t for t ≥ t1 . If g(t) ≥ t1 , we have from (2.12) that y(g(t)) ≥ R∗ (g(t), t1 ) 1/α R2 (g(t), t1 ) 1/α L1 y(g(t)). (3.16) Setting (3.16) into (3.15), we obtain Z Z ∞ (R∗ (g(t), t1 ))β ∞ p(s) β/α ds + L1 y(g(t)) β/α kq(s) ds, L2 y(t) ≥ L1 y(t) r1 (s) t R (g(t), t1 ) t 2 which can be written as w′ (t) − P (t)w(t) − Q1 (t)w(g(t)) ≥ 0, where w(t) = r2 (t)L1 y(t). Setting z(t) = µ(t)w(t) > 0 in the above inequality and noting that µ(t) ≥ µ(g(t)), we obtain z ′ (t) − (µ(g(t)))1−β/α Q1 (t)z β/α (g(t)) ≥ 0. By [2, Lemma 2.2.10], the corresponding differential equation (3.14) also possesses an eventually positive solution, which is a contradiction. Assume L1 y < 0. By Lemma 3.1, condition (4.1) ensures that any solution tends to zero as t → ∞. The proof is complete.  The following corollary is immediate. Corollary 3.4. Assume that g(t) ≥ t and α = β. Let all the hypotheses of Theorem 3.2 hold, except (3.5). If Z g(t) 1 (3.17) lim inf Q1 (s) ds > , t→∞ e t then any solution y of (1.1) is either oscillatory or converges to zero as t → ∞. 4. Oscillation of (1.1) For delay equations, we are able to ensure nonexistence of possible nonoscillatory solutions y with yL1 y < 0. Theorem 4.1. Assume that g(t) < t for all t ∈ I. Let the hypotheses of Theorem 3.2 hold. If, moreover, there exists c∗ > 0 such that Rt Z Z t 1/α 1/α q(x) dx R2 (s, t1 )  t u lim sup ds = c∗ , (4.1) du 1/α t→∞ s r2 (u)R2 (u, t1 ) g(t) r1 (s) then (1.1) is oscillatory. EJDE-2016/215 OSCILLATION FOR THIRD-ORDER DIFFERENTIAL EQUATIONS 11 Proof. Assume to the contrary that y is a nonoscillatory solution of (1.1), say y(t) > 0, y(g(t)) > 0 and L1 y(t) < 0 for t ≥ t1 , t1 ∈ I with limt→∞ y(t) = 0. As in the proof of Lemma 3.1, we obtain that y is a solution of the inequality (3.2) satisfying (3.3) on [t1 , ∞). Since α ≥ β, there exists t2 ≥ t1 such that y β−α (g(t)) ≥ cβ−α (4.2) for all t ≥ t2 and every c > 0. Using (4.2) in (3.2), we obtain ′  r1 r2 v 2 ( (y ′ )α )′ (t) + kcβ−α q(t)v(t)y α (g(t)) ≤ 0. v Integrating (4.3) twice from s to t, t > s, one obtains Z R 
 t ut q(x)v(x)y β (g(x)) dx 1/α ′ β−α v(s) 1/α . du − y (s) ≥ kc r1 (s) r2 (u)v 2 (u) s (4.3) (4.4) Using the property (2.3) of v, (4.4) becomes ′ β−α −y (s) ≥ kc  R (s, t ) 1/α  Z t R t q(x)y α (g(x)) dx 1/α 2 1 u . du r1 (s) r2 (u)R2 (u, t1 ) s Integrating the above inequality from g(t) to t, we obtain Rt Z t Z 1/α 1/α q(x) dx R2 (s, t1 )  t β−α u y(g(t)) ≥ kc y(g(t)) du ds, 1/α g(t) s r2 (u)R2 (u, t1 ) r (s) 1 which is a contradiction with (4.1). The proof is complete.  We propose one condition in which the function p(t) is directly included. Theorem 4.2. Assume that g(t) < t for all t ∈ I. Let the hypotheses of Theorem 3.2 hold. If, moreover, there exists a constant c∗ > 0 such that Z Z t nZ t
1/α o 1  t 1 lim sup Q(u) du dv ds > 1, (4.5) 1/α t→∞ s r2 (v) v g(t) r1 (s) where Q(t) = kcβ−α q(t) − ∗ p(t)R2 (t, t1 ) >0 r1 (t)(R∗ (t, g(t)))α for all t ≥ t1 , then (1.1) is oscillatory. Proof. Assume to the contrary that y is a nonoscillatory solution of (1.1), say y(t) > 0, y(g(t)) > 0 and L1 y(t) < 0 for t ≥ t1 , t1 ∈ I with limt→∞ y(t) = 0. We consider L2 y(t). The case L2 y(t) ≤ 0 cannot holds for all large t, say t ≥ t2 ≥ t1 , since by integrating this inequality, we see  L y(t ) 1/α  L y(t ) 1/α 1 2 1 2 y ′ (t) = ≤ for all t ≥ t2 , (4.6) r1 (t) r1 (t) which contradicts the positivity of y(t). Therefore, either L2 y(t) > 0 or L2 y(t) changes sign on [t2 , ∞). We claim that Q(t) > 0 implies L2 y(t) > 0 on [t2 , ∞). 12 M. BOHNER, S. R. GRACE, I. JADLOVSKÁ EJDE-2016/215 Similarly to the proof of Lemma 3.1, we obtain that y is a positive solution of (3.2) satisfying (3.3) on [t1 , ∞). Now, for x ≥ u ≥ t1 , we obtain Z x 1/α v(s) 1/α  r1 (s) ′ y(u) − y(x) = − (y (s))α ds r1 (s) v(s) u  r (x) 1/α Z x  v(s) 1/α 1 ds ≥ −y ′ (x) v(x) r1 (s) u Z x 1/α (4.7) L1 y(x) R2 (s, t1 ) 1/α ds ≥ − 1/α r1 (s) R (x, t1 ) u = 2 1/α L y(x)R∗ (x, u) − 1 1/α . R2 (x, t1 ) Using (4.7) with u = g(t), x = t and −L1 y(t) > 0, we obtain y(g(t)) ≥ R∗ (t, g(t)) 1/α R2 (t, t1 ) 1/α (−L1 y(t)), for t ≥ t1 , e.g., R2 (t, t1 ) y α (g(t)). (R∗ (t, g(t)))α Using this inequality in (2.1), we obtain  p(t)R2 (t, t1 )  α − L3 y(t) ≥ kq(t)y β−α (g(t)) − y (g(t)). r1 (t)(R∗ (t, g(t)))α L1 y(t) ≥ − (4.8) In view of (3.1) and the fact that α ≥ β, there exists t2 ≥ t1 such that y β−α (g(t)) ≥ cβ−α for every c > 0 and for all t ≥ t2 . Thus we have  p(t)R2 (t, t1 )  α y (g(t)) −L3 y(t) ≥ kcβ−α q(t) − r1 (t)(R∗ (t, g(t)))α = Q(t)y α (g(t)) > 0. (4.9) (4.10) Hence L3 y < 0 and similarly as in the proof of Lemma 2.4, we see that L2 y ≥ 0 on [t2 , ∞). Integrating (4.10) from s to t, t > s, we obtain Z t Q(u)y α (g(u)) du. L2 y(s) ≥ s Integrating again from s to t, we obtain Z t 1 Z t 1/α 1/α . Q(u)y α (g(u)) du dv −L1 y(s) ≥ s r2 (v) v Finally, integrating the above inequality from g(t) to t, we arrive at Z Z t Z t 1/α 1  t 1 ds, Q(u) du dv y(g(t)) ≥ y(g(t)) 1/α s r2 (v) v g(t) r1 (s) which in view of (4.5) results in contradiction. The proof is complete.  The following corollary is immediate. Corollary 4.3. Assume that g(t) < t for all t ∈ I. Let the hypotheses of Theorem 3.2 hold. If, moreover, there exists a constant c∗ > 0 such that (4.1) or (4.5) holds, then (1.1) is oscillatory. EJDE-2016/215 OSCILLATION FOR THIRD-ORDER DIFFERENTIAL EQUATIONS 13 Remark 4.4. Estimate (4.5) slightly differs from the one used in [9] but it correctly takes into account a class of nonoscillatory solutions with yL2 y oscillatory. 0.20 y (t) 0.15 0.10 0.05 0 20 40 60 80 100 120 ��� ��� t Figure 1. y(t) = 2 t − sin(t) t2 ����� -����� � ′ (�) -����� -����� -����� -����� � �� �� Figure 2. y ′ (t) = �� �� � 2 sin(t) t3 − 2 t2 − cos(t) t2 5. Examples We give a couple of examples to illustrate our main results. Example 5.1. Consider the equation of Euler type a b y ′′′ (t) + 2 y ′ (t) + 3 y(λt) = 0, t ≥ 1, λ > 0, a ≤ 1/4, (5.1) t t where a, b are some positive constants. Setting k = 1 and ρ(t) = t2 , we can conclude from Theorem 3.2 that any solution y of (5.1) is oscillatory or converges to zero as t → ∞ for (2 − a)2 (2 − a)2 b> for λ ∈ (0, 1); b > for λ ≥ 1. 2 4λ 4λ 14 M. BOHNER, S. R. GRACE, I. JADLOVSKÁ EJDE-2016/215 ������ � ′′ (�) ������ ������ ������ -������ -������ -������ � �� �� �� �� � Figure 3. y ′′ (t) = − 6 sin(t) + t4 4 t3 + 4 cos(t) t3 ��� + ��� sin(t) t2 If we take λ ∈ (0, 1) and, moreover, b(λ2 (1 − ln λ) − ln λ − 1) > 4 or b(1 − λ2 ) − a  ln λ λ2 3 λ− > 1, − − 2 (1 − λ ) 2 4 4 then it follows from Corollary 4.3 that (5.1) is oscillatory. We note that none of the results in [1, 3, 4, 8, 9, 14] can guarantee oscillation of (1.1). Example 5.2. We consider the equation  ′′ a 3 (y ′ (t))1/3 + 25/12 y 1/3 (λt) = 0, (5.2) t1/4 (y ′ (t))1/3 + 16t7/4 t for t ≥ 1, λ > 0. In [5], the authors deduced that (5.2) is oscillatory for λ = 0.4 provided that a > 16.1197. The same conclusion follows from Corollary 4.3 for a > 8.1263, which is a significantly better result. We also stress that in contrast to [5], we do not require any information about the auxiliary solution v of (1.2). On the other hand, if we set λ > 1 say λ = 2, then, from Theorem 3.2, any solution of (5.2) is either oscillatory or converges to zero as t → ∞ for a > 0.2589. 6. General Remarks The results of this note complement those obtained in a recent paper [9] and can be applied to both delayed and advanced third-order differential equations with damping. As is well known, it is only the delay in (1.1) that can generate oscillation of all solutions. The class of positive solutions with L2 y oscillatory has been eliminated under the essential assumption that (1.2) is nonoscillatory. It appears that the case when (1.2) is oscillatory is still open. For instance, the equation y ′′′ (t) + y ′ (t) + 2(t3 + 2t2 sin(t) + 6t − 12 sin(t) + 9t cos(t)) y(t) = 0 t3 (2t − sin(t)) (6.1) admits a nonoscillatory solution y satisfying (2.7) with L2 y oscillatory, as depicted on Figures 1–3. Eliminating such a case seems to be the major challenge. EJDE-2016/215 OSCILLATION FOR THIRD-ORDER DIFFERENTIAL EQUATIONS 15 It might be also interesting to extend results of this paper to higher-order differential equations of the form    α
α ′ ′ r2 r1 y (n−2) (t) + p(t) y (n−2) (t) + q(t)f (y(g(t))) = 0 for n odd. This would be left to further research. References [1] R. P Agarwal, M. F. Aktas, A. Tiryaki; On oscillation criteria for third order nonlinear delay differential equations. Arch. Math.(Brno) 45.1 (2009): 1–18. [2] R. P. Agarwal, S. R. Grace, D. O’Regan; Oscillation theory for difference and functional differential equations. Springer Science & Business Media, 2013. [3] M. F. Aktas, A. Tiryaki, A. Zafer; Oscillation criteria for third-order nonlinear functional differential equations. Applied Mathematics Letters 23.7 (2010): 756–762. [4] M. F. Aktas, A. Tiryaki; Oscillation criteria of a certain class of third order nonlinear delay differential equations. Proceedings of the 6th International ISAAC Congress, Ankara, Turkey, 13–18 August 2007, edited by H. G. W. Begehr, A. O. Celebi and R. P. Gilbert, World Scientific 2009, 507–514. [5] B. Baculı́ková, J. Džurina, I. Jadlovská; Properties of the third order trinomial functional differential equations. Electronic Journal of Qualitative Theory of Differential Equations 2015.34 (2015): 1–13. [6] J H. Barrett; Oscillation theory of ordinary linear differential equations. Advances in Mathematics 3.4 (1969): 415–509. [7] G. D. Birkhoff; One the Solutions of Ordinary Linear Homogeneous Differential Equations of the Third Order ; The Annals of Mathematics 12.3 (1911): 103–127. [8] S. R. Grace; Oscillation criteria for third order nonlinear delay differential equations with damping. Opuscula Mathematica 35.4 (2015): 485–497. [9] M. Bohner, S. R. Grace, I. Saǧer, E. Tunc; Oscillation of third-order nonlinear damped delay differential equations. Applied Mathematics and Computation, 278 (2016): 21–32. [10] M. Greguš; Third Order Linear Differential Equations. Reidel, Dordrecht, The Netherlands, 1982. [11] P. Hartman; Ordinary Differential Equations. John Wiley, New York, London, Sydney, 1964. [12] S. Padhi, S.Pati; Theory of third-order differential equations. Springer 2014. [13] Ch. A. Swanson; Comparison and oscillation theory of linear differential equations. Mathematics in Science and Engineering, New York: Academic Press, 1968. [14] A. Tiryaki, M. F. Aktaş; Oscillation criteria of a certain class ot third order nonlinear delay differential equations with damping. J. Math. Anal. Appl. 325(2007): 54–68. [15] W. F. Trench; Canonical forms and principal systems for general disconjugate equations. Trans. Amer. Math. Soc. 189 (1973), 319–327. Martin Bohner Missouri University of Science and Technology, Department of Mathematics and Statistics, Rolla, Missouri 65409-0020, USA E-mail address: bohner@mst.edu Said R. Grace Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221, Egypt E-mail address: srgrace@eng.cu.edu.eg Irena Jadlovská Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00 Košice, Slovakia E-mail address: irena.jadlovska@student.tuke.sk