# New Oscillation Criteria for Third-Order Nonlinear Functional Differential Equations

Authors: Quanxin Zhang; Li Gao; Shouhua Liu; Yuanhong Yu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper discusses oscillatory and asymptotic behavior of solutions of a class
of third-order nonlinear functional differential equations. By using the generalized Riccati
transformation and the integral averaging technique, three new sufficient conditions which
insure that the solution is oscillatory or converges to zero are established. The results obtained
essentially generalize and improve the earlier ones.


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## Body

## 1. Introduction

As is well known, the comparison and separation theory of zeros distribution for second-order homogeneous linear differential equations established by Ladde et al. lays a foundation of oscillation theory for differential equations. During one and a half century, oscillation theory of differential equations has developed quickly and played an important role in qualitative theory of differential equations and the theory of boundary value problem. Oscillation theory of differential equations has been widely used in areas of physics, mechanics, radio technology, control system, sciences of life, economic relations, and population growth. The oscillations are physical phenomena which widely exist in physics and technological sciences, such as the oscillation of building and machine, electromagnetic vibration in radio technology and optical science, self-excited vibration in control system, sound vibration, beam vibration in synchrotron accelerator, the vibration sparked for burning rocket engine, and the complicated oscillation in chemical reaction. All different phenomena can be unified into an oscillation theory through an oscillation equation. There are many books on the oscillation theory, about which we can refer to [1].The oscillation theory of third-order nonlinear functional differential equations has been widely applied in research of a lossless high-speed computer network and physical sciences. In this paper, we are concerned with oscillatory behavior of a third-order nonlinear functional differential equation as follows:(1)(r2(t)[(r1(t)x′(t))′]α)′+q(t)f(x[g(t)])=0,t≥t0,
where α is the ratio of positive odd integers. We have the following hypotheses:(A1)
r1(t),r2(t),q(t)∈C1([t0,∞),(0,∞)) and r1(t),r2(t) satisfy
(2)∫t0∞1r1(s)ds=∞,∫t0∞(1r2(s))1/αds=∞;(A2)
f(u)∈C(R,R), such that f(u)/uα≥K>0, u≠0;(A3)
g(t)∈C1([t0,∞),R), g′(t)≥0, and lim⁡t→∞⁡g(t)=∞.By a solution of (1), we mean a nontrivial function x(t) satisfying (1) which has the properties x(t)∈C1([Tx,∞),R) for Tx≥t0 and r2(t)[(r1(t)x′(t))′]α∈C1([Tx,∞),R). Our attention is paid to those solutions of (1) which satisfy sup⁡{|x(t)|:t≥T}>0 for all T≥Tx. A solution x of (1) is said to be oscillatory on [Tx,∞) if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.In recent years, there have been numerous researches or many research activities concerning the oscillation and nonoscillation of solutions of three-order functional differential equations, which are special cases of (1), and for recent contributions, we refer to [2–8]. Consider
(3)x′′′(t)+q(t)x[g(t)]=0,t≥t0;(4)(r2(t)[(r1(t)x′(t))′]α)′+q(t)xα(t)=0,t≥t0;(5)(r2(t)(r1(t)x′(t))′)′+q(t)f(x[g(t)])=0,t≥t0;(6)(r(t)[x′′(t)]α)′+q(t)f(x[g(t)])=0,t≥t0.Parhi and Padhi [2] studied asymptotic behavior of solutions of (3). By using the integral averaging technique, Baculíková et al. [3] obtained sufficient conditions which insured that the solution of self-liner ordinary differential equation (4) was oscillatory or converges to zero. Mojsej [4] established the comparison results which insured that the solution of (5) was oscillatory or converges to zero. By the integral averaging technique, Saker [5] gave some oscillatory results of (5) when the condition g(t)=t-σ holds. Several authors had proved some oscillatory results of (6) by method of comparison; see [6–8]. In this paper we intend to use Riccati transformation and the integral averaging technique to obtain some sufficient conditions which guarantee that every solution x(t) of (1) is oscillatory or converges to zero. Our results generalize and improve the corresponding theorems established in [3, 5].


## 2. Several Lemmas

Lemma 1.
Assume thatx(t) is a positive solution of (1). Then, there exists T∈[t0,∞) such that either(I)
x(t)>0, x′(t)>0, (r1(t)x′(t))′>0, t≥T,or(II)
x(t)>0, x′(t)<0, (r1(t)x′(t))′>0, t≥T.The proof is similar to that of [3, Lemma 1] or [7, Lemma 1].Throughout this paper, for sufficiently largeT, we denote
(7)R1(t,T)=∫Tt(1r2(s))1/αds,R2(t,T)=∫TtR1(s,T)r1(s)ds,Q(t)=(K∫t∞q(s)ds)1/α.In order to make the definition ofQ(t) meaningful, we denote
(8)∫t∞q(s)ds<∞.Lemma 2.
Assume thatx(t) is a positive solution of (1) which satisfies case (I) in Lemma 1. Then there exists T∈[t0,∞), such that
(9)x′(t)>R1(t,T)r1(t)r21/α(t)(r1(t)x′(t))′,t≥T,(10)x(t)>R2(t,T)r21/α(t)(r1(t)x′(t))′,t≥T.
Assume that (8) and g(t)≥t hold. Then
(11)x(t)<1Q(t)r21/α(t)(r1(t)x′(t))′,t≥T.Proof.
PickT∈[t0,∞) so that x[g(t)]>0 for t≥T. Using (1), we obtain
(12)(r2(t)[(r1(t)x′(t))′]α)′=-q(t)f(x[g(t)])≤-Kq(t)xα[g(t)]<0,t≥T.
Then, r2(t)[(r1(t)x′(t))′]α is strictly decreasing on [T,∞). We get
(13)r1(t)x′(t)>r1(t)x′(t)-r1(T)x′(T)=∫Tt(r2(s)[(r1(s)x′(s))′]α)1/αr21/α(s)ds≥(r2(t)[(r1(t)x′(t))′]α)1/α∫Tt(1r2(s))1/αds,
and, hence, we have
(14)x′(t)>R1(t,T)r1(t)r21/α(t)(r1(t)x′(t))′,t≥T.
By integrating both sides of the above inequality from T to t, it yields
(15)x(t)>R2(t,T)r21/α(t)(r1(t)x′(t))′,t≥T.
Furthermore, by integrating both sides of (1) from t to ∞ and noting that x′(t)>0, g′(t)≥0, g(t)≥t, we obtain
(16)r2(t)[(r1(t)x′(t))′]α>∫t∞q(s)f(x[g(s)])ds≥K∫t∞q(s)xα[g(s)]ds≥K∫t∞q(s)xα(s)ds≥(K∫t∞q(s)ds)xα(t).
Then,
(17)x(t)<1Q(t)r21/α(t)(r1(t)x′(t))′.
This completes the proof.Lemma 3.
Assume thatx(t) is a positive solution of (1) which satisfies case (II) in Lemma 1. Furthermore,
(18)∫t0∞1r1(t)∫t∞[1r2(s)∫s∞q(u)du]1/αdsdt=∞.
Then, lim⁡t→∞x(t)=0.Proof.
Assume thatx(t) is a positive solution of (1) which satisfies case (II) in Lemma 1. Then, x(t) is decreasing and lim⁡t→∞x(t)=l≥0. We assert that l=0. If not, then x[g(t)]≥l>0, t≥T. Integrating (1) from t to ∞, we get
(19)-r2(t)[(r1(t)x′(t))′]α≤-K∫t∞q(s)xα[g(s)]ds≤-Klα∫t∞q(s)ds,t≥T.
Hence, we have
(20)-(r1(t)x′(t))′≤-l[1r2(t)∫t∞Kq(s)ds]1/α.
Integrating the above inequality from t to ∞, we obtain
(21)r1(t)x′(t)≤-lK1/α∫t∞[1r2(s)∫s∞q(u)du]1/αds.
Integrating the last inequality again from T to t, we have
(22)x(t)-x(T)≤-lK1/α∫Tt1r1(s)∫s∞[1r2(u)∫u∞q(v)dv]1/αduds.
Since condition (18) holds, we obtain lim⁡t→∞x(t)=-∞, which contradicts x(t)>0. Hence, l=0. This completes the proof.


## 3. Main Results

In this section, we obtain three new oscillatory criteria for (1) by using the generalized Riccati transformation and integral averaging technique of Philos-type [9]. Let
(23)D={(t,s):t≥s≥T};D0={(t,s):t>s≥T}.A functionH(t,s)∈C′(D,R) is said to have the property of P and denote H∈P if it satisfies(i)
H(t,t)=0, t≥T; H(t,s)>0, (t,s)∈D0;(ii)
∂H(t,s)/∂s≤0 and it is continuous.The following are the main results of this paper.Theorem 4.
Let (8), (18), and g(t)≥t hold. Assume that there exist ρ,φ∈C1([t0,∞),(0,∞)), H(t,s)∈P, and m(t,s)∈C(D,R), such that
(24)-∂H(t,s)∂s-R*(s,T)ρ(s)H(t,s)=m(t,s)ρ(s)H(t,s),
and for arbitrary T0≥T, one has
(25)lim⁡sup⁡t→∞1H(t,T0)×∫T0t[H(t,s)Q(s,T)-m-2(t,s)r1(s)4αρ(s)R(s,T)]ds=∞,
where
(26)R(t,T)={R1(t,T)Q1-α(t),0<α≤1;R1(t,T)R2α-1(t,T),α≥1,R*(t,T)=ρ′(t)+2αR(t,T)r1(t)r2(t)φ(t),Q(t,T)=ρ(t)[Kq(t)-(r2(t)φ(t))′+αR(t,T)r1(t)(r2(t)φ(t))2],m-(t,s)=max⁡{0,-m(t,s)},m+(t,s)=max⁡{0,m(t,s)}.
Then, every solution x(t) of (1) is oscillatory or converges to zero.Proof.
Assume that (1) has a nonoscillatory solution x(t) on [t0,∞). Without loss of generality we may assume that there exists a sufficiently large t1≥t0, such that x(t)>0, t≥t1. By Lemma 1, we see that x(t) satisfies either case (I) or case (II).
If case (I) holds, thenx′(t)>0, t≥T. Define the function W(t) by
(27)W(t)=ρ(t)r2(t)[((r1(t)x′(t))′x(t))α+φ(t)].
Then,
(28)W′(t)=ρ′(t)ρ(t)W(t)-ρ(t)q(t)f(x[g(t)])xα(t)+ρ(t)(r2(t)φ(t))′-αρ(t)r2(t)[(r1(t)x′(t))′x(t)]αx′(t)x(t).
When 0<α≤1 holds, using (9) and (11), we get
(29)x′(t)x(t)>R1(t,T)r1(t)r2(t)[(r1(t)x′(t))′x(t)]α×[r21/α(t)(r1(t)x′(t))′x(t)]1-α>R1(t,T)Q1-α(t)r1(t)r2(t)[(r1(t)x′(t))′x(t)]α.
In view of (28) and (29), noting that g(t)≥t, we obtain
(30)W′(t)≤-Kρ(t)q(t)+ρ(t)(r2(t)φ(t))′+ρ′(t)ρ(t)W(t)-αρ(t)R1(t,T)Q1-α(t)r1(t)×[r2(t)((r1(t)x′(t))′x(t))α]2.
When α≥1 holds, using (9) and (10), we have
(31)x′(t)x(t)>R1(t,T)r1(t)r2(t)[(r1(t)x′(t))′x(t)]α×[x(t)r21/α(t)(r1(t)x′(t))′]α-1>R1(t,T)R2α-1(t,T)r1(t)r2(t)[(r1(t)x′(t))′x(t)]α.
In view of (28) and (31), which yields
(32)W′(t)≤-Kρ(t)q(t)+ρ(t)(r2(t)φ(t))′+ρ′(t)ρ(t)W(t)-αρ(t)R1(t,T)R2α-1(t,T)r1(t)×[r2(t)((r1(t)x′(t))′x(t))α]2.
From (30), (32), and the definition of R(t,T), we get
(33)W′(t)≤-Kρ(t)q(t)+ρ(t)(r2(t)φ(t))′+ρ′(t)ρ(t)W(t)-αρ(t)R(t,T)r1(t)[r2(t)((r1(t)x′(t))′x(t))α]2.
By the definition of W(t), we have
(34)[r2(t)((r1(t)x′(t))′x(t))α]2=(W(t)ρ(t)-r2(t)φ(t))2=1ρ2(t)W2(t)-2r2(t)φ(t)ρ(t)W(t)+(r2(t)φ(t))2.
From (33) and (34), noting the definition of Q(t,T) and R*(t,T), we obtain
(35)Q(t,T)≤-W′(t)+R*(t,T)ρ(t)W(t)-αR(t,T)ρ(t)r1(t)W2(t).
Multiplying both sides of (35), with t replaced by s, by H(t,s), integrating with respect to s from T0 to t≥T0, we get
(36)∫T0tH(t,s)Q(s,T)ds≤-∫T0tH(t,s)W′(s)ds+∫T0tH(t,s)R*(s,T)ρ(s)W(s)ds-∫T0tαH(t,s)R(s,T)ρ(s)r1(s)W2(s)ds.
By integrating parts and using H(t,s)∈P and (24), we obtain
(37)∫T0tH(t,s)Q(s,T)ds≤H(t,T0)W(T0)+∫T0t∂H(t,s)∂sW(s)ds+∫T0tH(t,s)R*(s,T)ρ(s)W(s)ds-∫T0tαH(t,s)R(s,T)ρ(s)r1(s)W2(s)ds=H(t,T0)W(T0)+∫T0t[-m(t,s)H(t,s)ρ(s)W(s)-αH(t,s)R(s,T)ρ(s)r1(s)W2(s)]ds≤H(t,T0)W(T0)+∫T0t[m-(t,s)H(t,s)ρ(s)W(s)-αH(t,s)R(s,T)ρ(s)r1(s)W2(s)]ds.
Using averaging technique, we have
(38)m-(t,s)H(t,s)ρ(s)W(s)-αH(t,s)R(s,T)ρ(s)r1(s)W2(s)≤m-2(t,s)r1(s)4αρ(s)R(s,T).
Combining (37) and (38), we get
(39)1H(t,T0)∫T0t[m-2(t,s)r1(s)4αρ(s)R(s,T)H(t,s)Q(s,T)-m-2(t,s)r1(s)4αρ(s)R(s,T)]ds≤W(T0),
which contradicts (25).
If case (II) holds, from (18), by Lemma 3, lim⁡t→∞x(t)=0. This completes the proof.Theorem 5.
Let (8), (18), and g(t)≥t hold. Assuming that there exist ρ,φ∈C1([t0,∞),(0,∞)) and, for all sufficiently large T, there exists a T0≥T, one has
(40)lim⁡sup⁡t→∞∫T0t[Q(s,T)-r1(s)R*2(s,T)4αρ(s)R(s,T)]ds=∞,
where R(t,T), Q(t,T), and R*(t,T) are defined in Theorem 4. Then every solution x(t) of (1) is oscillatory or converges to zero.Proof.
Assume that (1) has a nonoscillatory solution x(t) on [t0,∞). Without loss of generality, we may assume that there exists a sufficiently large t1≥t0, such that x(t)>0, t≥t1. By Lemma 1, we see that x(t) satisfies either case (I) or case (II).
If case (I) holds, we proceed in the proof of Theorem4 and get (34). Then, from the definition of Q(t,T) and R*(t,T), we obtain
(41)W′(t)<-Q(t,T)+R*(t,T)ρ(t)W(t)-αR(t,T)ρ(t)r1(t)W2(t).
By using the averaging technique, we find that
(42)R*(t,T)ρ(t)W(t)-αR(t,T)ρ(t)r1(t)W2(t)≤r1(t)R*2(t,T)4αρ(t)R(t,T).
Hence, we get
(43)W′(t)<-[Q(t,T)-r1(t)R*2(t,T)4αρ(t)R(t,T)].
Integrating (43) from T0 to t, we have
(44)-W(T0)≤W(t)-W(T0)≤-∫T0t[Q(s,T)-r1(s)R*2(s,T)4αρ(s)R(s,T)]ds.
It follows that
(45)∫T0t‍[Q(s,T)-r1(s)R*2(s,T)4αρ(s)R(s,T)]ds≤W(T0),
which contradicts (40).
If case (II) holds, from (18), by Lemma 3, lim⁡t→∞x(t)=0. This completes the proof.By applying Theorem5 with ρ(t)=t, φ(t)=0, we have the following result.Corollary 6.
Let (8), (18), and g(t)≥t hold, and for all sufficiently large T, there exists a T0≥T; then, one has
(46)lim⁡sup⁡t→∞∫T0t[sq(s)-r1(s)4αKsR(s,T)]ds=∞,
where R(t,T) is defined in Theorem 4. Then every solution x(t) of (1) is oscillatory or converges to zero.Theorem 7.
Let (18) and g(t)≥t hold. Assume that there exist ρ∈C1([t0,∞),(0,∞)), H(t,s)∈P, and h(t,s)∈C(D,R), such that
(47)∂H(t,s)∂s+ρ′(s)ρ(s)H(t,s)=-h(t,s)ρ(s)Hα/(α+1)(t,s),
and all sufficiently large T0≥T such that
(48)lim⁡sup⁡t→∞1H(t,T0)×∫T0t[h-α+1(t,s)r1α(s)(α+1)α+1[ρ(s)R1(s,T)]αKH(t,s)ρ(s)q(s)-h-α+1(t,s)r1α(s)(α+1)α+1[ρ(s)R1(s,T)]α]ds=∞.
Then, every solution x(t) of (1) is oscillatory or converges to zero.Proof.
Assume that (1) has a nonoscillatory solution x(t) on [t0,∞). Without loss of generality, we may assume that x(t) is a positive solution of (1). By Lemma 1, we see that x(t) satisfies either case (I) or case (II).
If case (I) holds, thenx′(t)>0, t≥T. Define the function W(t) by
(49)W(t)=ρ(t)r2(t)[(r1(t)x′(t))′x(t)]α.
Using (9), we have
(50)W′(t)=-ρ(t)q(t)f(x[g(t)])xα(t)+ρ′(t)ρ(t)W(t)-αρ(t)r2(t)[(r1(t)x′(t))′x(t)]αx′(t)x(t)<-Kρ(t)q(t)+ρ′(t)ρ(t)W(t)-αρ(t)R1(t,T)r1(t)r21+1/α(t)[(r1(t)x′(t))′x(t)]α+1.
Hence, by the definition of W(t), we have
(51)W′(t)≤-Kρ(t)q(t)+ρ′(t)ρ(t)W(t)-αR1(t,T)ρ1/α(t)r1(t)W1+1/α(t).
Multiplying both sides of (51), with t replaced by s, by H(t,s) and integrating with respect to s from T0≥T to t≥T0, we get
(52)∫T0tKH(t,s)ρ(s)q(s)ds≤-∫T0tH(t,s)W′(s)ds+∫T0tH(t,s)ρ′(s)ρ(s)W(s)ds-∫T0tH(t,s)αR1(s,T)ρ1/α(s)r1(s)W1+1/α(s)ds.
Integrating by parts and using (47), which yields
(53)∫T0tKH(t,s)ρ(s)q(s)ds≤H(t,T0)W(T0)+∫T0t∂H(t,s)∂sW(s)ds+∫T0tH(t,s)ρ′(s)ρ(s)W(s)ds-∫T0tH(t,s)αR1(s,T)ρ1/α(s)r1(s)W1+1/α(s)ds≤H(t,T0)W(T0)+∫T0t[-h(t,s)Hα/(α+1)(t,s)ρ(s)W(s)-H(t,s)αR1(s,T)ρ1/α(s)r1(s)W1+1/α(s)]ds≤H(t,T0)W(T0)+∫T0t[h-(t,s)Hα/(α+1)(t,s)ρ(s)W(s)-H(t,s)αR1(s,T)ρ1/α(s)r1(s)W1+1/α(s)]ds.
Define A and B as follows:
(54)Aλ=αH(t,s)R1(s,T)ρ1/α(s)r1(s)Wλ(s),Bλ-1=h-(t,s)r11/λ(s)λ[αρ(s)R1(s,T)]1/λ,
where λ=(α+1)/α>1, A≥0, and B≥0. Using the inequality [10, Theorem 41]
(55)λABλ-1-Aλ≤(λ-1)Bλ,
we obtain
(56)h-(t,s)H1/λ(t,s)ρ(s)W(s)-αH(t,s)R1(s,T)ρλ-1(s)r1(s)Wλ(s)≤h-α+1(t,s)r1α(s)(α+1)α+1[ρ(s)R1(s,T)]α.
Combining (53) and (56), we get
(57)1H(t,T0)∫T0t[h-α+1(t,s)r1α(s)(α+1)α+1[ρ(s)R1(s,T)]αKH(t,s)ρ(s)q(s)-h-α+1(t,s)r1α(s)(α+1)α+1[ρ(s)R1(s,T)]α]ds≤W(T0),
which contradicts (48).
If case (II) holds, from (18), by Lemma 3, lim⁡t→∞x(t)=0. This completes the proof.Remark 8.
If we letρ(t)=1, φ(t)=0 in Theorem 4 and the function m(t,s) is Q(t,s) of Theorem  3.3 in [5], then condition (25) is (3.12) in [5]. Therefore, the result of α=1 in [5] is generalized to the case that α is the ratio of positive odd integers. If we let ρ(t)=1 in Theorem 7, the function h-α+1(t,s) is Qα+1(t,s)/Hα(t,s) of Theorem  3.4 in [3], which condition (48) is converted to (3.19) in [3]. Then, the result of (1.3) in [3] is generalized to the one of (1) in this paper.Example 9.
Consider the three-order differential equation(58)([(1tx′(t))′]3/5)′+1t2x3/5(t)=0,t∈[1,+∞),
where
(59)r1(t)=1t,r2(t)=1,q(t)=1t2,f(u)=u3/5,g(t)=t,α=35.
Conditions (A1), (A2), and (A3) are clearly satisfied. It is easy to find that (8) and (18) hold. Let ρ(t)=t, φ(t)=1/t. Here
(60)R1(t,T)=∫Tt(1r2(s))1/αds=∫Ttds=t-T,Q(t)=(K∫t∞q(s)ds)1/α=(∫t∞1s2ds)5/3=t-5/3,R(t,T)=R1(t,T)Q1-α(t)=(t-T)t-2/3,R*(t,T)=ρ′(t)+2αR(t,T)r1(t)r2(t)φ(t)ρ(t)=1+65(t-T)t1/3,Q(t,T)=ρ(t)[αR(t,T)r1(t)Kq(t)-(r2(t)φ(t))′+αR(t,T)r1(t)(r2(t)φ(t))2]=2t+35(t-T)t-2/3.
From Theorem 5, we have
(61)lim⁡sup⁡t→∞∫1t[Q(s,T)-r1(s)R*2(s,T)4αρ(s)R(s,T)]ds=lim⁡sup⁡t→∞∫1t[1s-5121(s-T)s4/3]ds=∞,
so (40) is satisfied. Hence, by Theorem 5, every solution x(t) of (58) is oscillatory or converges to zero.



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