# Set of Oscillation Criteria for Second Order Nonlinear Forced Differential Equations with Damping

Authors: Ambarka Abdalla Salhin; Ummul Khair Salma Din; Rokiah Rozita Ahmad; Mohd Salmi Md Noorani
Journal: Discrete Dynamics in Nature and Society
 (2014)

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## Abstract

By employing a generalized Riccati technique and an integral averaging technique, some new oscillation criteria are established for the second order nonlinear forced differential equation with damping. These results extend, improve, and unify some known oscillation criteria in the existing literature.


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## Body

## 1. Introduction

The oscillatory problem for second order nonlinear forced differential equation with damping(1)(r(t)ψ(x(t))f(x′(t)))′+h(t)f(x′(t))+q(t)g(x(t))Φ(x′(t))=H(t,x(t),x′(t)),t≥t0,
is concerned, where r,h,q∈C([t0,∞),R) and f,ψ,g,Φ∈C(R,R) andH is a continuous function on [t0,∞)×R2.Throughout this paper we will also suppose that there are positive constantsk1,k2,k3,k4,k5,l, and k satisfying the following:(A1)
r(t)>0,t≥0,(A2)
0<k1≤ψ(x(t))≤k2forallx,(A3)
l>0andf2(y)≤lyf(y)forally∈R,(A4)
q(t)≥0and0<k3≤Φ(x′(t)),(A5)
xg(x)>0and0<k≤g′(x) for all x≠0,(A6)
p:[t0,∞)→R is continuous function such that H(t,x,y)/g(x)≤p(t)∀t∈[t0,∞);x,y∈R and x≠0,(A7)
0<k4≤(f(y)/y)≤k5forally≠0.We will consider only nontrivial solutions of (1) which are defined for all large t. A solution of (1) is said to be oscillatory if it has a sequence of zeros clustering at ∞ and nonoscillatory otherwise. Equation (1) is said to be oscillatory if all its solutions are oscillatory.In the late 19th century, some scholars focus on sufficient conditions for the oscillation theorems of different classes of differential equations with damping. We refer to the new published papers [1–11]. The oscillatory theory of second order nonlinear differential equations has been widely applied in research of lossless high-speed computer network and physical sciences.Recently, the oscillatory behavior for various particular cases of (1), such as the nonlinear differential equations
(2)(r(t)x′(t))′+h(t)x′(t)+q(t)g(x(t))Φ(x′(t))=0,(r(t)ψ(x(t))f(x′(t)))′+h(t)f(x′(t))+q(t)g(x(t))Φ(x′(t))=0,(r(t)ψ(x(t))f(x′(t)))′+h(t)f(x′(t))+φ(t,x(t))=H(t,x(t),x′(t)),
has been studied extensively by numerous authors with different methods; see, for example, [9–11] and the references quoted therein.In this paper, by using a generalized Riccati and integral averaging technique, several new oscillation criteria for (1) are established.A significant drawback of many oscillation results for differential equations with damping reported in the literature is a necessity to impose a variety of additional restrictions on the sign of the damping termh(t). We emphasize that our theorems are free of particular restrictions on h(t).


## 2. Main Results

For convenience, we introduce the class of the functionΩ. Let D={(t,s):t0≤s≤t}. A function H*∈(D,R) is said to belong to the class Ω, if(1)
H*(t,t)=0 for t≥t0 and H*(t,s)>0 for t>s≥t0,(2)
H*(t,s) has continuous and nonpositive partial derivatives on D with respect to the second variable,(3)
there exists a functionh1(t,s)∈C(D,R) such that -∂H*(t,s)/∂s=h1(t,s)H*(t,s).In this section, several oscillation conditions for (1) are established under the assumptions (A1)–(A7).Theorem 1.
Let assumptions(A1)–(A6) be fulfilled and H*∈Ω. If there exist functions R∈C([t0,∞),R+) and φ∈C1([t0,∞),R+) such that (rR)∈C1([t0,∞),R) and
(3)lim⁡t→∞sup⁡1H*(t,t0)×∫t0t[H*(t,s)Q(s)-lk24kφ(s)r(s)h22(t,s)]ds=∞,
where
(4)Q(t)=φ(t){k3q(t)-p(t)+klk2r(t)R2(t)-(R(t)r(t))′(1k1-1k2)h2(t)r(t)-1k2h(t)R(t)-l4k(1k1-1k2)h2(t)r(t)},h2(t,s)=h1(t,s)-H*(t,s)(φ′(s)φ(s)+2klk3R(s)-h(s)k2r(s)),
then (1) is oscillatory.Proof.
Letx(t) be a nonoscillatory solution of (1). Then there exists a T0≥t0 such that x(t)≠0 for all t≥T0. Without loss of generality, we may assume that x(t)>0 on interval [T0,∞). A similar argument holds also for the case when x(t) is eventually negative. Defining a generalized Riccati transformation by
(5)w(t)=φ(t)r(t)[ψ(x(t))f(x′(t))g(x(t))+R(t)],
for all t≥T0, then differentiating Equation (5), and using (1) and (A1)–(A6), it follows
(6)w′(t)=φ(t)H(t,x(t),x′(t))g(x(t))-φ(t)h(t)f(x′(t))g(x(t))-φ(t)q(t)Φ(x′(t))-φ(t)r(t)ψ(x(t))f(x′(t))g′(x(t))x′(t)g2(x(t))+φ(t)(r(t)R(t))′+w(t)φ′(t)φ(t),w′(t)≤φ′(t)φ(t)w(t)-φ(t)h(t)f(x′(t))g(x(t))-φ(t)(k3q(t)-p(t))-klφ(t)r(t)ψ(x(t))f2(x′(t))g2(x(t))+φ(t)(r(t)R(t))′=φ(t)(r(t)R(t))′-φ(t)(k3q(t)-p(t))+φ′(t)φ(t)w(t)+lφ(t)h2(t)4kr(t)ψ(x(t))-φ(t)ψ(x(t))×[kr(t)lψ(x(t))f(x′(t))g(x(t))+h(t)2lkr(t)]2≤φ(t)(r(t)R(t))′-φ(t)(k3q(t)-p(t))+φ′(t)φ(t)w(t)+lφ(t)h2(t)4r(t)ψ(x(t))-φ(t)k2×[kr(t)lψ(x(t))f(x′(t))g(x(t))+h(t)2lkr(t)]2=φ(t)(r(t)R(t))′-φ(t)(k3q(t)-p(t))+φ′(t)φ(t)w(t)+lφ(t)h2(t)4kr(t)ψ(x(t))-φ(t)k2×[kr(t)l(w(t)φ(t)r(t)-R(t))+h(t)2lkr(t)]2≤-Q(t)+(φ′(t)φ(t)+2kR(t)lk2-h(t)k2r(t))w(t)-klk2φ(t)r(t)w2(t),
for all t≥T0 with Q(t) defined as above. Then we obtain
(7)Q(t)≤-w′(t)+(φ′(t)φ(t)+2kR(t)lk2-h(t)k2r(t))w(t)-klk2φ(t)r(t)w2(t).
Multiplying both sides of (7) by H*(t,s), integrating it with respect to s from T to t, and using the properties of the function H*(t,s), we get, for all t≥T≥T0,
(8)∫TtH*(t,s)Q(s)ds≤-∫TtH*(t,s)w′(s)ds+∫TtH*(t,s)(φ′(s)φ(s)+2kR(s)lk2-h(s)k2r(s))w(s)ds-∫TtH*(t,s)klk2φ(s)r(s)w2(s)ds=-H*(t,s)w(s)|Tt+∫Ttw(s)dH*(t,s)+∫TtH*(t,s)(φ′(s)φ(s)+2kR(s)lk2-h(s)k2r(s))w(s)ds-∫TtH*(t,s)klk2φ(s)r(s)w2(s)ds=H*(t,T)w(T)-∫Ttw(s)h1(t,s)H*(t,s)ds+∫TtH*(t,s)(φ′(s)φ(s)+2kR(s)lk2-h(s)k2r(s))w(s)ds-∫TtH*(t,s)klk2φ(s)r(s)w2(s)ds=H*(t,T)w(T)-∫Tt[h2(t,s)H*(t,s)w(s)+kH*(t,s)lk2φ(s)r(s)w2(s)]ds.
Therefore, for all t≥T≥T0,
(9)∫Tt[Q(s)H*(t,s)-lk24kφ(s)r(s)h22(t,s)]ds≤H*(t,T)w(T)-∫Tt[kH*(t,s)lk2φ(s)r(s)w(s)lk2φ(s)r(s)kh2hhhhhhhhhhhhhhhhhhh+12lk2φ(s)r(s)kh2(t,s)]2ds≤H*(t,T)w(T)≤H*(t,T)|w(T)|≤H*(t,t0)|w(T)|.
Applying inequality (9), for T=T0, yields
(10)∫t0t[Q(s)H*(t,s)-lk24kφ(s)r(s)h22(t,s)]ds=∫t0T0[Q(s)H*(t,s)-lk24kφ(s)r(s)h22(t,s)]ds+∫T0t[Q(s)H*(t,s)-lk24kφ(s)r(s)h22(t,s)]ds≤H*(t,t0)∫t0T0|Q(s)|ds+H*(t,t0)|w(T0)|.
It follows that
(11)lim⁡t→∞sup⁡1H*(t,t0)×∫t0t[H*(t,s)Q(s)-lk24φ(s)r(s)h22(t,s)]ds≤∫t0T0|Q(s)|ds+|w(T0)|<∞,
which contradicts assumption (3), so (1) is oscillatory.Corollary 2.
If condition (3) is replaced by conditions
(12)lim⁡t→∞sup⁡1H*(t,t0)∫t0tH*(t,t0)Q(s)ds=∞,(13)lim⁡t→∞sup⁡1H*(t,t0)∫t0tφ(s)r(s)h22(t,s)ds<∞,
then (1) is oscillatory, where Q(s) and h2(t,s) are the same as defined in Theorem 1.Example 3.
Consider the nonlinear damped differential equation(14)[t-1(2-sinx(t))(x′(t)1+(x′(t))2)]′+cos⁡tx′(t)1+(x′(t))2+t2(1-cos⁡t)x(t)(1+x4(t))(1+x′2(t))=x5(t)sintcos⁡x′(t),t≥t0=1,
where x∈(-∞,∞) and t≥t0=1. Since k1=1/2, k2=k3=1, and l=1 the assumptions (A1)–(A6) hold. If we take R(t)=0 and (H(t,x,x′)/g(x(t)))≤1=p(t), then φ(t)=t and H*(t,s)=(t-s)2. A direct computation yields that the conditions of Theorem 1 are satisfied; Example 3 is oscillatory.Theorem 4.
Let assumptions (A1)–(A6) be fulfilled and H*∈Ω. Suppose that
(15)0<inf⁡s≥t0{lim⁡t→∞inf⁡H*(t,s)H*(t,t0)}≤∞.
If there exist functions R,χ∈C([t0,∞),R) and φ∈C1([t0,∞),R+) such that (rR)∈C1([t0,∞),R) and
(16)lim⁡t→∞sup⁡1H*(t,t0)∫t0tφ(s)r(s)h22(t,s)ds<∞,(17)lim⁡t→∞sup⁡1H*(t,t0)∫t0tχ+2(s)φ(s)r(s)ds=∞,
and for any T≥t0(18)lim⁡t→∞sup⁡1H*(t,T)×∫Tt[H*(t,s)Q(s)-lk24kφ(s)r(s)h22(t,s)]ds≥χ(T),
where Q(s) and h2(t,s) are the same as defined in Theorem 1, and(19)χ+(s)=max⁡{χ(s),0},
then (1) is oscillatory.Proof.
Letx(t) be a nonoscillatory solution of (1). Then there exists a T0≥t0 such that x(t)≠0 for all t≥T0. Without loss of generality, we may assume that x(t)>0 on interval [T0,∞). A similar argument holds also for the case when x(t) is eventually negative.
Define the functionw(t) as in (5). Similar to the proof of Theorem 1, we obtain inequality (9). Further, it follows
(20)1H*(t,T)∫Tt[Q(s)H*(t,s)-lk24kφ(s)r(s)h22(t,s)]ds≤w(T)-1H*(t,T)×∫Tt[kH*(t,s)lk2φ(s)r(s)w(s)+12lk2φ(s)r(s)kh2(t,s)]2ds,
for t>T≥T0 and therefore
(21)lim⁡t→∞sup⁡1H*(t,T)×∫Tt[Q(s)H*(t,s)-lk24kφ(s)r(s)h22(t,s)]ds≤w(T)-lim⁡t→∞inf⁡1H*(t,T)×∫Tt[kH*(t,s)lk2φ(s)r(s)w(s)+12lk2φ(s)r(s)kh2(t,s)]2ds.
Thus, by (18), we get
(22)w(T)≥χ(T)+lim⁡t→∞inf⁡1H*(t,T)×∫Tt[kH*(t,s)lk2φ(s)r(s)w(s)lk2φ(s)r(s)k+12lk2φ(s)r(s)kh2(t,s)]2ds,
for all t>T≥T0. This implies that
(23)w(T)≥χ(T),∀T≥T0,lim⁡t→∞inf⁡1H*(t,T0)×∫T0t[kH*(t,s)lk2φ(s)r(s)w(s)lk2φ(s)r(s)k+12lk2φ(s)r(s)kh2(t,s)]2ds≤w(T0)-χ(T0)<∞.
Define
(24)α(t)=1H*(t,T0)∫T0tkH*(t,s)lk2φ(s)r(s)w2(s)ds,β(t)=1H*(t,T0)∫T0th2(t,s)H*(t,s)w(s)ds,
for all t>T0. Then
(25)lim⁡t→∞inf⁡⁡[α(t)+β(t)]≤lim⁡t→∞inf⁡1H*(t,T0)×∫T0t[kH*(t,s)lk2φ(s)r(s)w(s)lk2φ(s)r(s)k+12lk2φ(s)r(s)kh2(t,s)]2ds<∞.
In order to show that
(26)∫T0∞w2(s)φ(s)r(s)ds<∞,
suppose that
(27)∫T0∞w2(s)φ(s)r(s)ds=∞.
By (15), there exists a positive constant λ such that
(28)inf⁡s≥t0{lim⁡t→∞inf⁡H*(t,s)H*(t,t0)}>λ>0.
On the other hand, according to (27) for any positive constant γ there exists a T1>T0 such that
(29)∫T0tw2(s)φ(s)r(s)ds≥γλ∀t≥T1.
For t≥T1,
(30)α(t)=1H*(t,T0)∫T0tkH*(t,s)lk2φ(s)r(s)w2(s)ds=klk2H*(t,T0)∫T0tH*(t,s)d[∫T0sw2(u)φ(u)r(u)du]=klk2H*(t,T0)[H*(t,s)∫T0sw2(u)φ(u)r(u)du|T0tH*(t,s)∫T0sw2(u)φ(u)r(u)du|T0t-∫T0t(∫T0sw2(u)φ(u)r(u)du)∂H*(t,s)∂sds]=klk2H*(t,T0)∫T0t(∫T0sw2(u)φ(u)r(u)du)(-∂H*(t,s)∂s)ds≥klk2H*(t,T0)∫T1t(∫T0sw2(u)φ(u)r(u)du)(-∂H*(t,s)∂s)ds≥γkλlk2H*(t,T0)∫T1t(-∂H*(t,s)∂s)ds=γkH*(t,T1)λlk2H*(t,T0).
By (28) we can easily see that
(31)lim⁡t→∞inf⁡H*(t,T1)H*(t,t0)>λ>0.
Then there exists T2≥T1 such that (H*(t,T1)/H*(t,t0))≥λ for all t≥T2. Therefore, by (30), α(t)≥γk/lk2 for all t≥T2, and since γ is an arbitrary constant, we can make a conclusion that
(32)lim⁡t→∞α(t)=∞.
Next, let us consider a sequence {tn}n=1,2,3,… in (T0,∞) with lim⁡n→∞tn=∞ and such that
(33)lim⁡n→∞[α(tn)+β(tn)]=lim⁡t→∞inf⁡[α(t)+β(t)].
Now, by (25), there exists a constant N such that
(34)α(Tn)+β(Tn)≤N(n=1,2,…),
and hence (32) leads to
(35)lim⁡n→∞β(tn)=-∞.
By taking into account (32), from (34), we derive
(36)1+β(tn)α(tn)≤Nα(tn)<εfornlargeenough,
where ε∈(0,1). Thus
(37)β(tn)α(tn)<ε-1<0.
The above inequality and (35) imply that
(38)β2(tn)α(tn)>(ε-1)β(tn),lim⁡n→∞β2(tn)α(tn)=∞.
Further, by Schwarz inequality, we have, for any positive integer n,
(39)β2(tn)=1H*2(tn,T0)[∫T0tnh2(tn,s)H*(tn,s)w(s)ds]2≤[1H*(tn,T0)∫T0tnH*(tn,s)r(s)φ(s)w2(s)ds]×[1H*(tn,T0)∫T0tnr(s)φ(s)h22(tn,s)ds]=lk2kα(tn)[1H*(tn,T0)∫T0tnr(s)φ(s)h22(tn,s)ds],
and therefore
(40)β2(tn)α(tn)≤lk2k[1H*(tn,T0)∫T0tnr(s)φ(s)h22(tn,s)ds]≤lk2λk[1H*(tn,t0)∫t0tnr(s)φ(s)h22(tn,s)ds].
It follows from (38) that
(41)lim⁡n→∞1H*(tn,t0)∫t0tnr(s)φ(s)h22(tn,s)ds=∞.
Consequently,
(42)lim⁡t→∞sup⁡1H*(t,t0)∫t0tr(s)φ(s)h22(t,s)ds=∞,
but the latter contradicts assumption (16). Hence, (27) fails to hold. Finally, by (23), we obtain
(43)∫T0∞χ+2(s)r(s)φ(s)ds≤∫T0∞w2(s)r(s)φ(s)ds<∞.
This contradicts the assumption (12). Therefore, (1) is oscillatory.Theorem 5.
Let assumptions (A1)–(A6) be fulfilled and H*∈Ω. Suppose that (15) holds. If there exist functions R,χ∈C([t0,∞),R) and φ∈C1([t0,∞),R+), such that (rR)∈C1([t0,∞),R), and (17) holds, and
(44)lim⁡t→∞inf⁡1H*(t,t0)∫t0tH*(t,s)Q(s)ds<∞,
and for every T≥t0(45)lim⁡t→∞inf⁡1H*(t,T)×∫Tt(H*(t,s)Q(s)-lk24kφ(s)r(s)h22(t,s))ds≥χ(T),
where Q(t)and h2(t,s) are defined as in Theorem 1, χ+(s)=max⁡{χ(s),0}, then (1) is oscillatory.Proof.
Without loss of generality, we assume that there exists a solutionx(t) of (1) such that x(t)>0 on [T0,∞) for some T0≥t0. The function w(t) is defined as in (5). Then, following the proof of Theorem 4, we have (20). Now it follows that
(46)lim⁡t→∞inf⁡1H*(t,T)×∫Tt[Q(s)H*(t,s)-lk24kφ(s)r(s)h22(t,s)]ds≤w(T)-lim⁡t→∞sup⁡1H*(t,T)×∫Tt[kH*(t,s)lk2φ(s)r(s)w(s)+12lk2φ(s)r(s)kh2(t,s)]2ds,
for every T≥T0. By (45), we know that (23) holds and
(47)lim⁡t→∞sup⁡1H*(t,T0)×∫T0t[kH*(t,s)lk2φ(s)r(s)w(s)+12lk2φ(s)r(s)kh2(t,s)]2ds≤w(T0)-χ(T0)<∞.
Then,
(48)lim⁡t→∞sup⁡[α(t)+β(t)]≤lim⁡t→∞sup⁡1H*(t,T0)×∫T0t[kH*(t,s)lk2φ(s)r(s)w(s)lk2φ(s)r(s)k+12lk2φ(s)r(s)kh2(t,s)]2ds<∞,
where α(t) and β(t) are defined as in the proof of Theorem 4 It follows from (44) and (45) that
(49)lim⁡t→∞inf⁡1H*(t,t0)∫t0tφ(s)r(s)h22(t,s)ds<∞.
Then there exists a sequence {tn}n=1∞ in (T0,∞) such that lim⁡t→∞tn=∞ and
(50)lim⁡n→∞1H*(tn,t0)∫t0tnφ(s)r(s)h22(tn,s)ds=lim⁡t→∞inf⁡1H*(t,t0)∫t0tφ(s)r(s)h22(t,s)ds<∞.
Now, suppose that (27) holds. With the same argument as in Theorem 4, we conclude that (32) is satisfied. By (48), there exists a constant N1 such that
(51)α(tn)+β(tn)≤N1∀n>N1.
Then, similar to the proof of Theorem 4, we obtain (41) which contradicts (50), and hence (27) fails. From (23) and (26) we have
(52)∫T0∞χ+2(s)r(s)φ(s)ds≤∫T0∞w2(s)r(s)φ(s)ds<∞,
which contradicts assumption (17).Theorem 6.
Let assumptions (A1)–(A6) be fulfilled and H*∈Ω. Suppose that (15) holds. If there exist functions R,χ∈C([t0,∞),R) and φ∈C1([t0,∞),R+), such that (rR)∈C1([t0,∞),R), and (17) and (45) hold, and
(53)lim⁡t→∞inf⁡1H*(t,t0)∫t0tφ(s)r(s)h22(t,s)ds<∞,
where Q(t) and h2(t,s) are defined as in Theorem 1 and χ+(s)=max⁡{χ(s),0}, then (1) is oscillatory.The proof of Theorem6 is similar to the proof of Theorem 5.Example 7.
Consider the nonlinear damped differential equation(54)[(2+cos⁡2t1+cos⁡2t)(1+x2(t)2+x2(t))(x′(t)1+α(x′(t))2)]′+2sint|cos⁡t|x′(t)1+αx′2(t)+9(1+cos⁡2t)10+cos⁡2tx5(t)(1+x2(t))(1+x′2(t))=2x7(t)sintcos⁡x′(t)t2.
Obviously, for all x∈(-∞,∞) and t≥t0=1, α≥0 is a constant. Since k1=1/2, k2=k3=1, and l=1, the assumptions (A1)–(A6) hold. If R(t)=0 and (Η(t,x,x′)/g(x(t)))≤2/t2=p(t), then φ(t)=t-2 and H(t,s)=(t-s)2, for all t≥1.
A direct computation yields(55)lim⁡t→∞inf⁡1H(t,T)∫Tt(H(t,s)Q(s)-le24kφ(s)r(s)h32(t,s))ds=lim⁡t→∞inf⁡1t2×∫Tt{(t-s)2s-2(2sins|cos⁡s|(1+cos⁡2s)2+cos⁡2s+2s)×(t-s)]2×((1+cos⁡2s)210+cos⁡2s-sin2scos⁡2s(1+cos⁡2s)2+cos⁡2s)-14s22+cos⁡2s1+cos⁡2s×[2+(2sins|cos⁡s|(1+cos⁡2s)2+cos⁡2s+2s)(2sins|cos⁡s|(1+cos⁡2s)2+cos⁡2s+2s)×(t-s)]2}ds,≥lim⁡t→∞1t2∫Tt{(t-s5s)2-2s2(ts+(t-s)64)2}ds,=23T3+62T2+71100T,lim⁡t→∞inf⁡1H(t,t0)∫t0tφ(s)r(s)h32(t,s)ds=lim⁡t→∞inf⁡1t2×∫1t{s-22+cos⁡2s1+cos⁡2s(2sins|cos⁡s|(1+cos⁡2s)2+cos⁡2s+2s)×(t-s)]2×[2+(2sins|cos⁡s|(1+cos⁡2s)2+cos⁡2s+2s)(2sins|cos⁡s|(1+cos⁡2s)2+cos⁡2s+2s)×(t-s)]2}ds≤lim⁡t→∞inf⁡1t2∫1t{s-232[1+(23+1s)(t-s)]2}=∞.
We conclude by Theorem 6 that all solutions of this equation are oscillatory.Remark 8.
If (5) is replaced by
(56)w(t)=φ(t)[r(t)ψ(x(t))f(x′(t))x(t)+r(t)R(t)],
and (A5) by (g(x)/x)≥k>0 for x≠0, we can obtain similar oscillation results that are derived in the present paper.Remark 9.
If we takeΦ(x′(t))=1, H(t,x,x′)=0, it is easy to see that Theorems 1–6 reduce to Theorems 1−4 of Wang [8]. If we take Φ(x′(t))=1, ψ(x(t))=1,f(y)=y, and H(t,x′(t),x(t))=0 for x,y∈R, then (1) reduces to (r(t)x′(t))′+h(t)x′(t)+q(t)g(x(t))=0, and by taking φ(t)=exp⁡(-2k∫t0tR(s)ds), Theorems 1–5 reduce to Theorems 1–3 of Rogovchenko and Tuncay [6].Remark 10.
Theorems1–6 and Corollary 2 are obtained by analogy with Theorems 1–4 from [9], and we do not require any restriction on the sign and differentiability of h(t).The following lemma will significantly simplify the proofs of next theorems. First recall class functions defined onD={(t,s):t0≤s≤t}. A function H∈(D,R) is said to belong to the class ξ if(i)
H(t,t)=0 for t≥t0and H(t,s)>0 when t≠s,(ii)
H(t,s) has partial derivatives on D such that
(57)∂H(t,s)∂s=-h1(t,s)H(t,s),∂H(t,s)∂t=h2(t,s)H(t,s),for some h1,h2∈Lloc⁡1(D,R).Lemma 11.
LetB1,B2,B3∈C([t0,∞),R) with B3>0, Z∈C1([t0,∞),R). If there exist (a,b)⊂[t0,∞) and c∈(a,b) such that
(58)Z′≤-B1(s)+B2(s)Z-B3(s)Z2,s∈(a,b),
then
(59)1H(c,a)∫ac[H(s,a)B1(s)-14B3(s)θ12(s,a)]ds+1H(b,c)∫cb[H(b,s)B1(s)-14B3(s)θ22(b,s)]ds≤0,
for all H∈ξ, where
(60)θ1(s,a)=[h1(s,a)+B2(s)H(s,a)],θ2(b,s)=[h2(b,s)-B2(s)H(b,s)].
The proof of this lemma is similar to that of [4] and hence will be omitted.In the next theorems we define the following functions that will be used in the proofs. Let(61)β(t)=k1k5ϕ′(t)r(t)-k4ϕ(t)h(t),δ(t)=lϕ(t)r(t)ψ(x(t)),γ[t,T]=δ(t)(∫Ttδ(s)ds)-1.Theorem 12.
Suppose that (A1)–(A7) hold. Assume that
(62)∫∞dug(u)<∞,∫-∞dug(u)<∞,(63)∫∞g′(u)g(u)du<∞,∫-∞g′(u)g(u)du<∞,(64)min⁡{inf⁡u>0g′(u)∫u∞g′(u)g(u)du,inf⁡u<0g′(u)∫u-∞g′(u)g(u)du}>0(65)β(t)≥0,β′(t)≤0,t≥t0,(66)∫∞δ(s)ds=∞.
If there exists a continuously differentiable function ϕ:[t0,∞)→(0,∞) such that ϕ′ is nonnegative and decreasing function, we have
(67)lim⁡t→∞inf⁡∫t0tϕ(s)(k3q(s)-p(s))ds>-∞.
There exists an interval (a,b)⊂[T,∞), and that there exists c∈(a,b),H∈ξ, and for any constant F>0, such that
(68)1H(c,a)∫ac[H(s,a)ϕ(s)(k3q(s)-p(s))14Jγ[s,t0]-14Jγ[s,t0]θ12(s,a)]ds+1H(b,c)∫cb[H(b,s)ϕ(s)(k3q(s)-p(s))14Jγ[s,t0]-14Jγ[s,t0]θ22(b,s)]ds>0,
where
(69)θ1(s,a)=[h1(s,a)+1k4δ(s)β(s)H(s,a)],θ2(b,s)=[h2(b,s)-1k4δ(s)β(s)H(b,s)].
Then (1) is oscillatory.Proof.
Without loss of generality, we may assume that there exists a nonoscillatory solutionx(t) of (1) such that x(t)>0 on [T,∞) for some 0≤t0≤T. The similar argument holds also for x(t)<0. Define the function w(t) as
(70)w(t)=ϕ(t)r(t)ψ(x(t))f(x′(t))g(x(t)).
Differentiating (70), using (1) and (A1)–(A7), we get
(71)w′(t)≤-ϕ(t)(k3q(t)-p(t))-k4ϕ(t)h(t)x′(t)g(x(t))+k1k5ϕ′(t)r(t)x′(t)g(x(t))-lϕ(t)r(t)ψ(x(t))w2(t)g′(x(t))=-ϕ(t)(k3q(t)-p(t))+(k1k5ϕ′(t)r(t)-k4ϕ(t)h(t))x′(t)g(x(t))-lϕ(t)r(t)ψ(x(t))w2(t)g′(x(t))=-ϕ(t)(k3q(t)-p(t))+β(t)x′(t)g(x(t))-δ(t)w2(t)g′(x(t)).
Integrating (71) from t0 to t we get that
(72)w(t)≤w(t0)-∫t0tϕ(s)(k3q(s)-p(s))ds+∫t0tβ(s)x′(s)g(x(s))ds-∫t0tδ(s)w2(s)g′(x(s))ds.
Since β′(s)≤0, then by Bonnet’s Theorem, there exist b1∈[t0,∞) for every t≥t0 such that
(73)∫t0tβ(s)x′(s)g(x(s))ds=β(t0)∫t0b1x′(s)g(x(s))ds=β(t0)∫x(t0)x(b1)dug(u)≤β(t0)∫x(t0)∞dug(u)=k6,
where k6>0 is a constant. Then, we have, for t≥t0,
(74)w(t)≤F-∫t0tϕ(s)(k3q(s)-p(s))ds-∫t0tδ(s)w2(s)g′(x(s))ds,
where F=w(t0)+k6.
Three cases of the oscillatory solutions are discussed below.Case 1.
Assume thatx′(t) is oscillatory; then there exists a sequence {tn}n=1,2,… such that lim⁡n→∞tn=∞ and x(tn)=0, n=1,2,… on [t0,∞). From (74) we get
(75)∫t0tnδ(s)w2(s)g′(x(s))ds≤F-∫t0tnϕ(s)(k3q(s)-p(s))ds,n=1,2,….
Using (67) we obtain
(76)∫t0tnδ(s)w2(s)g′(x(s))ds<∞.
Then there exists a constant A>0, such that
(77)∫t0tnδ(s)w2(s)g′(x(s))ds≤A,t≥t0.
Using Schwarz inequality, (A7), and (77) we have
(78)|∫t0tx′(s)g(x(s))g′(x(s))ds|2=1k4|∫t0tδ(s)(δ(s)w(s)g′(x(s)))ds|2≤1k4(∫t0tδ(s)ds)(∫t0tnδ(s)w2(s)g′(x(s))ds)≤Ak4∫t0tδ(s)ds,t≥t0.
Applying (64),
(79)g′(x(t))∫x(t)∞g′(u)g(u)du≥M,t≥t0,
where M is a positive constant.
LetM1=∫x(t0)∞(g′(u)/g(u))du>0; applying (78) we have
(80)g′(x(t))≥M2[∫x(t)∞g′(u)g(u)du]-2=M2[M1-∫x(t0)x(t)g′(u)g(u)du]-2=M2[M1-∫t0tx′(s)g(x(s))g′(x(s))ds]-2≥M2[M1+|∫t0tx′(s)g(x(s))g′(x(s))ds|]-2.
Using (78) in the above inequality leads to
(81)g′(x(t))≥M2[M1+(Ak4∫t0tδ(s)ds)1/2]-2.
Then, there exist constants J>0 and T>t0, such that
(82)g′(x(t))≥J(Ak4∫t0tδ(s)ds)-1,t≥T.
Substituting (82) in (71) we get
(83)w′(t)≤-ϕ(t)(k3q(t)-p(t))+1k4δ(t)β(t)w(t)-Jγ[t,t0]w2(t),t≥T.
From (83) and by Lemma 11, we conclude that for any c∈(a,b) and H∈ξ(84)1H(c,a)∫ac[H(s,a)ϕ(t)(k3q(t)-p(t))14Jγ[t,t0]-14Jγ[t,t0]θ12(s,a)]ds+1H(b,c)∫cb[H(b,s)ϕ(t)(k3q(t)-p(t))14Jγ[t,t0]-1JDγ[t,t0]θ22(b,s)]ds≤0,
which contradicts condition (68).Case 2.
Assuming thatx′(t)>0 for t0≤t1≤t, then w(t)>0 for t≥t1, and by (74) we have
(85)∫t1tδ(s)w2(s)g′(x(s))ds≤F-∫t1tϕ(s)(k3q(s)-p(s))ds,t≥t1.
From (67) we see that
(86)∫t1∞δ(s)w2(s)g′(x(s))ds<∞.
The following steps are similar to the proof of Case 1.Case 3.
Assume thatx′(t)<0 for t0≤t1≤t; if (86) holds, then we have similar discussion in Case 2. If the integration in (86) is divergent, we can get the following inequality from (74) and (67):
(87)F1+∫t1tδ(s)w2(s)g′(x(s))ds≤-w(t),t≥t1,
where F1 is a constant. By taking t2≥t1,
(88)F2=F1+∫t1t2δ(s)w2(s)g′(x(s))ds>1.
From (87) and (88) we have
(89)w(t)<0,
and from (86) we find
(90)δ(t)w2(t)g′(x(t))F1+∫t1tδ(s)w2(s)g′(x(s))ds≥-x′(t)g(x′(t))g(x(t)),t≥t2.
Integrating the above inequality, we obtain
(91)ln⁡[F1+∫t1tδ(s)w2(s)g′(x(s))ds]≥ln⁡g(x(t2))g(x(t)),t≥t2.
Therefore
(92)F1+∫t1tδ(s)w2(s)g′(x(s))ds≥g(x(t2))g(x(t)),t≥t2.
Applying (87) and (92), we get
(93)x′(t)≤-1k4δ(t)g(x(t2))<0.
Hence
(94)x(t)≤x(t2)-1k4g(x(t2))∫t2tδ(s)ds⟶-∞,t⟶∞.
The above inequality contradicts x(t)>0. This completes the proof.Theorem 13.
Suppose that (A1)–(A7) and (62)–(64) hold. Assume that there exists a function ϕ:[t0,∞)→(0,∞) such that (65)–(68) and there exists H∈ξ such that
(95)lim⁡t→∞sup⁡∫a1t[H(s,a1)ϕ(s)(k3q(s)-p(s))14Jγ[s,t0]-14Jγ[s,t0]θ12(s,a1)]ds>0,(96)lim⁡t→∞sup⁡∫a1t[14Jγ[s,t0]H(t,s)ϕ(s)(k3q(s)-p(s))-14Jγ[s,t0]θ22(b,s)]ds>014Jγ[s,t0]
hold for all a1∈[t1,∞), where θ1,θ2, and J are the same as in Theorem 12. Then (1) is oscillatory.Proof.
Suppose thatx(t)≠0 for all t∈[t2,∞) for some t2≥t1. Set a1=a≥t2 in (23). We can get c>a such that
(97)∫ac[H(s,a)ϕ(s)(k3q(s)-p(s))14Jγ[s,t0]-14Jγ[s,t0]θ12(s,a)]ds>0.
Similarly, with (96) by setting a1=c≥t2, it follows that there exists b>c such that
(98)∫cb[H(b,s)ϕ(s)(k3q(s)-p(s))14Jγ[s,t0]-14Jγ[s,t0]θ12(b,s)]ds>0.
Clearly, we can observe that (68) is satisfied. Then (1) is oscillatory.Example 14.
Consider the following second order nonlinear differential equation with damping:(99)[(11+t2)(x′(t)x′(t)1+(x′(t))2)]′-1t(3x′(t)(x′(t))71+(x′(t))6)+3+t24[2t-(6n-4)π+1+t2t]×x(t)(1+(x(t))2)(1+(x′(t))2)=x3(t)cos⁡tsinx′(t)t2,(6n-4)π≤t≤(6n-72)πforn=1,2,….
We can see thatr(t)=1/(1+t2), h(t)=-1/t, when t≥t0=π/2. Let ϕ(t)=1, and then
(100)β(t)=k1k5ϕ′(t)r(t)-k4ϕ(t)h(t)=11t,δ(t)=lϕ(t)r(t)ψ(x(t))=1+t2,γ[t,π2]=δ(t)(∫Ttδ(s)ds)-1=1+t2t+(t3/3)+(π/2)+(π3/24).
Moreover, β(t) and δ(t) satisfy the conditions (65) and (66) of Theorem 12, respectively.Remark 15.
Theorems12 and 13 are an extension of Theorems 3.1 and 3.2 of Lu and Meng [4] where they established several new oscillation criteria for (1) with ψ(x)=1, f(x′)=x′, g(x)=x, Φ(x′)=1, and H(t,x,x′)=0.Theorem 16.
Suppose that (A1)–(A6) hold and
(101)g′(x)ψ(x)≥K>0foraconstantK,x≠0.
Moreover, assume that there exists a continuous differentiable function ρ such that ρ(t)>0 for all t∈[t0,∞) and a constant λ∈(1,∞). Then (1) is oscillatory if
(102)lim⁡t→∞sup⁡1tλ×∫t0t{(t-s)λρ(s)(k3q(s)-p(s))-r(s)4Mρ(s)×[(t-s)ρ(s)h(s)k2r(s)+λρ(s)-(t-s)ρ′(s)]2r(s)4Mρ(s)×(t-s)λ-2}ds=∞,M=Kl.Proof.
On the contrary, we assume that (1) has a nonoscillatory solution x(t). Without loss of the generality we suppose that x(t)>0 for all t∈[t0,∞). A similar argument holds also for the case when x(t)<0. Define the function w(t) as
(103)w(t)=r(t)ψ(x(t))f(x′(t))g(x(t)).
This definition and (1) imply
(104)w′(t)=p(t)-Φ(x′(t))q(t)+h(t)f(x′(t))g(x(t))-r(t)ψ(x(t))f(x′(t))g2(x(t))x′(t)g′(x(t)).
From (A1)–(A6) and (101) we have
(105)w′(t)≤p(t)-k3q(t)-h(t)k2r(t)w(t)-Mr(t)w2(t).
Then, for all t≥t0,
(106)∫t0t(t-s)λρ(s)(k3q(s)-p(s))ds≤-∫t0t(t-s)λρ(s)w′(s)ds-∫t0t(t-s)λρ(s)h(s)k2r(s)w(s)ds-M∫t0t(t-s)λρ(s)r(s)w2(s)ds.
Note that
(107)-∫t0t(t-s)λρ(s)w′(s)ds=(t-t0)ρ(t0)w(t0)-∫t0t[λρ(s)-(t-s)ρ′(s)](t-s)λw(s)ds.
Substituting (107) in (102) we obtain
(108)∫t0t(t-s)λρ(s)(k3q(s)-p(s))ds≤(t-t0)ρ(t0)w(t0)-∫t0t[λρ(s)-(t-s)ρ′(s)](t-s)λw(s)ds-∫t0t(t-s)λρ(s)h(s)k2r(s)w(s)ds-M∫t0t(t-s)λρ(s)r(s)w2(s)ds.
Then,
(109)∫t0t{(t-s)λρ(s)(k3q(s)-p(s))-r(s)4Mρ(s)×[(t-s)ρ(s)h(s)k2r(s)+λρ(s)-(t-s)ρ′(s)]2×(t-s)λ-2r(s)4Mρ(s)}ds≤(t-t0)ρ(t0)w(t0)-∫t0t{Mρ(s)r(s)(t-s)λ/2w(s)+12r(s)Mρ(s)×[(t-s)ρ(s)h(s)k2r(s)+λρ(s)-(t-s)ρ′(s)]Mρ(s)r(s)×(t-s)(λ-2)/2}2ds≤(t-t0)ρ(t0)w(t0).
Dividing (109) by tλ and taking the upper limit as t→∞, it contradicts with (102). This completes the proof.Example 17.
Consider the following second order nonlinear differential equation with damping:(110)[(t+1t(t+2))x′(t)]′+1t(t+1)x′(t)+t(3+t)(t+1)3x(t)(3+x′2(t))=-2tx(t)cos⁡tsinx′(t)(t+1)3,t≥t0=1.
We note that g′(x)/ψ(x)=1>0,  l=k2=k3=1. Considering ρ(t)=t and λ=2, then
(111)lim⁡t→∞sup⁡1t2∫1t{(t-2)2s2(s+2)(s+1)3-s+14s2(s+2)×((t-2)s+2+3s-t)2}ds=lim⁡t→∞sup⁡1t2[13t3+t28-t2ln⁡⁡(1+t)-5t3-t16ln⁡⁡(2+t)+t16ln⁡t-516ln⁡t-716ln⁡⁡(2+t)+2924+2ln⁡⁡(1+t)t28+t2ln⁡2+716ln⁡3-2ln⁡2+t16ln⁡3]=∞,lim⁡t→∞sup⁡1tλ×∫t0t{(t-s)λρ(s)(k3q(s)-p(s))-r(s)4Mρ(s)r(s)4Mρ(s)×[(t-s)ρ(s)h(s)k2r(s)+λρ(s)-(t-s)ρ′(s)]2}ds=lim⁡t→∞sup⁡1t2×∫1t{(t-s)2(s)(s2(s+1)3+2s(s+1)3)-s+14Ms2(s+2)×((t-s)ss(s+1)+2s-(t-s)(1))2}ds.
Example 17 is oscillatory.Remark 18.
Ifr(t)=1, ψ(x)=1, f(x′)=x′, g(x)=x, Φ(x′)=1, and H(t,x,x′)=0, Theorem 16 reduces to Theorem 1 of Philos [5]. Furthermore, when ψ(x)=1, Φ(x′)=1, and H(t,x,x′)=0, then Theorem 16 reduces to Theorem 1 of Elabbasy and Elsharabasy [2].Corollary 19.
If condition (101) in Theorem 16 is replaced by
(112)lim⁡t→∞sup⁡1tλ×∫t0t{r(s)ρ(s)×[(t-s)ρ(s)h(s)k2r(s)+λρ(s)-(t-s)ρ′(s)]2×(t-s)λ-2r(s)ρ(s)}ds<∞,lim⁡t→∞sup⁡1tλ∫t0t(t-s)λρ(s)(k3q(s)-p(s))ds=∞,
then Theorem 16 is still valid.



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*Source: 545189-2014-09-02.xml*