# Oscillation Criteria of Second-Order Nonlinear Differential Equations with Variable Coefficients

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

---

## Abstract

Some new oscillation criteria are given for second-order nonlinear differential equations with variable coefficients. Our results generalize and extend some of the well-known results in the literatures. Some examples are considered to illustrate the main results.


---

## Body

## 1. Introduction

Since many problems in physics, chemically reacting systems, celestial mechanics, and others fields are modeled by second-order nonlinear differential equations, the oscillatory and asymptotic behaviors of solutions of such differential equations have been investigated by many authors [1–9]. Investigation of nonlinear differential equation in this paper is motivated by [1], where some of the conditions required in the theorems contain the unknown solution x(t). It seems that any verification of such conditions is questionable.In this paper, we consider the oscillation of second-order nonlinear forced differential equation(1)(r(t)ψ(x(t))f(x′(t)))′+q(t)g(x(t))=H(t,x′(t),x(t)),t∈[t0,∞),
where r,q∈C([t0,∞),ℝ) and f,ψ,g∈C(ℝ,ℝ) and H is a continuous function on [t0,∞)×ℝ2.Consider the following with respect to (1):(A1)
r(t)>0, t≥0;(A2)
ψ(x)∈C(ℝ), ψ(x)>0 for all x;(A3)
xg(x)>0, g∈C1(ℝ) for x≠0;(A4)
H(t,y,x)/g(x)≤p(t) for all t∈[t0,∞); x,y∈ℝ and x≠0.We say that a nonzero function x:[t0,t1)→(-∞,∞), t1>t0 is called a nontrivial solution of (1) if x(t) satisfies (1) for all t∈[t0,t1). A solution x(t) of (1) is called oscillatory if there exists a sequence {λn}n=1∞ of points in the interval [t0,∞), so that lim⁡n→∞λn=∞ and x(λn)=0, n∈ℕ; otherwise it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.We note that iff(x′(t))=x′(t) and H(t,x(t),x′(t))=0, then (1) becomes the differential equation of the form
(2)(r(t)ψ(x(t))x′(t))′+q(t)g(x(t))=0,111111111111111t∈[t0,∞),t0>0.
The oscillation of solutions of (2) has been studied by El-sheikh [1], Manojlovic [10], and Ohriska and Zulova [5] under different conditions. Also, if ψ(x(t))=1, f(x′(t))=x′(t) and q(t)g(x(t))=Q(t,x), then (1) becomes the differential equation of the form
(3)(r(t)x′(t))′+Q(t,x)=H(t,x′(t),x(t)),t∈[t0,∞),t0>0.The oscillation of solutions of (3) has been studied by Remili [7, 8], where new results with additional suitable weighted function are found. Recently, Temtek and Tiryaki [9] obtained several new oscillation criteria for (3) with x′(t)=|x′(t)|α-1x′(t) by using generalized Riccati transformation and well-known techniques, such as averaging technique.The purpose of this research is to remove the above mentioned conditions which depend on the solution and extend some results presented in [1, 5, 7, 8].Although Grace [2] considered the function H defined by H(t,s)=(∫t0t(ds/Θ(s)))α, t≥t0>0 for some constant α>1 and there is a positive continuous function Θ on [t0,∞) such that ∫t0∞(ds/Θ(s))=∞, Kirane and Rogovchenko [11] and Tiryaki and Ayanlar [12] also mentioned that they can obtain new oscillation criteria with the a particular choice of the functions H and h with H(t,s)=(∫t0t(ds/Θ(s)))α-1, t≥t0>0 and α>2, ∫t0∞(ds/Θ(s))=∞, and Li [13] defined H(t,s)=(R(t)-R(s))α, α>1, R(t)=∫t0∞(ds/r(s)),  t≥t0. However, according to Ohriska and Zulova [5], research which involves oscillatory criteria based on the weight function H(t,s) has increased. These results have great theoretical value but they are less effective in applications. On the other hand, the results which contain the requirements only on the functions occurring in differential equation are usually better applicable. In particular, our weight function H(t,s) is not strictly Philos-type function because in the present paper, we ignored Philos’s conditions (see [2, 6, 10, 11, 14–17]. This paper contains only results of the latter kind by restricting only the functions in the differential equation (1). The relevance of the theorems in this paper is illustrated by examples.


## 2. Main Results

In this section we give some new sufficient conditions for all solutions of (1) to be oscillatory. We introduce the following notation:
(4)R(t)=∫t0tdsr(s),t≥t0,Z(t)=ρ(t)[q(t)-p(t)]-14A(ρ′(t))2ρ(t)r(t).
By Theorems 1 and 3, the given conditions include Kamenev’s integral criteria.Theorem 1.
One assumes that there are positive constantsk and l satisfying
(5)g′(x)ψ(x)≥k>0,∀x≠0,f2(y)≤lyf(y)∀y∈ℝ.
Furthermore let, for some integer m≥3,
(6)lim⁡t→∞sup⁡1Rm-1(t)∫t0t[R(t)-R(s)]m-1(q(s)-p(s))ds=∞.
Then (1) is oscillatory.Proof.
Assume, for the sake of a contradiction, that (1) has an eventually positive solution x(t)>0, for all T≥t0>0. The case x(t) being eventually negative can be similarly discussed. Set
(7)w(t)=r(t)ψ(x(t))f(x′(t))g(x(t)),t≥T.
Then by (A4), (1), and (5), we have
(8)w′(t)=[r(t)ψ(x(t))f(x′(t))]′g(x(t))-r(t)ψ(x(t))f(x′(t))x′(t)g′(x(t))g2(x(t)),w′(t)+q(t)-p(t)+klw2(t)r(t)≤0,t≥T.
Hence, for all t≥s≥T, it can be written as
(9)∫Tt(R(t)-R(s))m-1[q(s)-p(s)]ds≤-∫Tt(R(t)-R(s))m-1w′(s)ds-kl∫Tt(R(t)-R(s))m-1w2(s)r(s)ds=w(T)(R(t)-R(T))m-1-∫Tt[k(R(t)-R(s))m-1lr(s)w2(s)](m-1)r(s)(R(t)-R(s))m-2w(s)+k(R(t)-R(s))m-1lr(s)w2(s)]ds.
Then (9) implies that
(10)∫Tt(R(t)-R(s))m-1[q(s)-p(s)]ds≤w(T)(R(t)-R(T))m-1-l(m-1)24k×∫Tt(R(t)-R(s))m-31r(s)ds-∫Tt(R(t)-R(s))m-3r(s)×{kl(R(t)-R(s))w(s)+m-12k/l}2ds.
Then we have
(11)1Rm-1(t)∫Tt(R(t)-R(s))m-1[q(s)-p(s)]ds≤[1-R(T)R(t)]m-1w(T)+l(m-1)24k(m-2)R(t)[1-R(T)R(t)]m-2,t≥T.
It is clear that, for t∈[t0,T], it holds
(12)1Rm-1(t)∫t0T(R(t)-R(s))m-1[q(s)-p(s)]ds≤1Rm-1(t)∫t0TRm-1(t)[q(s)-p(s)]ds=∫t0T[q(s)-p(s)]ds.
Combining (11) and (12), it follows that
(13)1Rm-1(t)∫t0t(R(t)-R(s))m-1[q(s)-p(s)]ds≤[1-R(T)R(t)]m-1w(T)+l(m-1)24k(m-2)R(t)[1-R(T)R(t)]m-2+∫t0T[q(s)-p(s)]ds,t≥t0.
It is observable that lim⁡t→∞(1/R(t))=N∈[0,∞). Thus
(14)lim⁡t→∞sup⁡1Rm-1(t)∫t0t(R(t)-R(s))m-1[q(s)-p(s)]ds≤[1-NR(T)]m-1w(T)+Nl(m-1)24k(m-2)R(t)[1-NR(T)]m-2+∫t0T[q(s)-p(s)]ds<∞,
which contradicts condition (6). The proof of the theorem is completed.Example 2.
Consider the differential equation(15)[1t(1x2(t)+1)x′(t)]′+3x(t)(2+11+x2(t))=x(t)4(2+11+x2(t))sin⁡(x′(t))2(x′(t))2+1.
Clearly,
(16)g′(x)ψ(x)=3+2x41+x2≥3=k,f2(y)=y2+2y21+y2+y2(1+y2)2≤4y2,14f2(y)≤y2≤y2+y2y2+1=yf(y),∀y∈ℝ,H(t,x′(t),x(t))g(x(t))=x(t)4(2+11+x2(t))sin⁡(x′(t))2(x′(t))2+1×1x(t)(1+x2(t))2≤14=p(t),∀x′∈ℝ,x∈ℝ,t≥t0>0.
and, for any integer m≥3, we have
(17)lim⁡t→∞sup⁡1Rm-1(t)∫t0t[R(t)-R(s)]m-1(q(s)-p(s))ds=4lim⁡t→∞sup⁡1t4∫t0t[t22-s22]2(3-14)ds=∞.
By Theorem 1, this equation is oscillatory.Theorem 3.
Assume that (5) holds and suppose that
(18)lim⁡t→∞sup⁡1tm∫t0t(t-s)m(q(s)-p(s))-lm24k(t-s)m-2r(s)ds=∞,
for some m≥2. Then (1) is oscillatory.Proof.
Assume that (1) has an eventually positive solution x(t)>0, for all T≥t0>0. Taking w(t) as it is defined in (7), we obtain (8).
Hence by (A3), for t≥s≥T, we can write
(19)∫Tt(t-s)m[q(s)-p(s)]ds≤-∫Tt(t-s)mw′(s)ds-kl∫Tt(t-s)mw2(s)r(s)ds.
That is,
(20)∫Tt(t-s)m[q(s)-p(s)]ds≤(t-T)mw(T)-m∫Tt(t-s)m-1w(s)ds-kl∫Tt(t-s)mw2(s)r(s)ds=(t-T)mw(T)-∫Tt[k(t-s)mlr(s)w(s)+m2lr(s)(t-s)m-2k]2ds+lm24k∫Ttr(s)(t-s)m-2ds.
Hence it follows that
(21)∫Tt((t-s)m[q(s)-p(s)]-lm24kr(s)(t-s)m-2)ds≤(t-T)mw(T)≤(t-t0)mw(T),t≥T.
Using the inequality (21), for t≥t0, we obtained
(22)∫Tt((t-s)m[q(s)-p(s)]-lm24kr(s)(t-s)m-2)ds=∫t0T((t-s)m[q(s)-p(s)]-lm24kr(s)(t-s)m-2)ds+∫Tt((t-s)m[q(s)-p(s)]-lm24kr(s)(t-s)m-2)ds,∫t0t((t-s)m[q(s)-p(s)]-lm24kr(s)(t-s)m-2)ds≤(t-t0)mw(T)∫t0t(q(s)-p(s))ds+(t-t0)mw(T).
Thus,
(23)lim⁡t→∞sup⁡1tm∫t0t(-lm24kr(s)(t-s)m-2)(t-s)m[q(s)-p(s)]1111111111111-lm24kr(s)(t-s)m-2)ds≤∫t0t(q(s)-p(s))ds+w(T)<∞,
and it contradicts (18).Example 4.
Consider the differential equation(24)[(1+(x(t))22+(x(t))2)x′(t)(1+2(x′(t))24+2(x′(t))2)]′+(1+cos⁡⁡t)x(t)=x5(t)x2(t)+1cos⁡⁡t(x′(t))2(x′(t))2+1,t>0,
which satisfies the conditions of Theorem 3, since we have
(25)g′(x)ψ(x)=2+(x(t))21+(x(t))2≥2=k∀x≠0,yf(y)=y21+2y24+2y2>0,f2(y)≤Lyf(y),L=1,H(t,x′(t),x(t))g(x(t))=x5(t)x2(t)+1cos⁡⁡t(x′(t))2(x′(t))2+1×1x3(t)≤cos⁡⁡t=p(t),∀x′∈ℝ,x∈ℝ,t≥t0;
and, for any integer m≥2, we have
(26)lim⁡t→∞sup⁡1tm∫t0t(m28(t-s)m[1+cos⁡⁡(s)-cos⁡⁡(s)]11111111111111-m28(t-s)m-2)ds=lim⁡t→∞sup⁡1tm[t(m+1)-m28(m-1)t]=∞.
It thus follows that every solution of the given equation oscillates.Theorem 5.
Suppose that(27)0<k2≤ψ(x(t))≤k3,∀x∈ℝ,0<k4≤f(y)y≤k5,∀y=x′(t)≠0.
Let ρ be a positive continuously differentiable function over [T,∞) such that ρ′(t)≥0 over [T,∞),
(28)lim⁡t→∞∫T0t1ρ(s)r(s)ds=∞,(29)lim⁡t→∞∫T0tZ(s)ds=∞.
Then all solutions of (1) are oscillatory.Proof.
Letx(t) be a nonoscillatory solution on [T,∞), T≥T0 of (1). We assume that x(t) is positive on [T,∞). A similar argument holds for the case when x(t) is negative. Let
(30)w(t)=r(t)ψ(x(t))f(x′(t))g(x(t)).
Then
(31)w′(t)=[r(t)ψ(x(t))f(x′(t))g(x(t))]′.
Using (1), we obtain
(32)w′(t)=H(t,x′(t),x(t))g(x(t))-q(t)-r(t)ψ(x(t))f(x′(t))g′(x(t))x′(t)g2(x(t)).
Since g′(x(t))>k and condition (A4) we obtained,(33)w′(t)≤p(t)-q(t)-kr(t)ψ(x(t))r2(t)ψ2(x(t))(f(x′(t)))2g2(x(t))×x′(t)f(x′(t)).
From (27) and (30),
(34)w′(t)≤-[q(t)-p(t)]-kk3k41r(t)w2(t).
Therefore,
(35)[r(t)ψ(x(t))f(x′(t))g(x(t))]′≤-[q(t)-p(t)]-kk3k41r(t)w2(t).
Multiplying by ρ(t) and integrating from T to t, we obtained
(36)ρ(t)[r(t)ψ(x(t))f(x′(t))g(x(t))]′≤-ρ(t)[q(t)-p(t)]-Aρ(t)r(t)w2(t);A=kk3k4,∫Ttρ(s)[r(s)ψ(x(s))f(x′(s))g(x(s))]′ds≤∫Tt-ρ(s)[q(s)-p(s)]ds-∫TtAρ(s)r(s)w2(s)ds.
Let
(37)CT=ρ(T)r(T)ψ(x(T))f(x′(T))g(x(T)).
By integrating by parts, we obtained
(38)ρ(t)r(t)ψ(x(t))f(x′(t))g(x(t))≤CT-∫Ttρ(s)[q(s)-p(s)]ds-∫TtAρ(s)r(s)w2(s)ds+∫Ttρ′(s)r(s)ψ(x(s))f(x′(s))g(x(s))ds=CT-∫Ttρ(s)[q(s)-p(s)]ds-∫TtAρ(s)r(s)w2(s)ds+∫Ttρ′(s)w(s)ds=CT-∫Ttρ(s)[q(s)-p(s)]ds+∫Tt[-Aρ(s)r(s)w2(s)+ρ′(s)w(s)]ds.
The inequality presented by (38) can be written as
(39)ρ(t)r(t)ψ(x(t))f(x′(t))g(x(t))≤CT-∫Ttρ(s)[q(s)-p(s)]ds+∫Tt-Aρ(s)r(s)×[(w(s)-ρ′(s)r(s)2Aρ(s))2-(ρ′(s))2r2(s)4A2ρ2(s)]ds.
Let w(s)-((ρ′(s)r(s))/(2Aρ(s)))=W(s),
(40)ρ(t)r(t)ψ(x(t))f(x′(t))g(x(t))≤CT-∫Ttρ(s)[q(s)-p(s)]ds-∫TtAρ(s)r(s)[W2(s)-(ρ′(s)r(s)2Aρ(s))2]ds=CT-∫Ttρ(s)[q(s)-p(s)]ds-∫Tt[Aρ(s)r(s)W2(s)-ρ′2(s)r(s)4Aρ(s)]ds=CT-∫Ttρ(s)[q(s)-p(s)]-14A(ρ′(s))2ρ(s)r(s)ds-∫TtAρ(s)r(s)W2(s)ds,ρ(t)r(t)ψ(x(t))f(x′(t))g(x(t))≤CT-∫TtZ(s)ds.
Taking the limit for both sides of (40) and using (29),
(41)lim⁡t→∞ρ(t)r(t)ψ(x(t))f(x′(t))g(x(t))⟶-∞.
Hence, there exists T1≥T such that
(42)f(x′(t))<0,∀t≥T1,x′(t)<0,∀t≥T1.
Condition (29) also implies that ∫T∞ρ(s)[q(s)-p(s)]ds=∞, and there exists T2≥T1 such that
(43)∫T1T2ρ(s)[q(s)-p(s)]ds=0,∫T2tρ(s)[q(s)-p(s)]ds≥0,∀t≥T2.
Multiplying (1) by ρ(t) and integrating by parts on [T2,t], we obtained
(44)ρ(t)[r(t)ψ(x(t))f(x′(t))]′+ρ(t)q(t)g(x(t))=ρ(t)H(t,x′(t),x(t)),
which implies
(45)ρ(t)[r(t)ψ(x(t))f(x′(t))]′g(x(t))≤-ρ(t)[q(t)-p(t)],ρ(t)[r(t)ψ(x(t))f(x′(t))]′≤-ρ(t)g(x(t))[q(t)-p(t)].
Now, integrating by parts, we get
(46)ρ(t)[r(t)ψ(x(t))f(x′(t))]-∫T2tρ′(s)r(s)ψ(x(s))f(x′(s))ds≤CT2-∫T2tρ(s)g(x(s))[q(s)-p(s)]ds=CT2-g(x(t))∫T2tρ(s)[q(s)-p(s)]ds+∫T2tx′(s)g′(x(s))∫T2sρ(u)[q(u)-p(u)]duds+∫T2tρ′(s)r(s)ψ(x(s))f(x′(s))ds≤CT2,∀t≥T1,
where CT2=(ρ(T2)r(T2)ψ(x(T2))f(x′(T2)))/(g(x(T2)))<0.
Therefore,(47)ρ(t)r(t)ψ(x(t))f(x′(t))≤CT2.
From (27), we have
(48)x′(t)≤CT2k2k41r(t)ρ(t),x(t)≤CT2k2k4∫T2t1r(s)ρ(s)ds.
Finally, from (28), x(t)→-∞ast→∞, which is a contradiction.Example 6.
Let us consider the following equation:(49)[1t(5+x6(t)x6(t)+1)(13x′(t)+x′(t)(x′(t))2+1)]′+(t+sin⁡tt)x5(t)=2x12sin⁡tcos⁡⁡(x′(t)+1)(x7+1)t3,HHHHHHHHHHHHHHHHHHHHt≥π2.
We note that
(50)H(t,x′(t),x(t))g(x(t))=2x12sin⁡tcos⁡⁡(x′(t)+1)(x7+1)t3×1x5(t)≤2t3=p(t),1111111111111111∀x′∈ℝ,x∈ℝ,t≥t0.
Hence,
(51)r(t)=1t>0,t>π2,5≤5+x6(t)x6(t)+1<6,∀x∈ℝ,13<13+1(x′(t))2+1≤14,∀y≠0.
Let
(52)ρ(t)=t⟹ρ′(t)=1,∫t∞dsr(s)ρ(s)=∫t∞ds=∞,Z(s)=ρ(s)[q(s)-p(s)]-14A(ρ′(s))2ρ(s)r(s)=s[s+sin⁡ss-2s3]-14A1s2.
For every t≥T=π/2, we obtained
(53)∫t0∞Z(s)ds=∫t0∞s2+sin⁡s-2s2-14As2ds=∞.
Then, Theorem 5 ensures that every solution in this example is oscillatory.Theorem 7.
If conditions (27)–(29) hold and
(54)∫T0∞ρ(s)[q(s)-p(s)]ds<∞,(55)lim⁡t→∞⁡inf⁡⁡[∫TtZ(s)ds]≥0,∀largeT,(56)lim⁡t→∞∫T0t1ρ(s)r(s)∫s∞Z(u)duds=∞,(57)∫ε∞dyg(y)<∞,∫-ε-∞dyg(y)<∞,foreveryε>0,
thus all solutions of (1) are oscillatory.Remark 8.
Condition (55) implies that ∫T∞Z(s)≥0 and lim⁡⁡inf⁡t→∞∫T∞Z(s)ds=∫T∞Z(s)ds; hence, (56) takes the form of ∫T∞Z(s)≥0, for all large T.Proof.
Letx(t) be a nonoscillatory solution on [T,∞), T≥T0 of (1). Let us assume that x(t) is positive on [T,∞) and consider the following three cases for the behavior of x′(t).
Case
1. Consider x′(t)>0 for T1≥T for some t≥T1; then, from (40), we have
(58)∫T1tR(s)ds≤r(T1)ρ(T1)ψ(x(T1))f(x′(T1))g(x(T1))-ρ(t)r(t)ψ(x(t))f(x′(t))g(x(t)),k4k2r(t)ρ(t)x′(t)g(x(t))≤r(T1)ρ(T1)ψ(x(T1))f(x′(t))g(x(T1))-∫T1tR(s)ds.
From (27), we obtained
(59)∫T1tZ(s)ds≤r(T1)ρ(T1)ψ(x(T1))f(x′(T1))g(x(T1))-k4k2r(t)ρ(t)x′(t)g(x(t)).
Hence, for all t≥T1,
(60)∫t∞R(s)ds≤k4k2r(t)ρ(t)x′(t)g(x(t)),∫T1t1r(s)ρ(s)∫s∞Z(u)duds≤k4k2∫T1∞x′(s)g(x(s))ds,∫T1t1r(s)ρ(s)∫s∞Z(u)duds≤k4k2∫T1(s)∞dyg(y).
Using (57), we obtained
(61)∫T1t1r(s)ρ(s)∫s∞Z(u)duds<∞.
This contradicts condition (56).
Case
2. If x′(t) is oscillatory, then there exists a sequence {αn}→∞ on [T,∞) such that x′(αn)<0. Let us assume that N is sufficiently large, so that
(62)∫αN∞Z(s)ds≥0.
Then, from (27) and (40), we have
(63)k3k5ρ(t)r(t)x′(t)g(x(t))≤CαN-∫αNtZ(s)ds.
Thus,
(64)k3k5lim⁡⁡sup⁡t→∞ρ(t)r(t)x′(t)g(x(t))≤CαN+lim⁡⁡sup⁡t→∞[-∫αNtZ(s)ds]=CαN-lim⁡⁡inf⁡t→∞[∫αNtZ(s)ds].
By (55), we obtained
(65)k3k5lim⁡⁡sup⁡t→∞ρ(t)r(t)x′(t)g(x(t))<0,
which contradicts the fact that x′(t) oscillates.
Case
3. Let x′(t)<0 for t≥T for some T1≥T; then, for any t0≥T0, there exists t1≥t0 such that ∫t1∞ρ(s)[q(s)-p(s)]ds≥0 for all t≥t1. Choosing t1≥T1, and multiplying (1) by ρ(t) and integrating by parts, we obtained
(66)ρ(t)[r(t)ψ(x(t))f(x′(t))]-∫t1tρ′(s)r(s)ψ(x(s))f(x′(s))ds≤Ct1-∫t1tρ(s)g(x(s))[q(s)-p(s)]ds=Ct1-g(x(t))∫t1tρ(s)[q(s)-p(s)]ds+∫t1tx′(s)g′(x(s))∫t1sρ(u)[q(u)-p(u)]duds+∫t1tρ′(s)r(s)ψ(x(s))f(x′(s))ds,
where Ct1=ρ(t1)r(t1)ψ(x(t1))f(x′(t1))<0.
Thus,(67)ρ(t)r(t)ψ(x(t))f(x′(t))≤Ct1.
From (27), we obtained
(68)x′(t)≤Ct1k2k41r(t)ρ(t),x(t)≤Ct1k2k4∫T2t1r(s)ρ(s)ds.
From (28), it follows that x(t)→-∞, as t→∞, which is a contradiction.Example 9.
Consider the following equation:(69)[t(8+x4(t)x4(t)+1)(7x′(t)+(x′(t))5(x′(t))4+1)]′+1t3x3(t)=x3cos⁡⁡xsin⁡2x′(t)t4,t>1.
We note that
(70)H(t,x′(t),x(t))g(x(t))=x3cos⁡⁡xsin⁡2x′(t)t4×1x3(t)≤1t4=p(t),∀x′∈ℝ,x∈ℝ,t≥t0,
hence,
(71)r(t)=t>0,8≤ψ(x(t))=8+x4(t)x4(t)+1<9,∀x∈ℝ,7≤f(y)y2=7+(x′(t))4(x′(t))4+1<8,∀y≠0.
Let ρ(t)=t⇒ρ′(t)=1; then.                           (72)lim⁡t→∞∫T0t1r(s)ρ(s)ds=lim⁡t→∞∫T0tds=∞,∫T0tρ(s)[q(s)-p(s)]ds=∫T0t(1s2-1s3)ds<∞.
Because Z(s)=ρ(s)[q(s)-p(s)]-(1/4A)((ρ′(s))2/ρ(s))r(s)=(1/s2)-(1/s3)-(1/4A),
(73)lim⁡t→∞inf⁡∫TtZ(s)ds=lim⁡t→∞inf⁡∫Tt(1s2-1s3-14A)ds=0,lim⁡t→∞∫T0t1r(s)ρ(s)∫s∞Z(u)duds=lim⁡t→∞∫T0t1s2∫s∞(1u2-1u3-14A)duds=∞,∫ε∞dyf(y)=∫ε∞dyy3=-2y2|ε∞=2ε2<∞,∫-ε-∞dyf(y)=2ε2<∞.
Thus, from Theorem 7, it follows that the equation is oscillatory.Remark 10.
If we letZ(t)=1 and f(x′(t))=x′(t) in our Theorems 5 and 7, we will get Theorems 1 and 2 of Remili [7] which extends the results of Graef et al. [3].



---

*Source: 279236-2014-02-18.xml*