# Oscillation Theorems for Second-Order Forced Neutral Nonlinear Differential Equations with Delayed Argument

Authors: Jing Shao; Fanwei Meng
Journal: International Journal of Differential Equations
 (2010)

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## Abstract

We are concerned with the oscillation of the forced second-order neutral nonlinear differential
equations with delayed argument in the form(r(t)(x(t)+a(t)x(σ(t)))′)′+p(t)f(x(τ(t)))+∑i=1nqi(t)|x(t)|λisngx(t)=e(t). No restriction is imposed on the potentials p(t), qi(t), ande(t) to be nonnegative. Our methodology is somewhat different from those of previous authors.


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## Body

## 1. Introduction

In this paper, we study the oscillatory behavior of the forced neutral nonlinear functional differential equation of the form(1.1)(r(t)(x(t)+a(t)x(σ(t)))′)′+p(t)f(x(τ(t)))+∑i=1nqi(t)|x(t)|λisgnx(t)=e(t),
where t≥t0. In this paper, we assume that(I1)
r(t)∈C([t0,∞),(0,∞)),r′(t)≥0,∫∞(1/r(t))dt=∞,(I2)
a(t)∈C([t0,∞),[0,1)),(I3)
σ(t)∈C([t0,∞),ℝ) is nondecreasing, σ(t)≤t for t≥t0, and limt→∞σ(t)=∞,(I4)
τ(t)∈C([t0,∞),ℝ),τ(t)≤t for t≥t0 and limt→∞τ(t)=∞,(I5)
p(t),qi(t), and e(t) are continuous functions defined on [0,∞), p(t)>0, λ1>⋯>λm>1>λm+1>⋯>λn>0(n>m≥1),(I6)
f(x) is nondecreasing, f(x)/x≥M>0, and x≠0.No restriction is imposed on the potentialsp(t),qi(t), and e(t) to be nonnegative. As usual, a solution of (1.1) is called oscillatory if it is defined on some ray [T,∞) with T≥0 and has unbounded set of zeros. (1.1) is called oscillatory if all of its solutions on some ray are oscillatory.In the last decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of second-order linear and nonlinear delay differential equations (see, for example, [1–22] and the references therein). Let us consider the familiar forced Emden-Fowler equation(1.2)x′′(t)+p(t)|x(t)|λsgnx(t)=e(t),t≥t0.
When λ1>1, (1.2) is known as the superlinear equation, and when 0<λ1<1, it is known as the sublinear equation. The oscillation of (1.2) has been the subject of much attention during the last 50 years; see the seminal book by Agarwal, et al. [23]. Here, we refer to the papers [1–3] and the references cited therein. In this case, one can usually establish oscillation criteria for more general nonlinear equations by using a technique introduced by Kartsatos [9] where it is additionally assumed that f is the second derivative of an oscillatory function. This approach has been expressed in [5, 6]. Sun [4] has extended these results to delay differential equations of the form of (1.2), where λ≥1 and the potentials p(t) and e(t) are allowed to change sign. However, Sun [4] does not say anything else for the oscillation of equation (1.2) with 0<λ<1. Later, employing the arguments in [4], Çakmak and Tiryaki [7] have established similar oscillation criteria for the equation of the form(1.3)x′′(t)+q(t)f(x(τ(t)))=e(t),
where f(x) is assumed to satisfy certain growth conditions.Very recently, Sun et al. [13, 14] obtained some new oscillation criteria for the equations in the form(1.4)(r(t)x′(t))′+p(t)x(t)+∑i=1nqi(t)|x(t)|λisgnx(t)=0,(r(t)x′(t))′+p(t)x(t)+∑i=1nqi(t)|x(t)|λisgnx(t)=e(t),
where λ1>⋯>λm>1>λm+1>⋯>λn>0(n>m≥1). He also established oscillation theorems when n>1. When n=1, this approach was initiated by Agarwal and Grace [1, pages 244–249] for higher-order equations and subsequently developed in papers of Ou and Wong [15], Q.Yang [18], X.Yang [19], as well as Sun and Agarwal [16, 17].In [24], Xu and Meng studied the oscillation of the equation(1.5)(r(t)(x(t)+a(t)x(σ(t)))′)′+∑i=1nqi(t)fi(y(τi(t)))=0,t≥t0,
by using the generalized Riccati technique and the function class 𝒴.The purpose of this paper is to give some new oscillation criteria for (1.1), which can be regarded as further investigation for the (1.1) including the papers of Sun and Wong [13], Xu and Meng [24]. These criteria do not assume that r(t),p(t),qi(t), and e(t) are of definite sign. Our methodology is somewhat different from those of previous authors, and the results we obtained are more general than those of Sun and Wong [13].


## 2. Main Results

We will need the following lemmas that have been proved in [13].Lemma 2.1 (see [13]).
Letλi,i=1,2,…,n, be n-tuple satisfying λ1>⋯>λm>1>λm+1>⋯>λn>0. Then there exists an n-tuple (k1,k2,…,kn) satisfying
(a)∑i=1nλiki=1,
which also satisfies either
(b)∑i=1nki<1,0<ki<1,
or
(c)∑i=1nki=1,0<ki<1.Lemma 2.2 (see [13]).
Letu,A,B,C,andD be positive real numbers. Then(i)
Auα+B≥α(α-1)1/α-1A1/αB1-1/αu,α>1,(ii)
Cu-Duα≥(α-1)αα/(1-α)Cα/(α-1)D1/(1-α),0<α<1.Remark 2.3.
For a given set of exponentsλi satisfying λ1>⋯>λm>1>λm+1>⋯>λn>0, Lemma 2.1 ensures the existence of an n-tuple (k1,k2,…,kn) such that either (a) and (b) hold or (a) and (c) hold. When n = 2 and λ1>1>λ2>0, in the first case, we have that(d)k1=1-λ2(1-k0)λ1-λ2,k2=λ1(1-k0)-1λ1-λ2,
where k0 can be any positive number satisfying 0<k0<(λ1-1)/λ1. This will ensure that 0<k1,k2<1, and conditions (a) and (b) are satisfied. In the second case, we simply solve (a) and (c) and obtain
(e)k1=1-λ2λ1-λ2,k2=λ1-1λ1-λ2.Theorem 2.4.
Suppose that, for anyT≥0, there exist constants a1,b1,a2,b2 such that T≤a1<b1≤a2<b2, and
(2.1)qi(t)≥0,t∈[τ(a1),b1]∪[τ(a2),b2],i=1,…,n,e(t)≤0,t∈[τ(a1),b1],e(t)≥0,t∈[τ(a2),b2].
Let D(aj,bj)={u∈C1[aj,bj]:uν+1>0,ν>0isaconstant,t∈(aj,bj),andu(aj)=u(bj)=0}, for j = 1, 2. Assume that there exists a positive, nondecreasing function ρ∈C1([t0,∞),ℝ) such that, for some H∈D(aj,bj) and for some θ≥1,
(2.2)∫ajbj[Hν+1(t)ρ(t)R(t)-θρ(t)r(t)Hν-1(t)A2(t)4]dt>0,
for j = 1, 2, then (1.1) is oscillatory, where
(2.3)R(t)=Mp(t)[1-a(τ(t))]τ(t)-τ(aj)t-τ(aj)+a0(1-a(t))|e(t)|k0∏i=1nqiki(t),A(t)=H(t)ρ′(t)ρ(t)+(ν+1)H′(t),a0=∏i=0nki-ki, and k0,k1,…,kn are positive constants satisfying (a) and (b) of Lemma 2.1.Proof.
Assume to the contrary that there exists a solutionx(t) of (1.1) such that x(t)>0,x(τ(t))>0,x(σ(t))>0, when t≥t0>0, for some t0 depending on the solution x(t). Set
(2.4)z(t)=x(t)+a(t)x(σ(t)).
By assumption, we have thatz(t)>0 for t≥t0≥0, and from (2.4) it follows that
(2.5)(r(t)z′(t))′=e(t)-p(t)f(x(τ(t)))-∑i=1nqi(t)|x(t)|λisgnx(t)≤0,t≥t0≥0.
It is not difficult to show that z′(t) is eventually positive. In fact, first, we know that z′(t)≢0 for sufficiently large t, since z(t) is nontrivial. Second, if there exists an t1≥t0 such that r(t1)z′(t1)=C<0, then r(t)z′(t)≤C for t≥t1≥t0, that is, z′(t)≤C/r(t), and hence, z(t)≤z(t1)+∫t1t(C/r(t))dt→-∞ as t→∞, which contradicts the fact that z(t)>0. Without loss of generality; say z′(t)>0, t≥t0≥0. Thus we have that
(2.6)x(t)≥z(t)-a(t)z(t),t≥t0≥0.
Define
(2.7)w(t)=ρ(t)r(t)z′(t)z(t),t≥t0.
It follows from (2.7) that w(t) satisfies the following differential equality:
(2.8)w′(t)=ρ′(t)ρ(t)w(t)-ρ(t)p(t)f(x(τ(t)))z(t)-w2(t)ρ(t)r(t)+ρ(t)e(t)z(t)-ρ(t)∑i=1nqi(t)xλi(t)z(t).
Using (2.6), we have that
(2.9)w′(t)≤ρ′(t)ρ(t)w(t)-ρ(t)p(t)f((1-a(τ(t)))z(τ(t)))z(t)-w2(t)ρ(t)r(t)+ρ(t)e(t)z(t)-ρ(t)∑i=1nqi(t)(1-a(t))λizλi-1(t),
and by the condition f(x)/x≥M>0, we have that
(2.10)w′(t)≤ρ′(t)ρ(t)w(t)-Mρ(t)p(t)(1-a(τ(t)))z(τ(t))z(t)-w2(t)ρ(t)r(t)+ρ(t)e(t)z(t)-ρ(t)∑i=1nqi(t)(1-a(t))λizλi-1(t).
By assumption, we can choose a1,b1≥t0 such that b1≥τ(a1),τ2(a1)=τ(τ(a1))≥t0,qi(t)≥0, i=1,2,…,n, for t∈[τ(a1),b1], and e(t)≤0 for t∈[τ(a1),b1]. Recall the arithmetic-geometric mean inequality (see [25])
(2.11)∑i=0nkiui≥∏i=0nuiki,ui≥0,
where k0=1-∑i=1nki and ki>0,i=1,2,…,n are chosen to satisfy (a) and (b) of Lemma 2.1 for the given λ1,λ2,…,λn>0. Now return to (2.10) and identify u0=k0-1|e(t)|z-1(t) and ui=ki-1qi(t)zλi-1(t)(1-a(t))λi in (2.11) to obtain
(2.12)w′(t)≤ρ′(t)ρ(t)w(t)-Mρ(t)p(t)(1-a(τ(t)))z(τ(t))z(t)-w2(t)ρ(t)r(t)-ρ(t)(1-a(t))k0-k0|e(t)|k0∏i=1nki-kiqiki(t).
From (1.1), we can easily obtain z′′(t)≤0, for t∈[τ(a1),b1]. Therefore, we have that, for t∈[τ(a1),b1],
(2.13)z(t)-z(τ(a1))=z′(s)(t-τ(a1))≥z′(t)(t-τ(a1)).
Noting that z(t)>0 for t≥τ(a1), we get by (2.13) that
(2.14)z(t)≥z′(t)(t-τ(a1)),t∈[τ(a1),b1],
that is,
(2.15)z′(t)z(t)≤1t-τ(a1),t∈[τ(a1),b1].
Integrating (2.15) from τ(t) to t>a1, we obtain
(2.16)z(τ(t))z(t)≥τ(t)-τ(a1)t-τ(a1),t∈(a1,b1].
By using (2.16) in (2.12), we have that, for t∈(a1,b1],
(2.17)w′(t)≤ρ′(t)ρ(t)w(t)-Mρ(t)p(t)[1-a(τ(t))]τ(t)-τ(a1)t-τ(a1)-w2(t)ρ(t)r(t)-ρ(t)(1-a(t))k0-k0|e(t)|k0∏i=1nki-kiqiki(t)=ρ′(t)ρ(t)w(t)-ρ(t)R(t)-w2(t)ρ(t)r(t).
Multiplying both sides of (2.17) by Hν+1(t) as given in the hypothesis of Theorem 2.4 and integrating (2.17) from a1 to b1, we obtain
(2.18)∫a1b1Hν+1(t)ρ(t)R(t)dt≤∫a1b1Hν+1(t)ρ′(t)ρ(t)w(t)dt-∫a1b1Hν+1(t)w′(t)dt-∫a1b1Hν+1(t)w2(t)ρ(t)r(t)dt.
Using the integration by parts formula, we have that
(2.19)∫a1b1Hν+1(t)w′(t)dt=Hν+1(t)w(t)∣a1b1-∫a1b1(ν+1)Hν(t)H′(t)w(t)dt=-∫a1b1(ν+1)Hν(t)H′(t)w(t)dt,
where H(a1)=H(b1)=0. Substituting (2.19) into (2.18), we obtain
(2.20)∫a1b1Hν+1(t)ρ(t)R(t)dt≤∫a1b1Hν+1(t)ρ′(t)ρ(t)w(t)dt+∫a1b1(ν+1)Hν(t)H′(t)w(t)dt-∫a1b1Hν+1(t)w2(t)ρ(t)r(t)dt=∫a1b1A(t)Hν(t)w(t)dt-∫a1b1Hν+1(t)w2(t)ρ(t)r(t)dt.
Then
(2.21)∫a1b1Hν+1(t)ρ(t)R(t)dt≤-∫a1b1[-A(t)Hν(t)w(t)+Hν+1(t)w2(t)ρ(t)r(t)]dt=-∫a1b1[Hν+1(t)θρ(t)r(t)w(t)-θρ(t)r(t)4Hν+1(t)Hν(t)A(t)]2dt+∫a1b1[θρ(t)r(t)4Hν+1(t)Hν(t)A(t)]2dt-∫a1b1(θ-1)Hν+1(t)θρ(t)r(t)w2(t)dt.
From the hypothesis of Theorem 2.4 and (2.21), we have that
(2.22)∫aibi[Hν+1(t)ρ(t)R(t)-(θρ(t)r(t)4Hν+1(t)Hν(t)A(t))2]dt=∫aibi[Hν+1(t)ρ(t)R(t)-θρ(t)r(t)Hν-1(t)A2(t)4]dt≤0,
which contradicts (2.2). When x(t) is eventually negative, we can obtain similar contradiction using the interval [τ(a2),b2] instead of [τ(a1),b1]. This completes the proof.Remark 2.5.
Letr(t)=1, qi(t)=0, and a(t)=0 for i=1,2,…,n. It is easy to see that Theorem 2.4 reduces to Theorem 1 of [7].In Theorem2.6, we do not impose any restriction on signs of those coefficients corresponding to sublinear terms of (1.1), that is, ql(t) for l=m+1,…,n. If it is nonpositive, we can easily see that Theorem 2.4 is invalid. However, the following theorem is valid for this case.Theorem 2.6.
Suppose that, for anyT≥0, there exist constants a1,b1,a2,b2 such that T≤a1<b1≤a2<b2 and
(2.23)qi(t)≥0,t∈[τ(a1),b1]∪[τ(a2),b2],i=1,…,m,e(t)<0,t∈[τ(a1),b1],e(t)>0,t∈[τ(a2),b2].
Assume that there exists a positive, nondecreasing function ρ∈C1([t0,∞),ℝ) such that, for some H∈D(aj,bj) and for some θ≥1,
(2.24)∫ajbj[Hν+1(t)ρ(t)R¯(t)-θρ(t)r(t)Hν-1(t)A2(t)4]dt>0,
for j = 1, 2, then (1.1) is oscillatory, where
(2.25)A(t)=H(t)ρ′(t)ρ(t)+(ν+1)H′(t),R¯(t)=p(t)M[1-a(τ(t))]τ(t)-τ(aj)t-τ(aj)+∑i=1mμi(βi|e(t)|)1-1/λiqi1/λi(t)-∑l=m+1nγl(δl|e(t)|)1-1/λlq¯l1/λl(t),
with ∑i=1mβi+∑l=m+1nδl=1 for βi>0,δl>0, μi=λi(λi-1)1/λi-1,i=1,…,m,γl=λl(1-λl)1/λl-1, and q¯l(t)=max{-ql(t),0},l=m+1,…,n.Proof.
Assume to the contrary that there exists a solutionx(t) of (1.1) such that x(t)>0,x(τ(t))>0, when t≥t0>0, for some t0 depending on the solution x(t). When x(t) is eventually negative, the proof follows the same argument using the interval [τ(a2),b2] instead of [τ(a1),b1]. Apply the assumption of βi and δl, then (1.1) is rearranged as
(2.26)(r(t)(x(t)+a(t)x(σ(t)))′)′+p(t)f(x(τ(t)))+∑i=1m[qi(t)|x(t)|λi-βie(t)]+∑l=m+1n[ql(t)|x(t)|λl-δle(t)]=0.
Noting the assumption (2.23) and applying Lemma 2.2(i) to the first summation term in (2.26), we get that
(2.27)(r(t)(x(t)+a(t)x(σ(t)))′)′+p(t)f(x(τ(t)))+x(t)∑i=1mμi(βi|e(t)|)1-1/λiqi1/λi(t)+∑l=m+1n(ql(t)|x(t)|λl-δle(t))≤0.
Introduce the Riccati substitution as (2.4) and apply Lemma 2.2(ii) to each of the nonlinear terms in the last sum in (2.27). Here u=x(t),α=λl,D=q¯l(t), and
(2.28)C=α(1-α)1/α-1(δl|e(t)|)1-1/αq¯l1/α(t).
We can obtain from (2.27) the following Riccati inequality:
(2.29)w′(t)≤ρ′(t)ρ(t)w(t)-ρ(t)p(t)M[1-a(τ(t))]τ(t)-τ(a1)t-τ(a1)-w2(t)ρ(t)r(t)-ρ(t)∑i=1mμi(βi|e(t)|)1-1/λiqi1/λi(t)+ρ(t)∑l=m+1nγl(δl|e(t)|)1-1/λlq¯l1/λl(t)=ρ′(t)ρ(t)w(t)-ρ(t)R¯(t)-w2(t)ρ(t)r(t),
where q¯l(t)=max{-ql(t),0}. The remaining argument is the same as that in Theorem 2.4 and this completes the proof.Remark 2.7.
Letf(x)=x,a(t)=0, and τ(t)=t,n=1. Theorems 2.4 and 2.6 reduce to Theorem 1 of [8, 11].We will use the function class𝒴 to study the oscillatory of (1.1). We say that a function Φ=Φ(t,s,l) belongs to the function class 𝒴, denoted by Φ∈𝒴 if Φ∈C(E,ℝ), where E={(t,s,l):t0≤l≤s≤t<∞}, which satisfies Φ(t,t,l)=0, andΦ(t,l,l)=0,Φ(t,s,l)≠0 for l<s<t, and has the partial derivative ∂Φ/∂s on E such that ∂Φ/∂s is locally integrable with respect to s in E.We defined the operatorB[·;l,t] by(2.30)B[g;l,t]=∫ltΦ2(t,s,l)g(s)ds,fort≥s≥l≥t0,g(s)∈C[t0,∞),
and the function φ=φ(t,s,l) is defined by(2.31)∂Φ(t,s,l)∂s=φ(t,s,l)Φ(t,s,l).
It is easy to verify that B[·;l,t] is a linear operator and satisfies(2.32)B[g′;l,t]=-2B[gφ;l,t],forg∈C1[t0,∞).Theorem 2.8.
Suppose that, for anyT≥0, there exist constants a1,b1,a2,b2 such that T≤a1<b1≤a2<b2 and satisfies (2.1). Assume that there exists a function Φ∈𝒴, such that, for each t≥t0 and for some θ≥1,
(2.33)limsupt→∞B[ρ(s)R(s)-θρ(s)r(s)4(2φ-ρ′(s)ρ(s))2;t0,t]>0,
for j=1, 2, then (1.1) is oscillatory, where the operator B is defined by (2.30), φ=φ(t,s,t0) is defined by (2.31),
(2.34)R(s)=Mp(s)[1-a(τ(s))]τ(s)-τ(aj)s-τ(aj)+a0(1-a(s))|e(s)|k0∏i=1nqiki(s),a0=∏i=0nki-ki, and k0,k1,…,kn are positive constants satisfying (a) and (b) of Lemma 2.1.Proof.
We proceed as in Theorem2.4. Assume to the contrary that there exists a solution x(t) of (1.1) such that x(t)>0,x(τ(t))>0, when t⩾t0>0, for some t0 depending on the solution x(t). From the proof of Theorem 2.4, we obtain (2.16) for all t∈[τ(a1),b1]. Applying B[·;t0,t],(t≥t0) to (2.17), we have that
(2.35)B[w′(s);t0,t]≤-B[ρ′(s)ρ(s)w(s);t0,t]-B[ρ(s)R(s);t0,t]-B[w2(s)ρ(s)r(s);t0,t].
By (2.32) and above inequality, we have, for t≥t0, that
(2.36)B[ρ(s)R(s);t0,t]≤2B[w(s)φ;t0,t]-B[ρ′(s)ρ(s)w(s);t0,t]-B[w2(s)ρ(s)r(s);t0,t]=B[-(w(s)θρ(s)r(s)-θρ(s)r(s)(φ-ρ′(s)2ρ(s)))2+θρ(s)r(s)4(2φ-ρ′(s)ρ(s))2-θ-1θw2(s)ρ(s)r(s);t0,t(w(s)θρ(s)r(s)-θρ(s)r(s)(φ-ρ′(s)2ρ(s)))2]≤B[θρ(s)r(s)4(2φ-ρ′(s)ρ(s))2;t0,t].
That is,(2.37)B[ρ(s)R(s)-θρ(s)r(s)4(2φ-ρ′(s)ρ(s))2;t0,t]≤0.
Taking the super limit in the above inequality, we have that
(2.38)limsupt→∞B[ρ(s)R(s)-θρ(s)r(s)4(2φ-ρ′(s)ρ(s))2;t0,t]≤0,
which contradicts assumption (2.33). When x(t) is eventually negative, the proof follows the same argument using the interval [τ(a2),b2] instead of [τ(a1),b1]. This completes the proof.Theorem 2.9.
Suppose that, for anyT≥0, there exist constants a1,b1,a2,b2 such that T≤a1<b1≤a2<b2 and satisfies (2.23). Assume that there exists a function Φ∈𝒴 such that, for each t≥t0 and for some θ≥1,
(2.39)limsupt→∞B[ρ(s)R¯(s)-θρ(s)r(s)4(2φ-ρ′(s)ρ(s))2;t0,t]>0,
for j = 1, 2, then (1.1) is oscillatory, where the operator B is defined by (2.30), φ=φ(t,s,t0) is defined by (2.31), and R¯(t) is defined by (2.25).The proof of Theorem2.9 can be completed by following the proofs of Theorems 2.6 and 2.8 with suitable changes, we omit it here.


## 3. Corollaries

As Theorems2.4–2.9 are rather general, it is convenient for applications to derive a number of oscillation criteria with the appropriate choice of the functions H,ρ, and Φ(t,s,l).With an appropriate choice of the functionsH(t),ρ(t), one can derive from Theorems 2.4 and 2.6 a number of oscillation criteria for (1.1). For example, we consider the simple case n=2; hence ,λ1>1>λ2>0. Let bj=aj+π/cj,bj=aj+π/4dj,j=1,2, where aj,bj are given in Theorem 2.4. This determines cj,dj,j=1,2. Choose H(t)=sin[cj(t-aj)], ρ(t)=sink[dj(t-aj)], and k being natural number. We obtain from Theorems 2.4 and 2.6 the following corollaries.Corollary 3.1.
Letk0,k1,k2 be chosen to satisfy (a), (b) of Lemma 2.1 for λ1>1>λ2>0. If for any T≥t0, there exist constants a1,b1,a2,b2 such that T≤a1<b1≤a2<b2, then it holds that
(3.1)q1(t)≥0,q2(t)≥0,t∈[τ(a1),b1]∪[τ(a2),b2],e(t)≤0,t∈[τ(a1),b1],e(t)≥0,t∈[τ(a2),b2],
and for cj=π2/(bj-aj)2,16dj=π2/(bj-aj)2, and θ≥1, one has that
(3.2)∫ajbj[(p(t)M(1-a(τ(t)))τ(t)-τ(aj)t-τ(aj)+k0-k0k1-k1k2-k2|e(t)|k0q1k1(t)q2k2(t)(1-a(t)))×sinν+1[cj(t-aj)]sink[dj(t-aj)]-θr(t)cjsinν-1[cj(t-aj)]4sink[dj(t-aj)]×(kdjsin[cj(t-aj)]sink-1[dj(t-aj)]cos[dj(t-aj)]+(ν+1)cjsink[dj(t-aj)]cos[cj(t-aj)])2]dt>0,
then all solutions of the equation
(3.3)(r(t)(x(t)+a(t)x(σ(t)))′)′+p(t)f(x(τ(t)))+q1(t)|x(t)|λ1sgnx(t)+q2(t)|x(t)|λ2sgnx(t)=e(t)
are oscillatory.Corollary 3.2.
Assume that, for anyT≥0, there exist constants a1,b1,a2,b2 such that T≤a1<b1≤a2<b2 and
(3.4)q1(t)≥0,t∈[τ(a1),b1]∪[τ(a2),b2],e(t)<0,t∈[τ(a1),b1],e(t)>0,t∈[τ(a2),b2].
Also assume that, for λ1>1>λ2>0, and that there exists some λ∈(0,1) such that
(3.5)∫ajbj[(p(t)M(1-a(τ(t)))τ(t)-τ(aj)t-τ(aj)+λ1(λ1-1)1/λ1-1(λ|e(t)|)1-1/λ1q11/λ1(t)(1-a(t))-λ2(1-λ2)1/λ2-1((1-λ)|e(t)|)1-1/λ2q¯21/λ2(t)(1-a(t))τ(t)-τ(aj)t-τ(aj))×sinν+1[cj(t-aj)]sink[dj(t-aj)]-θr(t)cjsinν-1[cj(t-aj)]4sink[dj(t-aj)]×(kdjsin[cj(t-aj)]sink-1[dj(t-aj)]cos[dj(t-aj)]+(ν+1)cjsink[dj(t-aj)]cos[cj(t-aj)])2τ(t)-τ(aj)t-τ(aj)]dt>0,
where q¯2(t)=max{-q2(t),0}, then all solutions of (3.3) are oscillatory.If we chooseΦ(t,s,l)=ρ(s)(t-s)α(s-l)β for α,β>1/2 and ρ(s)∈C1([t0,(0,∞))), then we have that(3.6)φ(t,s,l)=ρ′(s)ρ(s)+βt-(α+β)s+αl(t-s)(s-l).
Thus by Theorems 2.8 and 2.9, we have two new oscillation results.Corollary 3.3.
Suppose that, for anyT≥0, there exist constants a1,b1,a2,b2 such that T≤a1<b1≤a2<b2 and satisfies (2.1). For each t≥t0 and for some θ≥1,
(3.7)limsupt→∞∫t0tρ2(t)(t-s)2α(s-t0)2β[ρ(s)R(s)-θρ(s)r(s)4(2βt-(α+β)s+αt0(t-s)(s-t0)+ρ′(s)ρ(s))2]ds>0,
for j = 1, 2, then (1.1) is oscillatory, where
(3.8)R(s)=Mp(s)[1-a(τ(s))]τ(s)-τ(aj)s-τ(aj)+a0(1-a(s))|e(s)|k0∏i=1nqiki(s),a0=∏i=0nki-ki, and k0,k1,…,kn are positive constants satisfying (a) and (b) of Lemma 2.1.Corollary 3.4.
Suppose that, for anyT≥0, there exist constants a1,b1,a2,b2 such that T≤a1<b1≤a2<b2 and satisfies (2.23). Assume that there exists a function Φ∈Y such that, for each t≥t0 and for some θ≥1,
(3.9)limsupt→∞∫t0tρ2(t)(t-s)2α(s-t0)2β[ρ(s)R¯(s)-θρ(s)r(s)4(2βt-(α+β)s+αt0(t-s)(s-t0)+ρ′(s)ρ(s))2]ds>0,
for j = 1, 2, then (1.1) is oscillatory, where R¯(t) is defined by (2.25).We say that a functionH=H(t,s) belongs to the function class 𝒳, if H∈C(D,ℝ+), where D={(t,s):t0≤s≤t<∞}, which satisfies H(t,t)=0,H(t,s)>0 for t>s and has partial derivatives ∂H/∂s and ∂H/∂t on D such that(3.10)∂H∂t=h1(t,s)H(t,s),∂H∂s=-h2(t,s)H(t,s),
where h1(t,s),h2(t,s) are locally integrable with respect to t and s, respectively, in D.If we chooseΦ(t,s,l)=H1(s,l)H2(t,s), for H1,H2∈𝒳, then we have that(3.11)φ(t,s,l)=12(h1(1)(s,l)H1(s,l)-h2(2)(t,s)H2(t,s)),
where h1(1)(s,l),h2(2)(t,s) are defined as the following:(3.12)∂H1(s,l)∂s=h1(1)(s,l)H1(s,l),∂H2(t,s)∂s=-h2(2)(t,s)H2(t,s).
Thus by Theorems 2.8 and 2.9, we also have the two following oscillation results.Corollary 3.5.
Suppose that, for anyT≥0, there exist constants a1,b1,a2,b2 such that T≤a1<b1≤a2<b2 and satisfies (2.1). For each t≥t0 and for some θ≥1,
(3.13)limsupt→∞∫t0tH1(s,t0)H2(t,s)[ρ(s)R(s)-θρ(s)r(s)4(h1(1)(s,t0)H1(s,t0)-h2(2)(t,s)H2(t,s)-ρ′(s)ρ(s))2]ds>0,
for j = 1, 2, then (1.1) is oscillatory, where h1(1)(s,t0),h2(2)(t,s) are defined by (3.12),
(3.14)R(s)=Mp(s)[1-a(τ(s))]τ(s)-τ(aj)s-τ(aj)+a0(1-a(s))|e(s)|k0∏i=1nqiki(s),a0=∏i=0nki-ki, and k0,k1,…,kn are positive constants satisfying (a) and (b) of Lemma 2.1.Corollary 3.6.
Suppose that, for anyT≥0, there exist constants a1,b1,a2,b2 such that T≤a1<b1≤a2<b2 and satisfies (2.23). Assume that there exists a function Φ∈𝒴 such that, for each t≥t0 and for some θ≥1,
(3.15)limsupt→∞∫t0tH1(s,t0)H2(t,s)[ρ(s)R¯(s)-θρ(s)r(s)4(h1(1)(s,t0)H1(s,t0)-h2(2)(t,s)H2(t,s)-ρ′(s)ρ(s))2]ds>0,
for j = 1, 2, then (1.1) is oscillatory, where h1(1)(s,t0),h2(2)(t,s) is defined by (3.12) and R¯(t) are defined by (2.25).



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*Source: 181784-2010-02-02.xml*