# New Oscillation Criteria for Second-Order Delay Differential Equations with Mixed Nonlinearities

Authors: Yuzhen Bai; Lihua Liu
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

We establish new oscillation criteria for second-order delay differential equations with mixed nonlinearities of the form(p(t)x'(t))'+∑i=1npi(t)x(t-τi)+∑i=1nqi(t)|x(t-τi)|αisgnx(t-τi)=e(t),t≥0, where p(t), pi(t), qi(t), and e(t) are continuous functions defined on [0,∞), and p(t)>0, p′(t)≥0, and α1>⋯>αm>1>αm+1>⋯>αn>0. No restriction is imposed on the potentials pi(t), qi(t), and e(t) to be nonnegative. These oscillation criteria extend and improve the results given in the recent papers. An interesting example illustrating the sharpness of our results is also provided.


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## Body

## 1. Introduction

We consider the second-order delay differential equations containing mixed nonlinearities of the form(1.1)(p(t)x′(t))′+∑i=1npi(t)x(t-τi)+∑i=1nqi(t)|x(t-τi)|αisgnx(t-τi)=e(t),t≥0.In what follows we assume thatτi≥0, p∈C1[0,∞),  p(t)>0,p′(t)≥0,pi,qi,e∈C[0,∞), α1>⋯>αm>1>αm+1>⋯>αn>0(n>m≥1), and (1.2)∫0∞1p(t)dt=∞.As usual, a solutionx(t) of  (1.1) is called oscillatory if it is defined on some ray [T,∞) with T≥0 and has arbitrary large zeros, otherwise, it is called nonoscillatory. Equation (1.1) is called oscillatory if all of its extendible solutions are oscillatory.Recently, Mustafa [1] has studied the oscillatory solutions of certain forced Emden-Fowler like equations (1.3)x′′(t)+a(t)|x(t)|λsgnx(t)=e(t),t≥t0≥1.
Sun and Wong [2], as well as Sun and Meng [3] have established oscillation criteria for the second-order equation (1.4)(p(t)x′(t))′+q(t)x(t)+∑i=1nqi(t)|x(t)|αisgnx(t)=e(t),t≥0.
Later in [4], Li and Chen have extended  (1.4) to the delay differential equation (1.5)(p(t)x′(t))′+q(t)x(t-τ)+∑i=1nqi(t)|x(t-τ)|αisgnx(t-τ)=e(t),t≥0.
As it is indicated in [2, 3], further research on the oscillation of equations of mixed type is necessary as such equations arise in mathematical modeling, for example, in the growth of bacteria population with competitive species. In this paper, we will continue in the direction to study the oscillatory properties of  (1.1). We will employ the method in study of Kong in [5] and the arithmetic-geometric mean inequality (see [6]) to establish the interval oscillation criteria for the unforced  (1.1) and forced  (1.1), which extend and improve the known results. Our results are generalizations of the main results in [3, 4]. We also give an example to illustrate the sharpness of our main results.


## 2. Main Results

We need the following lemma proved in [2, 3] for our subsequent discussion.Lemma 2.1.
For any givenn-tuple {α1,α2,…,αn} satisfying α1>α2>⋯>αm>1>αm+1>⋯>αn>0, there corresponds an n-tuple {η1,η2,…,ηn} such that
(a)∑i=1nαiηi=1,
which also satisfies either
(b)∑i=1nηi<1,0<ηi<1,
or
(c)∑i=1nηi=1,0<ηi<1.For a given set of exponents{αi} satisfying α1>α2>⋯>αm>1>αm+1>⋯>αn>0, Lemma 2.1 ensures the existence of an n-tuple {η1,η2,…,ηn} such that either (a) and (b) hold or (a) and (c) hold. When n=2 and α1>1>α2>0, in the first case, we have (2.1)η1=1-α2(1-η0)α1-α2,η2=α1(1-η0)-1α1-α2,
where η0 can be any positive number satisfying 0<η0<(α1-1)/α1. This will ensure that 0<η1, η2<1, and conditions (a) and (b) are satisfied. In the second case, we simply solve (a) and (c) and obtain (2.2)η1=1-α2α1-α2,η2=α1-1α1-α2.Following Philos [7], we say that a continuous function H(t,s) belongs to a function class 𝒟a,b, denoted by H∈𝒟a,b, if H(b,s)>0,H(s,a)>0 for b>s>a, and H(t,s) has continuous partial derivatives ∂H(t,s)/∂t and ∂H(t,s)/∂s in [a,b]×[a,b]. Set (2.3)h1(t,s)=∂H(t,s)/∂t2H(t,s),h2(t,s)=-∂H(t,s)/∂s2H(t,s).Based on Lemma2.1, we have the following interval criterion for oscillation of  (1.1).Theorem 2.2.
If, for anyT≥0, there exist a1, b1, c1, a2,b2 and c2 such that T≤a1<c1<b1≤a2<c2<b2,(2.4)pi(t)≥0,t∈[a1-τi,b1]∪[a2-τi,b2],i=1,2,…,n,qi(t)≥0,t∈[a1-τi,b1]∪[a2-τi,b2],i=1,2,…,n,e(t)≤0,t∈[a1-τi,b1],e(t)≥0,t∈[a2-τi,b2],
and there exist Hj∈𝒟aj,bj such that
(2.5)1Hj(cj,aj)∫ajcj(Qj(s)Hj(s,aj)-p(s)hj12(s,aj))ds+1Hj(bj,cj)∫cjbj(Qj(s)Hj(bj,s)-p(s)hj22(bj,s))ds>0,
for j=1,2, where hj1,  hj2 are defined as in (2.3), η1,η2,…,ηn are positive constants satisfying (a) and (b) in Lemma 2.1, η0=1-∑i=1nηi, and
(2.6)Qj(t)=∑i=1npi(t)(t-ajt-aj+τi)+(η0-1|e(t)|)η0∏i=1n(ηi-1qi(t))ηi(t-ajt-aj+τi)αiηi,
then  (1.1) is oscillatory.Proof.
Letx(t) be a nonoscillatory solution of  (1.1). Without loss of generality, we may assume that x(t)>0 for all t≥t1-τ¯≥0, where t1 depends on the solution x(t) and τ¯=max{τi}, i=1,…,n. When x(t) is eventually negative, the proof follows the same argument by using the interval [a2,b2] instead of [a1,b1]. Choose a1,b1≥t1 such that pi(t),qi(t)≥0, e(t)≤0 for t∈[a1-τi,b1], and i=1,2,…,n.
From  (1.1), we have that x′(t)≥0 for t∈[a1-τi,b1]. If not, there exists t2∈[a1-τi,b1] such that x′(t2)<0. Because
(2.7)(p(t)x′(t))′≤0,
we have p(t)x′(t)≤p(t2)x′(t2). Integrating from t2 to t, we obtain
(2.8)x(t)≤x(t2)+p(t2)x′(t2)∫t2τ1p(s)ds.
Noting the assumption (1.2), we have x(t)≤0 for sufficient large t. This is a contradiction with x(t)>0. From (2.7) and the conditions p(t)>0, p′(t)≥0, we obtain x′′(t)≤0 for t∈[a1-τi,b1].
Employing the convexity ofx(t), we obtain
(2.9)x(t-τi)x(t)≥t-a1t-a1+τi,t∈[a1,b1].
Define(2.10)ω(t)=-p(t)x′(t)x(t).
Recall the arithmetic-geometric mean inequality(2.11)∑i=0nηiui≥∏i=0nuiηi,ui≥0,
where η0=1-∑i=1nηi and ηi>0, i=1,2,…,n, are chosen according to the given α1,α2,…,αn as in Lemma 2.1 satisfying (a) and    (b). Let
(2.12)u0(t)=η0-1|e(t)|,ui(t)=ηi-1qi(t)(x(t-τi))αi.
We have
(2.13)ω′(t)=-(p(t)x(t))′x(t)+ω2(t)p(t)=∑i=1npi(t)x(t-τi)+∑i=1nqi(t)(x(t-τi))αi-e(t)x(t)+ω2(t)p(t)≥∑i=1npi(t)(t-a1t-a1+τi)+(η0-1|e(t)|)η0∏i=1n(ηi-1qi(t))ηixαiηi(t-τi)x(t)+ω2(t)p(t)=∑i=1npi(t)(t-a1t-a1+τi)+(η0-1|e(t)|)η0∏i=1n(ηi-1qi(t))ηixαiηi(t-τi)∏i=1nxαiηi(t)+ω2(t)p(t)≥∑i=1npi(t)(t-a1t-a1+τi)+(η0-1|e(t)|)η0∏i=1n(ηi-1qi(t))ηi(t-a1t-a1+τi)αiηi+ω2(t)p(t)=Q1(t)+ω2(t)p(t).
Multiplying both sides of (2.13) by H1(b1,t)∈𝒟a1,b1 and integrating by parts, we find that
(2.14)-ω(c1)H1(b1,c1)≥∫c1b1(Q1(s)H1(b1,s)-p(s)h122(b1,s))ds.
That is,
(2.15)-ω(c1)≥1H1(b1,c1)∫c1b1(Q1(s)H1(b1,s)-p(s)h122(b1,s))ds.
On the other hand, multiplying both sides of (2.13) by H1(t,a1)∈𝒟a1,b1 and integrating by parts, we can easily obtain
(2.16)ω(c1)≥1H1(c1,a1)∫a1c1(Q1(s)H1(s,a1)-p(s)h112(s,a1))ds.
Equations (2.15) and (2.16) yield
(2.17)1H1(c1,a1)∫a1c1(Q1(s)H1(s,a1)-p(s)h112(s,a1))ds+1H1(b1,c1)∫c1b1(Q1(s)H1(b1,s)-p(s)h122(b1,s))ds≤0,
which contradicts (2.5) for j=1. The proof of Theorem 2.2 is complete.Remark 2.3.
Whenτ1=⋯=τn=0, Σi=1npi(t)=q(t), the conditions q(t)≥0 for t∈[a1,b1]∪[a2,b2], p′(t)≥0 and (1.2) can be removed. Therefore, Theorem 2.2 reduces to Theorem 1 in [3].Remark 2.4.
Whenτ1=⋯=τn=τ, Σi=1npi(t)=q(t), Theorem 2.2 reduces to Theorem 1 in [4] for which the conditions q(t)≥0 for t∈[a1-τ,b1]∪[a2-τ,b2],p′(t)≥0 and (1.2) are needed. There are some mistakes in the proof of Theorem 1 in [4].The following theorem gives an oscillation criterion for the unforced  (1.1).Theorem 2.5.
If, for anyT≥0, there exist a, b, and c such that T≤a<c<b,  pi(t)≥0, and qi(t)≥0 for t∈[a-τi,b],i=1,2,…,n, and there exists H∈𝒟a,b, such that
(2.18)1H(c,a)∫ac(H(s,a)Q(s)-p(s)h12(s,a))ds+1H(b,c)∫cb(H(b,s)Q(s)-p(s)h22(b,s))ds>0,
where
(2.19)Q(t)=∑i=1npi(t)(t-at-a+τi)+∏i=1n(ηi-1qi(t))ηi(t-at-a+τi)αiηi,η1,η2,…,ηn are positive constants satisfying (a) and (c) in Lemma 2.1, and h1,  h2 are defined as in (2.3), then the unforced  (1.1) is oscillatory.Proof.
Letx(t) be a nonoscillatory solution of  (1.1). Without loss of generality, we may assume that x(t)>0 for all t≥t1-τ¯≥0,  where t1 depends on the solution x(t) and τ¯=max{τi}, i=1,…,n. Similar to the proof in Theorem 2.2, we can obtain
(2.20)x(t-τi)x(t)≥t-at-a+τi,t∈[a,b].
Define
(2.21)ω(t)=-p(t)x′(t)x(t).
Recall the arithmetic-geometric mean inequality
(2.22)∑i=1nηiui≥∏i=1nuiηi,ui≥0,
where ηi>0, i=1,2,…,n, are chosen according to the given α1,α2,…,αn as in Lemma 2.1 satisfying (a) and (c). Let
(2.23)ui=ηi-1qi(t)(x(t-τi))αi.
We can obtain
(2.24)ω′(t)=-(p(t)x(t))′x(t)+ω2(t)p(t)=∑i=1npi(t)x(t-τi)+∑i=1nqi(t)(x(t-τi))αix(t)+ω2(t)p(t)≥∑i=1npi(t)(t-at-a+τi)+∏i=1n(ηi-1qi(t))ηixαiηi(t-τi)x(t)+ω2(t)p(t)=∑i=1npi(t)(t-at-a+τi)+∏i=1n(ηi-1qi(t))ηixαiηi(t-τi)∏i=1nxαiηi(t)+ω2(t)p(t)≥∑i=1npi(t)(t-at-a+τi)+∏i=1n(ηi-1qi(t))ηi(t-at-a+τi)αiηi+ω2(t)p(t)=Q(t)+ω2(t)p(t).
Multiplying both sides of (2.24) by H(b,t)∈𝒟a,b and integrating by parts, we obtain
(2.25)∫cbH(b,t)ω′(t)dt≥∫cbH(b,t)Q(t)dt+∫cbH(b,t)ω2(t)p(t)dt,-H(b,c)ω(c)≥∫cbH(b,t)Q(t)dt+∫cb(H(b,t)ω2(t)p(t)-2ω(t)h2(b,t)H(b,t))dt=∫cb(H(b,t)Q(t)-p(t)h22(b,t))dt+∫cb(H(b,t)p(t)ω(t)-p(t)h2(b,t))2dt≥∫cb(H(b,t)Q(t)-p(t)h22(b,t))dt.
It follows that
(2.26)-ω(c)≥1H(b,c)∫cb(H(b,t)Q(t)-p(t)h22(b,t))dt.
On the other hand, multiplying both sides of (2.24) by H(t,a)∈𝒟a,b and integrating by parts, we have
(2.27)ω(c)≥1H(c,a)∫ac(H(t,a)Q(t)-p(t)h12(t,a))dt.
Equations (2.26) and (2.27) yield
(2.28)1H(c,a)∫ac(H(t,a)Q(t)-p(t)h12(t,a))dt+1H(b,c)∫cb(H(b,t)Q(t)-p(t)h22(b,t))dt<0,
which contradicts (2.24). The proof of Theorem 2.5 is complete.Remark 2.6.
Whenτ1=⋯=τn=0, Σi=1npi(t)=q(t), the conditions q(t)≥0 for t∈[a,b], p′(t)≥0 and (1.2) can be removed. Therefore, Theorem 2.5 reduces to Theorem 2 in [3].Remark 2.7.
Whenτ1=⋯=τn=τ, Σi=1npi(t)=q(t), Theorem 2.5 reduces to Theorem 2 in [4] for which the conditions q(t)≥0 for t∈[a-τ,b], p′(t)≥0 and (1.2) are needed.


## 3. Example

In this section, we provide an example to illustrate our results.Consider the following equation:(3.1)x′′(t)+ksint|x(t-π8)|α1sgnx(t-π8)+lcost|x(t-π4)|α2sgnx(t-π4)=-mcos2t,t≥0,
where k, l, m are positive constants, α1>1, and 0<α2<1. Here (3.2)p(t)=1,p1(t)=p2(t)=0,q1(t)=ksint,q2(t)=lcost,τ1=π8,τ2=π4,e(t)=-mcos2t.According to the direct computation, we have(3.3)Qj(t)=k0|cos2t|η0(sint)η1(cost)η2(t-ajt-aj+τ1)α1η1(t-ajt-aj+τ2)α2η2,j=1,2,
where k0=(η0-1/m)η0(η1-1/k)η1(η2-1/l)η2,η0 can be any positive number satisfying 0<η0<(α1-1)/α1, and  η1,  η2 satisfy (2.1). For any T≥0, we can choose (3.4)a1=2iπ,a2=b1=2iπ+π4,b2=2iπ+π2,c1=2iπ+π8,c2=2iπ+3π8,
for i=0,1,…, and H1(t,s)=H2(t,s)=(t-s)2. By simple computation, we obtain hj1(t,s)=hj2(t,s)=1,  j=1,2. From Theorem 2.2, we have that  (3.1) is oscillatory if (3.5)∫2iπ2iπ+π/8Q1(s)(s-2iπ)2ds+∫2iπ+π/82iπ+π/4Q1(s)(2iπ+π4-s)2ds>π4,∫2iπ+π/42iπ+3π/8Q2(s)(s-2iπ-π4)2ds+∫2iπ+3π/82iπ+π/2Q2(s)(2iπ+π2-s)2ds>π4.
If H1(t,s)=H2(t,s)=sin2(t-s), by simple computation, we obtain hj1(t,s)=hj2(t,s)=cos(t-s) for j=1,2. From Theorem 2.2, we have that  (3.1) is oscillatory if (3.6)∫2iπ2iπ+π/8Q1(s)(s-2iπ)2ds+∫2iπ+π/82iπ+π/4Q1(s)(2iπ+π4-s)2ds>π16+28,∫2iπ+π/42iπ+3π/8Q2(s)(s-2iπ-π4)2ds+∫2iπ+3π/82iπ+π/2Q2(s)(2iπ+π2-s)2ds>π16+28.



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