# Generalized Variational Oscillation Principles for Second-Order Differential Equations with Mixed-Nonlinearities

Authors: Jing Shao; Fanwei Meng; Xinqin Pang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

Using generalized variational principle and Riccati technique, new oscillation criteria are established for forced second-order differential equation with mixed nonlinearities, which improve and generalize some recent papers in the literature.


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## Body

## 1. Introduction

In this paper, we consider the second-order forced differential equation with mixed nonlinearities:(1.1)(r(t)|y′(t)|α-1y′(t))′+p(t)|y(t)|α-1y(t)+∑j=1mqj(t)|y(t)|βj-1y(t)=e(t),t≥t0,
where r,p,qj(1≤j≤m),e∈C([t0,∞),ℝ) with r(t)>0 and 0<α<β1<β2<⋯<βm are real numbers, p,qj (1≤j≤m), and e might change signs.In this paper, we are concerned with the nonhomogeneous equation (1.1). By a solution of (1.1), we mean that a function y∈C1[Ty,∞)(Ty≥t0, where Ty≥t0 depends on the particular solution) which has the property p(t)|y′(t)|α-1y′(t)∈C1[Ty,∞) and satisfies (1.1). A nontrivial solution of (1.1) is called oscillatory if it has arbitrarily large zeros; otherwise, it is said to be nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.Whenm=0, we have the following second-order half-linear differential equation without or with forcing term:
(1.2)(r(t)|y′(t)|α-1y′(t))′+q(t)|y(t)|α-1y(t)=0,t≥t0,(1.3)(r(t)|y′(t)|α-1y′(t))′+q(t)|y(t)|α-1y(t)=e(t),t≥t0.
There are a lot of papers involved oscillation (see [1–6]) for these equations since the foundation work of Elbert [2]. In paper [1], using Leighton’s variational principle (see [3]) for (1.3), the following result was obtained by Li and Cheng.Theorem 1.1.
Suppose that for anyT≥t0, there exist T≤s1<t1≤s2<t2 such that e(t)≤0 for t∈[s1,t1] and e(t)≥0 for t∈[s2,t2]. Let D(si,ti)={u∈C1[si,ti]:u(t)≢0,u(si)=u(ti)=0} for i=1,2. If there exist H∈D(si,ti) and a positive, nondecreasing function ρ∈C1([t0,∞),ℝ) such that
(1.4)∫sitiH2(t)ρ(t)q(t)dt>(1α+1)α+1∫sitir(t)ρ(t)|H(t)|α-1(2|H′(t)|+|H(t)|ρ′ρ)α+1dt
for i=1,2. Then, (1.3) is oscillatory.Unfortunately, Theorem1.1 cannot be applied to the case where α>1, since for ρ(t)≡1, the term |H(t)|α-1 will appear as a denominator in (1.4) so that the requirement H(si)=H(ti)=0 will cause trouble. This certainly calls for investigation of oscillation criteria that can handle with such cases.Whenα=1, (1.2) and (1.3) are reduced to the linear differential equation:
(1.5)(r(t)y′(t))′+q(t)y(t)=0,t≥t0,(1.6)(r(t)y′(t))′+q(t)y(t)=e(t),t≥t0.
In paper [7], Wong proved the following result for (1.6).Theorem 1.2.
Suppose that for anyT≥t0, there exist T≤s1<t1≤s2<t2 such that e(t)≤0 for t∈[s1,t1] and e(t)≥0 for t∈[s2,t2]. Let D(si,ti)={u∈C1[si,ti]:u(t)≠0,u(si)=u(ti)=0} for i=1,2. If there exists u∈D(si,ti) such that
(1.7)Qi(u):=∫siti[q(t)u2(t)-r(t)(u′(t))2]dt>0,i=1,2,
then (1.6) is oscillatory.On the other hand, among the oscillation criteria, Komkov [8] gave a generalized Leighton’s variational principle, which also can be applied to oscillation for (1.5).Theorem 1.3.
Suppose that there exist aC1 function u(t) defined on [s1,t1] and a function G(u) such that G(u(t)) is not constant on [s1,t1], G(u(s1))=G(u(t1))=0, g(u)=G′(u) is continuous,
(1.8)∫s1t1[q(t)G(u(t))-r(t)(u′(t))2]dt>0,
and (g(u(t)))2≤4G(u(t)) for t∈[s1,t1]. Then, every solution of (1.5) must vanish on [s1,t1].We note that whenG(u)≡u2, the left-hand side of (1.8) is the energy functional related to (1.5).Whenp(t)≡0, m=1, (1.1) turns into the quasilinear differential equation:
(1.9)(r(t)|y′(t)|α-1y′(t))′+q(t)|y(t)|β-1y(t)=e(t),t≥t0,
where p,q,e∈C([t0,∞),ℝ) with p(t)>0 and 0<α≤β being constants. In paper [9], using the generalized variational principle, Shao proved the following result for (1.9).Theorem 1.4.
Assume that for anyT≥t0, there exist T≤s1<t1≤s2<t2 such that
(1.10)e(t){≤0,t∈[s1,t1],≥0,t∈[s2,t2].
Let u∈C1[si,ti] and nonnegative functions G1,G2 satisfying Gi(u(si))=Gi(u(ti))=0, gi(u)=Gi′(u) are continuous and (gi(u(t)))α+1≤(α+1)α+1Giα(u(t)) for t∈[si,ti], i=1,2. If there exists a positive function ϕ∈C1([t0,∞),ℝ) such that
(1.11)Qiϕ(u):=∫siti‍ϕ(t)[Qe(t)Gi(u(t))-r(t)(|u′(t)|+Gi1/(α+1)(u(t))|ϕ′(t)|(α+1)ϕ(t))α+1]dt>0
for i=1,2. Then (1.9) is oscillatory, where
(1.12)Qe(t)=α-α/ββ(β-α)(α-β)/β[q(t)]α/β|e(t)|(β-α)/β,
with the convention that 00=1.Recently, using Riccati transformation, the following oscillation criteria were given for (1.1) by Zheng et al. [10].Theorem 1.5.
Assume that for anyT≥t0, there exist T≤s1<t1≤s2<t2 such that qj(t)≥0(1≤j≤m) for t∈[s1,t1]∪[s2,t2] and
(1.13)e(t){≤0,t∈[s1,t1],≥0,t∈[s2,t2].
Let D(si,ti)={u∈C1[si,ti]:uα+1(t)>0,t∈(si,ti),u(si)=u(ti)=0} for i=1,2. If there exist H∈D(si,ti) and a positive function ϕ∈C1([t0,∞),ℝ) such that
(1.14)∫sitiϕ(t)[(p(t)+∑j=1mQj(t))Hα+1(t)-r(t)(|H′(t)|+|H(t)ϕ′(t)|(α+1)ϕ(t))α+1]dt>0
for i=1,2. Then (1.1) is oscillatory, where
(1.15)Qj(t)=α-α/βjβj[m(βj-α)](α-βj)/βj[qj(t)]α/βj|e(t)|(βj-α)/βj,1≤j≤m,
with the convention that 00=1.The purpose of this paper is to obtain new oscillation criteria for (1.1) based on generalized variational principles. Roughly, if the existence of a “positive” solution of a functional relation implies the “positivity” of an associated functional over a set of “admissible” functions, then we say that a variational oscillation principle is valid. For instance, in Theorem 1.1, H∈D(si,ti) is admissible, and the functional is
(1.16)∫siti‍{(1α+1)α+1p(t)ρ(t)|H(t)|α-1(2|H′(t)|+|H(t)|ρ′(t)ρ(t))α+1-H2(t)ρ(t)q(t)}dt.
Our emphasis will be directed towards oscillation criteria that are closely related to the generalized energy functional (the generalization of (α+1)-degree energy functional) for half-linear equations (see [4, 11–13] for more details on these functionals), which improve the results mentioned above. Examples will also be given to illustrate the effectiveness of our main results.


## 2. Main Results

Firstly, we give an inequality, which is a transformation of Young’s inequality.Lemma 2.1 (see [14]).
Suppose thatX and Y are nonnegative, then
(2.1)γXYγ-1-Xγ≤(γ-1)Yγ,γ>1,
where equality holds if and only if X=Y.Now, we will give our main results.Theorem 2.2.
Assume that for anyT≥t0, there exist T≤s1<t1≤s2<t2 such that
(2.2)e(t){≤0,t∈[s1,t1],≥0,t∈[s2,t2].
Let u∈C1[si,ti] and nonnegative functions G1,G2 satisfying Gi(u(si))=Gi(u(ti))=0, gi(u)=Gi′(u) are continuous and (gi(u(t)))α+1≤(α+1)α+1Giα(u(t)) for t∈[si,ti],i=1,2. If there exists a positive function ϕ∈C1([t0,∞),ℝ) such that
(2.3)Qiϕ(u):=∫sitiϕ(t)[Gi(u(t))(p(t)+∑j=1mQj(t))(2.4)-r(t)(|u′(t)|+Gi1/(α+1)(u(t))|ϕ′(t)|(α+1)ϕ(t))α+1]dt>0
for i=1,2, where Qj(t) is defined as (1.15) with the convention that 00=1. Then, (1.1) is oscillatory.Proof.
Suppose to the contrary that there is a nontrivial nonoscillatory solutiony=y(t). We assume that y(t)≠0 on [T0,∞) for some T0≥t0. Set
(2.5)w(t)=ϕ(t)r(t)|y′(t)|α-1y′(t)|y(t)|α-1y(t),t≥T0.
Then differentiating (2.5) and making use of (1.1), it follows that for all t≥T0,
(2.6)w′(t)=ϕ′(t)ϕ(t)w(t)-ϕ(t)p(t)+ϕ(t)e(t)|y(t)|α-1y(t)-α|w(t)|(α+1)/α(r(t)ϕ(t))1/α-ϕ(t)∑j=1mqj(t)|y|βj-α.
By the assumptions, we can choose si,ti≥T0 for i=1,2 so that e(t)≤0 on the interval I1=[s1,t1], with s1<t1 and y(t)≥0, or e(t)≥0 on the interval I2=[s2,t2], with s2<t2 and y(t)≤0. For given t∈I1 or t∈I2, set Fj(x)=qj(t)xβj-α-e(t)/mxα, 1≤j≤m, we have Fj′(xj*)=0, Fj′′(xj*)>0, where xj*=[-αe(t)/m(βj-α)qj(t)]1/βj. So, Fj(x) obtains it minimum on xj* and
(2.7)Fj(x)≥Fj(xj*)=Qj(t).
So on the interval I1 or I2, (2.6) and (2.2) imply that w(t) satisfies
(2.8)ϕ(t)(p(t)+∑j=1mQj(t))≤-w′(t)+ϕ′(t)ϕ(t)w(t)-α|w(t)|(α+1)/α(r(t)ϕ(t))1/α.
Multiplying Gi(u(t)) through (2.8) and integrating (2.8) from si to ti, using the fact that Gi(u(s1))=Gi(u(t1))=0, we obtain
(2.9)∫sitiϕ(t)(p(t)+∑j=1mQj(t))Gi(u(t))dt≤∫sitiGi(u(t)){-w′(t)+ϕ′(t)ϕ(t)w(t)-α|w(t)|(α+1)/α(r(t)ϕ(t))1/α}dt=-Gi(u(t))w(t)|siti+∫sitigi(u(t))u′(t)w(t)dt+∫sitiGi(u(t)){ϕ′(t)ϕ(t)w(t)-α|w(t)|(α+1)/α(r(t)ϕ(t))1/α}dt=∫siti[gi(u(t))u′(t)+Gi(u(t))ϕ′(t)ϕ(t)]w(t)dt-α∫sitiGi(u(t))|w(t)|(α+1)/α(r(t)ϕ(t))1/αdt≤∫siti[|gi(u(t))||u′(t)|+Gi(u(t))|ϕ′(t)|ϕ(t)]|w(t)|dt-α∫sitiGi(u(t))|w(t)|(α+1)/α(r(t)ϕ(t))1/αdt≤(α+1)∫siti[Giα/(α+1)(u(t))|u′(t)|+Gi(u(t))|ϕ′(t)|(α+1)ϕ(t)]|w(t)|dt-α∫sitiGi(u(t))|w(t)|(α+1)/α(r(t)ϕ(t))1/αdt.
Let
(2.10)X=[α(r(t)ϕ(t))1/α]α/(α+1)Giα/(α+1)|w(t)|,γ=1+1α,Y=(αϕ(t)r(t))α/(α+1)[|u′(t)|+Gi1/(α+1)|ϕ′(t)|(α+1)ϕ(t)]α,
by Lemma 2.1 and (2.9), we have
(2.11)∫sitiϕ(t)(p(t)+∑j=1mQj(t))Gi(u(t))dt≤∫sitiϕ(t)r(t)[|u′(t)|+Gi1/(α+1)(u(t))|ϕ′(t)|(α+1)ϕ(t)]α+1dt,
which contradicts with (2.3). This completes the proof of Theorem 2.2.Corollary 2.3.
Ifϕ(t)≡1 in Theorem 2.2, and (2.3) is replaced by
(2.12)Qi(u):=∫siti[(p(t)+∑j=1mQj(t))Gi(u(t))-r(t)|u′(t)|α+1]dt>0,
for i=1,2. Then, (1.1) is oscillatory.If we chooseG1(u)=G2(u)=uα+1 in Corollary 2.3, then we have the following corollary.Corollary 2.4.
Suppose that for anyT≥t0, there exist T≤s1<t1≤s2<t2 such that (2.2) is true. Let D(si,ti)={u∈C1[si,ti]:u(t)≢0,u(si)=u(ti)=0} for i=1,2. If there exist u∈D(si,ti) such that
(2.13)Q~i(u):=∫siti[(p(t)+∑j=1mQj(t))|u(t)|α+1-r(t)|u′(t)|α+1]dt>0,
for i=1,2. Then, (1.3) is oscillatory.Remark 2.5.
Corollary2.4 is closely related to the (α+1)-degree functional (1.8), so Theorem 2.2, Corollaries 2.3, and 2.4 are generalizations of Theorem 1.2, and improvement of Theorem 1.1 since the positive constant α in Theorem 2.2 and Corollary 2.3 can be selected as any number lying in (0,∞). We note further that in most cases, oscillation criteria are obtained using the same auxiliary function on [s1,t1] and [s2,t2], we note that such functions can be selected differently.Remark 2.6.
IfG(u)≡uα+1, then Theorem 2.2 reduces to Theorem 1.5, and if p(t)≡0, j=1, Theorem 2.2 reduces to Theorem 1.4. So Theorem 2.2 and Corollary 2.3 are generalizations of the papers by Zheng et al. [10] and Shao [9].Remark 2.7.
The hypothesis (2.2) in Theorem 2.2 and Corollary 2.3 can be replaced by the following condition:
(2.14)e(t){≥0,t∈[s1,t1],≤0,t∈[s2,t2].
The conclusion is still true for these cases.Example 2.8.
Consider the following forced mixed nonlinearities differential equation:(2.15)(γt-λ/3y′(t))′+p(t)y(t)+q(t)|y(t)|2y(t)=-sin⁡3t,t≥2π,
where γ,λ>0 are constants, q(t)=t-λexp(3sint), p(t)=t-λ/3exp(sint), for t∈[2nπ,(2n+1)π), and q(t)=t-λexp(-3sint), p(t)=t-λ/3exp(-sint), for t∈[(2n+1)π,(2n+2)π), n>0 is an integer, Shao [9] obtain oscillation for (2.15) when K(t)≡0. Using Theorem 2.2, we can easily verify that Q1(t)=(3/2)23t-λ/3exp(sint)sin2t for t∈[2nπ,(2n+1)π), and Q1(t)=(3/2)23t-λ/3exp(-sint)sin2t for t∈[(2n+1)π,(2n+2)π). For any T≥1, we choose n sufficiently large so that nπ=2kπ≥T and s1=2kπ and t1=(2k+1)π, we select u(t)=sint≥0, G1(u)=u2exp(-u) (we note that (G1′(u))2≤4G1(u) for u≥0), ϕ(t)=tλ/3, then we have
(2.16)∫s1t1ϕ(t)(p(t)+Q1(t))G1(u(t))dt=∫0πsin⁡2tdt+3223∫0πsin⁡4tdt=π2+9823,∫s1t1ϕ(t)p(t)[|u′(t)|+G11/(α+1)(u(t))|ϕ′(t)|(α+1)ϕ(t)]α+1dt=γ∫2kπ(2k+1)π[|cos⁡⁡t|+λ|sin⁡⁡t|exp⁡⁡(3sin⁡⁡t/2)2t]2dt<γ∫2kπ(2k+1)π‍(1+λe3/22)2dt=γ(1+λe3/22)2π.
So we have Q1ϕ(u)>0 provided, 0<γ<(4π+923)/2(2+λe3/2)2π. Similarly, for s2=(2k+1)π and t2=(2k+2)π, we select u(t)=sint≤0, G2(u)=u2exp(u) (we note that (G2′(u))2≤4G2(u) for u≤0), we can show that the integral inequality Q2ϕ(u)>0 for 0<γ<(4π+923)/2(2+λe3/2)2π. So (2.15) is oscillatory for 0<γ<(4π+923)/2(2+λe3/2)2π by Theorem 2.2.Example 2.9.
Consider the following forced mixed nonlinearities differential equation:(2.17)(t-λ|y′(t)|α-1y′(t))′+p(t)|y(t)|α-1y(t)+q(t)y3(t)=-sin⁡1/3t,
for t≥2π, where p(t)=Kt-λexp(sint), q(t)=t-9λ/5exp(9sint/5), for t∈[2nπ,(2n+1)π), and p(t)=Kt-λexp(-sint), q(t)=t-9λ/5exp(-9sint/5), for t∈[(2n+1)π,(2n+2)π), n>0 is an integer, K,λ>0 are constants and α=5/3>1, β=3. Obviously, Theorem 1.1 cannot be applied to this case. However, we conclude that (2.17) is oscillatory for K>(3/4)(1+3λe/8)8/3π-9/55/944/9. Since the zeros of the forcing term -sin1/3t are nπ, let u(t)=sint and ϕ(t)=tλ. Using Theorem 2.2, we can easily verify that Q(t)=(9/55/944/9)t-λexp(sint)sin4/27t for t∈[2nπ,(2n+1)π), and Q(t)=(9/55/944/9)t-λexp(-sint)sin4/27t for t∈[(2n+1)π,(2n+2)π). For any T≥1, choose n sufficiently large so that nπ=2kπ≥T and s1=2kπ and t1=(2k+1)π. For t∈[s1,t1], we select G1(u)=u8/3exp(-u) (we note that (G1′(u))8/3≤(8/3)8/3(G1(u))5/3 for u≥0). It is easy to verify the following estimations:
(2.18)∫s1t1ϕ(t)(p(t)+Q(t))G1(u(t))dt=∫2kπ(2k+1)πsin⁡8/3t(K+955/944/9sin⁡4/27t)dt>(K+955/944/9)∫2kπ(2k+1)π‍sin⁡3tdt=43(K+955/944/9),∫s1t1‍ϕ(t)r(t)[|u′(t)|+G11/(α+1)(u(t))|ϕ′(t)|(α+1)ϕ(t)]α+1dt=∫2kπ(2k+1)π[|cos⁡⁡t|+3λe-3sin⁡t/8|sin⁡⁡t|8t]8/3dt<∫2kπ(2k+1)π‍(1+3λe8)8/3dt=(1+3λe8)8/3π.
So we have Q1ϕ(u)>0. Similarly, for s2=(2k+1)π and t2=(2k+2)π, we select u(t)=sint<0, G2(u)=u8/3exp(u) (we note that (G2′(u))8/3≤(8/3)8/3(G2(u))5/3 for u≤0), we can show that the integral inequality Q2ϕ(u)>0. So (2.17) is oscillatory for K>(3/4)(1+3λe/8)8/3π-9/55/944/9 by Theorem 2.2.



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