# Generalized Variational Principles on Oscillation for Nonlinear Nonhomogeneous Differential Equations

Authors: Jing Shao; Fanwei Meng
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Using the generalized variational principle and the Riccati technique, new oscillation
criteria are established for the forced second-order nonlinear differential equation,
which improves and generalizes some of the new results in literature.


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## Body

## 1. Introduction

In this paper, we consider the oscillatory behavior of the nonlinear nonhomogeneous differential equation of the form(1.1)(p(t)Ψ(y(t))|y′(t)|α-1y′(t))′+q(t)f(y(t))=e(t),t≥t0,
where α is a positive constant, p,q,e∈C([t0,∞),ℝ) with p(t)>0, Ψ∈C(ℝ,(0,∞)), f∈C(ℝ,ℝ) satisfying uf(u)>0 for u≠0.As usual, by a solution of (1.1) we mean a function y∈C1[Ty,∞), Ty≥t0, where Ty≥t0 depends on the particular solution, which has the property p(t)Ψ(y(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.Recently, many research works have been done on the oscillatory and asymptotic behavior of solutions of the nonlinear nonhomogeneous differential equation of the form (1.1) or its special cases (see [1–7] and references cited therein). Using the variational method, oscillation criteria are obtained by Wong [1] for the forced linear differential equation, by Li and Cheng [2] for the forced half-linear differential equation, by Zheng and Meng [3] for the forced quasilinear differential equation, by Çakmak and Tiryaki [4] as well as Zheng and Cheng [5] for the forced nonlinear differential equation (some deficiencies in [2, 4] are pointed out by Zheng and Meng [3]), and Erbe et al. [7], as well as Saker [8, 9] for the dynamic equation on time scales.Meanwhile, in [6], Komkov gave a generalized Leighton's variational principle, which can also be used to obtain the oscillation criterion.Theorem 1.1.
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,
(1.2)∫s1t1[q(t)G(u(t))-p(t)(u′(t))2]dt>0,
and (g(u(t)))2≤4G(u(t)) for t∈[s1,t1]. Then each solution of the equation
(1.3)(p(t)y′(t))′+q(t)y(t)=0
vanishes at least once on [s1,t1].The purpose of our paper is to use the generalized variational principle to study the oscillation for (1.1). These oscillation criteria are closely related to the generalized variational formulae (1.2), which improve the results mentioned above. Examples will also be given to illustrate the effectiveness of our main results.Before going into the main results, let us state three sets of conditions commonly used in the literature which we rely on:(1.4)(S1)0<Ψ(u)≤M,|f′(u)|≥K|f(u)|(β-1)/β>0,foru≠0,(1.5)(S2)|f′(u)|[Ψ(u)|f(u)|β-1]1/β≥γ>0,foru≠0,(1.6)(S3)0<Ψ(u)≤M,|f(u)||u|β≥δ>0,foru≠0.
Here, M,K>0, 0<α≤β, and γ,δ>0 are constants. It is clear that assumption (S1) implies (S2), but not conversely. For example, the function f(u)=u3, Ψ(u)=u2 and β=1 do not satisfy (S1), but (S2) holds. In (S1) and (S2), we need f to be differentiable. Clearly, this condition is not required in (S3). These differences force us to study (1.1) under the assumptions (S1), (S2), and (S3) in separate manners.


## 2. The Case Whereβ=α

Firstly, we give an inequality, which is a transformation of Young's inequality.Lemma 2.1 (see [10]).
Suppose thatX and Y are nonnegative. Then
(2.1)λXYλ-1-Xλ≤(λ-1)Yλ,λ>1,
where the equality holds if and only if X=Y.Now, we will give our main results.Theorem 2.2.
Assume (S2) holds. Suppose further 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 ρ∈C1([t0,∞),ℝ) such that
(2.3)Qiρ(u):=∫sitiρ(t)[q(t)Gi(u(t))-(αγ)αp(t)(|u′(t)|+Gi1/(α+1)(u(t))|ρ′(t)|(α+1)ρ(t))α+1]dt>0,
for i=1,2. Then (1.1) is oscillatory.Proof.
Suppose to the contrary that there is a nonoscillatory solutiony(t) of (1.1). First, we consider the case when y(t)>0 eventually. Assume that y(t)>0 on [T0,∞) for some T0≥t0. Set
(2.4)w(t)=ρ(t)p(t)Ψ(y(t))|y′(t)|α-1y′(t)f(y(t)),t≥T0.
Then differentiating (2.4) and making use of (1.1), it follows that for all t≥T0,
(2.5)w′(t)=-ρ(t)q(t)+ρ(t)e(t)f(y(t))+ρ′(t)ρ(t)w(t)-|w(t)|(α+1)/αf′(y(t))[p(t)ρ(t)Ψ(y(t))|f(y(t))|α-1]1/α.
By assumptions, we can choose s1,t1≥T0 with s1<t1 so that e(t)≤0 on the interval I1=[s1,t1]. For t∈I1 and in view of (1.5) and (2.5), w(t) satisfies the inequality
(2.6)ρ(t)q(t)≤-w′(t)+ρ′(t)ρ(t)w(t)-γ|w(t)|(α+1)/αp1/α(t)ρ1/α(t).
Multiplying G1(u(t)) through (2.6) and integrating (2.6) from s1 to t1, using the fact that G1(u(s1))=G1(u(t1))=0, we obtain
(2.7)∫s1t1G1(u(t))ρ(t)q(t)dt≤∫s1t1G1(u(t))[-w′(t)+ρ′(t)ρ(t)w(t)-γ|w(t)|(α+1)/αp1/α(t)ρ1/α(t)]dt=-G1(u(t))w(t)|s1t1+∫s1t1[Gi(u(t))]′w(t)dt+∫s1t1G1(u(t))ρ′(t)ρ(t)w(t)dt-∫s1t1G1(u(t))γ|w(t)|(α+1)/αp1/α(t)ρ1/α(t)dt≤∫s1t1[|g1(u(t))||u′(t)|+G1(u(t))|ρ′(t)|ρ(t)]|w(t)|dt-γ∫s1t1G1(u(t))|w(t)|(α+1)/αp1/α(t)ρ1/α(t)dt≤(α+1)∫s1t1[|G1α/(α+1)(u(t))||u′(t)|+G1(u(t))|ρ′(t)|(α+1)ρ(t)]|w(t)|dt-γ∫s1t1G1(u(t))|w(t)|(α+1)/αp1/α(t)ρ1/α(t)dt.
Let(2.8)X=γα/(α+1)p1/(α+1)(t)ρ1/(α+1)(t)G1α/(α+1)(u(t))|w(t)|,λ=1+1α,Y=ααpα/(α+1)(t)ρα/(α+1)(t)γα2/(α+1)[|u′(t)|+G11/(α+1)(u(t))|ρ′(t)|(α+1)ρ(t)]α.
By Lemma 2.1 and (2.7), we have
(2.9)∫s1t1G1(u(t))ρ(t)q(t)dt≤∫s1t1(αγ)αp(t)ρ(t)(|u′(t)|+G11/(α+1)(u(t))|ρ′(t)|(α+1)ρ(t))α+1dt,
which contradicts (2.3) with i=1.
Wheny(t)<0 holds eventually, we assume y(t)<0 for t≥T0>t0. Defining the Riccati transformation as (2.4), we get that (2.5) is true. In this case, we choose t2>s2≥T0 so that e(t)≥0 on the interval I2=[s2,t2]. For a given t∈I2, (1.5) and (2.5) imply (2.6). Multiplying G2(u(t)) through (2.6) and integrating (2.6) from s2 to t2, using the fact that G2(u(s2))=G2(u(t2))=0, we obtain a similar contradiction to (2.3) with i=2. This completes the proof.Corollary 2.3.
Ifρ(t)≡1 in Theorem 2.2 and (2.3) is replaced by
(2.10)Qi(u):=∫siti[q(t)Gi(u(t))-(αγ)αp(t)|u′(t)|α+1]dt>0,
for i=1,2, then (1.1) is oscillatory.Remark 2.4.
The left side of inequality (2.10) is closely related to the generalized variational formulae (1.2). Particularly, we obtain α=γ for the linear equation; thus, (2.10) reduces to (1.2) for the linear differential equation. So our Theorem 2.2 and Corollary 2.3 are generalizations of the paper by Zheng and Cheng [5] and improvement of papers by Li and Cheng [2] and by Çakmak and Tiryaki [4].Theorem 2.5.
Assume that (S3) holds. Suppose further that for anyT≥t0, there exist T≤s1<t1≤s2<t2 such that (2.2) holds. 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 ρ∈C1([t0,∞),ℝ) such that
(2.11)Qiρ(u):=∫sitiρ(t)[δq(t)Gi(u(t))-Mp(t)(|u′(t)|+Gi1/(α+1)(u(t))|ρ′(t)|(α+1)ρ(t))α+1]dt>0,
for i=1,2. Then (1.1) is oscillatory.Proof.
Suppose to the contrary that there is a nonoscillatory solutiony(t). Firstly, we assume that y(t)>0 on [T0,∞) for some T0≥t0. Set
(2.12)w(t)=ρ(t)p(t)Ψ(y(t))|y′(t)|α-1y′(t)|y(t)|α-1y(t),t≥T0.
Then differentiating (2.12) and making use of (1.1) and (S3), we see that for all t≥T0, we have
(2.13)w′(t)=-q(t)ρ(t)f(y(t))|y(t)|α-1y(t)+ρ(t)e(t)|y(t)|α-1y(t)+ρ′(t)w(t)ρ(t)-α|w(t)|(α+1)/α[ρ(t)p(t)Ψ(y(t))]1/α≤-δq(t)ρ(t)+ρ′(t)w(t)ρ(t)+e(t)ρ(t)|y(t)|α-1y(t)-αM1/α|w(t)|(α+1)/αρ1/α(t)p1/α(t).
By assumptions, we choose s1,t1≥T0 so that e(t)≤0 on the interval I1=[s1,t1]. For t∈I1, (2.13) implies that w(t) satisfies the inequality
(2.14)δρ(t)q(t)≤-w′(t)+ρ′(t)w(t)ρ(t)-αM1/α|w(t)|(α+1)/αp1/α(t)ρ(t)1/α(t).
A similar method as to that of Theorem 2.2 implies a contradiction to (2.11) with i=1.
Wheny(t)<0 holds eventually, we assume y(t)<0 for t≥T0>t0. Defining the Riccati transformation as (2.12), we get that (2.13) is true. In this case, we choose t2>s2≥T0 so that e(t)≥0 on the interval I2=[s2,t2]. For a given t∈I2, we get that (2.14) holds. A similar method reaches a similar contradiction to (2.11) with i=2. This completes the proof.Now we give two examples to illustrate the efficiency of our results.Example 2.6.
Consider the following forced half-linear differential equation:(2.15)(tλ|y′(t)|α-1y′(t))′+Ktλ|y(t)|α-1y(t)=-sin⁡t,
for t≥1, where K,λ>0 are constants and α=1. We may show that (2.15) is oscillatory for K>2e(1+λ/2)2 using Theorem 2.2. Indeed, since the zeros of the forcing term -sint are nπ, the constant γ in (1.5) is α, that is, γ=α. For any T≥1, we choose n sufficiently large so that nπ=2kπ≥T and s1=2kπ and t1=(2k+1)π. Selecting u(t)=sint≥0, G1(u)=u2exp(-u) (we note that (G1′(u))2≤4G1(u) for u≥0), ρ(t)=t-λ, then we have
(2.16)∫s1t1ρ(t)q(t)G1(u(t))dt=∫2kπ(2k+1)πt-λKtλ(sin⁡t)2exp⁡(-sin⁡t)dt=K∫0π(sin⁡t)2exp⁡(-sin⁡t)dt≥Ke∫0π1-cos⁡⁡2t2dt=Kπ2e,∫s1t1ρ(t)(αγ)αp(t)(|u′(t)|+Gi1/(α+1)(u(t))|ρ′(t)|(α+1)ρ(t))α+1dt=∫2kπ(2k+1)πt-λ(αα)αtλ(|cos⁡⁡t|+λsin⁡texp⁡(-sin⁡t/2)t)2dt<∫0π(1+λ2)2dt=(1+λ2)2π.
So we have Q1ρ(u)>0 provided K>2e(1+λ/2)2. Similarly, for s2=(2k+1)π and t2=(2k+2)π, we select u(t)=sint≤0, G2(u)=u2exp(u), and we note that (G2′(u))2≤4G2(u) for u≤0; we can show that the integral inequality Q2ρ(u)>0 for K>2e(1+λ/2)2. So (2.15) is oscillatory for K>2e(1+λ/2)2 by Theorem 2.2.Example 2.7.
Consider the following forced half-linear differential equation:(2.17)[(2+cos⁡⁡t)t-λ|y′(t)|α-1y′(t)]′+Kt-λexp⁡(sin⁡t)|y(t)|α-1y(t)=-sin⁡t,
for t≥1, where K,λ>0 are constants and α=1. We may show that (2.17) is oscillatory for K>3(1+λ)2 using Theorem 2.2. Indeed, since the zeros of the forcing term -sint are nπ, the constant γ in (1.5) is α, that is, γ=α. In fact, for any T≥1, we choose n sufficiently large so that nπ=2kπ≥T and s1=2kπ and t1=(2k+1)π. Selecting u(t)=sint≥0, G1(u)=u2exp(-u) (we note that (G1′(u))2≤4G1(u) for u≥0), ρ(t)=tλ, then we have
(2.18)∫s1t1ρ(t)q(t)G1(u(t))dt=∫2kπ(2k+1)πtλKt-λ(sin⁡t)2exp⁡(-sin⁡t)dt=K∫0π(sin⁡t)2exp⁡(sin⁡t-sin⁡t)dt≥K∫0π1-cos⁡⁡2t2dt=Kπ2,∫s1t1ρ(t)(αγ)αp(t)(|u′(t)|+Gi1/(α+1)(u(t))|ρ′(t)|(α+1)ρ(t))α+1dt=∫2kπ(2k+1)πtλ12(αα)α(2+cos⁡⁡t)t-λ(|cos⁡⁡t|+λsin⁡texp⁡(-sin⁡t/2)t)2dt<∫0π32(1+λ)2dt=32(1+λ)2π.
So, we have Q1ρ(u)>0 provided K>3(1+λ)2. Similarly, for s2=(2k+1)π and t2=(2k+2)π, we select u(t)=sint≤0, G2(u)=u2exp(u), and we note that (G2′(u))2≤4G2(u) for u≤0; we can show that the integral inequality Q2ρ(u)>0 for K>3(1+λ)2. So, (2.17) is oscillatory for K>3(1+λ)2 by Theorem 2.2.


## 3. The Case Whereβ>α

We now handle the case whereβ>α.Theorem 3.1.
Assume that (S3) holds. Suppose further that for anyT≥t0, there exist T≤s1<t1≤s2<t2 such that (2.2) holds. Let u∈C1[si,ti], and the 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 ρ∈C1([t0,∞),ℝ) such that
(3.1)Qiρ(u):=∫sitiρ(t)[Qe(t)Gi(u(t))-Mp(t)(|u′(t)|+Gi1/(α+1)(u(t))|ρ′(t)|(α+1)ρ(t))α+1]dt>0,
for i=1,2, then (1.1) is oscillatory, where
(3.2)Qe(t)=α-α/ββ(β-α)(α-β)/β[δq(t)]α/β|e(t)|(β-α)/β.Proof.
Suppose to the contrary that there is a nontrivial nonoscillatory solution. Firstly, we assume thaty(t)>0 on [T0,∞) for some T0≥t0. Set
(3.3)w(t)=ρ(t)p(t)Ψ(y(t))|y′(t)|α-1y′(t)|y(t)|α-1y(t),t≥T0.
Then differentiating (3.3) and making use of (1.1), it follows that for all t≥T0,
(3.4)w′(t)=-q(t)ρ(t)f(y(t))|y(t)|α-1y(t)+ρ(t)e(t)|y(t)|α-1y(t)+ρ′(t)w(t)ρ(t)-α|w(t)|(α+1)/α[ρ(t)p(t)Ψ(y(t))]1/α=-q(t)ρ(t)f(y(t))|y(t)|β-1y(t)|y(t)|β-α+e(t)ρ(t)|y(t)|α-1y(t)+ρ′(t)w(t)ρ(t)-α|w(t)|(α+1)/α[p(t)ρ(t)Ψ(y(t))]1/α≤-δq(t)ρ(t)|y(t)|β-α+e(t)ρ(t)|y(t)|α-1y(t)+ρ′(t)w(t)ρ(t)-αM1/α|w(t)|(α+1)/αp1/α(t)ρ1/α(t).
By assumptions, we can choose t1>s1≥T0 so that e(t)≤0 on the interval I1=[s1,t1]. For a given t∈I1, set F(x)=δq(t)xβ-α-e(t)/xα, and we have F′(x*)=0, F′′(x*)>0, where x*=[-αe(t)/(β-α)δq(t)]1/β. So, F(x) attains its minimum at x* and
(3.5)F(x)≥F(x*)=Qe(t).
So (3.4) and (3.5) imply that w(t) satisfies
(3.6)ρ(t)Qe(t)≤-w′(t)-αM1/α|w(t)|(α+1)/αp1/α(t)ρ1/α(t)+ρ′(t)w(t)ρ(t).
The remaining argument is the same as in the proof of Theorem 2.2, so we obtain a desired contradiction to (3.1) with i=1 when y(t)>0 eventually.
On the other hand, ify(t) is a negative solution for t≥T0≥t0, we define the Riccati transformation (3.3) to yield (3.4). In this case, we choose t2>s2≥T0 so that e(t)≥0 on the interval I2=[s2,t2]. For a given t∈I2, set F(x)=δq(t)xβ-α-e(t)/xα, and we have F(x)≥F(x*)=Qe(t). The remaining proof is similar to that of Theorem 2.2; a desired contradiction to (3.1) with i=2 can be obtained. This completes the proof.Corollary 3.2.
Ifρ(t)≡1 in Theorem 3.1 and the hypothesis (3.1) is replaced by
(3.7)Q̃i(u):=∫siti[Qe(t)Gi(u(t))-Mp(t)|u′(t)|α+1]dt>0,
for i=1,2, then (1.1) is oscillatory.We remark that Corollary3.2 is closely related to the generalized variational formulae. Furthermore, in Theorem 3.1, there is no restriction on the positive constant α, so Theorem 2.5 can be treated as its limiting case when β→α+0 with the convention that 00=1.Example 3.3.
Consider the following forced quasilinear differential equation:(3.8)(γtλ/3y′(t))′+tλ|y(t)|2y(t)=-sin3t,t≥1,
where γ,λ>0 are constants. We see that Ψ(u)≡1, which implies that M=1, and α=1, β=3 in Theorem 3.1. Since α<β, Theorem 2.2 cannot be applied. However, we can obtain oscillation for (3.8) using Theorem 3.1. For any T≥1, choose n sufficiently large so that nπ=2kπ≥T and s1=2kπ and t1=(2k+1)π. We choose u(t)=sint≥0, G1(u)=u2exp(-u) (we note that (G1′(u))2≤4G1(u), for u≥0), ρ(t)=tλ. In fact, we can easily verify that Qe(t)=(3/2)23tλ/3sin2t,
(3.9)∫s1t1Qe(t)q(t)G1(u(t))dt=∫2kπ(2k+1)πt-λ/33223sin⁡t4exp⁡(-sin⁡t)dt=3223∫0π(sin⁡t)4exp⁡(-sin⁡t)dt≥92316eπ,∫s1t1ρ(t)(αγ)αMp(t)(|u′(t)|+Gi1/(α+1)(u(t))|ρ′(t)|(α+1)ρ(t))α+1dt=∫s1t1t-λ/3γtλ/3(|cos⁡⁡t|+(λ/3)sin⁡texp⁡(-sin⁡t/2)2t)2dt<γ(1+λ6)2π.
So, we have that (3.1) is true for i=1 provided 0<γ<923/16e(1+λ/6)2. Similarly, for s2=(2k+1)π and t2=(2k+2)π, we can show that (3.1) is true for i=2. So (3.8) is oscillatory for 0<γ<923/16e(1+λ/6)2 by Theorem 3.1.



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