By using the generalized variational principle and Riccati technique, a new oscillation criterion is established for second-order quasilinear differential equation with an oscillatory forcing term, which improves and generalizes some of new results in the literature.

1. Introduction

In this paper, we are concerned with oscillation for the second order forced quasilinear differential equation(p(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.

By a solution of (1.1), we mean a function y(t)∈C1([Ty,∞),ℝ), 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). We restrict our attention to the nontrivial solutions y(t) of (1.1) only, that is, to solutions y(t) such that sup{|y(t)|:t≥T}>0 for all T≥Ty. 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 nontrivial solutions are oscillatory.

We note that when 0<α=β, (1.1) turns into the half-linear differential equation(p(t)|y′(t)|α-1y′(t))′+q(t)|y(t)|α-1y(t)=e(t),t≥t0,
while 0<α<β, (1.1) is a super-half-linear differential equation.

Oscillation for (1.2) and its special case (without a forcing term)(p(t)|y′(t)|α-1y′(t))′+q(t)|y(t)|α-1y(t)=0,t≥t0,
are widely studied by many authors (see [1–6] and references cited therein) since the foundation work of Elbert [7]. Manojlović [8], Ayanlar and Tiryaki [9] obtained some Kamenev type oscillation criteria for (1.3) by using integral averaging techniques; a more detailed description of oscillation criteria can be found in the book in [10].

The half-linear differential equation (1.3) has similar properties to linear differential equation; for example, Sturm's comparison theorem and separation theorem (see Elbert [7], Jaros and Kusano [5], Li and Yeh [11] for details) are still true for (1.3). In particular, the Riccati type equation (related to (1.3) by the substitution w=p|y′|α-1y′/|y|α-1y)w′+q(t)+αp-1/α(t)|w|(α+1)/α=0
and the (α+1)-degree functionalI(y):=∫ab[q(t)|y|α+1-p(t)|y′|α+1]dt
play the same role as the classical Riccati equation and quadratic functional in the linear oscillation theory. (The proof of the relationship between (1.3), (1.4), and (1.5) is based on the recently established Picone's identity, see [5].)

When α=1, (1.2) is reduced to the forced linear differential equation(p(t)y′(t))′+q(t)y(t)=e(t),t≥t0.

In paper [12], based on the oscillatory behavior of the forcing term, Wong proved the following result for (1.6).

Theorem 1.1.

Suppose that, for any T≥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
Qi(u):=∫siti[q(t)u2(t)-p(t)(u′(t))2]dt>0,i=1,2,
then (1.6) is oscillatory.

This result involves the “oscillatory interval” of e(t) and Leighton's variational principle for (1.6). Recently, using a similar method, Wong's result was extended to (1.2) by Li and Cheng [1] with a positive and nondecreasing function ϕ∈C1([t0,∞),ℝ).

Theorem 1.2.

Suppose that, for any T≥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 ϕ∈C1([t0,∞),ℝ+) such that
∫sitiH2(t)ϕ(t)q(t)dt>K∫sitip(t)ϕ(t)|H(t)|α-1(2|H′(t)|+|H(t)|ϕ′(t)ϕ(t))α+1dt
for i=1,2, where K=(1/(α+1))α+1, then (1.2) is oscillatory.

However, inequality (1.8) has no relation to the (α+1)-degree functional (1.5), and Theorem 1.2 cannot be applied to oscillation when α>1, since |H(t)|α-1 is the denominator of the fraction in the right-side integral of (1.8), and H(si)=H(ti)=0.

Zheng and Meng [4] improved the paper [1] and gave an oscillation criterion without any restriction on the signs of q, e, and ϕ′, which is closely related to the (α+1)-degree functional (1.5). Here we list the main result of Zheng and Meng (Jaros et al. [6] obtained a similar result using the generalized Pocone's formulae with ϕ(t)≡1).

Theorem 1.3.

Suppose that, for any T≥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 function ϕ∈C1([t0,∞),ℝ+) such that
Qiϕ(H):=∫sitiϕ[QeHα+1-p(|H′|+H|ϕ′|(α+1)ϕ)α+1](t)dt>0
for i=1,2, then (1.1) is oscillatory, where Qe(t) is the same as (2.4).

Meanwhile, among the oscillation criteria, Komkov [13] gave a generalized Leighton's variational principle, which also can be applied to oscillation for linear equation.

Theorem 1.4.

Suppose that there exist a C1 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,
∫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 every solution of the equation (p(t)y′(t))′+q(t)y(t)=0 must vanish on [s1,t1].

As far as we know, there is no oscillation criterion which relates to the generalized Leighton's variational principle for (1.1). The question then arises as to whether a more general result can be established to this equation. In this paper, we give an affirmative answer for this question and give a new oscillation criterion for the quasilinear differential equation (1.1); this oscillation criterion is closely related to the generalized Leighton's variational principle, which generalizes and improves the results mention above.

2. Main Results

Firstly, we give an inequality, which is a transformation of Young's inequality.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B17">14</xref>]).

Supposing that X and Y are nonnegative, then
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 any T≥t0, there exist T≤s1<t1≤s2<t2 such that
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
Qiϕ(u):=∫sitiϕ[QeGi(u)-p(|u′|+Gi1/(α+1)(u)|ϕ′|(α+1)ϕ)α+1](t)dt>0
for i=1,2, then (1.1) is oscillatory, where
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.

Proof.

Suppose on the contrary that there is a nontrivial nonoscillatory solution. We assume that y(t)>0 on [T0,∞) for some T0≥t0. Set
w(t)=ϕ(t)p(t)|y′(t)|α-1y′(t)|y(t)|α-1y(t),t≥T0.
Then for all t≥T0, we obtain the Riccati type equation
w′(t)=ϕ′(t)ϕ(t)w(t)-ϕ(t)[q(t)|y|β-α-e(t)|y(t)|α-1y(t)]-α|w(t)|(α+1)/α(p(t)ϕ(t))1/α.
By the assumption, we can choose t1>s1≥T0 so that e(t)≤0 on the interval I1=[s1,t1]. For given t∈I1, setting F(x)=q(t)xβ-α-e(t)/xα, we have F′(x*)=0, F′′(x*)>0, where x*=[-αe(t)/(β-α)q(t)]1/β. So F(x) obtains its minimum on x*, and
F(x)≥F(x*)=Qe(t).
So (2.6) and (2.7) imply that w(t) satisfies
ϕ(t)Qe(t)≤-w′(t)+ϕ′(t)ϕ(t)w(t)-α|w(t)|(α+1)/α(p(t)ϕ(t))1/α.
Multiplying G1(u(t)) through (2.8) and integrating (2.8) from s1 to t1, using the fact that G1(u(s1))=G1(u(t1))=0, we obtain
∫s1t1ϕ(t)Qe(t)G1(u(t))dt≤∫s1t1G1(u(t)){-w′(t)+ϕ′(t)ϕ(t)w(t)-α|w(t)|(α+1)/α(p(t)ϕ(t))1/α}dt=-G1(u(t))w(t)|s1t1+∫s1t1g1(u(t))u′(t)w(t)dt+∫s1t1G1(u(t)){ϕ′(t)ϕ(t)w(t)-α|w(t)|(α+1)/α(p(t)ϕ(t))1/α}dt=∫s1t1[g1(u(t))u′(t)+G1(u(t))ϕ′(t)ϕ(t)]w(t)dt-α∫s1t1G1(u(t))|w(t)|(α+1)/α(p(t)ϕ(t))1/αdt≤∫s1t1[|g1(u(t))||u′(t)|+G1(u(t))|ϕ′(t)|ϕ(t)]|w(t)|dt-α∫s1t1G1(u(t))|w(t)|(α+1)/α(p(t)ϕ(t))1/αdt≤(α+1)∫s1t1[G1α/(α+1)(u(t))|u′(t)|+G1(u(t))|ϕ′(t)|(α+1)ϕ(t)]|w(t)|dt-α∫s1t1G1(u(t))|w(t)|(α+1)/α(p(t)ϕ(t))1/αdt.
Let
X=[α(p(t)ϕ(t))1/α]α/(α+1)G1α/(α+1)|w(t)|,γ=1+1α,Y=(αϕ(t)p(t))α/(α+1)[|u′(t)|+G11/(α+1)|ϕ′(t)|(α+1)ϕ(t)]α.
By Lemma 2.1 and (1.2), we have
∫s1t1ϕ(t)Qe(t)G1(u(t))dt≤∫s1t1ϕ(t)p(t)[|u′(t)|+G11/(α+1)(u(t))|ϕ′(t)|(α+1)ϕ(t)]α+1dt,
which contradicts (2.3) with i=1.

When y(t) is a negative solution for t≥T0>t0, we define the Riccati transformation as (2.5), and we obtain (2.8) also. In this case, we choose t2>s2≥T0 so that e(t)≥0 on the interval I2=[s2,t2]. For given t∈I2, setting F(x)=q(t)xβ-α-(e(t)/xα), we have F(x)≥F(x*)=Qe(t). We use u∈C1[s2,t2] and G2 on [s2,t2] to reach a similar contradiction. The proof is complete.

Corollary 2.3.

If ϕ(t)≡1 in Theorem 2.2 and (2.3) is replaced by
Qi(u):=∫siti[Qe(t)Gi(u(t))-p(t)|u′(t)|α+1]dt>0
for i=1,2, then (1.1) is oscillatory.

Remark 2.4.

Corollary 2.3 is closely related to the generalized variational formulae similar to (1.10), so Theorem 2.2, Corollary 2.3 are generalizations of Theorems 1.1 and 1.3 (if we choose G1(u)=G2(u)=uα+1 in Corollary 2.3, then we obtain Corollary 2.3 of paper [4]), and when 0<α=β, we have Qe(t)=q(t), so Corollary 2.3 improves the main result of paper [1], since the positive constant α in Theorem 2.2, Corollary 2.3 can be selected as any number lying in (0,∞).

Remark 2.5.

Hypothesis (2.2) in Theorem 2.2, Corollary 2.3 can be replaced by the following condition
e(t){≥0,t∈[s1,t1],≤0,t∈[s2,t2].
The conclusion is still true for this case.

Now we give an example to illustrate the efficiency of our result.

Example 2.6.

Consider the following forced quasilinear differential equation:
(γt-λ/3y′(t))′+q(t)|y(t)|2y(t)=-sin3t,t≥2π,
where γ,λ>0 are constants, q(t)=t-λexp(3sint) for t∈[2nπ,(2n+1)π), and q(t)=t-λexp(-3sint) for t∈[(2n+1)π,(2n+2)π), n>0 is an integer. Here α=1, β=3 in Theorem 2.2. Since α<β, Theorems 1.1 and 1.2 cannot be applied to this case. However, we can obtain oscillation for (2.14) using Theorem 2.2. In fact, we can easily verify that Qe(t)=(3/2)23t-λ/3exp(sint)sin2t for t∈[2nπ,(2n+1)π) and Qe(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
∫s1t1ϕ(t)Qe(t)G1(u(t))dt=3223∫2kπ(2k+1)πsin4tdt=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[|cost|+λ|sint|exp(3sint/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<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), and we can show that the integral inequality Q2ϕ(u)>0 for 0<γ<923/2(2+λe3/2)2π. So (2.14) is oscillatory for 0<γ<923/2(2+λe3/2)2π by Theorem 2.2. We note further that Theorem 1.3 cannot be applied to (2.14) with G(u)=u2.

Acknowledgments

Thanks are due to the referees for giving the author some useful comments which improve the main result. This paper was partially supported by the NSF of China (Grant 10801089), NSF of Shandong (Grant ZR2009AM011, ZR2009AQ010), and NSF of Jining University (Grant 2010QNKJ09).

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