# Oscillation Properties for Second-Order Partial Differential Equations with Damping and Functional Arguments

Authors: Run Xu; Yuhua Lu; Fanwei Meng
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Using an integral averaging method and generalized Riccati technique, by introducing a parameterβ≥1, we derive new oscillation criteria for second-order partial differential equations with damping. The results are of high degree of generality and sharper than most known
ones.


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## Body

## 1. Introduction

Consider the second-order partial delay differential equation(1.1)∂∂t(r(t)∂∂tu(x,t))+p(t)∂u(x,t)∂t=a(t)Δu(x,t)+∑k=1sak(t)Δu(x,t-ρk(t))-q(x,t)f(u(x,t))-∑j=1mqj(x,t)fj(u(x,t-σj)),(x,t)∈Ω×R+≡G,
where Δ is the Laplacian in RN,R+=[0,∞) and Ω is a bounded domain in RN with a piecewise smooth boundary ∂Ω.Throughout this paper, we assume that(H1)
r(t)∈C1(R+,(0,∞)),p(t)∈C(R+,R);(H2)
q(x,t),qj(x,t)∈C(G¯,R+),q(t)=min⁡x∈G¯q(x,t),qj(t)=min⁡x∈G¯⁡qj(x,t),j∈Im={1,2,…,m};(H3)
a(t),ak(t),ρk(t)∈C(R+,R+),lim⁡t→∞(t-ρk(t))=∞, and σj are nonnegative constants, j∈Im, k∈Is={1,2,…,s};(H4)
f(u)∈C1(R,R),fj(u)∈C(R,R) are convex in R+ with ufj(u)>0,uf(u)>0, and f′(u)≥μ>0, (u≠0).We say that a continuous functionH(t,s) belongs to the function class ω, denoted by H∈ω, if H∈C(D,R+), where D={(t,s):-∞<s≤t<+∞}, satisfy(1.2)H(t,t)=0,H(t,s)>0,-∞<s<t<+∞.Furthermore, the continuous partial derivative∂H/∂S exists on D, and there is h∈Lloc⁡(D,R), such that(1.3)∂H∂s=-h(t,s)H(t,s).Various results on the oscillation for the partial functional differential equation have been obtained recently. We refer the reader to [1–3] for parabolic equations and to [4–11] for hyperbolic equations.Recently, Li and Cui [12] studied the equation of the form(1.4)∂∂t[p(t)∂∂t(u(x,t)+∑i=1lλi(t)u(x,t-τi))]=a(t)Δu(x,t)+∑k=1sak(t)Δu(x,t-ρk(t))-q(t)u(x,t)-∑j=1mqj(x,t)u(x,t-σj),(x,t)∈Ω×R+≡G
with Robin boundary condition(1.5)∂u(x,t)∂γ+g(x,t)u(x,t)=0,(x,t)∈∂Ω×R+,
where γ is the unit exterior vector to ∂Ω and g(x,t) is a nonnegative continuous function on ∂Ω×R+ and obtained the following result.Theorem A (see [12, Theorem 2.2]).
Suppose thatH∈ω, let(C1)
0<inf⁡s≥t0{liminf⁡t→∞(H(t,s)/H(t,t0))}≤∞, suppose that there exists some j0∈Im and there exist two functions ϕ∈C1[t0,∞),A∈C[t0,∞) satisfying,(C2)
limsup⁡t→∞⁡(1/H(t,t0))∫t0tp(s-σj0)ϕ(s)h2(t,s)ds<∞,(C3)
∫t0∞(A+2(s)/p(s-σj0)ϕ(s))ds=∞, and for every t1≥t0,(C4)
limsup⁡t→∞(1/H(t,t1))∫t1t[H(t,s)ψ(s)-(1/4)ϕ(s)p(s-σj0)h2(t,s)]ds≥A(t1),where ϕ(s)=exp⁡{-2∫sϕ(ξ)dξ},A+(s)=max⁡{A(s),0}, and ψ(s)=ϕ(s){αj0qj0(s)[1-∑i=1lλi(s-σj0)]+p(s-σj0)ϕ2(s)-[p(s-σj0)ϕ(s)]′}. Then every solution u(x,t) of the problem (1.4), (1.5) is oscillatory in G.In 2008, Rogovchenko and Tuncay [13] established new oscillation criteria for second-order nonlinear differential equations with damping term(1.6)(r(t)x′(t))′+p(t)x′(t)+q(t)f(x(t))=0,
without an assumption that has been required in related results reported in the literature over the last two decades. Motivated by the ideas in [12, 13], by introducing a Parameter β≥1, we will further improve Theorems A and derive new interval criteria for oscillation of (1.1). We suggest two different approaches which allow one to remove condition (C2) in Theorem A. A modified integral averaging technique enables one to simplify essentially the proofs of oscillation criteria.


## 2. Main Results

Theorem 2.1.
Suppose that there exists a functiony∈C1[t0,∞) such that for some β≥1 and for some H∈ω,
(2.1)lim⁡t→∞sup⁡1H(t,t0)∫t0t(H(t,s)ψ(s)-β4μv(s)r(s)h2(t,s))ds=∞,t≥0,
where
(2.2)v(t)=exp⁡(-2∫t(μy(s)-p(s)2r(s))ds),(2.3)ψ(t)=v(t)[q(t)+μr(t)y2(t)-p(t)y(t)-(r(t)y(t))′],
then every solution u(x,t) of the problem (1.1), (1.5) is oscillatory in G.Proof.
Suppose to the contrary that there is a nonoscillatory solutionu(x,t) of the problem (1.1), (1.5) which has no zero on Ω×[t0,∞) for some t0>0. Without loss of generality, we assume that u(x,t)>0,u(x,t-ρk(t))>0 and u(x,t-σj)>0 in Ω×[t1,∞),t1≥t0,k∈Is,j∈Im. Integrating (1.1) with respect to x over the domain Ω, we have
(2.4)ddt(r(t)ddt∫Ωu(x,t)dx)+p(t)ddt∫Ωu(x,t)dx=a(t)∫ΩΔu(x,t)dx+∑k=1sak(t)∫ΩΔu(x,t-ρk(t))dx-∫Ωq(x,t)f(u(x,t))dx-∑j=1m∫Ωqj(x,t)fj(u(x,t-σj))dx,t≥t1.
From Green’s formula and the boundary condition (1.5), we have
(2.5)∫ΩΔu(x,t)dx=∫∂Ω∂u(x,t)∂γds=-∫∂Ωg(x,t)u(x,t)ds≤0,∫ΩΔu(x,t-ρk(t))dx=∫∂Ω∂u(x,t-ρk(t))∂γds=-∫∂Ωg(x,t-ρk(t))u(x,t-ρk(t))ds≤0,t≥t1,k∈Is,
where ds denotes the surface element on ∂Ω. Moreover, from (H2), (H4) and Jensen’s inequality, we have
(2.6)∫Ωq(x,t)f(u(x,t))dx≥q(t)∫Ωf(u(x,t))dx≥|Ω|q(t)f(1|Ω|∫Ωu(x,t)dx),∫Ωqj(x,t)fj(u(x,t-σj))dx≥qj(t)∫Ωfj(u(x,t-σj))dx≥|Ω|qj(t)fj(1|Ω|∫Ωu(x,t-σj))dx,
where |Ω|=∫Ωdx.
Set(2.7)U(t)=1|Ω|∫Ωu(x,t)dx,t≥t1.
In view of (2.5)–(2.7), (2.4) yields that
(2.8)(r(t)U′(t))′+p(t)U′(t)+q(t)f(U(t))+∑j=1mqj(t)fj(U(t-σj))≤0,t≥t1.
Note that (H4), (2.8) yields that
(2.9)(r(t)U′(t))′+p(t)U′(t)+q(t)f(U(t))≤0,t≥t1.
Put
(2.10)w(t)=v(t)r(t)[U′(t)f(U(t))+y(t)],t≥t1,
where v(t) is given by (2.2), then
(2.11)w′(t)=[-2μy(t)+p(t)r(t)]w(t)+v(t)[(r(t)U′(t))′f(U(t))-r(t)f′(U(t))(U′(t))2f2(U(t))+(r(t)y(t))′]≤[-2μy(t)+p(t)r(t)]w(t)+v(t)[(r(t)U′(t))′f(U(t))-μr(t)(U′(t))2f2(U(t))+(r(t)y(t))′]≤[-2μy(t)+p(t)r(t)]w(t)-v(t)[p(t)U′(t)f(U(t))+q(t)+μr(t)(U′(t)f(U(t)))2-(r(t)y(t))′]=[-2μy(t)+p(t)r(t)]w(t)-v(t)[p(t)(w(t)v(t)r(t)-y(t))+q(t)+μr(t)[w(t)v(t)r(t)-y(t)]2-(r(t)y(t))′]=[-2μy(t)+p(t)r(t)]w(t)-v(t)[p(t)v(t)r(t)w(t)-p(t)y(t)+q(t)+μr(t)w2(t)v2(t)r2(t)-2μw(t)y(t)v(t)+μr(t)y2(t)-(r(t)y(t))′]=-2μy(t)w(t)+p(t)r(t)w(t)-p(t)r(t)w(t)-v(t)[-p(t)y(t)+q(t)+μr(t)y2(t)-(r(t)y(t))′]+2μy(t)w(t)-μw2(t)v(t)r(t)=-v(t)[-p(t)y(t)+q(t)+μr(t)y2(t)-(r(t)y(t))′]-μw2(t)v(t)r(t)=-ψ(t)-μw2(t)v(t)r(t),
that is,
(2.12)ψ(t)≤-w′(t)-μw2(t)v(t)r(t),
where ψ(t) is defined by (2.3). Multiplying (2.12) by H(t,s) and integrating from T to t, we have, for some β≥1 and for all t≥T≥t1,
(2.13)∫TtH(t,s)ψ(s)ds≤-∫TtH(t,s)w′(s)ds-∫TtH(t,s)μv(s)r(s)w2(s)ds=H(t,T)w(T)-∫Tt(h(t,s)H(t,s)w(s)+H(t,s)μw2(s)v(s)r(s))ds=H(t,T)w(T)-∫Tt[μH(t,s)βv(s)r(s)w(s)+βv(s)r(s)4μh(t,s)]2ds+β4μ∫Ttv(s)r(s)h2(t,s)ds-∫Tt(β-1)μβv(s)r(s)H(t,s)w2(s)ds.
Writing the latter inequality in the form
(2.14)∫Tt[H(t,s)ψ(s)-β4μv(s)r(s)h2(t,s)]ds≤H(t,T)w(T)-∫Tt[μH(t,s)βv(s)r(s)w(s)+βv(s)r(s)4μh(t,s)]2ds-∫Tt(β-1)μβv(s)r(s)H(t,s)w2(s)ds.
Using the properties of H(t,s), we have
(2.15)∫t1t[H(t,s)ψ(s)-β4μv(s)r(s)h2(t,s)]ds≤H(t,t1)|w(t1)|≤H(t,t0)|w(t1)|,t≥t1,
and for all t≥t1≥t0,
(2.16)∫t0t[H(t,s)ψ(s)-β4μv(s)r(s)h2(t,s)]ds≤H(t,t0)[∫t0t1|ψ(s)|ds+|w(t1)|].
By (2.16),
(2.17)lim⁡t→∞sup⁡1H(t,t0)∫t0t(H(t,s)ψ(s)-β4μv(s)r(s)h2(t,s))ds≤∫t0t1|ψ(s)|ds+|w(t1)|<∞,
which contradicts (2.1). This proves Theorem 2.1.Consider a Kamenev-type functionH(t,s) defined by H(t,s)=(t-s)n-1,(t,s)∈D, where n>2 is an integer. Obviously, H belongs to the class ω, and h(t,s)=(n-1)(t-s)(n-3)/2,(t,s)∈D. Then, we can get the following results.Corollary 2.2.
Suppose that there exists a functiony(t)∈C1([t0,∞);R) such that for some integer n>2 and some β≥1,
(2.18)lim⁡t→∞sup⁡1tn-1∫t0t(t-s)n-3[(t-s)2ψ(s)-β(n-1)24μv(s)r(s)]ds=∞,
where v(t) and ψ(t) are as defined in Theorem 2.1. Then every solution u(x,t) of the problem (1.1), (1.5) is oscillatory in G.Theorem 2.3.
Suppose that(2.19)0<inf⁡s≥t0(lim⁡t→∞inf⁡H(t,s)H(t,t0))≤∞.
Assume that there exist functions f∈C1([t0,∞);R) and ϕ∈C([t0,∞);R) such that, for all t≥T≥t0 and for some β>1,
(2.20)lim⁡t→∞sup⁡1H(t,T)∫Tt(H(t,s)ψ(s)-β4μv(s)r(s)h2(t,s))ds≥ϕ(T),
where v(t),ψ(t) are as defined in Theorem 2.1 and suppose further that
(2.21)lim⁡t→∞sup⁡∫t0tϕ+2(s)v(s)r(s)=∞,
where ϕ+(t)=max⁡(ϕ(t),0). Then every solution u(x,t) of the problem (1.1), (1.5) is oscillatory in G.Proof.
Suppose to the contrary that there is a nonoscillatory solutionu(x,t) of the problem (1.1), (1.5) which has no zero on Ω×[t1,∞) for some t1>t0, without loss of generality, we assume that u(x,t)>0,u(x,t-ρk(t))>0 and u(x,t-σj)>0 in Ω×[t1,∞),t≥t1≥t0,k∈Is,j∈Im.
As in the proof of Theorem2.1, (2.14) holds for all t≥T≥t1, we have
(2.22)1H(t,T)∫Tt[H(t,s)ψ(s)-β4μv(s)r(s)h2(t,s)]ds≤w(T)-1H(t,T)∫Tt[μH(t,s)βv(s)r(s)w(s)+βv(s)r(s)4μh(t,s)]2ds-1H(t,T)∫Tt(β-1)μβv(s)r(s)H(t,s)w2(s)ds≤w(T)-1H(t,T)∫Tt(β-1)μβv(s)r(s)H(t,s)w2(s)ds.
Therefore, for t>T≥T0,
(2.23)lim⁡t→∞sup⁡1H(t,T)∫Tt[H(t,s)ψ(s)-β4μv(s)r(s)h2(t,s)]ds≤w(T)-lim⁡t→∞inf⁡1H(t,T)∫Tt(β-1)μβv(s)r(s)H(t,s)w2(s)ds.
It follows from (2.20) that
(2.24)w(T)≥ϕ(T)+lim⁡t→∞inf⁡1H(t,T)∫Tt(β-1)μβv(s)r(s)H(t,s)w2(s)ds
for all T≥t1 and for any β>1. Then, for all T≥t1,
(2.25)w(T)≥ϕ(T),(2.26)lim⁡t→∞inf⁡1H(t,t1)∫t1tH(t,s)v(s)r(s)w2(s)ds≤βμ(β-1)(w(t1)-ϕ(t1)).
Now, we claim that
(2.27)∫t1∞w2(s)v(s)r(s)ds<∞.
Suppose the contrary, that is,
(2.28)∫t1∞w2(s)v(s)r(s)ds=∞.
By (2.19), there is a positive constant M1, satisfying
(2.29)inf⁡s≥t0(lim⁡t→∞inf⁡H(t,s)H(t,t0))>M1>0.
Let M be any arbitrary positive number, then from (2.28) we get that there exists a T1>t1 such that, for all t≥T1,
(2.30)∫t1tw2(s)v(s)r(s)ds≥MM1.
Using integration by parts, for all t≥T1, we get
(2.31)1H(t,t1)∫t1tH(t,s)w2(s)v(s)r(s)ds=1H(t,t1)∫t1t(-∂H(t,s)∂s)(∫t1sw2(τ)v(τ)r(τ)dτ)ds≥1H(t,t1)∫T1t(-∂H(t,s)∂s)(∫t1sw2(τ)v(τ)r(τ)dτ)ds≥MM11H(t,t1)∫T1t(-∂H(t,s)∂s)ds=MM1H(t,T1)H(t,t1).
By (2.29), there exists a T2>T1 such that, for all t≥T2,
(2.32)H(t,T1)H(t,t1)≥M1.
It follows from (2.31) that for all t≥T2,
(2.33)1H(t,t1)∫t1tH(t,s)w2(s)v(s)r(s)ds≥M.
Since M is an arbitrary positive constant,
(2.34)lim⁡t→∞inf⁡1H(t,t1)∫t1tH(t,s)w2(s)v(s)r(s)ds=∞,
which contradicts (2.26). Consequently, (2.27) holds. And from (2.25), we obtain
(2.35)∫t1∞ϕ+2(s)v(s)r(s)ds≤∫t1∞w2(s)v(s)r(s)ds<∞,
which contradicts (2.21). This completes the proof of Theorem 2.3.ChoosingH as in Corollary 2.2, by Theorem 2.3, we can obtain the following corollary.Corollary 2.4.
Letv(t) and ψ(t) be as in Theorem 2.1, assume further that there exist functions f∈C1([t0,∞);R) and ϕ∈C([t0,∞);R) such that, for all T≥t0, for some integer n>2, and for some β>1,
(2.36)lim⁡t→∞sup⁡1tn-1∫Tt(t-s)n-3[(t-s)2ψ(s)-β(n-1)24μv(s)r(s)]ds≥ϕ(T)
and (2.21) hold. Then every solution u(x,t) of the problem (1.1), (1.5) is oscillatory in G.Theorem 2.5.
Suppose that there exists a functionf∈C1([t0,∞);R) such that for some β≥1 and for some H∈ω,
(2.37)lim⁡t→∞sup⁡1H(t,t0)∫t0t[H(t,s)(ψ¯(s)-p2(s)v¯(s)2μr(s))-μβ2v¯(s)r(s)h2(t,s)]ds=∞,
where
(2.38)v¯(t)=exp⁡(-2μ∫ty(s)ds),ψ¯(t)=v¯(t)(q(t)+μr(t)y2(t)-p(t)y(t)-(r(t)y(t))′).
Then every solution u(x,t) of the problem (1.1), (1.5) is oscillatory in G.Proof.
As in Theorem2.1, without loss of generality, we assume that a nonoscillatory solution u(x,t) of the problem (1.1), (1.5) satisfies u(x,t)>0,u(x,t-ρk(t))>0 and u(x,t-σj)>0 in Ω×[t1,∞), t1≥t0, k∈Is,j∈Im. Define a generalized Riccati transformation
(2.39)w¯(t)=v¯(t)r(t)[U′(t)f(U(t))+y(t)],t≥t1,
where v¯(t) is given by (2.38). Then
(2.40)w¯′(t)≤-2μy(t)w¯(t)+v¯(t){-q(t)+(r(t)y(t))′-p(t)[w¯(t)v¯(t)r(t)-y(t)]-μr(t)[w¯(t)v¯(t)r(t)-y(t)]2}=-ψ¯(t)-p(t)r(t)w¯(t)-μ1r(t)v¯(t)w¯2(t),t≥t1.
Using an elementary inequality
(2.41)-ax2+bx≤-a2x2+b22a,
for all a>0 and for all b,x∈R, we conclude from (2.40) that
(2.42)ψ¯(t)-p2(t)v¯(t)2μr(t)≤-w¯′(t)-μ2v¯(t)r(t)w¯2(t),t≥t1.
Multiplying (2.42) by H(t,s) and integrating from T<t, we obtain, for some β≥1 and for all t≥T≥t1,
(2.43)∫TtH(t,s)(ψ¯(s)-p2(s)v¯(s)2μr(s))ds≤H(t,T)w¯(T)-∫Tth(t,s)H(t,s)w¯(s)ds-∫Ttμw¯2(s)2v¯(s)r(s)ds≤H(t,T)w¯(T)+μβ2∫Ttv¯(s)r(s)h2(t,s)ds-μ∫Tt(β-1)H(t,s)2βv¯(s)r(s)w¯2(s)ds-μ2∫Tt(H(t,s)βv¯(s)r(s)w¯(s)+βv¯(s)r(s)h(t,s))2ds.
Therefore, for all t≥T≥t1, we have
(2.44)∫Tt[H(t,s)ψ¯(s)-H(t,s)p2(s)v¯(s)2μr(s)-μβ2v¯(s)r(s)h2(t,s)]ds≤H(t,T)w¯(T)-μ∫Tt(β-1)H(t,s)2βv¯(s)r(s)w¯2(s)ds-μ2∫Tt(H(t,s)βv¯(s)r(s)w¯(s)+βv¯(s)r(s)h(t,s))2ds.
Following the same lines as in the proof of Theorem 2.1, we have
(2.45)lim⁡t→∞sup⁡1H(t,t0)∫t0t(H(t,s)ψ¯(s)-H(t,s)p2(s)v¯(s)2μr(s)-μβ2v¯(s)r(s)h2(t,s))ds≤∫t0t1|ψ¯(s)|ds+|w¯(t1)|<∞
which contradicts the assumption (2.19).
This completes the proof.Theorem 2.6.
Let (2.19) holds. Assume that there exist functions f∈C1([t0,∞);R) and ϕ∈C([t0,∞),R) such that, for all t≥t0, any T≥t0, and for some β>1,
(2.46)lim⁡t→∞sup⁡1H(t,T)∫Tt[H(t,s)(ψ¯(s)-p2(s)v¯(s)2μr(s))-μβ2v¯(s)r(s)h2(t,s)]ds≥ϕ(T)
and (2.21) holds, where ψ(t),v¯(t) are defined as in Theorem 2.6 and ϕ+(t)=max⁡(ϕ(t),0), then every solution u(x,t) of the problem (1.1), (1.5) is oscillatory in G.Theorem 2.7.
Let all assumptions of Theorem2.6 be satisfied except that condition (2.46) be replaced by
(2.47)lim⁡t→∞inf⁡1H(t,T)∫Tt[H(t,s)(ψ¯(s)-p2(s)v¯(s)2μr(s))-μβ2v¯(s)r(s)h2(t,s)]ds≥ϕ(T).
Then every solution u(x,t) of the problem (1.1), (1.5) is oscillatory in G.Remark 2.8.
By introducing the parameterβ in Theorem 2.3, we derive new oscillation criteria of the problem (1.1), (1.5) which are simpler than that in Theorem A; furthermore, modifications of the proofs through the refinement of the standard integral averaging method allowed us to shorten significantly the proofs of Theorem 2.3. We can also derive a number of oscillation criteria with the appropriate choice of the function H and ρ, here, we omit the details.


## 3. Examples

Now, we consider these following examples.Example 3.1.
Consider the partial differential equation(3.1)∂∂t[1t∂∂t(u(x,t)+12u(x,t-π))]+2cos⁡t∂u(x,t)∂t=Δu(x,t)+1t2Δu(x,t-32π)-(2t3+tcos⁡2t+sin⁡t)f(u(x,t))-1t2f1(u(x,t-π)),(x,t)∈(0,π)×(0,∞),
with the boundary condition
(3.2)ux(0,t)=ux(π,t)=0,t>0,
where f(u)=u3+u,f1(u)=ueu+u.
Here,N=1,l=1,s=1,m=1,μ=1, r(t)=1/t, p(t)=2cos⁡t, q(x,t)=q(t)=(2/t3+tcos⁡2t+sin⁡t), q1(x,t)=1/t2, f(u)=fj(u)=u,a(t)=1,a1(t)=1/t2, ρ1(t)=(3/2)π,σ1=π,τ1=π.
Let(3.3)y(t)=1t+tcos⁡t,
then
(3.4)v(t)=t2,ψ(t)=t-1.
Let n=3, for any β≥1,
(3.5)lim⁡t→∞sup⁡1t2∫1t[(t-s)2s-1-βs21s]ds=lim⁡t→∞sup⁡1t2∫1t[(t-s)2s-1-βs]ds=∞.
Therefore, Corollary 2.2 holds, then every solution u(x,t) of the problem (3.1), (3.2) oscillates in (0,π)×(0,∞).Example 3.2.
Consider the partial differential equation(3.6)∂∂t[(1+12t3)(2+sin⁡t)∂u(x,t)∂t]+3t(1+12t3)(2+sin⁡t)∂u(x,t)∂t=3Δu(x,t)+(2-cos⁡t)Δu(x,t-32π)-t-3((1-t3+2t2-6t)sin⁡t+12t)f(u(x,t))-(2t+sin⁡t)f1(u(x,t-π))-2f2(u(x,t-π2)),(x,t)∈(0,π)×(0,∞)
with the boundary condition (3.2), where f(u)=u5+u,f1(u)=usin⁡2u,f2(u)=u3cos⁡2u.
HereN=1,s=1,m=2,μ=1, r(t)=(1+1/2t3)(2+sin⁡t), p(t)=(3/t)(1+1/2t3)(2+sin⁡t), q(t)=t-3[(1-t3+2t2-6t)sin⁡t+12t], a(t)=3,a1(t)=2-cos⁡t,q1(x,t)=2+sin⁡t,q2(x,t)=2, ρ1(t)=(3/2)π, σ1=π,σ2=π/2.
Lety(t)=0, then v(t)=t3 and ψ(t)=v(t)q(t)=(1-t3+2t2-6t)sin⁡t+12t.
Chooseβ=2, n=3, a straightforward computation yields
(3.7)limsup⁡t→∞1t2∫Tt[(t-s)2((1-s3+2s2-6s)sin⁡s+12s)-(2s3+1)(2+sin⁡s)]ds=16-T3cos⁡T+T2(2cos⁡T-6+3sin⁡T)-4Tsin⁡T-3cos⁡T=ϕ(T).
Let ϕ+(t)=max⁡(ϕ(t),0). It is not difficult to see that
(3.8)limsup⁡t→∞∫1tϕ+2(s)(s3+(1/2))(2+sin⁡s)ds≥limsup⁡t→∞∫1tϕ+2(s)3(s3+(1/2))ds=∞.
By Corollary 2.4, we obtain that every solution of problem (3.6), (3.2) oscillates in (0,π)×(0,∞).
Note that in this example,(3.9)limsup⁡t→∞1t2∫1t4(s3+12+(2+sin⁡s))ds=∞,
so the condition (C2) would not have been satisfied with the same choices of v(t).



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*Source: 901631-2011-12-08.xml*