# Unbounded Solutions of Asymmetric Oscillator

Authors: Tieguo Ji; Zhenhui Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We obtain sufficient conditions for the existence of unbounded
solutions of the following nonlinear differential equation(φp(x′))′+(p−1)[αφp(x+)−βφp(x−)]=(p−1)f(t,x,x′), where φp(u)=|u|p−2u,p>1,x+=max{x,0},x−=max{−x,0},α,β are positive constants, and f is continuous, bounded, and T-periodic in t for some T>0.


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## Body

## 1. Introduction

In 1960s, Lazer and Leach [1] considered the existence of 2π-periodic solution of the following differential equation:
(1)x′′+n2x=g(x)+e(t),
where n∈ℕ,  e is continuous and 2π-periodic, and g is bounded and continuous. If g(±∞)=limx→±∞g(x) exist, they showed that (1) has at least one 2π-periodic solution provided:
(2)2|g(+∞)-g(-∞)|>|∫02π‍e(t)eintdt|.
Assume α,β are positive constants satisfying
(3)1α+1β=2n∈ℚ.
Fabry and Mawhin [2] considered the following asymmetric oscillator:
(4)x′′+αx+-βx-=f(x)+g(x)+e(t),
where f,g are continuous and bounded, e is continuous and 2π-periodic, x+=max{x,0}, and x-=max{-x,0}. Moreover, the limits limx→±∞f(x)=:f(±∞) exist; g has a sublinear primitive; that is,
(5)G(x)x=∫0x‍g(s)dsx⟶0as|x|⟶∞.
They introduced the function
(6)ϕ(θ)=n(f(+∞)α-f(-∞)β)+∫02π‍e(t)ψ(t+θ)dt,
where ψ is the solution of the following initial value problem:
(7)x′′+αx+-βx-=0,x(0)=0,x′(0)=1,
and gave approximations of solutions of (4) with large initial values. Moreover, they proved that (4) has unbounded solutions if the function ϕ(θ) has only simple zeros.Later, Kunze et al. [3] considered the following resonant oscillator:
(8)x′′+f(x)x′+n2x+g(x)=p(t),
where f,g,p∈C0, g is bounded, p is 2π-periodic, and n∈N. They showed that if
(9)4n|F|∞+2(supg-infg)<|∫02π‍p(t)eintdt|,
where F(x)=∫0x‍f(s)ds is bounded, |F|∞=supx∈ℝ|F(x)|, then every solution of (8) is unbounded.Wang [4] considered the unboundedness of solutions of the equation:
(10)x′′+f(x)x′+αx+-βx-=p(t),
where α-1/2+β-1/2∈ℝ+∖ℚ and p is continuous and 2π-periodic. By using the well-known Birkhoff Ergodic theorem [5], he showed that if the function F(x)=∫0x‍f(s)ds is bounded and the limits limx→±∞F(x)=F(±∞) exist, then F(+∞)F(-∞)<0 is a sufficient condition for the existence of unbounded solutions of (10).Liu [6] discussed the boundedness of all solutions of the following nonlinear equation:
(11)(φp(x′))′+αφp(x+)-βφp(x-)=f(t,x),
where
(12)πpα1/p+πpβ1/p=2πn,
with πp=2π/psin(π/p),  n∈ℕ. Let n=1; he defined a 2π-periodic function Z(θ) as
(13)Z(θ)=∫I+‍f+(t+θ)v(t)dt+∫I-‍f-(t+θ)v(t)dt,
where f±(t)=limx→±∞f(t,x)∈L∞[0,2π] exist, I+={t∈[0,2π]:v(t)>0},  I-={t∈[0,2π]:v(t)<0}, and v(t) is the solution of
(14)(φp(v′))′+αφp(v+)-βφp(v-)=0,
with the initial value (v(0),v′(0))=(0,1). Under some smoothness conditions, he showed that all solutions of (11) are bounded if the function Z(θ) has no zero for all θ∈ℝ.Recently, Li and Zhang [7] studied the unboundedness of solutions of the following asymmetric oscillator:
(15)x′′+f(x′)+ax+-bx-=ϕ(t,x),
where a,b are positive constants satisfying nonresonance condition:
(16)a-1/2+b-1/2∈ℝ+∖ℚ,f and ϕ are bounded and ϕ is 2π-periodic in t. By using the similar method used in [4], they obtained some sufficient conditions for the existence of unbounded solutions of (15). For more recent results on the boundedness or unboundedness of solutions of differential equations of second order, we refer to [6, 8–18] and the references therein.Assumeα,β are positive constants. Let S(t) be the solution of the following initial value problem:
(17)(φp(u′))′+(p-1)[αφp(u+)-βφp(u-)]=0,u(0)=0,u′(0)=1;
then it is well-known that S is τ-periodic with
(18)τ=πp(α-1/p+β-1/p),πp=2πpsin(π/p).In this paper, we consider the following asymmetric oscillator:(19)(φp(x′))′+(p-1)[αφp(x+)-βφp(x-)]=(p-1)f(t,x,x′),
where φp(u)=|u|p-2u,  p>1,x±=max{±x,0}, f is continuous, bounded, and T-periodic in t, and the limits limx→±∞,y→±∞f(t,x,y)=f(t,±∞,±∞)∈L∞[0,T] exist. We assume that T satisfies the following resonance condition:
(20)Tτ=nm∈ℚ+.
A solution x(t) of (19) is called to have large initial value if |x(0)|+|x′(0)|≫1. We will give some sufficient conditions for the unboundedness of solutions of (19). Especially, if f(t,x,y)=f(t), we will obtain higher order approximation and corresponding sufficient conditions for the unboundedness of solutions of (19) with large initial values. The results of this paper are new which improve some relative results on the literature in some sense. Throughout this paper, we assume T/τ=n/m with m,n∈ℕ.The main results of this paper are the following.Theorem 1.
Introduce aτ-periodic function
(21)λ(θ)=-∫Iθ++‍f(t,+∞,+∞)S(θ+t)dt-∫Iθ+-‍f(t,+∞,-∞)S(θ+t)dt-∫Iθ-+‍f(t,-∞,+∞)S(θ+t)dt-∫Iθ--‍f(t,-∞,-∞)S(θ+t)dt,
where
(22)Iθ++={t∈[0,mT]:S(θ+t)>0,S′(θ+t)>0},Iθ+-={t∈[0,mT]:S(θ+t)>0,S′(θ+t)<0},Iθ-+={t∈[0,mT]:S(θ+t)<0,S′(θ+t)>0},Iθ--={t∈[0,mT]:S(θ+t)<0,S′(θ+t)<0}.
Then all solutions of (19) with large initial values are unbounded provided that the function λ(θ) has only finite number of zeros in [0,τ] and all its zeros are simple.Corollary 2.
Assumef(t,x,y)=f(t)∈L∞[0,T] is T-periodic and define a τ-periodic function λ(θ) as
(23)λ(θ)=-∫0mT‍S(θ+t)f(t)dt.
If λ(θ) has only finite number of zeros in [0,τ] and all its zeros are simple, then all solutions of (19) with large initial values are unbounded.Theorem 3.
Consider (19) with f(t,x,y)=f(t)∈L∞[0,T] is T-periodic. Let λ(θ) be given in Corollary 2 and assume λ(θ)≡0. Define τ-periodic functions λ¯(θ) and μ¯(θ) as
(24)λ¯(θ)=(p-2)∫0mT‍S(θ+t)f(t)000000000×∫0t‍S′(θ+u)f(u)dudt,μ¯(θ)=-(p-1)∫0mT‍S′′(θ+t)f(t)0000000000×∫0t‍S(θ+u)f(u)dudt.
Then all solutions of (19) with large initial values are unbounded provided one of the following conditions holds: (i)
p≠2 and λ¯(θ) has only finite number of zeros in [0,mT]=[0,nτ] and all of them are simple;(ii)
p=2 and the function μ¯(θ) has no zero in [0,mT]=[0,nτ].


## 2. Generalized Polar Coordinate Transformation

Letu=u(t) be the solution of the following initial value problem:
(25)(φp(u′))′+(p-1)φp(u)=0,u(0)=0,u′(0)=1,
and define a function S(t) as
(26)S(t)={α-1/pu(α1/pt),t∈[0,α-1/pπp],-β-1/pu[β1/p(t-α-1/pπp)],t∈[α-1/pπp,τ];
then it is easy to verify that S∈C2; is τ-periodic and satisfies the identity:
(27)|S′(t)|p+α(S+(t))p+β(S-(t))p≡1,∀t∈ℝ.Moreover, one can verify thatS is the solution of (17).Forr>0,   θ∈ℝ, define the generalized polar coordinate transformation TS:(r,θ)→(x,x′) by
(28)TS:x=r1/(p-1)S(θ),x′=r1/(p-1)S′(θ).Under the mappingTS and by using (27), it is not difficult to verify that (19) can be transformed into the following planar system:
(29)r′=(p-1)S′(θ)f(t,x,x′),θ′=1-r-1S(θ)f(t,x,x′),
where x,x′ are given by (28).Lemma 4.
Let(r(t),θ(t)) be the solution of (29) satisfying initial condition (r(0),θ(0))=(r0,θ0), then for r0≫1; the Poincaré mapping P:(r0,θ0)→(r1,θ1)=:(r(mT),θ(mT)) satisfies
(30)r1=r0-(p-1)λ′(θ0)+o(1),θ1=θ0+mT+r0-1λ(θ0)+o(r0-1),
where λ(θ) is given by (21).Proof.
Forr(0)=r0≫1, t∈[0,mT], by the boundedness of S,S′ and f, we obtain from (29)
(31)r(t)=r0+O(1),
which implies
(32)r-1(t)=r0-1+O(r0-2).
Substituting (32) into (29) and integrating it over [0,t]⊂[0,mT], we obtain
(33)θ(t)=θ0+t+O(r0-1),x(t)=r01/(p-1)[S(θ0+t)+O(r0-1)],x′(t)=r01/(p-1)[S′(θ0+t)+O(r0-1)].
Substituting (32) and (33) into (29) and integrating it over [0,mT], we obtain
(34)r1=r0+μ(θ0)+o(1),θ1=θ0+mT+λ(θ0)r0-1+o(r0-1),
where
(35)μ(θ)=(p-1)∫0mT‍S′(θ+t)00000000×limr0→+∞f(t,x(t),x′(t))dt=(p-1)[∫Iθ++‍S′(θ+t)f(t,+∞,+∞)dt000000..00.+∫Iθ+-‍S′(θ+t)f(t,+∞,-∞)dt]=(p-1)[∫Iθ-+‍S′(θ+t)f(t,-∞,+∞)dt00000..000+∫Iθ--‍S′(θ+t)f(t,-∞,-∞)dt],Iθ++,Iθ+-,Iθ-+,Iθ--, and λ(θ) are given by (21). Moreover, it is easy to verify from (21) and (35) that for all θ∈ℝ,
(36)μ(θ)=-(p-1)λ′(θ).Lemma 5.
Letf(t,x,x′)=f(t)∈L∞[0,T] and be T-periodic. Assume λ(θ)≡0, where λ(θ) is given in Corollary 2 and P:(r0,θ0)→(r1,θ1)=:(r(mT),θ(mT)) is the Poincaré mapping of the solution of (29). Then for r0≫1, there exists the following higher order approximation:
(37)r1=r0+μ¯(θ0)r0-1+o(r0-1),θ1=θ0+mT+λ¯(θ0)r0-2+o(r0-2),
where μ¯(θ) and λ¯(θ) are given in Theorem 3.Proof.
Sincef(t,x,x′)=f(t) in (29), we integrate the first equation of (29) from t=0 to t∈(0,mT]. For r0≫1, we obtain
(38)r(t)=r0+O(1),r-1(t)=r0-1+O(r0-2).
Substituting (38) into the second equation of (29) and integrating it over [0,t], we get
(39)θ(t)=θ0+t+λ1(θ0,t)r0-1+o(r0-1),
where
(40)λ1(θ,t)=-∫0t‍S(θ+u)f(u)du.
Substituting (38) and (40) into (29) and integrating over [0,t]  we get
(41)r(t)=r0+μ1(θ0,t)+o(1),θ(t)=θ0+t+λ1(θ0,t)r0-1+o(r0-1),
where
(42)μ1(θ,t)=(p-1)∫0t‍S′(θ+u)f(u)du.
By (41) we get the following approximation
(43)r-1(t)=r0-1-μ1(θ0,t)r0-2+o(r0-2),S(θ(t))=S(θ0+t)+S′(θ0+t)λ1(θ0,t)r0-1+o(r0-1).
Substituting (43) into (29) and integrating it over [0,t], we get
(44)r(t)=r0+μ1(θ0,t)+μ2(θ0,t)r0-1+o(r0-1),θ(t)=θ0+t+λ1(θ0,t)r0-1+λ2(θ0,t)r0-2+o(r0-3),
where
(45)μ2(θ,t)=(p-1)∫0t‍λ1(θ,u)S′′(θ+u)f(u)du,λ2(θ,t)=∫0t‍[S′(θ+u)λ1(θ,u)μ1(θ,u)S(θ+u)000000-S′(θ+u)λ1(θ,u)μ1]f(u)du.
Substituting (40) and (42) into (45) and then integrating it over [0,mT], we obtain
(46)r(mT)=r0+μ1(θ0,mT)+μ2(θ0,mT)r0-1+o(r0-1),θ(mT)=θ0+mT+λ1(θ0,mT)r0-1+λ2(θ0,mT)r0-2+o(r0-3),
where
(47)λ1(θ,mT)=λ(θ)≡0,μ1(θ,mT)=-(p-1)λ1′(θ)≡0,(48)μ2(θ,mT)=-(p-1)∫0mT‍S′′(θ+t)f(t)×∫0t‍S(θ+u)f(u)dudt,λ2(θ,mT)=(p-1)∫0mT‍S(θ+t)f(t)×∫0t‍S′(θ+u)f(u)dudt+∫0mT‍S′(θ+t)f(t)×∫0t‍S(θ+u)f(u)dudt=(p-2)∫0mT‍S(θ+t)f(t)×∫0t‍S′(θ+u)f(u)dudt+λ1(θ)λ1′(t).
Let
(49)λ1(θ,mT)=λ(θ)≡0,λ2(θ,mT)=λ¯(θ),μ2(θ,mT)=μ(θ);
then (46) reduces to (37).Lemma 6.
Ifλ(θ)≡0, then
(50)(p-2)μ¯(θ)=(p-1)λ¯′(θ).Proof.
It follows from the second equation of (48) that
(51)λ2′(θ,mT)=p2(λ1′(θ,mT))2+(p-1)λ1(θ,mT)λ1′′(θ,mT)+p-2p-1μ2(θ,mT),
which, together with λ1(θ,mT)≡0,   λ2(θ,mT)=λ¯(θ),  μ2(θ,mT)=μ¯(θ) implies (50).


## 3. Unboundedness Motions of Planar Mappings

In this section, we adopt the notations used in [9]. Given σ>0, let the set Eσ be the exterior of the open ball Bσ centered at the origin with radius σ; that is,
(52)Eσ=ℝ2-Bσ;
then Eσ={(θ,r)∣r≥σ}.DefineS1=ℝ∖ℤτ; then the points and the group distance in S1 can be described, respectively, by
(53)θ-=θ+kτ,k∈ℤ,θ∈ℝ,∥θ-∥=min{|θ+kτ|,k∈ℤ}.
Let P-:Eσ→ℝ2 be a one to one and continuous mapping. We denote its lift by P in the form
(54)r1=r0-cpλ′(θ0)+F(r0,θ0),θ1=θ0+nτ+r0-1λ(θ0)+G(r0,θ0),
where λ∈C1[0,τ] is τ-periodic, n∈ℕ,cp≠0 is a constant, and F,G are continuous, τ-periodic and satisfy F(r,θ)=o(1),G(r,θ)=o(r-1).Given a point(θ0,r0)∈Eσ, let {(θk,rk)}k∈I be the unique solution of the initial value problem for the differential equation:
(55)(rk+1,θk+1)=P(rk,θk).
This solution is defined in a maximal interval
(56)I={k∈ℤ∣ka<k<kb},
where ka,kb are certain numbers in the set ℤ∪{+∞,-∞} satisfying
(57)-∞≤ka<0<kb≤+∞.
The solution {(θk,rk)} is said to be defined in the future if kb=+∞ and is said to be defined in the past if ka=-∞.Let{θi}i=1k be the ordered sequence of zeros of λ(θ) in [0,τ) such that
(58)0≤θ1<θ2<⋯<θk<τ,Ω={θ1,…,θk}≠∅.Lemma 7.
Ifλ has a simple zero θj; that is, λ′(θj)≠0,θj∈Ω, then there exists orbits of (54) which are to be defined in the future and satisfy
(59)rn⟶∞asn⟶+∞,
or they are defined in the past and satisfy
(60)rn⟶∞asn⟶-∞.
Moreover, if Ω≠∅ and all zeros of λ(θ) are simple, then every orbit of (54) with large initial value is either to be defined in the future satisfying
(61)rn⟶∞asn⟶+∞,
or is defined in the past satisfying
(62)rn⟶∞asn⟶-∞.
The proof of the above lemma is similar to that of Proposition  3.1 in [9].Lemma 8.
Assume that forr0≫1,  θ0∈ℝ,  P has the following expression:
(63)r1=r0+μ(θ0)r0-k+F(r0,θ0),θ1=θ0+nτ+λ(θ0)r0-m+G(r0,θ0),
where k,m,n∈ℕ,   μ,λ∈C[0,τ], and F,G are continuous, τ-periodic in θ and satisfy F(r,θ)=o(r-k),   G(r,θ)=o(r-m) uniformly in θ∈S1. Moreover If μ(θ)>0,   for  all  θ∈ℝ, then all orbits of (63) with large initial values are defined in the future and satisfy limn→+∞rn=∞. If μ(θ)<0,   for  all  θ∈ℝ, then all orbits of (63) with large initial values are defined in the past and satisfy limn→-∞rn=∞.Proof.
Letμ(θ)>0,  for  all  θ∈ℝ, then 2C0=maxθ∈ℝμ(θ)≥minθ∈ℝμ(θ)=2c0>0 and it follows from (63) that for r0≫1,
(64)r1>r0+c0r0-k,(65)r1<r0+3C0r0-k<r0+3C0.
By induction, we get from (65)
(66)rn<r0+3nC0,
and replacing r0 by rn,r1 by rn+1 we get from (64)
(67)rn+1>rn+c0rn-k.
Obviously, (66) implies that the orbit rn is defined in the future. Next we claim that the orbit rn is unbounded in the future. In fact, (67) implies that rn is monotone increasing, hence the limit limn→+∞rn=r*≤∞ exists. If r*<∞, taking the limit on both sides of (67) we obtain
(68)r*≥r*+c0r*k>r*,
which is a contradiction. Hence the orbit of (63) is defined in the future and satisfies limn→+∞rn=∞. Similarly, if μ(θ)<0,   for  all  θ∈ℝ, we can prove that the orbit of (63) is defined in the past and satisfies limn→-∞rn=∞.


## 4. Proofs of Theorems and an Example

Proof of Theorem1.
SincemT=nτ, by Lemma 4, for r0≫1,θ0∈ℝ, the mapping P:(r0,θ0)→(r1,θ1):=(r(mT),θ(mT)) has the approximate expression of (30) with mT=nτ, where λ(θ) is given by Theorem 1 and μ(θ)=-(p-1)λ′(θ). Now Lemma 7 implies that all solutions of (30) with r0≫1 go to infinity either in the future or in the past, which implies that the solutions of (19) with large initial values are unbounded.Proof of Theorem3.
By Lemma5, for r0≫1,   θ0∈ℝ, the mapping P′:(r0,θ0)→(r1,θ1)=(r(mT),θ(mT)) has the form (37) with mT=nτ and λ¯ and μ¯ are given by Theorem 3. If p≠2, then it follows from Lemma 6 that
(69)μ¯(θ)=p-1p-2λ¯′(θ).
Now, Lemma 7 implies that all orbits of (37) with r0≫1 go to infinity either in the future or in the past, which means that all solutions of (19) with large initial values are unbounded.
Ifp=2, then by assumption of Theorem 3, λ¯(θ)≡0 and the function μ¯(θ) has no sign-changing. Lemma 8 implies that all orbits of (37) with r0≫1 go to infinity either in the future or in the past, which implies that all solutions of (19) with large initial values are unbounded.Example 1.
For (19), we assume p>1,   τ=T=2πp,   m=n=1,  α≠β, and f(t,x,y)=f(t)=sin(kπt/πp) for some k∈ℕ. We write S(t) in Fourier series form as
(70)S(t)=a02+∑k=1+∞‍[akcoskπtπp+bksinkπtπp],
where
(71)a0=2∫02πp‍S(t)dtπp,ak=1πp∫02πp‍S(t)coskπtπpdt,bk=1πp∫02πp‍S(t)sinkπtπpdt,k∈ℕ.
It follows from Corollary 2 that
(72)λ1(θ)=λ(θ)=∫02πp‍S(θ+t)sinkπtπpdt=πp[aksinkπθπp-bkcoskπθπp],
and it is not difficult to show that ak·bk≠0. Then λ(θ) has only finite number of zeros in [0,2πp] and all of them are simple. Corollary 2 implies that all solutions of (19) with large initial values are unbounded.



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