Multiple periodic solutions for second order Josephson-type differential systems

R E S E A R C H

Open Access

Multiple periodic solutions for second

order Josephson-type differential systems

Shengui Zhang

*_{Correspondence:}

[email protected] College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, Gansu 730030, P.R. China

Abstract

In this paper, some new existence theorems are obtained for multiple periodic solutions of second order Josephson-type differential systems with partially periodic potential by using the minimax methods in critical point theory, which generalize and improve some known results in the literature.

MSC: 34C25; 58E50

Keywords: periodic solution; critical point; second order Josephson-type differential systems; periodicity; minimax methods; the generalized saddle point theorem

1 Introduction

In this paper, we study the second order Josephson-type differential systems ⎧

⎨ ⎩ ¨

u(t) +Au(t) –∇F(t,u(t)) =h(t), a.e.t∈[,T],

u() –u(T) =u() –˙ u(T˙ ) = ,

(.)

whereAis an (N×N)-symmetric matrix,h(t)∈L_([,T];RN_{),T> ,∇F(t,x) denotes its} gradient with respect to the second variable, that is,

∇F(t,x) =∇xF(t,x) = (∂F/∂x, . . . ,∂F/∂xN), x(t) =

x(t), . . . ,xN(t)

,

andF: [,T]×RN_{→Rsatisfies the following assumptions:}

(H) F(t,x)is measurable intfor eachx∈RN _{and continuously differentiable inxfor a.e.} t∈[,T], and there exista∈C(R+,R+),b∈L([,T];R+)such that

F(t,x)≤a|x|b(t), ∇F(t,x)≤a|x|b(t)

for allx∈RN and a.e.t∈[,T].

(H) dimN(A) =m≥and matrixAhas no eigenvalue of the formk_υ_{(k∈N/{}), where}

υ= π/T.

(H) There exist linearly independent vectorsej∈RN (≤j≤m) such that

N(A) =span{e,e, . . . ,em}

and T



h(t),ej

dt= .

This problem (.) occurs in various branches of mathematical physics, for example, whenA=N_D_{and –∇F(t,u(t)) =f(u(t)) = (a}

sinu, . . . ,aNsinuN), problem (.) reduces to the nonlinear systems of the form

⎧ ⎨ ⎩ ¨

u(t) +NDu(t) +f(u(t)) =h(t), a.e.t∈[,T],

u() –u(T) =u() –˙ u(T˙ ) = , (.)

whereDis an (N×N)-symmetric matrix. This type of problem can be applied to describe the motion of forced linearly coupled pendulums.

During the past two decades, the existence of periodic solutions for second order differ-ential systems have been studied extensively, and many solvability conditions have been obtained via variational methods and critical point theory. In this direction we mention the papers [–], and we refer the reader to [–] for a broad introduction to varia-tional methods and critical point theory. It might be also interesting to study the above mentioned abstract equations with more general potentials, see the paper [].

In the classical monograph [], Mawhin and Willem proved that problem (.) has at least one solution by using the saddle point theorem under the following bounded condi-tion: there existsg∈L_([,T];R+_{) such that}

F(t,x)+∇F(t,x)≤g(t) (.)

for allx∈RN _{and a.e.t∈[,T]. They obtained the following result.}

Theorem A([]) Suppose that F satisfies(H)-(H), (.)and

(H) there existsTj> such that

F(t,x+Tjej) =F(t,x), ≤j≤m,

for allx∈RN _{and a.e.t∈[,T].Then Eq. (.)has at least one solution inH}

T,where the Sobolev spaceH

Tis defined by

H_T = u: [,T]→RN |u is absolutely continuous, u() =u(T)andu˙∈L[,T];RN

andH_T is a Hilbert space with the norm

u=

T



u(t)_˙ dt+ T



u(t)dt 



, u∈H_T.

When the nonlinearity∇F(t,x) is sublinear, that is, there existf,g∈L([,T];R+) and

α∈[, ) such that

for allx∈RN_{and a.e.t∈[,T], Tang [, ] researched the existence of periodic solutions} for system (.) in the caseA≡.

Subsequently, Feng and Han [] generalized Theorem A to the sublinear case, and they assume that the following assumptions hold:

(H) There existTj> ,≤r≤msuch that

F(t,x+Tjej) =F(t,x), ≤j≤r,

for allx∈RN and a.e.t∈[,T]. (H) limx→∞

T

 F(t,x)dt

xα = –∞,asx∈N(A)span{e,e, . . . ,er}.

Theorem B([]) Suppose that F satisfies(H)-(H), (H)-(H)and(.).Then Eq. (.) has at least r+ distinct solutions in H

T.

In [], the author obtained the following result.

Theorem C([]) Suppose that F satisfies(H)-(H), (H), (.)and the following gener-alized Ahmad-Lazer-Paul type coercive conditions:

(H) limx→∞ T

F(t,x)dt

xα < –L,asx∈N(A)span{e,e, . . . ,er},whereLis a positive con-stant.Then Eq. (.)has at leastr+ distinct solutions inH

T.

In this paper, we use a more general control function instead of|x|α_{in (.). By using the}

generalized saddle point theorem due to Liu [], we can prove the existence of multiple periodic solutions for the second order Josephson-type differential systems for a new and large range of the nonlinear term.

2 Preliminaries

In [], Mawhin and Willem established a variational structure which enables us to reduce the existence of solutions for problem (.) to the existence of critical points of the fol-lowing energy functional. Define the energy functional associated with problem (.) on H_T

ϕ(u) =  

T



u(t)_˙ dt– 

T



Au(t),u(t)dt+ T



Ft,u(t)dt

+ T



h(t),u(t)dt.

It follows from assumption (H) that the functionalϕis continuously differentiable. More-over, one has

ϕ(u),v= T



˙

u(t),v(t)˙ dt– T



Au(t),v(t)dt+ T



∇Ft,u(t),v(t)dt

+ T



h(t),v(t)dt, ∀u,v∈H_T.

Let

q(u) = T



_u(t)˙ 

–Au(t),u(t)dt.

Therefore, we can see that

q(u) =u– T



(A+I)u(t),u(t)dt

=(I–K)u,u,

whereIdenotes the identity operator onH

TandK:HT →HT is the linear self-adjoint operator defined, using Riesz representation theorem, by

T



(A+I)u(t),v(t)dt= (Ku,v), ∀u,v∈H_T.

It is easy to see thatKis compact. By classical spectral theory, we can decomposeH_T into the orthogonal sum of invariant subspaces forI–K

H_T =H–⊕H⊕H+,

whereH_{= Ker(I–K) =N(A) anddimH}–_{< +∞, for someδ> , we have}

q(u)≤–δu, ∀u∈H–, (.)

q(u)≥δu, ∀u∈H+. (.)

Lemma .([]) There is a continuous embedding H

T→C([,T],RN),and the embed-ding is compact.Then there exists C> such that

u∞:= max

≤t≤T

u(t)≤Cu, ∀u∈HT. (.)

Define

Y=span{e,e, . . . ,er}, Y=N(A)Y=span{er+,er+, . . . ,em}, then

u(t) =u–(t) +u+(t) +Pu+Qu, whereu–_∈H–_,u+_∈H+_,Pu_∈Y

andQu=

r

j=cjej. Let

G= _r

i=

kiTieiki∈Z, ≤i≤r

be a discrete subgroup ofH_T, whereZis the set of all integers, and letπ:H_T →H_T/Gbe the canonical surjection. Let

whereW=H+_,Z=H–_⊕Y

,V=Y/G, thendimZ< +∞,dimV< +∞, andV is

isomor-phic to the torusTr. The element inVcan be represented as

Quˆ= r

j=

ˆ cjej,

wherecˆj=cj–kjTj, ≤ ˆcj<Tj. Let

ˆ

u(t) =u–(t) +u+(t) +Pu+Quˆ.

By (H) and (H), we have

Ft,u(t)=F

t,u(t) +ˆ r

i=

kiTiei

=Ft,u(t)ˆ ,

∇Ft,u(t)=∇F

t,u(t) +ˆ r

i=

kiTiei

=∇Ft,u(t)ˆ ,

and

T



h(t),u(t)dt= T



h(t),u(t) +ˆ r

i=

kiTiei

dt= T



h(t),u(t)ˆ dt.

Thus,ϕ(u) =ϕ(ˆu),ϕ(u) =ϕ(u). Defineˆ ψ :X×V→R:ψ(π(u)) =ϕ(u), thenψ is well defined. Moreover,ψis continuously differentiable and

ψπ(u)=ψπ(u)ˆ , ψπ(u)=ψπ(ˆu).

Definition .([]) Suppose thatψ satisfies the (PS) condition, that is, every sequence {xn}ofX×Vsuch thatψ{xn}is bounded andψ{xn} → asn→ ∞possesses a conver-gent subsequence.

Lemma .(The generalized saddle point theorem []) Let X be a Banach space with a

decomposition X=Z+W,where Z and W are two subspaces of X withdimZ< +∞.Let V be a finite-dimensional,compact C_{-manifold without boundary.Letψ:X×V→R}

be a C_{-function satisfying the(PS)condition.Suppose that there exist constantsρ> and}

γ <βsuch that

(a) inf

x∈W×Vψ(x)≥β,

(b) sup

x∈S×V

ψ(x)≤γ,

3 Main results

Here are our main results.

Theorem . Suppose that assumptions(H)-(H), (H)hold and there exist constants

Mi> ,i= , , ,and a nonnegative functionω∈C([,∞), [,∞))with the properties:

(ω) ω(s)≤ω(t),∀s≤t,s,t∈[,∞);

(ω) ω(s+t)≤M(ω(s) +ω(t)),∀s,t∈[,∞);

(ω) ≤ω(s)≤Ms+M,∀s,t∈[,∞);

(ω) ω(s)→+∞ass→+∞.

Moreover,there exist constant a> and f,g∈L_([,T];R+_)with

T



f(t)dt< δ ( +a)MMC

(.)

such that

_∇F(t,x)≤f(t)ω|x|+g(t) (.)

for all x∈RN _{and a.e.t∈[,T],and}

lim sup

|x|→∞ T

 F(t,x)dt

ω_(|x|) < –



  a– +

 

M

M

T



f(t)dt (.)

as x∈N(A)span{e,e, . . . ,er}.Then Eq. (.)has at least r+ distinct solutions in HT.

Theorem . Suppose that assumptions(H)-(H), (H), (.), (.)hold and

lim inf

|x|→∞

T

 F(t,x)dt

ω_(|x|) >



  a– +

 

M

M

T



f(t)dt

as x∈N(A)span{e,e, . . . ,er}.Then Eq. (.)has at least r+ distinct solutions in HT. By Theorems . and ., it is easy to obtain the following corollary.

Corollary . Suppose that assumptions(H)-(H), (H), (.), (.)hold and

lim

|x|→∞ T

 F(t,x)dt

ω_(|x|) = +∞ (or–∞)

as x∈N(A)span{e,e, . . . ,er}.Then Eq. (.)has at least r+ distinct solutions in HT.

Remark . (i) WhenA≡, assumptions (ω)-(ω) and condition (.) were introduced

(ii) To show that our Theorem . is new, we give an example to illustrate our result. For example, let ≤r≤m,x= (x,x, . . . ,xN)T∈RN, and

F(t,x) =

 T–t

ln

 +

r+  +

r

j=

sinxj+  

N

j=r+

x_j

+T–tln

 +

r+  +

r

j=

sinxj+  

N

j=r+

x_j

,

letω(|x|) =ln[ + (r+  +|x|_{)]. ThenFsatisfies all the conditions of Theorem ., but not}

covered by the results of [–].

For example, letx= (x,x, . . . ,xN)T∈RN,ω(|x|) =|x|and

F(t,x) =

 T–t

r+  +

r

j=

sinxj+  

N

j=r+

x_j

+T–t

r+  + r

j=

sinxj+  

N

j=r+

x_j 

 ,

where ≤r≤m. ThenF satisfies all the conditions of Theorem ., but not covered by the results of [–].

For the sake of convenience, we denote byCi (i= , , , . . . , ) various positive con-stants.

Proof of Theorem. First, we prove thatψ satisfies the (PS) condition. Letπ:W_T,p(t)→ W_T,p(t)/Gbe the canonical surjection. Defineψ:X×V→Rbyψ(π(u)) =ϕ(u). Assume that (π(un)) is a (PS) sequence forψ, that is,ψ(π(un)) is bounded andψ(π(un))→. Thenϕ(un) is bounded andϕ(un)→.

We can get from (ω), (ω), and (ω) that

ωu(t)ˆ =ωu+_{(t) +u}–_{(t) +Quˆ}_+Pu

≤ωu++u–+Quˆ+Pu ≤M

ωu++u–+Qˆu+ωPu ≤M

Mu++u–+Qˆu+M

+MωPu

≤MMu+_∞+u–_∞

+MMQˆu+MM

+MωPu. (.)

By (.) and the boundedness of|Qˆu_{|, we have}

_T∇Ft,u(t)ˆ ,u+(t)dt

≤ T



f(t)ωu(t)ˆ u+(t)dt+ T



≤MM

T



f(t)dtu+_∞+u–_∞u+_∞

+M

T



f(t)dtωPuu+_∞+Cu+_∞

≤MMC

T



f(t)dtu++u–u+

+MC

T



f(t)dtωPuu++Cu+

≤MMC

T



f(t)dtu++u

–_+u+



+MC

T



f(t)dt

ω(|Pu_|)

MC +MCu

+

 +Cu

+

=MMC

T



f(t)dt

u++ u

–

+ 

M

M

T



f(t)dtωPu+Cu+. (.)

From (H) and (.), we obtain that

_Th(t),u+(t)dt≤u+_∞ T



h(t)dt

≤Cu+

T



h(t)dt. (.)

It follows from (.), (.), and (.) that

u+_n≥ϕ(un),u+n

=ϕ(ˆun),u+n

= T



u_˙+_n(t)dt– T



Aun(t),u+n(t)

dt

+ T



∇Ft,uˆn(t)

,u+_n(t)dt+ T



h(t),u+_n(t)dt

≥

δ– MMC

T



f(t)dtu+_n

– 

MMC

 

T



f(t)dtu–_n

–  

M

M

T



f(t)dtωPu_n+Cu+n (.)

for largen. So we have



MMC

 

T



f(t)dtu–_n+ω

_(|Pu

n|) M

C

≥

δ– MMC

T



=

δ– ( +a)MMC

T



f(t)dtu+_n

+aMMC

T



f(t)dtu+_n–Cu+n

≥aMMC

T



f(t)dtu+_n+C,

whereC=mins∈[,+∞){[δ– ( +a)MMC

T

 f(t)dt]s–Cs}.

From (.), one has that ( +a)MMC

T

 f(t)dt<δ, thenC< . Hence

u+_n≤  a

u–_n+ω

_(|Pu

n|) M

C

+C.

In a similar way, we have

_u– n  ≤  a

u+_n+ω

_(|Pu

n|) M

C

+C.

Combining the above two inequalities, one has that

u+_n≤  a

u–_n+ω

_(|Pu

n|) M

C

+C

≤  au

+ n  +  a  a+ 

ω_(|Pu

n|) M_C_ +

C

a+C. Consequently,

u+_n≤  a– 

ω_(|Pu

n|) M

C

+C. (.)

Using similar arguments, we can prove that

_u–

n



≤ 

a– 

ω(|Pu

n|) M

C

+C. (.)

By (.), (.), (.), (.), and (.), we have

T



u_˙+_n(t)dt– T



Aun(t),u+n(t)

dt

≤u+_n– T



∇Ft,uˆn(t)

,u+_n(t)dt

– T



h(t),u+_n(t)dt

≤MMC

T



f(t)dt

u+_n



+  u–_n

 +  M M T 

≤

 

 a– +

 

M

M

T



f(t)dtωPu_n

+CωPun+C. (.)

In a similar way, we can obtain T



u_˙–_n(t)dt– T



Aun(t),u–n(t)

dt

≤

 

 a– +

 

M

M

T



f(t)dtωPu_n

+CωPun+C. (.)

By (.) and (ω), one has

T



Ft,uˆn(t)

dt– T



Ft,Pu_ndt

= T



 

∇Ft,Pu_n+sQuˆ_n+u+_n+u–_n,Quˆ_n+u+_n+u–_nds dt

≤ T



 

f(t)ωPu_n+Quˆ_n+u+_n+u–_nQuˆ_n+u+_n+u–_nds dt

+ T



 

g(t)Qˆu_n+u+_n+u–_nds dt.

From (.), (.), and the boundedness of|Qˆu_{|, we have}

_TFt,uˆn(t)

dt– T



Ft,Pu_ndt ≤

T



f(t)dtMMQuˆn+u+n∞+u–n∞



+ T



f(t)dtMMQˆun+u+n∞+u–n∞

+ T



f(t)dtMωPunQˆun+u+n∞+u–n∞

+ T



g(t)dtQuˆ_n+u+_n∞+u–_n∞ ≤MMC

T



f(t)dtu+_n+u–_n

+MC

T



f(t)dtωPu_nu+_n+u–_n +Cu+n+u–n+CωPun+C.

Hence, we have

T



Ft,uˆn(t)

dt– T



Ft,Pu_ndt ≤MMC

T



+MC

T



f(t)dt



MCω

_(|Pu

n|) +MCu+n 

+MC

T



f(t)dt



MCω

_(|Pu

n|) +MCu–n 

+Cu+n+u–n+CωPun+C

≤

MMC

 

T



f(t)dtu+_n+u–_n

+M M

T



f(t)dtωPu_n

+Cu+n+u–n+CωPun+C. (.)

By (.), (.), and (.), we have

T



Ft,uˆn(t)

dt– T



Ft,Pu_ndt

≤

 a– + 

M

M

T



f(t)dtωPu_n

+CωPun+C. (.)

From (H), (.), (.), and (.), one has

T



h(t),uˆn(t)

dt≤ T



h(t)dtu+_n∞+u–_n∞

≤C

T



h(t)dtu+_n+u–_n

≤CωPun+C. (.)

It follows from (.), (.), (.), and (.) that

ϕ(un) =ϕ(uˆn)

=  

T



u_˙+_n(t)dt– T



Aun(t),u+n(t) dt +  T 

u_˙–_n(t)dt– T



Aun(t),u–n(t) dt + T 

Ft,uˆn(t)

dt– T



Ft,Pu_ndt

+ T



Ft,Pu_ndt+ T



h(t),uˆn(t) dt ≤    a– +

  M M T 

f(t)dt+ T

 F(t,Pu 

n)dt

ω(|Pu

n|)

which implies|Pu

n|is bounded. Otherwise, we assume|Pun| → ∞asn→ ∞. From (ω), we obtain that

ωPu_n→+∞ asn→ ∞.

By (.), we conclude that

ϕ(un)→–∞ asn→ ∞,

this contradicts the boundedness of{ϕ(un)}, so |Pun|is bounded. Combining (.) and (.), we obtain thatu+_nandu–_nare bounded. Furthermore,|Qˆu|is bounded, so{ˆun} is bounded inH

T. Arguing then as in Proposition . in [],{ˆun}has a convergent subse-quence. Byπ(uˆn) =π(un), we conclude thatψ satisfies the (PS) condition.

Next, we only need to verify the linking conditions of the generalized saddle point the-orem:

(a) Forπ(u)∈W×V,u(t) =u+_{(t) +Qu}_{. By the proof of (.), we have}

T



Ft,u+(t) +Qudt– T



F(t, )dt

≤

MMC

 

T



f(t)dtu++Cu++C.

Hence

ψπ(u)=ψπu++Qu =ϕu++Qu =

 T



_u˙+_(t)_dt+

 T



Au(t),u+(t)dt

+ T



Ft,u+(t) +Qudt– T



F(t, )dt

+ T



h(t),u+(t) +Qudt+ T



F(t, )dt

≥

δ

– 

MMC

 

T



f(t)dtu+–Cu+–C.

Note the boundedness of|Qu_{|and (.), we obtain thatψ(π(u))→+∞asu →–∞for}

allπ(u)∈W×V, which implies that there existsβ∈Rsuch thatψ(π(u))≥βonW×V. (b) In a way similar to the proof of (.), we have

T



Ft,u–+Pu+Qudt– T



Ft,Pudt

≤

MMC

 

T



f(t)dtu–+M M

T



Consequently,

ψπ(u)=ψπu–+Pu+Qu =ϕu–+Pu+Qu =

 T



u_˙–dt+ 

T



Au,u–dt+ T



Ft,Pudt

+ T



Ft,u–+Pu+Qudt– T



Ft,Pudt

+ T



h(t),u–+Pu+Qudt

≤ 

–δ+ MMC

T



f(t)dtu–+Cu–

+

M

M

T



f(t)dt+ T

 F(t,Pu)dt

ω_(|Pu_|)

ωPu +CωPu+C.

Froma>  and (.), one has that –δ+ MMC

T

 f(t)dt< .

By (.), we deduce that

lim sup

|Pu_|→∞ T

 F(t,Pu)dt

ω(|Pu|) < –

M

M

T



f(t)dt,

so we obtain thatψ(π(u))→+∞asu →–∞for allπ(u)∈Z×V, which implies that there existsγ <βsuch thatψ(π(u))≤γ onZ×V.

The functional ψ satisfies all the assumptions of Lemma ., so it has at least

cuplength(V) +  critical points, and sinceVis the torusTr, thencuplength(V) =r. Henceϕ

has at leastr+  critical points. Therefore, problem (.) has at leastr+  distinct solutions

inH_T. The proof of Theorem . is completed.

Proof of Theorem. The proof of Theorem . is similar to the proof of Theorem ., so

we omit the discussions here.

Remark . Using the parallel arguments with little change as in the proofs of Theorems . and ., the conclusions of Theorems . and . hold if we replace (ω) with

(ω) ≤ω(s)≤Msα+M,∀s,t∈[,∞), where≤α< .

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 31260098).

Competing interests

The author declares that they have no competing interests.

Author’s contributions

The author read and approved the final manuscript.

Publisher’s Note

Received: 26 September 2016 Accepted: 3 May 2017 References

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