A min max theorem and its applications to nonconservative systems

PII. S0161171203108010 http://ijmms.hindawi.com © Hindawi Publishing Corp.

A MIN-MAX THEOREM AND ITS APPLICATIONS

TO NONCONSERVATIVE SYSTEMS

LI WEIGUO and LI HONGJIE

Received 2 August 2001 and in revised form 4 February 2002

A nonvariational generation of a min-max principle by A. Lazer is made. And it is used to prove a new existence results for a nonconservative systems of ordinary differential equations with resonance.

2000 Mathematics Subject Classification: 34C25, 49J35.

1. Introduction and lemmas. LetXandY be two subspaces of a real Hilbert spaceHsuch thatH=X⊕Y. Letf:H→Rbe of classC2_{and denote by∇f}

and∇2_{f the gradient and the Hessian off, respectively. In 1975, Lazer et al.}

[3] under the following conditions:

∇2_{f (u)h, h≤ −m1h}2_{, m1>0,∀h∈X,∀u∈H;}

∇2_{f (u)k, k≥m2k}2_{, m2>0,∀k∈Y ,∀u∈H} (1.1)

proved thatfhas a unique critical point, that is, there exists a uniquev0∈H

such that ∇f (v0)= 0. Moreover, this critical point is characterized by the equality

fv0=max

x∈Xminy∈Yf (x+y). (1.2)

In [5], with the following conditions:

∇2_{f (x+y)h, h≤ −βxh}2_{, ∞}

1 β(s)ds= ∞, ∀x, h∈X,∀y∈Y;

∇2_{f (x+y)k, k≥αyk}2_{, ∞}

1 α(s)ds= ∞, ∀x∈X, ∀y, k∈Y ,

(1.3)

whereα(s)andβ(s)are two continuous nonincreasing functions from[0,∞)

to(0,∞), it is proved thatf has a unique critical pointv0such thatf (v0)=

maxx∈Xminy∈Yf (x+y). These results were generalized in [6] and especially for a nonselfadjoint extension of the results of Lazer. This extension was ap-plied in [6] to prove that if the following conditions hold:

whereNis a nonnegative integer andIis ann×nmatrix, then the following differential equations system has a unique 2π-periodic solution:

u(t)+Au(t)+∇G(u)=e(t), (1.5)

whereAis a constant symmetric matrix. System (1.5) is included in the follow-ing nonconservative system (1.6), and assume the followfollow-ing:

u(t)+Au(t)+∇G(u, t)=e(t). (1.6)

With the use of a nonvariational version of a max-min principle inspired by [5,6], inSection 2we generalize these unique existence results of system (1.6) to a more general case. To be more precise, we apply a min-max lemma to the periodic boundary value problem of the nonconservative system (1.6) and assume that the following conditions hold:

B1+αuI≤ ∇2_{G(u, t)≤B2−βuI,} +∞

1

minα(s), β(s)ds= +∞, (1.7)

whereu∈Rn_{,B1andB2are two real symmetric matrices, and the eigenvalues} ofB1andB2areN2

i and(Ni+1)2,i=1, . . . , n, respectively; here,Ni,i=1, . . . , n are nonnegative integers andα(s) and β(s) are two positive nonincreasing functions fors∈[0,∞).

InSection 3, we show with some examples that our main results extend the results known so far.

We firstly employ the following lemma from [9].

Lemma1.1(see [9]). Assume thatH is a Hilbert space. Let T ∈C1_{(H, H),} T(u)∈Isom(H;H), for allu∈H. Then,T is a global diffeomorphism ontoH if there exists a continuous mapω:R+→R+\{0}such that

+∞

1 ds

ω(s)= +∞, T

_(u)−1_{≤ωu. (1.8)}

With this lemma, we can prove the following lemma.

Lemma1.2(see [4]). LetXandY be two closed subspaces of a real Hilbert spaceH, andH=X⊕Y. Suppose thatT:H→His aC1_{-mapping. If there exist} two continuous functionsα:[0,∞)→(0,∞)andβ:[0,∞)→(0,∞)such that

T(u)x, x≤ −αux2_,

T(u)y, y≥βuy2_, (1.9)

for arbitraryu∈H,x∈X,y∈Y, and

+∞

1

minα(s), β(s)ds= +∞, (1.11)

thenT is a diffeomorphism fromHontoH.

The following lemma is required in the proof ofTheorem 2.2.

Lemma1.3(see [2]). LetHbe a vector space such that for subspacesY and Z, H=Z⊕Y. IfZ is finite dimensional andX is a subspace of H such that X∩Y= {0}anddimensionX=dimensionZ, thenH=X⊕Y.

2. Unique existence. Assume thatG(u, t)is continuous for(u, t)∈Rn_×

[0,2π ]and twice continuously differiable aboutu. Denote by ∇G(u, t) and

∇2_{G(u, t)the gradient and the Hessian ofG(u, t), respectively. We will}

inves-tigate the unique existence of periodic solutions for system (1.6). Firstly, we introduce the following definition.

Definition2.1. The real symmetric matrixAis called admissible with two real symmetric matricesB1andB2if there exist orthogonal matricesP1andP2

such thatPT

1B1P1,P2TB2P2, andP1TAP2are simultaneously diagonal matrices.

Theorem2.2. If conditions (1.7) hold for allt∈[0,2π ], allu∈Rn_{, and forA} is admissible with the matricesB1andB2, then there exists a unique2π-periodic solution to system (1.6).

Proof. BecauseAis admissible with the matricesB1andB2, and conditions (1.7) hold for allt∈[0,2π ] and allu∈Rn_{, we can get orthogonal matrices}

P1=(a1, a2, . . . , an),P2=(b1, b2, . . . , bn),P1TB1P1=diag(N12, . . . , Nn2),P2TB2P2=

diag((N1+1)2_{, . . . , (N}

n+1)2), andP1TAP2=diag(γ1, γ2, . . . , γn). Clearly,aiand

biare the eigenvectors ofB1andB2, respectively, corresponding to the eigen-valuesN2

i and(Ni+1)2, which satisfy

aT

iaj=bTibj=δij, i, j=1,2, . . . , n, (2.1)

whereδij=0,i=j;δij=1,i=j. Define

V=v(t)=v1(t), . . . , vn(t)

T

|vi(0)=vi(2π ), i=0,1, . . . , n;

v(t)absolutely continuous andv(t)∈L2_{[0,2π ],} (2.2)

and it is easy to see thatVis a Hilbert space with the following inner product:

u, v = 2π

0

Denote by·V the norm induced by this inner product, and define subspaces ofVas follows:

X=   x(t)=

n

i=1

fi(t)ai|fi(t)=ci0+

N_i

m=1

cimcosmt+dimsinmt

 ;

Y=   y(t)=

n

i=1

gi(t)bi|gi(t)= ∞

m=Ni+1

pimcosmt+qimsinmt

 ;

Z=   z(t)=

n

i=1

hi(t)bi|hi(t)=pi0+ Ni

m=1

pimcosmt+qimsinmt

  ,

(2.4)

whereNi,i=1, . . . , nare as in (1.7) andcim,dim,pim, andqimare constants. Obviously,V=Z⊕Y. Using the Riesz representation theorem, define a map-pingT:V→Vby

T (u), v= 2π

0 u

T_(t)v(t)−vT_(t)Au(t)−vT_{(t)∇Gu(t), tdt (2.5)}

for arbitraryv∈V. We observe thatTis defined implicitly. From (2.5) and the fact thatGisC2_{, it can be proved thatT is aC}1_{-mapping and that}

T(u)w, v= 2π

0

wTv(t)−vT_(t)AwT_(t)−vT_(t)∇2_{G(u, t)w(t)dt (2.6)}

for all v(t), u(t), w(t)∈V. Again, from the Riesz representation theorem, there exists an elementd∈Vsatisfying

d, v = − 2π

0 v

T_{(t)e(t)dt. (2.7)}

It can be proved thatuis a 2π-periodic solution to (1.6) if and only ifusatisfies the operator equation

T (u)=d. (2.8)

We will next show thatT satisfies the conditions ofLemma 1.2. This will, in turn, imply that (1.6) has a unique 2π-periodic solution. For any x∈X and

u∈V, we have that

T(u)x, x= 2π

0

xT_(t)x(t)−xT_(t)Ax(t)−xT_(t)∇2_{G(u, t)x(t)dt,}

(2.9) where

2π

0

xT(t)x(t)dt= 2π

0

n

i=1

f_i2(t)dt≤

n

i=1 N_i2

2π

0

f_i2(t)dt;

2π

0

xT_(t)Ax(t)dt=1 2x

T_(t)Ax(t)|2π

0 =0.

By (1.7), we have

2π

0

xT_(t)∇2_{G(u, t)x(t)dt}

≥ 2π

0 x

T_{(t)B1x(t)dt+αu} V 2π 0 x T_(t)x(t)dt = 2π 0 n i=1 n j=1

fi(t)fj(t)aTiB1ajdt+αuV

2π

0

xT_(t)x(t)dt

=

n

i=1 N2

i

2π

0 f2

i(t)dt+α

uV

2π

0

xT_(t)x(t)dt;

x2

V=

2π

0

xT_(t)x(t)dt+

2π

0

xT(t)x(t)dt

≤M2+1

2π

0 x

T_(t)x(t)dt,

(2.11)

whereM=max1≤i≤n{Ni}, therefore

T(u)x, x≤ −α

uV

M2₊₁ x 2

V. (2.12)

Similarly, from

2π

0 y

T_(t)y(t)dt≥ n

i=1

Ni+12

2π 0 g 2 i(t)dt, − 2π 0 y

T_(t)∇2_{G(u, t)y(t)dt≥ −} 2π

0 y

T_(t)B2y(t)dt

+βuV

2π

0

yT_(t)y(t)dt,

(2.13)

we can get that for ally∈Y and allu∈V,

2π

0

1+(M+1)2 _yT_(t)y(t)−yT_(t)∇2_{G(u, t)y(t)}

−βuV yTy(t)+yT(t)y(t)

dt

= 1+(M+1)2−βuV

2π

0 y

T_(t)y(t)dt

− 2π

0

1+(M+1)2_yT_(t)∇2_{G(u, t)y(t)dt−βu}

V

2π

0

yT_(t)y(t)dt

≥ 1+(M+1)2−βuV

n

i=1

Ni+1

22π

0

g2_i(t)dt

− 2π

0

1+(M+1)2_yT_{(t)B2y(t)dt+(M+1)}2_βu

V

2π

0

= 1+(M+1)2−βuV

n

i=1

Ni+1

22π

0

g2_i(t)dt

− 1+(M+1)2

n

i=1

Ni+1

22π

0 g 2

i(t)dt+(M+1)

2_βu

V

n

i=1 2π

0 g 2

i(t)dt

=βuV

n

i=1

(M+1)2−Ni+1

22π

0 g 2

i(t)dt≥0,

(2.14)

and from

2π

0

yT_(t)Ay(t)dt=1 2y

T_(t)Ay(t)|2π

0 =0, (2.15)

we can prove that for ally∈Y and allu∈V,

T(u)y, y= 2π

0

yT(t)y(t)−yT_(t)Ay(t)−yT_(t)∇2_{G(u, t)y(t)dt}

≥ β

uV

(M+1)2₊₁y 2

V.

(2.16)

Obviously, for allx∈Xand ally∈Y, we have the following:

T(u)x, y−x, T(u)y

= 2π

0

xT_(t)Ay(t)−yT_(t)Ax(t)dt

= 2π

0 2

f1(t), . . . , fn(t)

P1TAP2

g1(t), . . . , gn(t)

T

dt

= 2π

0

n

i=1

2γifi(t)gi(t)dt=0.

(2.17)

Letα1=α(s)/(M2_{+1)andβ1(s)=β(s)/((M+1)}2_{+1), then}

c(s)=minα1(s), β1(s)≥minα(s), β(s)/(M+1)2_{+1. (2.18)}

Based on conditions (1.7),₁+∞c(s)ds= +∞. SinceT(u)is positive definite on

Y and negative definite onX, we see thatX∩Y = {0}. Moreover, it is readily seen that

dimensionX=dimensionZ=

n

i=1

2Ni+1

. (2.19)

If we setV= {v(t)=(v1(t), . . . , vn(t))T|vi(0)=vi(π )=0, i=1, . . . , n;v(t) to be absolutely continuous andv(t)∈L2_{[0, π ]}, it is easy to know thatV is}

a Hilbert space about the following inner product:

u, v = π

0 u

T_(t)v(t)+uT_{(t)v(t)dt. (2.20)}

Again, define the norm induced by this inner product and subspacesX,Y, and

Z, correspondingly; we can prove the following theorem similarly.

Theorem2.3. Assume thatG(u, t)is continuous andC2_{-mapping with} re-spect touand that conditions (1.7) hold for allt∈[0, π ], allu∈Rn_{, and forA} is admissible with matricesB1andB2. Lete(t)be a continuous function. Then, there exists a unique solution to (1.6), which satisfies boundary value condition u(0)=u(π )=0.

Especially, when B1 =N2_{I and B2=(N+1)}2_{I, where N is natrual and I}

is n×nidentity matrix, Ais admissible withB1 andB2as long asAis real symmetric. So, we have the following corollary.

Corollary2.4. Assume thatAis real symmetric and there exist two positive continuous functionsδ1andδ2:Rn_{→Rsuch that for allu∈R}n _{and allt∈}

[0,2π ],

N2I < δ1(u)I≤ ∇2_{G(u, t)≤δ2(u)I < (N+1)}2_{. (2.21)}

Letρ(r )=min{1−maxn

i=1|ξi|≤r(δ2(ξ)/(N+1)

2_),maxn

i=1|ξi|≤r(δ1(ξ)/N 2₎₋

1}, and if₁+∞ρ(r )dr = +∞, then the system (1.6) has a unique2π-periodic solution.

If we setA=0, system (1.6) becomes a conservative system and admissibility is trivial. So, the main conclusion in [7] (the method there is different from ours) is a corollary ofTheorem 2.2.

Corollary2.5. Assume that there exist integersNi≥0such that for all

u∈Rn_{and allt∈[0,2π ],}

N_i2< λi(u, t) <

Ni+1

2

, i=1, . . . , n;

δu, t= max v≤u

min

1≤i≤n

λi(u, t)−Ni2,

Ni+1

2

−λi(u, t)

, (2.22)

whereλi(u, t),i=1,2, . . . , ndenote the eigenvalues of∇2G(u). If

+∞

1 δ(s, t)ds = +∞for allt∈[0,2π ], then there exists a2π-periodic solution to (1.5).

LetG(u, t)=G(u)and

c(s)=minα(s), β(s)≥c0>0; (2.23)

Corollary2.6. If the real symmetric matrixAis admissible with real sym-metric matricesB1andB2, then assume that

B1≤ ∇2G(u)≤B2, Ni2< λi≤µi<Ni+12, (2.24)

whereλiandµiare eigenvalues of B1andB2, respectively, andNi,i=1, . . . , n are nonnegative integers; there exists a unique2π-periodic solution to system (1.5).

3. Examples. It should be pointed out that conditions (1.9) and (1.11) are not completely the same as (1.3). In fact, from (1.3), we know that α(x)

depends on subspaceXandβ(y)depends upon subspaceY. So, condition (1.3) is more strict than conditions (1.9). But, from (1.11), we can deduce the following conditions:

+∞

1

α(s)ds= +∞,

+∞

1

β(s)ds= +∞. (3.1)

Conversely, note that (3.1) does not imply (1.11). Now, we give an example to illustrate it.

Example3.1. First of all, we define two nondecreasing functions as follows:

α(x)=2−(2i+1)2_{, x2}

i−1≤x < x2i+1;

β(x)=2−(2i+2)2_{, x2}

i≤x < x2i+2,

(3.2)

wherexi=ik=02k 2

,i=0,1, . . .,x−1=0, andβ(x)=1, when 0≤x <1.

It is easy to see thatα(x)andβ(s)are two nondecreasing positive functions for all x ∈[0,+∞); and the number of noncontinuous points is countable infinite. We also have

+∞

1

α(x)dx=1+

+∞

i=1

x2i+1−x2i−1

2(2i+1)2

=1+

+∞

i=1

2(2i+1)2₊₂(2i)2 2(2i+1)2 = +∞, +∞

1 β(x)dx=

+∞

i=1

x2i−x2i−2

2(2i)2 =

+∞

i=1

2(2i)2₊₂(2i−1)2 2(2i)2 = +∞, _+∞

1 min

α(x), β(x)dx≤

+∞

i=0

x2i+1−x2i 2(2i+2)2 +

+∞

i=0

x2i−x2i−1

2(2i+1)2

≤

+∞

i=0

2(2i+1)2

2(2i+2)2+

+∞

i=0

2(2i)2

2(2i+1)2 <+∞.

(3.3)

We can state, from the following example, thatTheorem 2.3is more general than the results of [1,2,3,4,5,6,7,8].

Example3.2. Assume thatf (t)is continuous and 2π-periodic in (1.6). Let

G(u, t)=5

4

1+4

5sin

2_tu2 1+u22

+3

2

1+2

3sin

2_tu1u2

+u1ln

u1+

1+u21

+u2ln

u2+1+u22

−1+u2 1−

1+u2

2+C1u1+C2u2,

(3.4)

then

∇2_{G(u, t)=}        5 2

1+4₅sin2t

+ 1

1+u2 1

3 2

1+2₃sin2(t)

3 2

1+2

3sin

2_(t)

₅

2

1+4

5sin

2_t

+ 1

1+u2 2       ,     5 2 3 2 3 2 5 2    +       1

1+u21

0

0 1 1+u22

     ≤ ∇

2_{G(u, t)}

≤     13 2 5 2 5 2 13 2    −      

1− 1

1+u2 1

0

0 1− 1

1+u2 2      . (3.5)

It is easy to see thatG(u, t)satisfies (1.7). Therefore, there exists a unique 2π-periodic solution to (1.6) byTheorem 2.2, but we cannot make this conclu-sion from [1,2,3,4,5,6,7,8].

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[4] W. Li,Periodic solutions for2kth order ordinary differential equations with reso-nance, J. Math. Anal. Appl.259(2001), no. 1, 157–167.

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Li Weiguo: Department of Applied Mathematics, University of Petroleum, Dongying 257061, Shandong Province, China

E-mail address:[email protected]

Li Hongjie: Department of Mathematics, Linyi Teacher’s College, Linyi 276005, Shan-dong Province, China.