The Solution of Two-Point Boundary Value Problem of a Class of Duffing-Type Systems with Non- Perturbation Term

Boundary Value Problems

Volume 2009, Article ID 287834,12pages doi:10.1155/2009/287834

Research Article

The Solution of Two-Point Boundary Value

Problem of a Class of Duffing-Type Systems with

Non-

C

1

Perturbation Term

Jiang Zhengxian and Huang Wenhua

School of Sciences, Jiangnan University, 1800 Lihu Dadao, Wuxi Jiangsu 214122, China

Correspondence should be addressed to Huang Wenhua,[email protected]

Received 14 June 2009; Accepted 10 August 2009

Recommended by Veli Shakhmurov

This paper deals with a two-point boundary value problem of a class of Duffing-type systems with non-C1_{perturbation term. Several existence and uniqueness theorems were presented.}

Copyrightq2009 J. Zhengxian and H. Wenhua. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Minimax theorems are one of powerful tools for investigation on the solution of differential equations and differential systems. The investigation on the solution of differential equations and differential systems with non-C1 perturbation term using minimax theorems came into being in the paper of Stepan A.Tersian in 19861. Tersian proved that the equationLut ft, ut L −d2/dt2_{exists exactly one generalized solution under the operatorsB}

jj

1,2 related to the perturbation term ft, ut being selfadjoint and commuting with the operatorL−d2/dt2_{and some other conditions in1. Huang Wenhua extended Tersian’s}

theorems in1in 2005 and 2006, respectively, and studied the existence and uniqueness of solutions of some differential equations and differential systems with non-C1 _perturbation

term 2–4, the conditions attached to the non-C1 _{perturbation term are that the operator}

Bu related to the term is self-adjoint and commutes with the operatorA where A is a selfadjoint operator in the equationAuft, u. Recently, by further research, we observe that the conditions imposed uponBucan be weakened, the self-adjointness ofBucan be removed andBuis not necessarily commuting with the operatorA.

In this note, we consider a two-point boundary value problem of a class of Duffi ng-type systems with non-C1_{perturbation term and present a result as the operatorBurelated}

to the perturbation term is not necessarily a selfadjoint and commuting with the operator

2. Preliminaries

LetHbe a real Hilbert space with inner product·,·and norm · , respectively, letXand

Ybe two orthogonal closed subspaces ofHsuch thatHX⊕Y. LetP:H → X, Q:H → Y

denote the projections fromH toX and fromH toY, respectively. The following theorem will be employed to prove our main theorem.

Theorem 2.12. LetHbe a real Hilbert space,f : H → Ran everywhere defined functional with Gˆateaux derivative∇f : H → Heverywhere defined and hemicontinuous. Suppose that there exist two closed subspacesX andY such thatHX⊕Yand two nonincreasing functionsα:

0, ∞ → 0, ∞, β:0, ∞ → 0, ∞satisfying

s·αs−→ ∞, s·βs−→ ∞, ass−→ ∞ 2.1

and

∇fh1 y

− ∇fh2 y

, h1−h2

≤ −αh1−h2h1−h22, 2.2

for allh1, h2∈X, y∈Y, and

∇fx k1− ∇fx k2, k1−k2

≥βk1−k2k1−k22, 2.3

for allx∈X, k1, k2∈Y. Then

afhas a unique critical pointv0 ∈Hsuch that∇fv0 0;

bfv0 maxx∈X miny∈Yfx y miny∈Y maxx∈Xfx y.

We also need the following lemma in the present work. To the best of our knowledge, the lemma seems to be new.

Lemma 2.2. LetAandBbe two diagonalizationn×nmatrices, letμ1 ≤ μ2 ≤ · · · ≤ μn andλ1 ≤

λ2 ≤ · · · ≤λnbe the eigenvalues ofAandB, respectively, where each eigenvalue is repeated according

to its multiplicity. IfAcommutes withB, that is,ABBA, thenA Bis a diagonalization matrix andμ1 λ1≤μ2 λ2 ≤ · · · ≤μn λnare the eigenvalues ofA B.

Proof. SinceAis a diagonalizationn×nmatrix, there exists an inverse matrixP such that P−1_{AP diagμ}

1E1, μ2E2, . . . , μsEs, whereμ1 < μ2 < · · · < μs1 ≤ s ≤ nare the distinct

eigenvalues ofA,Eii1,2, . . . , sare theri×rir1 r2 · · · rsnidentity matrices. And

sinceABBA, that is,

P diag μ₁E1, μ2E2, . . . , μsEs

P−1_{BBP diagμ}

1E1, μ2E2, . . . , μsEs

P−1_{, 2.4}

we have

diag μ₁E1, μ₂E2, . . . , μ_sEs

P−1_BPP−1_{BP diagμ}

1E1, μ2E2, . . . , μsEs

DenoteP−1_{BP C}

ij, whereCij are the submatrices such thatEiCijandCijEii1,2, . . . , s

are defined, then, by2.5,

μ_iCijμ_jCij

i, j1,2, . . . , s. 2.6

Noticed thatμ_i/μ_j i /j, we haveCijOi /j, and hence

P−1_{BPdiag C}

11,C22, . . . ,Css, 2.7

where Cii and Eii 1,2, . . . , s are the same order square matrices. Since B is a

diagonalizationn×nmatrix, there exists an invertible matrixQdiagQ1,Q2, . . . ,Qssuch

that

Q−1_P−1_BPQdiagQ−1

1 ,Q−21, . . . ,Q−s1

·diagC11,C22, . . . ,Css·diagQ1,Q2, . . . ,Qs

diagQ−₁1C11Q1,Q2−1C22Q2, . . . ,Q−s1CssQs

diagλ1, λ2, . . . , λn,

2.8

whereλ1≤λ2≤ · · · ≤λnare the eigenvalues ofB.

LetRPQ, thenRis an invertible matrix such thatR−1_BRdiagλ

1, λ2, . . . , λnand

R−1_{A BRR}−1_{AR R}−1_BRQ−1_P−1_{APQ R}−1_BR

diagQ−₁1,Q−₂1, . . . ,Q_s−1·diagμ₁E1, μ2E2, . . . , μsEs

·diagQ1,Q2, . . . ,Qs

diagλ1, λ2, . . . , λn

diagμ₁E1, μ2E2, . . . , μsEs

diagλ1, λ2, . . . , λn

diagμ1, μ2, . . . , μn

diag λ1, λ2, . . . , λn

diagμ1 λ1, μ2 λ2, . . . , μn λn

.

2.9

A Bis a diagonalization matrix andμ1 λ1 ≤μ2 λ2 ≤ · · · ≤μn λnare the eigenvalues of

A B.

The proof ofLemma 2.2is fulfilled.

Let·,· denote the usual inner product onRn _{and denote the corresponding norm}

by |u| {n_i1u2

i}

1/2

, where u u1, u2, . . . , unT. Let ·,· denote the inner product on

L2_{0, π,R}n_{. It is known very well thatL}2_{0, π,R}n_{is a Hilbert space with inner product}

u,v

π

0

ut,vtdt, u,v∈L20, π,Rn 2.10

Now, we consider the boundary value problem

⎧ ⎨ ⎩

u _{Au gt,u ht, t∈0, π,}

u0 a, uπ b, 2.11

whereu:0, π → Rn_{,Ais a real constant diagonalizationn×nmatrix with real eigenvalues}

μ1 ≤ μ2 ≤ · · · ≤ μn each eigenvalue is repeated according to its multiplicity,g : 0, π×

Rn _{→ R}n_{is a potential Carath´eodory vector-valued function ,h:0, π → R}n_{is continuous,}

a a1, a2, . . . , anT,b b1, b2, . . . , bnT,ai, bi∈R,i1,2, . . . , n.

Letut vt ωt,ωt 1−t/πa t/πb, t ∈ 0, π, then2.11may be written in the form

⎧ ⎨ ⎩

v _{Av g}∗_{t,v h}∗_t,

v0 vπ 0,

2.12

whereg∗t,v gt,v ω,h∗t ht−Aωt. Clearly,g∗t,vis a potential Carath´eodory vector-valued function,h∗:0, π → Rn_{. Clearly, ifv}

0is a solution of2.12,u0v0 ωwill

be a solution of2.11.

Assume that there exists a real bounded diagonalization n× n matrix Bt,u t ∈

0, π,u∈Rn_{such that for a.e.t∈0, πandξ,η∈L}2_{0, π,R}n

gt,η−gt,ξ Bt,ξ τη−ξη−ξ, 2.13

where τ diagτ1, τ2, . . . , τn, τi ∈ 0,1 i 1,2, . . . , n, Bt,u commutes withA and is

possessed of real eigenvaluesλ1t,u ≤ λ2t,u ≤ · · · ≤ λnt,u. In the light ofLemma 2.2,

A Bt,uis a diagonalizationn×nmatrix with real eigenvaluesμ1 λ1t,u≤μ2 λ2t,u≤

· · · ≤ μn λnt,u each eigenvalue is repeated according to its multiplicity. Assume that

there exist positive integersNi i1,2, . . . , nsuch that foru∈L20, π,Rn

N_i2−μi< λit,u<Ni 12−μi i1,2, . . . , n. 2.14

Letξ_i i1,2, . . . , nbenlinearly independent eigenvectors associated with the eigenvalues

μi λit,u i1,2, . . . , nand letγ_i i1,2, . . . , nbe the orthonormal vectors obtained by

orthonormalizing to the eigenvectorsξ_i i 1,2, . . . , nofμi λit,u i 1,2, . . . , n. Then

for everyu∈Rn

A Bt,uγ_iμi λit,u

γ_i i1,2, . . . , n. 2.15

And let the set{γ₁,γ₂, . . . ,γ_n}be a basis for the spaceRn_{, then for everyu∈R}n_,

It is well known that eachv ∈ L2_{0, π,R}n _{can be represented by the absolutely}

convergent Fourier series

v 2 π n

i1

∞

k1

Ckisinktγ_i, Cki

2

π

π

0

vitsinktdt i1,2, . . . , n;k1,2, . . .. 2.17

Define the linear operatorL−d2/dt2_:DL⊂L2_{0, π,R}n _{→ L}2_{0, π,R}n_,

DL ⎧ ⎨

⎩v∈L20, π,Rn|v0 vπ 0, vt

2

π

n

i1

∞

k1

Ckisinktγi,

Cki 2 π π 0

vitsinktdt, i1,2, . . . , n, n

i1

∞

k1

C_ki2k4< ∞ ⎫ ⎬ ⎭, Lv 2 π n

i1

∞

k1

k2Ckisinktγi, σL

n2|n∈N.

2.18

Clearly, L −d2/dt2is a selfadjoint operator and DLis a Hilbert space for the inner product

u,v

π

0

u_{t,vt ut,vtdt, u,v∈ DL, 2.19}

and the norm induced by the inner product is

v2 π 0

v_{t,vt vt,vtdt, v∈ DL. 2.20}

Define

X ⎧ ⎨

⎩x∈L20, π,Rn|xt

2

π

n

i1

Ni

k1

Ckisinktγi, t∈0, π,

Cki 2 π π 0

xitsinktdt

⎫ ⎬ ⎭, 2.21 Y ⎧ ⎨

⎩y∈L20, π,Rn|yt

2

π

n

i1

∞

kNi 1

Ckisinktγi, t∈0, π,

Cki 2 π π 0

yitsinktdt, n

i1

∞

kNi 1

C2_kik4 < ∞ ⎫ ⎬ ⎭.

2.22

Define two projective mappingsP :DL → XandQ:DL → Y byPvx∈Xand

Qvy∈Y,vx y∈ DL, thenSP−Qis a selfadjoint operator.

Using the Riesz representation theorem , we can define a mappingT :L2_{0, π,R}n _→

L2_{0, π,R}n_by

Tu,v

π

0

u_{,v−Au,v−gt,u,v ht,vdt, ∀v∈L}2_{0, π,R}n_{. 2.23}

We observe thatT in2.23is defined implicity. LetTu ∇Fuin2.23, we have

∇Fu,v

π

0

u_{,v−Au,v−gt,u,v ht,vdt, ∀v∈ DL⊂L}2_{0, π,R}n_.

2.24

Clearly,∇Fand henceFis defined implicity by2.24. It can be proved thatuis a solution of

2.11if and only ifusatisfies the operator equation

∇Fu 0. 2.25

3. The Main Theorems

Now, we state and prove the following theorem concerning the solution of problem2.11.

Theorem 3.1. Assume that there exists a real diagonalization n × n matrix Bt,u u ∈ L2_{0, π,R}n _{with real eigenvalues λ}

1t,u ≤ λ2t,u ≤ · · · ≤ λnt,u satisfying 2.14 and

commuting withA. Denote

αu min

u≤umin1≤i≤n0min≤t≤π

λit,u μi−Ni2>0

, 3.1

βu min

u≤u1min≤i≤n 0min≤t≤π

Ni 12−μi−λit,u>0

. 3.2

If

α:0, ∞−→0, ∞, β:0, ∞−→0, ∞,

s·αs−→ ∞, s·βs−→ ∞, as s−→ ∞, 3.3

problem2.11has a unique solutionu0, andu0satisfies∇Fu0 0,and

Fu0 max_x

∈X miny∈Y Fx y ω miny∈Y maxx∈X Fx y ω, 3.4

Proof. First, by virtue of2.21and2.22, we have

π

0

x_,xdt π 0

−x_,xdt

≤ π 0 ⎛ ⎝ 2 π n

i1

N_i2

Ni

k1

Ckisinktγ_i,

2

π

n

i1

Ni

k1

Ckisinktγ_i

⎞ ⎠_dt

≤

max

1≤i≤nNi

2 π

0

x,xdt,

3.5

π

0

y_,ydt π 0

−y_,ydt

π 0 ⎛ ⎝ 2 π n

i1

∞

kNi 1

k2Ckisinktγi,

2

π

n

i1

∞

kNi 1

Ckisinktγi

⎞ ⎠_dt,

3.6

1

max1≤i≤nNi 12 π

0

y_,ydt

π 0 ⎛ ⎝ 2 π n

i1

∞

kNi 1

k2

max1≤i≤nNi 12

Ckisinktγ_i,y

⎞ ⎠_dt

≥ π

0

y,ydt.

3.7

Denote∇Fu ∇Fv ω ∇F∗v.

By2.24,2.13,3.5,3.6,3.7,3.1, and3.2, for allx1,x2 ∈ X, y ∈ Y, let v1

x1 y∈ DL,v2 x2 y ∈ DL,vv1−v2x1−x2 x∈X,x1Pv1 ∈X,x2 Pv2∈X,

yQv1Qv2∈Y, we have

∇F∗v1− ∇F∗v2,x1−x2∇Fu1− ∇Fu2,x1−x2∇Fu1,x − ∇Fu2,x

π

0

u₁,x−Au1,x−gt,u1,x ht,x dt

− π

0

u₂,x−Au2,x−gt,u2,x ht,x dt

π

0

u1−u2,x

−Au1−u2,x−gt,u1−gt,u2,x

dt

π

0

−v_{,x−Av,x−Bt,v}

2 ω τvv,x

dt

π

0

−x_{,x−Ax,x−Bt,v}

2 ω τvx,x

≤ π

0

⎡ ⎣

⎛ ⎝n

i1

N_i2·

2

π

Ni

k1

Ckisinktγi,x

⎞ ⎠

− ⎛ ⎝n

i1

2

π

Ni

k1

CkisinktA Bt,vγi,x

⎞ ⎠ ⎤ ⎦dt

≤ π

0

⎛ ⎝n

i1

N_i2−μi−λit,v

₂ π

Ni

k1

Ckisinktγi,x

⎞ ⎠_dt

≤ −αv

π

0

x,xdt

−αv1−v2

1

max1≤i≤nNi2 1

× π

0

max1≤i≤nNi2

x,x x,x dt

≤ −α∗v1−v2x1−x22,

%

α∗v1−v2 α

v1−v2

max1≤i≤nNi2 1

& ,

3.8

for allx∈X,y1,y2∈Y, letv1x y1∈ DL,v2 x y2∈ DL,vv1−v2y1−y2y∈Y,

y1Qv1∈Y,y2Qv2∈Y,xPv1Pv2∈X, we have

∇F∗v1− ∇F∗v2,y1−y2

∇Fu1− ∇Fu2,y1−y2

π

0

u1−u2,y

−Au1−u2,y−gt,u1−gt,u2,y

dt

π

0

v_{,y−Av,y−Bt,vv,ydt}

π

0

y_{,y−A Bt,vy,ydt}

≥ π

0

⎡

⎣y_,y−

⎛ ⎝ 2 π n

i1

∞

kNi 1

k2

max1≤i≤nNi 12

Ckisinkt

μi λit,v

γ_i,y

⎞ ⎠ ⎤ ⎦dt

π

0

⎡ ⎢

⎣−y_,y−

⎛ ⎜ ⎝ 1

max

1≤i≤nNi 1

2 2 π n

i1

∞

kNi 1

k2Ckisinkt

μi λit,v

γ_i,y

≥ π 0 ⎡ ⎣ ⎛ ⎝ 2 π n

i1

∞

kNi 1

k2Ckisinktγi,y

⎞ ⎠

− ⎛

⎝ 1

max1≤i≤nNi 12

2

π

n

i1

∞

kNi 1

k2Ckisinkt

μi λit,v

γ_i,y

⎞ ⎠ ⎤ ⎦dt

≥ 1

max1≤i≤nNi 12 π 0 ⎛ ⎝ 2 π n

i1

∞

kNi 1

k2Ckisinkt

Ni 12−

μi λit,v

γ_i,y

⎞ ⎠_dt

≥ minv≤vmin1≤i≤nmint∈0,π

Ni 12−μi−λit,v>0

max1≤i≤nNi 12 1

· %

1 1

max1≤i≤nNi 12

& π

0

y_,ydt

≥ βv

max1≤i≤nNi 12 1 π

0

y_{,y y,ydt}

β∗vy2β∗v1−v2y1−y22,

%

β∗v1−v2

βv1−v2

max1≤i≤nNi 12 1

& .

3.9

By 3.3, s · α∗s → ∞, s · β∗s → ∞,ass → ∞. Clearly, α∗ and β∗

are nonincreasing. Now, all the conditions in the Theorem 2.1 are satisfied. By virtue of

Theorem 2.1, there exists a uniquev0 ∈ DLsuch that∇F∗v0 ∇Fv0 ω ∇Fu0 0

andF∗v0 Fv0 ω Fu0 maxx∈Xminy∈YFx y ω miny∈Y maxx∈XFx y ω,

whereFis a functional defined implicity in2.24andωt 1−t/πa t/πb, t∈0, π. v0tis just a unique solution of2.12andu0t v0t ωtis exactly a unique solution of

2.11. The proof ofTheorem 3.1is completed.

Now, we assume that there exists a positive integerNsuch that

N2−μi< λit,u<N 12−μi i1,2, . . . , n 3.10

foru∈L2_{0, π,R}n_{, t∈0, π.Define}

X ⎧ ⎨

⎩x∈L20, π,Rn|xt

2

π

n

i1

N

k1

Ckisinktγi, t∈0, π,

Cki 2 π π 0

xitsinktdt

⎫ ⎬ ⎭,

Y ⎧ ⎨

⎩y∈L20, π,Rn|yt

2

π

n

i1

∞

kN 1

Ckisinktγi, t∈0, π,

Cki

2

π

π

0

yitsinktdt, n

i1

∞

kN 1

C2

kik4< ∞

⎫ ⎬ ⎭,

3.12

αu min

u≤umin1≤i≤n0min≤t≤π

λit,u μi−N2 >0

, 3.13

βu min

u≤umin1≤i≤n0min≤t≤π

N 12−μi−λit,u>0

. 3.14

Replace the condition 2.14 by 3.10 and replace 2.21, 2.22, 3.1, and 3.2 by

3.11, 3.12, 3.11, and 3.14, respectively. Using the similar proving techniques in the

Theorem 3.1, we can prove the following theorem.

Theorem 3.2. Assume that there exists a real diagonalizationn×nmatrixBt,u t ∈0, π,u∈ Rn _{with real eigenvalues λ}

1t,u ≤ λ2t,u ≤ · · · ≤ λnt,u satisfying 2.13 and 3.10 and

commuting withA. If the functionsαandβdefined in (3.11) and3.14satisfy3.3, problem2.11

has a unique solutionu0, andu0satisfies∇Fu0 0and3.4.

It is also of interest to the case ofAO.

Corollary 3.3. Let ht, gt,u, a and b be as in 2.11. Assume that there exists a real diagonalizationn×nmatrixBt,u t∈0, π,u∈Rnwith real eigenvaluesλ1t,u≤λ2t,u≤

· · · ≤λnt,usatisfying2.13andN2i < λit,u<Ni 12 Ni∈Z , i1,2, . . . , n.Denote

αu min

u≤u1min≤i≤n0min≤t≤π

λit,u−N2i >0

,

βu min

u≤u1min≤i≤n0min≤t≤π

Ni 12−λit,u>0

.

3.15

Ifαandβsatisfy3.3, the problem

⎧ ⎨ ⎩

u _{gt,u ht, t∈0, π,}

u0 a, uπ b

3.16

has a unique solutionu0, andu0satisfies∇Fu0 0and3.4, whereFis a functional defined in

∇Fu,v

π

0

Corollary 3.4. Let ht, gt,u, a, b, and Bt,u be as in Corollary 3.3. The eigenvalues of

Bt,uλ1t,u≤λ2t,u≤ · · · ≤λnt,usatisfyN2< λit,u<N 12N∈Z .Denote

αu min

u≤u1min≤i≤n0min≤t≤π

λit,u−N2 >0

,

βu min

u≤u1min≤i≤n0min≤t≤π

N 12−λit,u>0

.

3.18

Ifαand βsatisfy3.3, problem3.16has a unique solutionu0, andu0satisfies∇Fu0 0and

3.4, whereFis a functional defined in3.17.

If there exists aC2 _{functionalG :0, π×R}n _{→ Rsuch thatgt,u ∇Gt,u, then}

2.13should be

gt,η−gt,ξ ∇Gt,η− ∇Gt,ξ

1

0

D2_{Gt,ξ τη−ξη−ξdτ, 3.19}

where D2_{G is just a Hessian of G. In this case, the following corollary follows from}

Theorem 3.1.

Corollary 3.5. Let the eigenvalues of1₀D2_{Gt,ξ τη−ξdτλ}

1t,u≤λ2t,u≤ · · · ≤λnt,u

satisfy 2.14. If αand β defined in3.1 and 3.2 satisfy 3.3, problem2.11(where gt,u ∇Gt,u) has a unique solutionu0, andu0satisfies∇Fu0 0and3.4.

Using the similar techniques of the present paper, we can also investigate the two-point boundary value problem

⎧ ⎨ ⎩

u _{Au gt,u ht, t∈0,2π,}

u0 a, u2π b,

3.20

where u, A, ht, gt,u, a and b are as in problem2.11. The corresponding results are similar to the results in the present paper.

The special case ofAO andn1 in problem3.20has been studied by Zhou Ting and Huang Wenhua5. Zhou and Huang adopted the techniques different from this paper to achieve their research.

References

1 S. A. Tersian, “A minimax theorem and applications to nonresonance problems for semilinear equations,”Nonlinear Analysis: Theory, Methods & Applications, vol. 10, no. 7, pp. 651–668, 1986. 2 H. Wenhua and S. Zuhe, “Two minimax theorems and the solutions of semilinear equations under the

asymptotic non-uniformity conditions,”Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 8, pp. 1199–1214, 2005.

4 H. Wenhua, “A minimax theorem for the quasi-convex functional and the solution of the nonlinear beam equation,”Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 8, pp. 1747–1756, 2006. 5 Z. Ting and H. Wenhua, “The existence and uniqueness of solution of Duffing equations with

non-C2_{perturbation functional at nonresonance,”Boundary Value Problems, vol. 2008, Article ID 859461, 9}