Volume 2010, Article ID 796165,17pages doi:10.1155/2010/796165
Research Article
Periodic Systems Dependent on Parameters
Min He
Department of Mathematical Sciences, Kent State University at Trumbull, Warren, OH 44483, USA
Correspondence should be addressed to Min He,[email protected]
Received 1 January 2010; Revised 5 March 2010; Accepted 14 March 2010
Academic Editor: Shusen Ding
Copyrightq2010 Min He. 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.
This paper is concerned with a periodic system dependent on parameter. We study differentiability with respect to parameters of the periodic solution of the system. Applying a fixed point theorem and the results regarding parameters forC0-semigroups, we obtained some convenient conditions for determining differentiability with parameters of the periodic solution. The paper is concluded with an application of the obtained results to a periodic boundary value problem.
1. Introduction
One of the fundamental subjects in dynamic systems is the boundary value problem. When studying boundary value problems of differential and integrodifferential equations, we often encounter the problems involving parameters. Take, for example, a periodic boundary value problem
utuxx, fort≥0,
ux,0 u0x, forx∈0,1,
k1u0, t−h1ux0, t f1t, ki, hi≥0 i1,2,
k2ut1, t h2ux1, t f2t, kihi>0 i1,2,
1.1
on the Banach space L20,1, where f
1t and f2t are both ρ-periodic and continuously
differentiable. It appears that the boundary conditions contain four scalarsk1, k2, h1,andh2.
as parameters. Reforming1.1 for details, seeSection 5, we have the periodic boundary value problem
wtwxxFt, ε, ε k1, k2, h1, h2∈R4, fort≥0,
wx,0 w0x, forx∈0,1,
k1w0, t−h1wx0, t 0,
k2w1, t h2wx1, t 0,
1.2
whereFt, εx 1/k1k2h2k2h1k1f2t−k2f1txh2k2f1th1f2t. Clearly
Ft, εisρ-periodic.
Furthermore, when1.2is written as a matrix equationfor details, seeSection 5, its associated abstract Cauchy problem has the following form:
dzt
dt Aεzt ft, zt, ε,
z0 z0.
1.3
This example motivates the discussion on the parameter properties of the general abstract periodic Cauchy Problem 1.3. Since the periodic system 1.3 depends on parameters, it is a natural need for investigating continuity and differentiability with respect to parameters of the solution of the system. Moreover, in applications, the differentiability with respect to parameter is often a typical and necessary condition for studying problems such as bifurcation and inverse problem1. It is worth mentioning that1.1indicates that the occurrence of parameters in the boundary conditions leads to the dependence of the domain of the operatorAεon the parameters. We have developed some effective methods for dealing with this tricky phenomenon.
In our previous work2, we have obtained results on continuity in parameters of
1.3. In this paper, we will discuss the differentiability with respect to parameters of solutions of1.3.
According to the semigroup theory, whenAεgenerates aC0-semigroupTt, ε, the
weak solution of1.3can be expressed in terms of theC0-semigroupTt, ε:
zt, ε Tt, εz0
t
0
Tt−s, εFs, zs, ε, εds. 1.4
It is clear that the differentiability with respect to parameterεof semigroupTt, εwill be the key for determining the differentiability with respect to parameterε of the solution zt, εof 1.3. Some recent works 3,4, and reference thereinhave obtained fundamental results on the differentiability with respect to parameters ofC0-semigroup. Applying these
differentiability with respect to the parameterεof the solution of1.1. The paper begins with the preliminary section, which presents some differentiability results, a fixed point theorem, and related theorems. These results will be used in proving our theorems in later sections. In order to obtain results for1.3, we, inSection 3, first study a special case of1.3
zAεzft, ε,
z0 z0,
1.5
where ft ρ, ε ft, ε for some ρ > 0, and ft, ε is continuous in t, ε ∈ R × P. After obtaining the differentiability results for1.5, we, in Section 4, employ a fixed point theorem to attain the differentiability results of1.3. Lastly, inSection 5, we will apply the obtained abstract results to the periodic boundary value problem1.1and use this example to illustrate the obtained results. One will see that the assumptions of the abstract theorems are just natural properties of1.1.
2. Preliminaries
In this section, we state some existing theorems that will be used in later proofs. We start by giving the results on differentiability with respect to parameters.
Consider the abstract Cauchy problem1.3, whereAε is a closed linear operator on a Banach spaceX, · andε ∈ P is a multiparameterP is an open subset of a finite-dimensional normed linear spacePwith norm|·|. LetTt, εbe theC0-semigroup generated
by the operatorAε. For further information onC0-semigroup, see5.
In3, we obtained a general theorem on differentiability with respect to the parameter ε of C0-semigroup Tt, ε on the entire space X. It is noticed that a major assumption of
the theorem is that the resolventλI−Aε−1 is continuouslyFrech´etdifferentiable with respect toε. In a recent paper, Grimmer and He4have developed several ways to determine differentiability with respect to parameterεof λI−Aε−1. Here, we include one of such theorems for reference.
Assumption Q. Let ε0 ∈ P be given. Then for each ε ∈ P there exist bounded operators
Q1ε, Q2ε : X → X with bounded inverses Q1−1ε and Q−21ε, such that Aε
Q1εAε0Q2ε.
Note that ifAε1 Q1ε1Aε0Q2ε1, then
Aε Q1εAε0Q2ε
Q1εQ1−1ε1Q1ε1Aε0Q2ε1Q2−1ε1Q2ε
Q1εAε1Q2ε.
2.1
Thus, having such a relationship for some ε0 implies a similar relationship at any
otherε1 ∈ P. Without loss of generality then, we may just consider the differentiating of
the semigroupTt, εatεε0 ∈P.
Define Rε λλI−Aε0−1I −Q−11εQ2−1ε, for λ ∈ ρAε0∩ρAε, and
Theorem 2.1see3. Assume Assumption Q and that
1there are constantsM≥1andω∈Rsuch that
λI−Aε−1≤ M
λ−ω, forλ > ω, n∈N, and allε∈P. 2.2
2There isK1>0such thatQ−21εxX≤K1xX,for allε∈P.
3There isK2>0such thatI−Rε−1xX ≤K2xX,for allε∈P.
4For each x ∈ X, Q−1
i εxi 1,2 and I−Rε−
1x
are (Frech´et) differentiable with respect toεatεε0.
Then for eachx∈X,λI−Aε−1xis (Frech´et) differentiable with respect toεatεε0.
Theorem 2.2see3. Assume the following
1For some0< δ < π/2, ρAε⊃δ{λ:|argλ|< π/2δ} ∪ {0},for allε∈P.
2For eachε∈P, there exists a constantMεsuch that
λI−A−1≤ Mε
|λ| forλ∈
δ
, λ /0. 2.3
3for each x ∈ X and each λ ∈ δ\{0},λI−Aε−1x is continuously (Frech´et) differentiable with respect toε onP. Moreover, for any ε0 ∈ P, there exists some ball
centered atε0, sayBε0, δ0,δ0>0such thatε∈Bε0, δ0implies
DελI−Aε−1x≤ηλ, x, 2.4
whereηλ, x, λ∈Γ, is measurable and fort >0
Γ
eλtηλ, x|dλ|<∞. 2.5
Then for eachx∈X, Tt, εxis continuously (Frech´et) differentiable with respect toεonPfort >0. In particular, fort >0
DεTt, εx 1
2πi
Γe
λtD
ελI−Aε−1x dλ, 2.6
whereΓis a smooth curve inδrunning from∞e−iθto∞eiθfor someθ,π/2< θ < π/2δ.
Definition 2.3see6, page 6. Suppose thatFis a subset of a Banach spaceX,| · |,Gis a subset of a Banach space Y, and {Ty, y ∈ G}is a family of operators taking F → X. The
operatorTyis said to be a uniform contraction onFifTy :F → Fand there is aλ,0≤λ <1
such that
Tyx−Tyx≤λ|x−x| ∀yinG, x, xinF. 2.7
Theorem 2.4see6, page 7. IfFis a closed subset of a Banach spaceX,Gis a subset of a Banach spaceY,Ty : F → F, y inGis a uniform contraction on F, andTyxis continuous inyfor each
fixedxinF, then the unique fixed pointgyofTy, yinG, is continuous iny. Furthermore, ifF,G
are the closures of open setsF◦,G◦andTyxhas continuous first derivativesAx, y, Bx, yiny, x,
respectively, forx∈ F◦, y∈ G◦, thengyhas a continuous first derivative with respect toyinG◦.
Theorem 2.5see7, page 167. Letfbe a continuous mapping of an open subsetΩofEintoF.
f is continuously (Frech´et) differentiable inΩif and only if f is (Frech´et) differentiable at each point with respect to theith (i1,2, . . . , n) variable, and the mappingx1, . . . , xn → Difx1, . . . , xn(of
ΩintoBEi, F) is continuous inΩ. Then at each pointx1, . . . , xnofΩ, the derivative offis given
by
Dfx1, . . . , xn·t1, . . . , tn n
i1
Difx1, . . . , xn·ti, t1, . . . , tn∈E. 2.8
Theorem 2.6 see 4. Let X, · X and Y, · Y be Banach spaces, and let {Bε}ε∈P ⊂
BX, Y.Assume that
Afor eachx∈X, Bεxis continuously (Frech´et) differentiable inP. In particular, forε0 ∈P,
DεBεx|εε0 ∈ BP, Yis the (Frech´et) derivative ofBεxatε ε0, andDεBεxis
continuous inP.
Then, for eachε0∈P, there is a constantHε0>0such that
DεBεx|εε 0
hY ≤Hε0xX· |h| forx∈X, h∈ P. 2.9
Lemma 2.7. LetB∈ BX, Y. IfB ≤1/2, thenI−B−1exists, and
I−B−1
∞
k0
Bk. 2.10
Moreover,I−B−1 ≤2.
Proof. The proof is standard and is omitted here.
3. Differentiability Results of
1.5
We first state a theorem from2. This result shows that1.5has a unique periodic solution which is continuous in parameterε.
Theorem 3.1see2. Assume that
1Tt, εzis continuous inεfor eachz∈X, and
Tt, ε ≤Mt0 3.1
for someMt0>0and allε∈P, t∈0, t0, and
2TNρ, ε ≤k <1for some integerNwithNρ < t0and allε∈P.
Then there exists a uniqueρ-periodic solution of 1.5, sayzt, ε, which is continuous inεforε∈P.
Now we will discuss differentiability with respect to parameter ε of the periodic solution of1.5. The following lemma presents a general result on the differentiability with respect to parameter of the fixed point of a parameter dependent operator.
Lemma 3.2. Let Kε ∈ BXfor each ε ∈ P, and letzεbe the fixed point of Kεfor each
ε∈P, which is continuous inε. Also, letQz, ε Kεz. IfQz, εhas the first partial derivatives
DzQz, εandDεQz, εwhich satisfy
1Dzzε, εxis continuous inεfor eachx∈XandDzQz, ε ≤ α <1for allz, ε∈
X×P, and
2DεQz, εis continuous inz, ε∈X×P,
thenzεis continuously (Frech´et) differentiable with respect toε∈P.
Proof. We begin by noting that the equation
yDzQzε, εyDεQzε, εh, h∈ P, 3.2
has a unique solution, sayyε, h, which is linear inh. It follows fromLemma 3.21andTheorem 2.4that
yε, h I−DzQzε, ε−1DεQzε, εh 3.3
is the unique solution of3.2forε, h∈P× P, which is continuous inε, h. From the uniqueness, one observes that
yε, αh1βh2
αyε, h1 βyε, h2, 3.4
for all scalarsα, βandh1, h2 ∈ P. That is,yε, his linear inhand may be written asCεh,
Letwzεh−zε−Cεh. Sincezε Qzε, εby hypothesis, one sees that
wQzεh, εh−Qzε, ε−Cεh
Qzεh, εh−Qzεh, ε Qzεh, ε−Qzε, ε−Cεh
DεQzεh, εh◦h DzQzε, εzεh−zε
◦zεh−zε−Cεh.
3.5
Note that there is a functionkε, hcontinuous inhand approaching zero ash → 0 such that
◦zεh−zε kε, hzεh−zε. 3.6
Now from3.5and sinceCεhis a solution of3.2, we have
wDεQzεh, εh◦h DzQzε, εzεh−zε
kε, hzεh−zε−Cεh
DεQzεh, ε−DεQzε, εh◦h
DzQzε, ε kε, hwkε, hCεh.
3.7
Thus
I−DzQzε, ε−kε, hw DεQzεh, ε−DεQzε, εh◦h kε, hCεh.
3.8
SinceDεQzε, εandzεare continuous, and{p∈ P | |p|1}is compact, the
right-hand side of this expression is◦|h|as|h| → 0. Also, there is aγ0>0 such thatDzQzε, ε
kε, h ≤β <1 for|h| ≤γ0, soI−DzQzε, ε−kε, h−1 is bounded. Thus|w|◦|h|as
|h| → 0.
Remark 3.3. This proof is based on that of Theorem 3.2 from6, page 7.
Now we prove the main theorem of the section.
Theorem 3.4. Assume that
1Tρ, ε ≤α <1for allε∈P.
2Tt, εzis continuously (Frech´et) differentiable with respect toεfor eachz∈X. Moreover for anyε0∈Pthere is someδε0>0such thatε∈Bε0, δε0
DεTt, εz ≤Hε0z for someHε0>0, t∈
0, ρ. 3.9
Then there exists a uniqueρ-periodic solution of 1.5, sayzt, ε, which is continuously (Frech´et) differentiable with respect toεforε∈P.
Proof. First note that fromTheorem 3.1, we have that
zt, ε Tt, εz0ε
t
0
Tt−s, εfs, εds 3.10
is the uniqueρ-periodic solution of1.5.
Now we want to show thatz0εis continuouslyFrech´etdifferentiable with respect
toεby applyingLemma 3.2. To this end, we need to prove the following two claims first.
Claim 1. Tt−s, εfs, εis continuouslyFrech´etdifferentiable with respect toεfort−s, s∈
0, ρ. In particular, for anyε0∈P,
DεTt−s, εfs, εεε0
DεTt−s, εfs, ε0εε0Tt−s, ε0
Dεfs, εεε0. 3.11
In fact, for anyε0∈Pandh∈ Pwithε0h∈P,
1
|h|Tt−s, ε0hfs, ε0h−Tt−s, ε0fs, ε0
−DεTt−s, εfs, ε0εε0Tt−s, ε0
Dεfs, εεε0
h
≤ 1
|h|
Tt−s, ε0h
fs, ε0h−fs, ε0−
Dεfs, εεε0
h
|1
h|
Tt−s, ε0hfs, ε0−Tt−s, ε0fs, ε0−
DεTt−s, εfs, ε0εε0h
|1
h|
{Tt−s, ε0h−Ts, ε0}
Dεfs, εεε0h.
3.12
The first two terms on the right go to 0 as|h| → 0 byTheorem 3.42and3. The last term on the right goes to 0 becauseTt−s, εzis continuous atε0and the set{Dεfs, ε|εε0p| |p|1} is compact, so3.11holds.
Now for each fixedtands, and anyε0∈P, andε∈Bε0, δε0, fromTheorem 3.42
3and3.11, it is clear thatDεTt−s, εfs, εis continuous atε0. This completes the proof
ofClaim 1.
Based onClaim 1, we have the following claim.
Claim 2. Dε
ρ
0Tρ−s, εfs, εds
ρ
0DεTρ−s, εfs, εds.
In fact, fromTheorem 2.5it suffices to show that
Dεi ρ
0
Tρ−s, εfs, εds
ρ
0
DεiT
W.l.o.g. assume thatP Rn. Let ε
0 εi0, . . . , ε0n be any point inP. Sincefs, εis
continuous on0, ρ×Bε0, δε0, there existsL >0 such that
fs, ε≤L fors, ε∈
0, ρ×Bε0, δε0. 3.14
Now from3.9, we have
εi
ε0 i
ρ
0
DεiT
ρ−s, τfs, τds dτ τ ε1, . . . , τ, . . . , εn
≤ εi ε0 i ρ 0
DεT
ρ−s, τ·fs, τds dτ≤
εi
ε0 i
ρ
0
L·Hε0ds dτ <∞.
3.15
Thus by a theorem from8, page 86, we have
εi
ε0 i
ρ
0
DεiT
ρ−s, τfs, τds dτ
ρ
0
εi
ε0 i
DεiT
ρ−s, τfs, τdτ ds. 3.16
Furthermore,
Dεi εi
ε0 i
ρ
0
eλtDεiλI−Aτ
−1x dλ dτD
εi ρ 0 εi ε0 i
eλtDεiλI−Aτ
−1x dτ dλ.
3.17
Now the left-hand side of3.17is
Dεi εi
ε0 i
ρ
0
eλtDεiλI−Aτ
−1x dλ dτ
ρ
0
eλtDεiλI−Aε
−1x dλ,
3.18
and the right-hand side of3.17is
Dεi ρ
0
εi
ε0 i
eλtDεiλI−Aτ
−1x dτ dλD
εi ρ
0
eλt
λI−Aε−1x−λI−Aε0−1x
dλ
Dεi ρ
0
eλtλI−Aε−1x dλ,
3.19
whereε0 ε1, . . . , εi−1, ε0i, εi1, . . . , εn.
That is,
Dεi ρ
0
eλtλI−Aε−1x dλ
ρ
0
eλtDεiλI−Aε
−1x dλ. 3.20
Next, consider the operator
KεzTρ, εz
ρ
0
Tρ−s, εFs, εds. 3.21
We will applyLemma 3.2and Claims1-2to show that the operatorKεhasz0εas the fixed
point andz0εis continuouslyFrech´etdifferentiable with respect toε.
Note that the operatorKεhas the following properties.
iKεis defined on the Banach spaceX, · .
iiKεis a uniform contraction onX. In fact, for allε∈Pandz1, z2∈X,
Kεz1−Kεz2T
ρ, εz1−z2
≤Tρ, ε· z1−z2 ≤αz1−z2
sinceTρ, ε≤α <1.
3.22
iiiKεzis continuous inεfor each fixedz∈X.For the detailed proof, see Theorem 3.2 from2.
ApplyingTheorem 2.4, we have thatz0εis the fixed point ofKε.
Furthermore, it is clear that the first derivative ofKεzwith respect toz
DzKεzT
ρ, ε 3.23
satisfiesLemma 3.21. It is also clear fromClaim 2that the first derivative ofKεzwith respect toε
DεKεz Dε
Tρ, εz
ρ
0
DεT
ρ−s, εfs, εds 3.24
is continuous inz, εby Theorem 3.42and 3.Note that using the same argument as in the proof ofClaim 1, we can show thatDεTρ−s, εfs, εis continuous atε ε0and
thereby0ρDεTρ−s, εfs, εdsis continuous atε0.
Finally applying Lemma 3.2, we have that z0ε is continuously Frech´et diff
eren-tiable with respect to ε. Using a similar argument as that in Claim 1 and Claim 2 we can show thatTρ, εz0εand
t
0Tρ−s, εfs, εdsare continuouslyFrech´etdifferentiable with
respect toε. Thus,
zt, ε Tt, εz0ε
t
0
Tt−s, εfs, εds 3.25
is continuouslyFrech´etdifferentiable with respect toεforε∈P.
Theorem 3.5. AssumeTheorem 3.4(1) and (3) and that
1for some0< δ < π/2, ρAε⊃δ{λ:|argλ|< π/2δ} ∪ {0}for allε∈P,
2there exists a constantMsuch thatλI−Aε−1 ≤M/|λ| forλ∈δ, λ /0and all
ε∈P,
3for each z ∈ X and each λ ∈ δ\{0},λI−Aε−1z is continuously (Frech´et) differentiable with respect toε on P. Moreover, for any ε0 ∈ P there exists δε0 > 0
such thatε∈Bε0, δε0implies
DελI−Aε−1z≤ηλ, z, 3.26
whereηλ, z, λ∈Γ,is measurable and fort >0
Γηλ, z
eλt|dλ|<∞. 3.27
Then there exists a uniqueρ-periodic solution of 1.5, sayzt, ε, which is continuously (Frech´et) differentiable with respect toεforε∈P.
Proof. First note that fromTheorem 2.2we have, for eachz∈X,
DεTt, εz 1
2πi
Γe
λtD
ελI−Aε−1z dλ, 3.28
whereΓis a smooth curve inδrunning from∞e−iθto∞eiθfor someθ,π/2< θ < π/2δ.
Moreover, since DελI−Aε−1z is continuous in ε, it is clear from 3.28 that
DεTt, εzis continuous inε, soTheorem 3.42is satisfied. Also it is clear from3.28that
DεTt, εzis continuous int, ε. ByTheorem 2.6, we have
DεTt, εz ≤Ht, εz, for someHt, ε>0. 3.29
Now by the Principle of Uniform Boundedness, there is aHε0>0 such that
DεTt, εz ≤Hε0z ∀t, ε∈
0, ρ×Bε0, δε0, 3.30
thus3.9is satisfied. Now the desired result follows fromTheorem 3.4.
4. Differentiability Results of
1.3
In this section, we discuss the general1.3. LetPB0,1∈ P.
Proof. First note that fromTheorem 3.41, we see that I−Tρ, ε−1 exists byLemma 2.7. Also,
I−Tρ, ε−1≤ 1 1−α
.
H. 4.1
Next consider the operator defined onX:
KεzTρ, εzy, wherey is a given point inX. 4.2
Then we have
Kεz1−Kεz2 ≤T
ρ, ε· z1−z2 ≤αz1−z2, 4.3
so Kε is a uniform contraction. Also it is obvious that Kεz is continuous in ε by
Theorem 3.42. Therefore fromTheorem 2.4it follows that there is a unique fixed point of Kε, sayzε.
Furthermore, since
DεKεzDεT
ρ, εz,
DzKεzT
ρ, ε, 4.4
which clearly satisfyLemma 3.21and2, so byLemma 3.2, we have that
zε I−Tρ, ε−1y 4.5
is continuouslyFrech´etdifferentiable w.r.t.ε.
LetP CR, ρ {g∈CR|gtρ gt}. Consider the equation
ztAεzt ft, gt, ε
z0 z0
4.6
on a Banach spaceX, · , whereftρ, g, ε ft, g, εfor someρ >0 andg ∈P CR, ρ, andft, g, εis continuous int, g, ε∈R×P CR, ρ×P.
Lemma 4.2. Assume thatLemma 3.2(1) andTheorem 3.4(2) and3.9are satisfied and
Then there exists a uniqueρ-periodic solution of 4.6, sayzt, ε, g, which is continuously (Frech´et) differentiable with respect toεforε∈P. Also
z0, ε, gI−Tρ, ε−1
ρ
0
Tρ−s, εfs, gs, εds. 4.7
which is continuously (Frech´et) differentiable with respect toε.
Proof. LetFt, ε ft, gt, ε. ThenFtρ, ε Ft, ε. Also it is obvious thatFt, εsatisfies
Theorem 3.43 . Therefore by Theorem 3.4, there is a unique ρ-solution zt, ε, g of 4.6 which is continuously Frech´et differentiable with respect to ε. In particular, z0, ε, g is continuouslyFrech´etdifferentiable with respect toε. Moreover, using the same argument as that in the proof ofTheorem 3.4we see that
z0, ε, gTρ, εz0, ε, g
ρ
0
Tρ−s, εfs, gs, εds. 4.8
Thus
z0, ε, gI−Tρ, ε−1
ρ
0
Tρ−s, εfs, gs, εds, 4.9
which is continuouslyFrech´etdifferentiable w.r.t.εbyLemma 4.1.
DefineKε:P CR, ρ → P CR, ρby
Kεgt Tt, εz0, ε, g
t
0
Tt−s, εfs, gs, εds. 4.10
Lemma 4.3. Assume thatTheorem 3.4(1)-(2) and3.9andLemma 4.2(K) are satisfied. In addition, assume that
1Tt, εzis continuous inεfor eachz∈X, and
Tt, ε ≤Mt0 4.11
for someMt0>0and allε∈P, t∈0, t0.
2ft, z1, ε−ft, z2, ε ≤Lεz1−z2,whereLεis continuous inε∈PandL0 0.
3f2t, g, ε ∂/∂gft, g, εis continuous int, g, ε.
Then the operatorKεhas a unique fixed pointg·, ε∈P CR, ρwhich is continuously (Frech´et) differentiable with respect toε.
Proof. It is clear thatP CR, ρ, · ∞is a Banach space. SinceL0 0, then, by the continuity ofLε, there isδ0such thatε∈B0, δ0implies
Lε≤ 1
4Mt0HMt0H1
Now forε∈P,
Tt, ε ·z0, g1, ε
−z0, g2, ε
Tt, ε ·I−Tρ, ε−1
ρ
0
Tρ−s, εfs, g1, ε
−fs, g2, ε
ds by4.7
≤Mt0
I−Tρ, ε−1
×
ρ
0
T
ρ−s, ε·fs, g1, ε
−fs, g2, εds
by Lemma 4.31
≤Mt0·H·Mt0
ρ
0
fs, g1, ε
−fs, g2, εds
by Lemma 4.1
≤H·M2t0·ρLεg1−g2≤
1
4g1−g2 by4.12
,
4.13 t
0
Tt−s, ε ·fs, g1s, ε
−fs, g2s, εds
≤Lε
t
0
g1s−g2sds
by Lemma 4.31-2
≤M·ρLεg1−g2≤ 1
4g1−g2 by4.12
.
4.14
Hence,
Kεg1−Kεg2≤ Tt, εz 0, g1, ε
−z0, g2, ε
t
0
Tt−s, ε ·fs, g1s, ε
−fs, g2s, εds
≤ 1
2g1−g2 by4.13and4.14
.
4.15
ThereforeKεis a uniform contraction.
Furthermore,Kεgis continuous inεfor fixedg, and also
DgKεgTt, εDgz
0, ε, g
t
0
Tt−s, εf2
s, gs, εds,
Tt, εI−Tρ, ε−1
ρ
0
f2
s, gs, εds
t
0
Tt−s, εf2
s, gs, εds,
DεKεg
DεTt, εz
0, ε, g
t
0
DεTt−s, εf
s, gs, εds
are continuous ing, ε. Therefore fromTheorem 2.4it follows thatKεhas a unique fixed point, say g·, ε ∈ P CR, ρ, which is continuously Frech´et differentiable with respect toε.
Now we present the main theorem for1.3.
Theorem 4.4. Assume that Theorem 3.4(1)-(2) and 3.5, Lemmas 4.2(K), and 4.3(1)–(3) are satisfied.
Then there exists a unique ρ-periodic solution of 1.3, sayzt, ε, which is continuously (Frech´et) differentiable with respect toεforε∈P.
Proof. This is an immediate result from Lemmas4.2and4.3.
5. Application to a Periodic Boundary Value Problem
Consider the periodic boundary value problem 1.1 on the Banach space L20,1, where
f1tρ f1t, f2tρ f2t, for someρ >0 andf1, f2∈C1R.
Let
wu−mx−b, 5.1
where
m k1f2−k2f1 k1k2h2 k2h1,
b k2h2f1h1f2 k1k2h2 k2h1
.
5.2
Then1.1becomes
wtwxxFt, ε, ε k1, k2, h1, h2∈R4, fort≥0,
wx,0 w0x forx∈0,1,
k1w0, t−h1wx0, t 0,
k2w1, t h2wx1, t 0,
5.3
whereFt, εx 1/k1k2h2 k2h1k1f2t−k2f1tx h2k2f1t h1f2t.
Assumek1, k2 > 0 and letα h1/k1 and β h2/k2. Then the associated abstract
Cauchy problem is
dwt
dt Aεwt Ft, ε,
w0 f,
onX L20,1, ·
L2, t∈R, where
Aε d
2
dx2, ε
α, β∈R2α, β∈R2 | α, β≥0,
DAε w∈H20,1|w0−αw0 0, w1 βw1 0.
5.5
We now show that5.4satisfies all assumptions ofTheorem 3.5.
It is well known that the operatorAεgenerates an analytic semigroup. The resolvent
λI−Aε−1ofAεsatisfies, for allε∈R2
andλ∈π/4 {λ| |argλ|<3π/4},
λI−Aε−1≤ M|
λ|, whereM
√
2. 5.6
Thus AssumptionsTheorem 3.51and2are satisfied. Also, refer to3,Section 4, we have shown thatλI−Aε−1xis continuouslyFrech´etdifferentiable with respect to ε. So AssumptionTheorem 3.53 is satisfied. Furthermore, from the expression of F, it is obvious thatFt, εis continuous int, ε and is continuouslyFrech´etdifferentiable with respect toε, soTheorem 3.43is satisfied. Now to applyTheorem 3.5, we only need to show thatTheorem 3.41is satisfied.
In fact, forw0∞n1anφnx,
Tt, εw0
∞
n1
aneλntφnx, 5.7
whereλn<0, andan, λn, andφndepend onε. Moreover,
T
ρ, εw02
∞
n1
aneλnρφnx
2 ≤
eλ1ρ
∞
n1
aneλn−λ1ρφnx
2
e2λ1ρ
∞
n1
|an|2e2λn−λ1ρ
≤e2λ1ρ
∞
n1 |an|2
sincee2λn−λ1ρ≤1
e2λ1ρw02α2w02 sinceα. eλ1ρ<1.
5.8
References
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