Impulsive Integrodifferential Equations Involving Nonlocal Initial Conditions

doi:10.1155/2011/634701

Research Article

Impulsive Integrodifferential Equations Involving Nonlocal Initial Conditions

Rong-Nian Wang and Jun Xia

Department of Mathematics, NanChang University, NanChang, JiangXi 330031, China

Correspondence should be addressed to Rong-Nian Wang,[email protected] Received 23 November 2010; Revised 22 February 2011; Accepted 7 March 2011 Academic Editor: Jin Liang

Copyrightq2011 R.-N. Wang and J. Xia. 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.

We focus on a Cauchy problem for impulsive integrodifferential equations involving nonlocal initial conditions, where the linear part is a generator of a solution operator on a complex Banach space. A suitable mild solution for the Cauchy problem is introduced. The existence and uniqueness of mild solutions for the Cauchy problem, under various criterions, are proved. In the last part of the paper, we construct an example to illustrate the feasibility of our results.

1. Introduction

LetX, · denote a complex Banach space and denoteLXby the space of all bounded linear operators from X into X with the usual operator norm · LX. Let us recall the following definitions.

Definition 1.1 see1. Let f : R → R be a continuous function and γ ≥ 1. Then the expression

I^γf t

_t

0

t−s^γ−2 Γ

γ−1fsds 1.1

is called the Riemann-Liouville integral of orderγ−1.

Definition 1.2see2. LetAbe a linear and closed operator with domainDAdefined on X. By a solution operator associated withAinX, we mean a family{Qα :R → LX}of

strongly continuous operators satisfying 1{λ^α: Reλ > θ} ⊂ρAand 2

λ^α−1Rλ^α, Ax _∞

0

e^−λtQαtx dt Reλ > θ, x∈X, 1.2

whereθ∈Ris a constant andRλ, A λI−A⁻¹stands for the resolvent ofA. In this case, we also say thatQ_αtis a solution operator generated byA.

Remark 1.3. It is to be noted that in the border caseα 1, the familyQ_αtcorresponds to a classical strongly continuous semigroup, whereas in the case α 2 a solution operator corresponds to the concept of a cosine family. Moreover, according to3, one can find that solution operators are a particular case ofa, k-regularized families and a solution operator Qαtcorresponds to a1, t^α−1/Γα-regularized family.

Remark 1.4. Note that solution operatorQ_αtdoes not satisfy the semigroup property.

Remark 1.5. Various solution operators are usually key tools in dealing with the abstract Cauchy problems and related issues. For more information, please see, for example,4–11 and references therein.

Starting from some speculations of Leibniz and Euler, the fractional calculus such as the Riemann-Liouville fractional integral which allows us to consider integration and differentiation of any order, not necessarily integer, have been the object of extensive study for analyzing not only stochastic processes driven by fractional Brownian motion, but also nonrandom fractional phenomena in physics and optimal controlcf. e.g.,1,12,13. One of the emerging branches of the study is the Cauchy problems of abstract differential equations involving fractional integration or fractional differentiationsee, e.g.,1,14–17. Let us point out that many phenomena in engineering, physics, economy, chemistry, aerodynamics, and electrodynamics of complex medium can be modeled by this class of equations.

In the present paper we study the existence and uniqueness of mild solutions for the Cauchy problem for impulsive integrodifferential equations involving nonlocal initial conditions in the form

ut− _t

0

t−s^α−2

Γα−1AusdsFt, ut, 0≤t≤a, t /ti,

u0 Hu,

u t_i

u t⁻_i

T_i u

t⁻_i

, i1, . . . , n,

1.3

where 1 < α < 2,A : DA ⊂ X → X is a generator of a solution operatorQ_αt, 0 < t₁ <

· · · < tn < a,ut_i limδ→0uti δandut⁻_i lim_δ→0⁻utiδstand for the right and left limits ofutatt t_i, respectively, andF : 0, a×X → X,T_i : X → X, i 1, . . . , n are appropriate functions to be specified later. As can be seen, the convolution integral in 1.3is the Riemann-Liouville fractional integral, and the functionHconstitutes a nonlocal condition.

As usual, the solutiont → utwith the points of discontinuity at the momentsti i 1, . . . , nfollows thatuti ut⁻_i, that is, at which it is continuous from the left.

We mention that in recent years, the theory of various integrodifferential equations in Banach spaces has been studied deeply due to their important values in sciences and technologies, and many significant results have been establishedsee, e.g., 2,18–23and references therein.

Interest in impulsive nonlocal Cauchy problems stems mainly from the observation that on one side, nonlocal initial conditions have better effects in treating physical problems than the usual ones see 21, 22, 24–27 and the references therein for more detailed information about the importance of nonlocal initial conditions in applications; on the other side, the dynamics of many evolutionary processes from some research fields are subject to abrupt changes of states at certain moments of time between intervals of continuous evolution, such changes can be well approximated as being instantaneous changes as state, that is, in the form of “impulses” cf. 20, 28 and the references therein. This class of equations has been the object of extensive study in recent years, see29–31and the references therein for more comments and citations. It is worth mentioning that in 31, Liang et al.

considered the following impulsive nonlocal Cauchy problem ut Aus ft, ut, 0≤t≤a, t /ti,

u0 gu u₀, u

t_i

−u t⁻_i

I_iuti, i1, . . . , n, 0< t₁ <· · ·< t_n< a,

1.4

where A is the generator of a strongly continuous semigroup in a Banach space and the existence and uniqueness of mild and classical solutions for the Cauchy problem, under various criterions, are proved. Also, Wang et al.32proved the existence and uniqueness of mild and classical solutions for the nonlocal Cauchy problem in the form

ut Aus ht, ut, t >0,

u0 H

t1, . . . , tp, u

u0, 1.5 where 0< t1<· · ·< t_p−1< tp<∞ p∈N,Ais aω-almost sectorial operatornot necessarily densely defined.

In this work, motivated by the above contributions, we shall combine these earlier work and extend the study to the Cauchy problem 1.3. New existence and uniqueness results in the case whenA is a generator of a solution operator, under various criterions, are proved. In the last part of paper, we construct an example to illustrate the feasibility of our results.

2. Preliminaries

Throughout this paper, we takeC0, a;Xto be the Banach space of allX-valued continuous functions from0, aintoXendowed with the uniform norm topology

u_asup{ut; t∈0, a}. 2.1

Put

I0 0, t1, Ii ti, t_i1, i1, . . . , n, 2.2

witht₀0,t_n1a, and letu_ibe the restriction of a functionutoI_i i0,1, . . . , n.

Consider the set of functions

PC0, a;X {u:0, a−→X; ui∈CIi;X, i0,1, . . . , n, u

t_i , u

t⁻_i

exist, and satisfyuti u t⁻_i

fori1, . . . , n

, 2.3

endowed with the norm

u_PCmax

sup

t∈Ii

uit; i0,1, . . . , n

. 2.4

It is easy to see PC0, a;Xis a Banach space.

Let 1< α <2. It follows from33that ifAis sectorial of typeθ ∈R, that is,Ais a closed linear operator, and there exist constantsϕ∈0, π/2andC>0 such thatC− {θλ: λ∈C,|arg−λ|< ϕ} ⊂ρAand

Rλ, A_LX≤ C

|λ−θ|, λ∈C−

θλ:λ∈C, arg−λ < ϕ

, 2.5

thenAis a generator of a solution operatorQ_αt, which is given by

Qαt 1 2πi

Γe^λtλ^α−1λ^α−A⁻¹dλ, 2.6 provided that 0≤ϕ <1−α/2π, whereΓis a suitable path lying outside the sector{θλ: λ∈C,|arg−λ|< ϕ}. And Cuesta18, Theorem 1, has proved that ifAis a sectorial operator of typeθ <0 and there is a positive constantC_αwhich depends onCsuch that the estimate

Qαt_LX≤ Cα

1|θ|t^α 2.7

holds for allt≥0.

We recall that the Laplace transform of a abstract functiong∈L¹R, Xis defined by

gζ:

_∞

0

e^−ζtgtdt. 2.8

We first treat the following problem:

ut _t

0

t−s^α−2

Γα−1Ausdsft, t >0, 1< α <2, u0 u₀.

2.9

Formally applying the Laplace transform in2.9, we obtain

λuζ −u0λ^1−αAuζ fλ, 2.10

which establishes the following result:

uζ λ^α−1Rλ^α, Au0λ^α−1Rλ^α, Afλ. 2.11 This means that

ut Qαtu0 _t

0

Qαt−sfsds. 2.12

Motivated by the above consideration, we give the following definition.

Definition 2.1. Let 1< α <2. A solutionu∈C0, a;Xof the integral equation

ut QαtHu

_t

0

Qαt−sFs, usds, t∈0, a, 2.13

is called a mild solution of the following problem:

ut− _t

0

t−s^α−2

Γα−1AusdsFt, ut, t∈0, a,

u0 Hu,

2.14

whereQ_αis the solution operator generated byA.

We list the following basic assumptions of this paper.

H1F : 0, a×X → X is continuous inton0, aand there exists a constantL_F > 0 such that

Ft, u1−Ft, u2 ≤L_Fu1−u₂ 2.15

for allt, u1,t, u2∈0, a×X.

H₁F : 0, a×X → X is continuous and there exists a functionρt ∈ L¹0, a;R such that

Ft, u1−Ft, u2 ≤ρtu1−u₂ 2.16 for allt∈0, a, u1, u₂∈X.

H2H : PC0, a;X → X is completely continuous and there exists a continuous nondecreasing functionΦ:R → Rsuch that for eachr >0,

sup

uPC≤rHu ≤Φr, lim inf

r→∞

Φr

r η <∞.

2.17

H₂H: PC0, a;X → Xis Lipschitz continuous with Lipschitz constantLH. H3Fori1, . . . , n,Ti:X → Xis Lipschitz continuous with Lipschitz constantLi. H₃Fori1, . . . , n,T_i:X → Xis completely continuous and there exists a continuous

nondecreasing functionΥi :R → Rsuch that for eachr >0,

sup

u≤rTiu ≤Υir, lim inf

r→∞

Υir

r λi<∞. 2.18

The following fixed-point theorem plays a key role in the proof of our main results.

Lemma 2.2see34. LetY be a convex, bounded, and closed subset of a Banach spaceX and let Ψ:Y → Y be a condensing map. Then,Ψhas a fixed point inY.

3. Main Results

To set the framework for our main existence results, we will make use of the following lemma.

Lemma 3.1. Let 1< α <2. Assume thatAis a sectorial operator of typeθ <0 andQ_αtis a solution operator generated byA. Suppose in addition thatF :0, a×X → X is a continuous function. If u∈C0, a;Xis a mild solution of the Cauchy problem2.14in the sense ofDefinition 2.1, then, usatisfying the following impulsive integral equation:

ut Φ^α_itHu _t

ti

Q_αt−sFs, usds

1≤j≤i

_tj

tj−1

Φ^α_i,jtQα

tj−s

Fs, usds

1≤j≤i

Φ^α_i,jtTj

u

t⁻_j

, t∈I_i, i0,1, . . . , n,

3.1

is a mild solution of problem1.3, where

Φ^α_it:

⎧⎪

⎨

⎪⎩

Qαt ifi0,

1≤j≤i

Qαt−tiQα

tj−t_j−1

ifi≥1, 3.2

Φ^α_i,jt:

⎧⎪

⎨

⎪⎩

Q_αt−t_i ifji,

j<k≤i

Q_αt−t_iQαtk−t_k−1 ifj < i. 3.3

Proof. Assume thatu∈C0, a;Xis a mild solution of2.14in the sense ofDefinition 2.1.

Obviously, ift∈ I₀, then one sees fromDefinition 2.1, that the assertion of theorem remains true. Thus, the rest proof of the theorem is done undert∈I_i i1, . . . , n.

ByDefinition 2.1, note that

ut QαtHu

_t

0

Qαt−sFs, usds 3.4

for allt∈I0. Takingtt1, then we get

ut1 Qαt1Hu _t1

0

Qαt1−sFs, usds. 3.5

Hence, it follows formut₁ ut⁻₁ T₁ut⁻₁that

u t₁

Qαt1Hu _t1

0

Qαt1−sFs, usdsT1

u t⁻₁

. 3.6

Ift∈I1, then combiningDefinition 2.1and the result above, we deduce that

ut Qαt−t1u t₁

_t

t1

Qαt−sFs, usds Qαt−t1Qαt1Hu Qαt−t1T1

u t⁻₁

_t1

0

Q_αt−t₁Qαt1−sFs, usds

_t

t1

Q_αt−sFs, usds.

3.7

This proves, for the casei1, that the conclusion of theorem holds.

Now takingtt2in3.7, one has

ut2 Qαt2−t1Qαt1Hu Qαt2−t1T1

u t⁻₁

_t1

0

Q_αt2−t₁Qαt1−sFs, usds

_t2

t1

Q_αt2−sFs, usds,

3.8

which implies that

u t₂

Qαt2−t1Qαt1Hu Qαt2−t1T1

u t⁻₁ T₂

u t⁻₂

_t1

0

Q_αt2−t₁Qαt1−sFs, usds

_t2

t1

Qαt2−sFs, usds,

3.9

provided thatut₂ ut⁻₂ T2ut⁻₂. Then, again making use ofDefinition 2.1, we get for allt∈I₂,

ut Qαt−t2u t₂

_t

t2

Qαt−sFs, usds Qαt−t2Qαt2−t1Qαt1Hu

Q_αt−t₂Qαt2−t₁T1

u t⁻₁

Q_αt−t₂T2

u t⁻₂

_t1

0

Q_αt−t₂Qαt2−t₁Qαt1−sFs, usds

_t2

t1

Q_αt−t₂Qαt2−sFs, usds

_t

t2

Qαt−sFs, usds,

Φ^α₂tHu _t

t2

Q_αt−sFs, usds

1≤j≤2

_tj

tj−1

Φ^α_2,jtQα

tj−s

Fs, usds

1≤j≤2

Φ^α_2,jtTj

u

t⁻_j ,

3.10

hereΦ^α₂tandΦ^α_2,jtare given by3.2and3.3withi2, respectively. A continuation of the same process shows that for anyt∈I_i i1, . . . , n, the assertion of theorem holds.

In this work, we adopt the following concept of mild solution for the problem1.3.

Definition 3.2. Let 1 < α < 2. Assume that A is a sectorial operator of typeθ < 0, Qαt is a solution operator generated by A, and Φ^α_it and Φ^α_i,jt are given by 3.2 and 3.3, respectively. A solutionu∈PC0, a;Xof the integral equation

ut Φ^α_itHu _t

ti

Qαt−sFs, usds

1≤j≤i

_tj

t_j−1

Φ^α_i,jtQα

t_j−s

Fs, usds

1≤j≤i

Φ^α_i,jtTj

u

t⁻_j

, t∈Ii,

3.11

herei0,1, . . . , n, is called a mild solution of the Cauchy problem1.3.

Remark 3.3. Note that if there is no discontinuity, that is, ifT_iut⁻_i 0, i 1, . . . , n, then Definition 2.1is equivalent toDefinition 3.2.

Now we present and prove our main results.

Theorem 3.4. Let 1 < α < 2. Assume thatA is a sectorial operator of typeθ < 0 andQ_αt is a solution operator generated byA. Suppose in addition that assumptionsH1–H3are fulfilled.

Then the Cauchy problem1.3admits at least one mild solution, provided Cⁿ¹_α ηaCαLFaCⁿ¹_α LFCⁿ_α

1≤j≤n

Lj<1 ifCα≥1, C_αηaC_αL_FaC²_αL_FC_α

1≤j≤n

L_j<1 ifC_α<1. 3.12

Proof. Consider the mappingΓ^α : PC0, a;X → PC0, a;X, which is defined for each u∈PC0, a;Xby

Γ^αut Φ^α_itHu _t

ti

Q_αt−sFs, usds

1≤j≤i

_tj

tj−1

Φ^α_i,jtQα

tj−s

Fs, usds

1≤j≤i

Φ^α_i,jtTj

u

t⁻_j

, t∈I_i, i0,1, . . . , n.

3.13

Then it is clear thatΓ^αis well defined.

To prove the theorem, it is sufficient to prove thatΓ^αhas a fixed point in PC0, a;X.

Put

W_r :{v∈PC0, a;X;vt ≤r, ∀t∈0, a} 3.14

forr >0 as selected below.

We first show that there exists an integerr >0 such thatΓ^αmapsW_r intoW_r. For the caset ∈ I0, by assumptionH1and the estimate2.7, a straightforward calculation yields that

Γ^αut

≤ QαtHu _t

0

Qαt−sFs, usds

≤ Qαt_LXHu _t

0

Qαt−s_LXFs,0ds

_t

0

Qαt−s_LXFs, us−Fs,0ds

≤C_αHuC_αL_{F t}

0

usdst₁C_αsup

s∈I0

Fs,0.

3.15

We claim that there exists an integerr >0 such thatΓ^αut ≤r provided thatu∈W_r. In fact, if this is not the case, then for eachN > 0, there would existu∈W_Nandt_N ∈ I₀such thatΓ^αuNtN> N. Thus, by3.15and assumptionH2we obtain

N <Γ^αu_NtN ≤C_αΦN t₁NC_αL_Ft₁C_αsup

s∈I0

Fs,0. 3.16

Dividing on both sides byNand taking the lower limit asN → ∞, we get

C_αηt₁C_αL_F≥1, 3.17

which contradicts3.12.

Since the interval0, ais divided into finite subintervals byt_i, i 1, . . . , n, we only need to prove that for a fixedi∈ {1, . . . , n},

Γ^α_iu

t: Γ^αut|_t∈Ii 3.18

mapsWrintoWr, herer >0 is a positive number yet to be determined, as the cases for other subintervals are the same.

From the HypothesesH1–H3, we infer for anyu∈Wr, Γ^α_iu

t≤Φ^α_itHu

1≤j≤i

_tj

tj−1

Φ^α_i,jtQα

t_j−s

Fs, usds

_t

ti

Qαt−sFs, usds

1≤j≤i

Φ^α_i,jtTj

u

t⁻_j

≤Φ^α_it

LXHu

_t

ti

Qαt−s_LXFs,0ds

_t

ti

Qαt−s_LXFs, us−Fs,0ds

1≤j≤i

_tj

tj−1

Φ^α_i,jtQα

tj−s

LXFs, us−Fs,0ds

1≤j≤i

_tj

tj−1

Φ^α_i,jtQα

t_j−s

LXFs,0ds

1≤j≤i

Φ^α_i,jtTj0

1≤j≤i

Φ^α_i,jt

LX

Tj

u

t⁻_j

−Tj0

≤C_αⁱ¹Φr t−tiCαsup

τ∈Ii

Fτ,0 t−tiCαLFr t_iCⁱ¹_α L_FrCⁱ_α

1≤j≤i

T_j0Cⁱ_αr

1≤j≤i

L_j Cⁱ¹_α

1≤j≤i

tj−t_j−1 sup

τ∈^tj−1,tjFτ,0.

3.19

Now, an application of the same idea with above discussion yields that there exists ar > 0 such thatΓ^α_iut ≤r. Indeed, if this is not the case, then we would deduce that

Cⁱ¹_α η t−tiCαLFtiCⁱ¹_α LFCⁱ_α

1≤j≤i

Lj≥1. 3.20

This is a contradiction to3.12. Thus, we prove that there exists an integerr > 0 such that Γ^αWr⊂Wr.

Fori0,1, . . . , n, we decompose the mappingΓ^α Γ^α₁ Γ^α₂ as follows:

Γ^α₁u

t Φ^α_itHu, t∈I_i, Γ^α₂u

t _t

ti

Q_αt−sFs, usds

1≤j≤i

Φ^α_i,jtTj

u t⁻_j

1≤j≤i

_tj

tj−1

Φ^α_i,jtQα

tj−s

Fs, usds

: _t

ti

Qαt−sFs, usds Γ^α_Tu

t Γ^α_Inu

t, t∈Ii.

3.21

Next, we show that for each ii 0,1, . . . , n, Γ^α₁ is completely continuous, while Γ^α₂ut|_t∈Ii is a contraction. In fact, it follows from assumptionH2and the estimate2.7 thatΓ^α₁|_Ii,i0,1, . . . , nis completely continuous. Note also that

Γ^α_Inu t

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

0, t∈I₀,

_t1

0

Q_αt−t₁Qαt1−sFs, usds, t∈I₁,

· · ·

1≤j≤n

_tj

tj−1

Φ^α_n,jtQα

tj−s

Fs, usds, t∈In,

Γ^α_Tu t

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

0, t∈I₀,

Q_αt−t₁T1

u t⁻₁

, t∈I₁,

· · ·

1≤j≤n

Φ^α_n,jtTj

u

t⁻_j

, t∈In.

3.22

For the casei0, it is clear that the conclusion holds in view of3.12. Fort∈Ii i1, . . . , n, byH1,H3and2.7we get

Γ^α₂u t−

Γ^α₂w t

≤ _t

ti

Qαt−sFs, us−Fs, wsds

1≤j≤i

Φ^α_i,jt T_j

u t⁻_j

−T_j w

t⁻_j

1≤j≤i

_tj

tj−1

Φ^α_i,jtQα

tj−s

Fs, us−Fs, wsds

≤C_αL_{F t}

ti

us−wsdsCⁱ_α

1≤j≤i

L_ju t_j

−w t_j Cⁱ¹_α LF

1≤j≤i

_tj

tj−1

us−wsds

≤t−tiCαLFus−ws_PCtiCⁱ¹_α LFus−ws_PC Cⁱ_αus−ws_PC

1≤j≤i

L_j

≤

⎛

⎝t−tiCαLFtiCⁱ¹_α LFCⁱ_α

1≤j≤i

Lj

⎞

⎠us−ws_PC,

3.23

provided thatu, w∈Wr. Hence, we deduce that Γ^α₂u−Γ^α₂w

PC≤

t−t_iCαL_Ft_iC_αⁱ¹L_F Cⁱ_α

1≤j≤iL_j

u−w_PC, 3.24

which means thatΓ^α₂ is a contraction due to3.12.

Thus,Γ^α Γ^α₁ Γ^α₂ is a condensing map onWr. Then, it follows fromLemma 2.2that the Cauchy problem1.3admits at least one mild solution. This completes the proof.

Theorem 3.5. Let 1< α <2. Assume thatAis a sectorial operator of typeθ <0,Qαtis a solution operator generated byA, and the HypothesesH1,H2,H₃are satisfied. Then the Cauchy problem 1.3admits at least one mild solution, provided

Cⁿ¹_α ηaC_αL_FaC_αⁿ¹L_FC_αⁿ

1≤j≤n

λ_j <1 ifC_α≥1,

C_αηaC_αL_FaC_α²L_FC_α

1≤j≤n

λ_j <1 ifC_α<1.

3.25

Proof. Assume that the mapΓ^α: PC0, a;X → PC0, a;Xand the setW_rare defined the same as inTheorem 3.4. First we claim that there exists an positive numberr >0 such that Γ^αWr⊂Wr. For the caset∈I0, the proof of the assertion follows fromTheorem 3.4. For the caset ∈ I_i i 1, . . . , n, if the conclusion is not true, then for each positive integerr, there would existu_r· ∈ W_r andt_r ∈ I_i such thatΓ^α_iu_rtr > r withΓ^α_iut: Γ^αut|_t∈Ii, wheretrdenotestdepending uponr. Thus, by assumptionsH1,H2,H₃, we have

r <Γ^α_iu_r

tr≤Φ^α_itrHur

_tr

ti

Qαtr−sFs, ursds

1≤j≤i

_tj

tj−1

Φ^α_i,jtrQα

t_j−s

Fs, ursds

1≤j≤i

Φ^α_i,jtrTj

ur

t⁻_j

≤Cⁱ¹_α Φr tr −t_iCαsup

τ∈Ii

Fτ, θ t_r −t_iCαL_Fr

t_iC_αⁱ¹L_FrCⁱ¹_α

1≤j≤i

t_j−t_j−1 sup

τ∈tj−1,tjFτ, θ C_αⁱ

1≤j≤i

Υjr.

3.26

Dividing on both sides byrand taking the lower limit asr → ∞, we have C_αⁱ¹ηaCαLFaC_αⁱ¹LFCⁱ_α

1≤j≤i

λj ≥1. 3.27

This is a contradiction to3.25.

Fori0,1, . . . , n, decompose the mappingΓ^α Γ^α₁ Γ^α₂as follows:

Γ^α₁u

t Φ^α_itHu

1≤j≤i

Φ^α_i,jtTj

u

t⁻_j

, t∈Ii

Γ^α₂u t

_t

ti

Q_αt−sFs, usds

1≤j≤i

_tj

tj−1

Φ^α_i,jtQα

t_j−s

Fs, usds.

3.28

Next, we will verify that for eacht∈I_i i0,1, . . . , n,Γ^α₁ is a completely continuous operator, while,Γ^α₂ is a contraction. Obviously, by assumptionsH2,H₃, it easily seen that Γ^α₁is a completely continuous operator. Moreover, by a similar proof with that inTheorem 3.4, we can prove thatΓ^α₂ is a contraction.

As a consequence of the above discussion andLemma 2.2, we can conclude that the problem1.3admits at least one mild solution. The proof is completed.

Theorem 3.6. Let 1 < α < 2. Assume that Ais a sectorial operator of typeθ < 0 andQ_αtis a solution operator generated byA. Then, under assumptionsH₁,H₂,H3, the Cauchy problem 1.3has a unique mild solution, provided

Cⁿ¹_α LH

CαC_αⁿ¹

a 0

ρsdsCⁿ_α

1≤j≤n

Lj<1 ifCα≥1,

C_αL_H C_αC²_α

a 0

ρsdsC_α

1≤j≤n

L_j<1 ifC_α<1.

3.29

Proof. Assume that the map Γ^α : PC0, a;X → PC0, a;X is defined the same as in Theorem 3.4. Now, we prove thatΓ^αis a contraction. Take anyu, w ∈ PC0, a;X. For the caset ∈ I0, the conclusion follows from assumptionsH₁,H₂, and3.29. Fort ∈ Ii i 1, . . . , n, a direct calculation yields

Γ^αut−Γ^αwt

≤Φ^α_itHu−Hw

_t

ti

Qαt−sFs, us−Fs, wsds

1≤j≤i

_tj

tj−1

Φ^α_i,jtQα

tj−s

Fs, us−Fs, wsds

1≤j≤i

Φ^α_i,jt T_j

u t⁻_j

−T_j w

t⁻_j

≤Cⁱ¹_α L_Hu−w_PCC_αu−w_{PC t}

ti

ρsds

Cⁱ¹_α u−w_{PC t}

0

ρsdsC_αⁱ

1≤j≤i

L_ju t⁻_j

−w t⁻_j

≤

⎛

⎝Cⁱ¹_α L_HC_{α t}

ti

ρsdsCⁱ¹_{α t}

0

ρsdsCⁱ_α

1≤j≤i

L_j

⎞

⎠u−w_PC

≤

⎛

⎝Cⁱ¹_α L_H

C_αCⁱ¹_α

a 0

ρsdsC_αⁱ

1≤j≤i

L_j

⎞

⎠u−w_PC.

3.30

in view of assumptionsH₁,H₂,H3. Hence, we deduce that

Γ^αu−Γ^αw_PC

≤

⎛

⎝Cⁱ¹_α L_H

C_αCⁱ¹_α

a 0

ρsdsCⁱ_α

1≤j≤i

L_j

⎞

⎠u−w_PC,

3.31

which impliesΓ^αis a contractive mapping on PC0, a;Xdue to3.29. ThusΓ^αhas a unique fixed pointu∈PC0, a;X, this means thatuis a mild solution of1.3. This completes the proof of the theorem.

4. Example

In this section, we present an example to illustrate the abstract results of this paper, which do not aim at generality but indicate how our theorems can be applied to concrete problems.

Consider the BVP of partial differential equation in the form

∂ut, x

∂t − 1

Γα−1 _t

0

t−s^α−2L_xus, xds |ut, x|

C1|ut, x|, 0≤t≤a, t /t_i, 0≤x≤π, ut,0 ut, π 0, 0≤t≤a,

u0, x 1 C u

t₀, x , 0≤x≤π, u

t_i, x u

t⁻_i, x

|uti, x|

inC t_i|uti, x|, i1, . . . , n,

4.1

where 1< α <2,t₀is a constant in0, a,C >0 is a constant yet to be determined,Lxstands for the operator with respect to the spatial variablexwhich is given by

Lx ∂²

∂^x2 −v v >0. 4.2 In what follows we consider the spaceX L²0, πwith norm · 2 and the operatorA : L_x:DA⊂X → Xwith domain

u∈X; u, uare absolutely continuous, u∈X, and u0 uπ 0

. 4.3

Clearly Ais densely defined in X and is sectorial of type θ −ν < 0. HenceA is a generator of a solution operator satisfying the estimate 2.7on X. Here, without lost of generality, we takeC_α≥1.

Set

utx ut, x,

Ft, utx |ut, x|

C1|ut, x|,

Hux 1

C u t₀, x , T_iutix |uti, x|

inC ti|uti, x|, i1, . . . , n.

4.4

Then we have

Ft, ut−Ft, vt₂≤ 1

Cut−vt₂, 0≤t≤a, Hu−Hv₂≤ 1

Cu−v₂, Tiu−T_iv₂≤ 1

inCu−v₂, i1, . . . , n.

4.5

Note that the problem4.1also can be reformulated as the abstract problem1.3, and due to4.5, it is not difficult to see that assumptionsH₁,H₂, andH3hold with

ρt 1

C t∈0,a, LH 1

C, Li 1

inC, i1, . . . , n, 4.6 which implies that one can choose large enough Csuch that the first inequality of 3.29 is satisfied. Hence, according toTheorem 3.6, the Cauchy problem4.1has a unique mild solution.

Acknowledgments

This research was supported in part by the NSF of JiangXi Province of China2009GQS0018 and the Youth Foundation of JiangXi Provincial Education Department of ChinaGJJ10051.

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