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Fractional Cauchy problems with almost sectorial operators
Applied Mathematics and Computation xxx (2014) xxx–xxx
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Fractional Cauchy problems with almost sectorial operators q Lu Zhang, Yong Zhou ⇑ Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, PR China
a r t i c l e
i n f o
a b s t r a c t This paper concerns the abstract Cauchy problem of fractional evolution equations with almost sectorial operators. The suitable mild solutions of evolution equations with Riemann–Liouville derivative and Caputo derivative are introduced respectively. The existence theorems of mild solutions of Cauchy problems are established. The results obtained here improve and generalize some known results. Finally, an example is given for demonstration. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Fractional evolution equations Almost sectorial operator Riemann–Liouville derivative Caputo derivative Measure of noncompactness Cauchy problem
1. Introduction Fractional differential equations have proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic. There has been a significant development in fractional differential equations in recent years, see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3], Lakshmikantham et al. [4], Tarasov [5], Zhou [6] and the references therein. In this paper, we assume that X is a Banach space with the norm j j. Let a 2 Rþ ; J ¼ ½0; a and J 0 ¼ ð0; a. Denote CðJ; XÞ be the Banach space of continuous functions from J into X with the norm kxk ¼ supt2½0;a jxðtÞj, where x 2 CðJ; XÞ, and BðXÞ be the space of all bounded linear operators from X to X with the norm kQ kBðXÞ ¼ supfjQ ðxÞj : jxj ¼ 1g, where Q 2 BðXÞ and x 2 X. Consider the following Cauchy problem of fractional evolution equations with Riemann–Liouville derivative
(
ðL Dq0þ xÞðtÞ ¼ AxðtÞ þ f ðt; xðtÞÞ; ðI1q 0þ xÞð0Þ
almost all t 2 ½0; a;
¼ x0 ;
ð1Þ
where L Dq0þ is Riemann–Liouville derivative of order q, I1q 0þ is Riemann–Liouville integral of order 1 q; 0 < q < 1, A is an almost sectorial operator on a complex Banach space, and f : J X ! X is a given function. The existence of mild solutions for the Cauchy problem of fractional evolution equations has been considered in several recent papers (see [7–25,34]), much less is known about the fractional evolution equations with Riemann–Liouville derivative. As pointed out in [15], the results in papers [7–14] are incorrect since the considered concept of mild solution is not appropriate. In [16], we introduce a concept of mild solution for fractional evolution equations and we study the existence and uniqueness of mild solutions. Our results have been extended and applied to Cauchy problems of many kinds of fractional evolution equations (see papers [17–22], etc.).
q Project supported by National Natural Science Foundation of PR China (11271309), the Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001) and Hunan Provincial Natural Science Foundation of China (12JJ2001). ⇑ Corresponding author. E-mail addresses: [email protected] (L. Zhang), [email protected] (Y. Zhou).
http://dx.doi.org/10.1016/j.amc.2014.07.024 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: L. Zhang, Y. Zhou, Fractional Cauchy problems with almost sectorial operators, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.07.024
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In most of the existed articles, Schauder’s fixed point theorem, Krasnoselskii’s fixed point theorem or Darbo’s fixed point theorem, Kuratowski measure of noncompactness are employed to obtain the fixed points of the solution operator of the Cauchy problems under some restrictive conditions. In order to show that the solution operator is compact, a very common approach is to use Arzela–Ascoli’s theorem. However, it is difficult to check the relatively compactness of the solution operator and the equicontinuity of certain family of functions which is given by the solution operator. In this paper, by using the theory of Hausdorff measure of noncompactness, we study the Cauchy problems of fractional evolution equations with almost sectorial operators. In the next Section, we recall some notations and useful concepts from fractional calculus and theory of measure of noncompactness. In Section 3, we introduce some basic propositions of almost sectorial operators. In Section 4, we give a appropriate definition on the mild solution and existence theorems of the problem (1). Finally, an example is given for demonstration. 2. Fractional derivatives and measure of noncompactness In this section, we introduce preliminary facts which are used throughout this paper. Firstly, we recall some basic definitions and properties of the fractional calculus. For more details, see Kilbas et al. [1]. Definition 2.1 [1]. The fractional integral Iq0þ f of order q for a function f 2 AC½0; 1Þ is defined as
ðIq0þ f ÞðtÞ ¼
1 CðqÞ
Z
0
t
f ðsÞ ðt sÞ1q
ds;
t > 0; 0 < q < 1
provided the right side is point-wise defined on ½0; 1Þ, where CðÞ is the gamma function. Definition 2.2 [1]. Riemann–Liouville derivative L Dq0þ f of order q for a function f 2 AC½0; 1Þ can be written as
ðL Dq0þ f ÞðtÞ ¼
1 d Cð1 qÞ dt
Z
t
0
f ðsÞ ds; ðt sÞq
t > 0; 0 < q < 1:
Remark 2.1. (i) If f ðtÞ 2 C 1 ½0; 1Þ, then
ðC Dq0þ f ÞðtÞ ¼
1 Cð1 qÞ
Z 0
t
0
f ðsÞ 0 ds ¼ ðI1q 0þ ½f ðsÞÞðtÞ; ðt sÞq
t > 0; 0 < q < 1:
(ii) If f is an abstract function with values in X, then integrals which appear in Definitions 2.1 and 2.2 are taken in Bochner’s sense.
Lemma 2.1 [26]. (i) Let n; g 2 R such that g > 1. If t > 0, then
In0þ
( nþg t ; sg ðtÞ ¼ Cðnþgþ1Þ Cðg þ 1Þ 0;
if n þ g – n if n þ g ¼ n
ðn 2 Nþ Þ:
(ii) Let n > 0 and g 2 Lðð0; aÞ; XÞ. Define
Gn ðtÞ ¼ In0þ g;
for t 2 ð0; aÞ;
then g ðIg0þ Gn ÞðtÞ ¼ ðInþ 0þ gÞðtÞ;
g > 0; ppt 2 ½0; a;
where ðppt 2 ½0; aÞ is to be read as ‘‘almost all t 2 ½0; a’’. Next, we recall some definitions and properties of measure of noncompactness. The measure of noncompactness a is said to be: (i) Monotone if for all bounded subsets B1 ; B2 of X; B1 # B2 implies aðB1 Þ 6 aðB2 Þ; Please cite this article in press as: L. Zhang, Y. Zhou, Fractional Cauchy problems with almost sectorial operators, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.07.024
L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
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(ii) Nonsingular if aðflg [ BÞ ¼ aðBÞ for every l 2 X and every nonempty subset B # X; (iii) Regular if aðBÞ ¼ 0 if and only if B is relatively compact in X. One of the most important examples of measure of noncompactness is the Hausdorff measure of noncompactness a defined on each bounded subset B of X by
(
aðBÞ ¼ inf e > 0 : B
m [
) Be ðxj Þ where xj 2 X ;
j¼1
where Be ðxj Þ is a ball of radius 6 e centered at xj ; j ¼ 1; 2; . . . ; m. Without confusion, Kuratowski measure of noncompactness a1 defined on each bounded subset B of X by
(
a1 ðBÞ ¼ inf
)
m [ e > 0 : B Mj
and diam ðM j Þ 6 e ;
j¼1
where the diameter of M j is defined by diamðM j Þ ¼ supfkx yk : x; y 2 M j g; j ¼ 1; 2; . . . ; m. It is well known that the Hausdorff measure of noncompactness a and Kuratowski measure of noncompactness a1 enjoy the above properties (i)–(iii) and other properties (see [27–30]). (iv) aðB1 þ B2 Þ 6 aðB1 Þ þ aðB2 Þ, where B1 þ B2 ¼ fx þ y : x 2 B1 ; y 2 B2 g; (v) aðB1 [ B2 Þ 6 maxfaðB1 Þ; aðB2 Þg; (vi) aðkBÞ 6 jkjaðBÞ for any k 2 R; In particular, the relationship of Hausdorff measure of noncompactness a and Kuratowski measure of noncompactness a1 is given by (vii) aðBÞ 6 a1 ðBÞ 6 2aðBÞ. For any W CðJ; XÞ, we define
Z
t
WðsÞds ¼
0
Z
t
uðsÞds : u 2 W ;
for t 2 J;
0
where WðsÞ ¼ fuðsÞ 2 X : u 2 Wg. Proposition 2.1. If W CðJ; XÞ is bounded and equicontinuous, then coW CðJ; XÞ is also bounded and equicontinuous. Proposition 2.2 [31]. If W CðJ; XÞ is bounded and equicontinuous, then t ! aðWðtÞÞ is continuous on J, and
Z
aðWÞ ¼ maxaðWðtÞÞ; a t2J
t
Z t WðsÞds 6 aðWðsÞÞds;
0
for t 2 J:
0
~ Proposition 2.3 [32]. Let fun g1 n¼1 be a sequence of Bochner integrable functions from J into X with kun ðtÞk 6 mðtÞ for almost all 1 þ ~ 2 L1 ðJ; Rþ Þ, then the function wðtÞ ¼ aðfun ðtÞg1 t 2 J and every n P 1, where m Þ belongs to L ðJ; R Þ and satisfies n¼1
Z
a
t
un ðsÞds : n P 1
0
62
Z
t
wðsÞds: 0
Proposition 2.4 [33]. If W is bounded, then for each
e > 0, there is a sequence fun g1 n¼1 W, such that
aðWÞ 6 2aðfun g1 n¼1 Þ þ e: 3. Almost sectorial operators We firstly introduce some special functions and classes of functions which will be used in the following, for more details, we refer to [35,36]. Let 1 < p < 0, and let S0l with 0 < l < p be the open sector
fz 2 C n f0g : j arg zj < lg and Sl be its closure, that is
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Sl ¼ fz 2 C n f0g : j arg zj 6 lg
[ f0g:
As in [37], we state the concept of almost sectorial operators as follows. Definition 3.1. Let 1 < p < 0 and 0 < x < p2 . By Hpx ðXÞ we denote the family of all linear closed operators A : DðAÞ X ! X which satisfy S (i) rðAÞ Sx ¼ fz 2 C n f0g : j arg zj 6 xg f0g and (ii) for every x < l < p there exists a constant C l such that
kRðz; AÞkBðXÞ 6 C l jzjp ;
for all z 2 C n Sl :
ð2Þ
where Rðz; AÞ ¼ ðzI AÞ1 ; z 2 qðAÞ, which are bounded linear operators the resolvent of A. A linear operator A will be called an almost sectorial operator on X if A 2 Hpx ðXÞ.
Remark 3.1. Let A 2 Hpx ðXÞ. Then the definition implies that 0 2 qðAÞ. We denote the semigroup associated with A by fQ ðtÞgtP0 . For t 2 S0px
QðtÞ ¼ e
tz
1 ðAÞ ¼ 2pi
Z
2
e
tz
Rðz; AÞdz;
Ch
where the integral contour Ch ¼ fRþ eih g lytic semigroup of growth order 1 þ p.
S þ ih fR e g is oriented counter-clockwise and x < h < l < p2 j arg tj, forms an ana-
Remark 3.2. We say that the estimate (2) in Definition 3.1 is ‘‘deficient’’ since p > 1. From [36], note in particular that if A 2 Hpx ðXÞ, then A generates a semigroup Q ðtÞ with a singular behavior at t ¼ 0 in a sense, called semigroup of growth 1 þ p. Moreover, the semigroup Q ðtÞ is analytic in an open sector of the complex plane C, but the strong continuity fails at t ¼ 0 for data which are not sufficiently smooth. Hence, it is impossible to apply to A the general results and techniques on generation of strongly continuous operator semigroup, as it is developed in [38]. Proposition 3.1 [36]. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . Then the following properties remain true. (i) Q ðtÞ is analytic in S0px and 2
dn dtn
Q ðtÞ ¼ ðAÞn Q ðtÞðt 2 S0px Þ; 2
(ii) the functional equation Q ðs þ tÞ ¼ Q ðsÞQ ðtÞ for all s; t 2 S0px holds; 2
(iii) there is a constant C 0 ¼ C 0 ðpÞ > 0 such that kQ ðtÞkBðXÞ 6 C 0 tp1 ðt > 0Þ; (iv) if b > 1 þ p, then DðAb Þ RQ ¼ fx 2 X : limt!0þ Q ðtÞx ¼ xg; R1 (v) Rðk; AÞ ¼ 0 ekt Q ðtÞdt for every k 2 C with Rek > 0. Consider the function of Wright-type (see [39])
Mq ðhÞ ¼
1 X
ðhÞn1 ; ðn 1Þ!Cð1 qnÞ n¼1
h2C
with 0 < q < 1. For 1 < d < 1; k > 0, the following results hold (see [39]). R1 ð1þdÞ (i) 0 hd M q ðhÞdh ¼ CCð1þqdÞ ; R 1 q kh 1 q (ii) 0 hqþ1 e M q hq dh ¼ ek . Define operator families fSq ðtÞgjt2S0p ; fP q ðtÞgjt2S0p
Sq ðtÞ ¼ Pq ðtÞ ¼
Z
1
2
q
M q ðhÞQ ðt hÞdh;
Z0 1
x
for t 2
2
x
by
S0px ; 2
for t 2 S0px :
qhMq ðhÞQðt q hÞdh;
2
0
Lemma 3.1 [20]. For each fixed t 2 S0px ; Sq ðtÞ and P q ðtÞ are linear and bounded operators on X. Moreover, for all t > 0 2
jSq ðtÞj 6 C 1 t qð1þpÞ
and jPq ðtÞj 6 C 2 t qð1þpÞ ;
CðpÞ ð1pÞ where C 1 ¼ C 0 Cð1qð1þpÞÞ ; C 2 ¼ qC 0 CCð1qpÞ :
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Lemma 3.2. For t > 0; Sq ðtÞ and Pq ðtÞ are strongly continuous, which means that, for any x 2 X and 0 < t 0 < t00 6 a, we have
jSq ðt00 Þx Sq ðt 0 Þxj ! 0;
jPq ðt00 Þx Pq ðt 0 Þxj ! 0;
as t 00 ! t0 :
Lemma 3.3. Let b > 1 þ p. For all x 2 DðAb Þ, we have
lim Sq ðtÞx ¼ x and lim P q ðtÞx ¼
t!0þ
t!0þ
x
CðqÞ
:
Proof. For any x 2 X, we have
Sq ðtÞx x ¼
Z
1
M q ðhÞðQðt q hÞx xÞdh
0
and
Pq ðtÞx
x ¼ CðqÞ
Z
1
qhM q ðhÞðQ ðt q hÞx xÞdh:
0
On the other hand, (iv) of Proposition 3.1 it follows that DðAb Þ RQ in view of b > 1 þ p. Therefore, we deduce, using (iii) of Proposition 3.1, that for any x 2 DðAb Þ, there exists a function gðhÞ 2 Lð0; 1Þ depending on M q ðhÞ such that
jMq ðhÞðQ ðtq hÞx xÞj 6 gðhÞ: Hence, by means of the Lebesgue dominated convergence theorem we obtain
Sq ðtÞx x ! 0 and Pq ðtÞx The proof is complete.
x
CðqÞ
! 0 as t ! 0 þ :
h
4. Existence of problem (1) Before presenting the definition of mild solution of problem (1), we firstly prove the following lemmas. Lemma 4.1. The Cauchy problem (1) is equivalent to the integral equation
xðtÞ ¼
x0 q1 1 t þ CðqÞ CðqÞ
Z
t
ðt sÞq1 ½AxðsÞ þ f ðs; xðsÞÞds;
for t 2 ð0; a
ð3Þ
for t > 0
ð4Þ
0
provided that the integral in (3) exists. Lemma 4.2. If
xðtÞ ¼
x0 q1 1 t þ CðqÞ CðqÞ
Z
t
ðt sÞq1 ½AxðsÞ þ f ðs; xðsÞÞds;
0
holds, then we have
xðtÞ ¼ tq1 Pq ðtÞx0 þ
Z
t
ðt sÞq1 Pq ðt sÞf ðs; xðsÞÞds;
for t > 0:
0
Due to Lemma 4.2, we give the following definition of the mild solution of (1). Definition 4.1. By the mild solution of the Cauchy problem (1), we mean that the function x 2 CðJ 0 ; XÞ which satisfies
xðtÞ ¼ tq1 Pq ðtÞx0 þ
Z
t
ðt sÞq1 Pq ðt sÞf ðs; xðsÞÞds;
for t 2 ð0; a:
0
Define
X ðqÞ ðJ 0 Þ ¼ ðqÞ
x 2 CðJ 0 ; XÞ : lim t 1þqp xðtÞ exists and is finite : t!0þ
0
For any x 2 X ðJ Þ, let the norm k kq defined by
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L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
kxkq ¼ sup t1þqp jxðtÞj : t2ð0;a
Then ðX ðqÞ ðJ 0 Þ; k kq Þ is a Banach space. For r > 0, define a closed subset BrðqÞ ðJ 0 Þ X ðqÞ ðJ 0 Þ as follows
n o 0 ðqÞ 0 BðqÞ r ðJ Þ ¼ x 2 X ðJ Þ : kxkq 6 r :
0 ðqÞ 0 Thus, BðqÞ ðJ Þ. r ðJ Þ is a bounded closed and convex subset of X Let BðJÞ be the closed ball of the space CðJ; XÞ with radius r and center at 0, that is
BðJÞ ¼ fy 2 CðJ; XÞ : kyk 6 rg: Thus BðJÞ is a bounded closed and convex subset of CðJ; XÞ. We introduce the following hypotheses: (H0) Q ðtÞðt > 0Þ is equicontinuous, i.e., Q ðtÞ is continuous in the uniform operator topology for t > 0; (H1) for each t 2 J 0 , the function f ðt; Þ : X ! X is continuous and for each x 2 CðJ 0 ; XÞ, the function f ð; xÞ : J 0 ! X is strongly measurable; (H2) there exists a function m 2 LðJ 0 ; Rþ Þ such that qp 0 Dt m
2 CðJ 0 ; Rþ Þ;
lim t1þqp 0 Dqp t mðtÞ ¼ 0
t!0þ
and 0 jf ðt; xÞj 6 mðtÞ for all x 2 BðqÞ and almost all t 2 ½0; a; r ðJ Þ
(H3) there exists a constant r > 0 such that
C2
! Z t ð1þqpÞ 1þqp 6 r: jx0 j þ sup t ðt sÞ mðsÞds t2ð0;a
0
For any x 2 Bqr ðJ 0 Þ, we define an operator T as follows
ðTxÞðtÞ ¼ t q1 Pq ðtÞx0 þ
Z
t
ðt sÞq1 P q ðt sÞf ðs; xðsÞÞds;
for t 2 ð0; a:
0
It is easy to see that limt!0þ t 1þqp ðTxÞðtÞ ¼ 0. For any y 2 BðJÞ, set
xðtÞ ¼ tð1þqpÞ yðtÞ; Then, x 2
BrðqÞ ðJ 0 Þ.
ðTyÞðtÞ ¼
for t 2 ð0; a:
Define T as follows
(
t 1þqp ðTxÞðtÞ; if t 2 ð0; a; 0;
if t ¼ 0:
Before giving the main results, we firstly prove the following lemmas. Lemma 4.3. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . Assume that (H0)–(H3) hold. Then the operator T : BðJÞ ! BðJÞ is equicontinuous provided x0 2 DðAb Þ with b > 1 þ p. Proof. For any y 2 BðJÞ, and t1 ¼ 0; 0 < t2 6 a, we get
Z
qð1þpÞ jðTyÞðt2 Þ ðTyÞð0Þj ¼
t 2 Pq ðt 2 Þx0 þ t 1þqp 2
t2
0
Z
qð1þpÞ
6 t 2 Pq ðt2 Þx0 þ
t21þqp
ðt2 sÞq1 Pq ðt 2 sÞf ðs; xðsÞÞds
t2
0
Z
qð1þpÞ
Pq ðt2 Þx0 þ C 2 t 21þqp 6 t 2
ðt 2 sÞq1 Pq ðt2 sÞf ðs; xðsÞÞds
t2
ðt 2 sÞð1þqpÞ mðsÞds ! 0;
as t 2 ! 0:
0
For 0 < t 1 < t 2 6 a, we have
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L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
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Z t2
qð1þpÞ
qð1þpÞ jðTyÞðt2 Þ ðTyÞðt 1 Þj 6 t 2 Pq ðt 2 Þx0 t 1 Pq ðt 1 Þx0 þ
t 21þqp ðt 2 sÞq1 Pq ðt 2 sÞf ðs; xðsÞÞds 0
Z t1
qð1þpÞ
qð1þpÞ q1
6 t t1þqp ðt sÞ P ðt sÞf ðs; xðsÞÞds Pq ðt2 Þx0 t 1 P q ðt1 Þx0
1 q 1 2 1
0
Z t2
Z t1
q1
þ
t 1þqp ðt sÞ P ðt sÞf ðs; xðsÞÞds t 1þqp ðt 2 sÞq1 P q ðt2 sÞf ðs; xðsÞÞds þ
2 q 2 2 2
t1 t1
0
Z t1
sÞ Pq ðt 2 sÞf ðs; xðsÞÞdsj þ
t 11þqp ðt 1 sÞq1 Pq ðt2 sÞf ðs; xðsÞÞds 0 0 Z t1
qð1þpÞ
qð1þpÞ t 11þqp ðt 1 sÞq1 P q ðt1 sÞf ðs; xðsÞÞdsj 6 t2 Pq ðt 2 Þx0 t1 Pq ðt 1 Þx0
0
Z t2
Z t1 h
i
q1 q1 q1 1þqp 1þqp
þ
t 1þqp ðt sÞ P ðt sÞf ðs; xðsÞÞds þ t ðt sÞ t ðt sÞ ðt sÞf ðs; xðsÞÞds P 2 q 2 2 1 q 2 2 2 1
Z
t1þqp ðt1 1
q1
t
0
Z 1t1
þ
t 1þqp ðt 1 sÞq1 ½Pq ðt 2 sÞf ðs; xðsÞÞ Pq ðt1 sÞf ðs; xðsÞÞds
¼: I0 þ I1 þ I2 þ I3 ; 1 0
By Lemma 3.2, it is easy to see that limt2 !t1 I0 ¼ 0. Since
Z
I1 6 C 2 t1þqp 2 þ C2
t1
Z
t1
lim
Z
t 2 !t 1
Thus, by
R t1
h
0
qp 0 Dt m
I2 6 C 2
t2
ðt 2 sÞð1þqpÞ mðsÞds t1þqp 1
Z 0
t1
ðt1 sÞð1þqpÞ mðsÞds
i
t1þqp ðt 1 sÞð1þqpÞ t 21þqp ðt 2 sÞð1þqpÞ mðsÞds; 1
t 1þqp ðt 1 sÞð1þqpÞ mðsÞds exists ðs 2 ð0; t1 Þ, then by the Lebesgue dominated convergence theorem, we have 1
0
t1
0
h
0
noting that
Z
ðt 2 sÞð1þqpÞ mðsÞds 6 C 2
t1þqp 2
t2
Z
i t1þqp ðt1 sÞð1þqpÞ t 21þqp ðt 2 sÞð1þqpÞ mðsÞds ¼ 0: 1
2 CðJ 0 ; Rþ Þ, one can deduce that limt2 !t1 I1 ¼ 0. Since
1þqp q1 q1
qð1þpÞ 1þqp mðsÞds;
t2 ðt2 sÞ t 1 ðt 1 sÞ ðt2 sÞ
t1
0
noting that
h i
1þqp q1 q1
qð1þpÞ 1þqp mðsÞ 6 t21þqp ðt 2 sÞð1þqpÞ þ t 1þqp ðt 1 sÞq1 ðt2 sÞqð1þpÞ mðsÞ
t 2 ðt 2 sÞ t1 ðt1 sÞ ðt 2 sÞ 1 h i ðt 1 sÞð1þqpÞ mðsÞ 6 t21þqp ðt 2 sÞð1þqpÞ þ t 1þqp 1 6 2t 11þqp ðt 1 sÞð1þqpÞ mðsÞ and
R t1 0
ð5Þ
t 1þqp ðt 1 sÞð1þqpÞ mðsÞds exists ðs 2 ð0; t1 Þ, then by Lebesgue’s dominated convergence theorem, we have 1
Z 0
t1
1þqp q1 q1
qð1þpÞ 1þqp mðsÞds ! 0 as t 2 ! t 1 ;
t 2 ðt 2 sÞ t 1 ðt 1 sÞ ðt 2 sÞ
then one can deduce that limt2 !t1 I2 ¼ 0. For e > 0 be enough small, we have
I3 6
Z
t 1 e
0
þ
Z
t 11þqp ðt 1 sÞq1 kPq ðt 2 sÞ Pq ðt 1 sÞkBðXÞ jf ðs; xðsÞÞjds
t1
t 1 e
6 t1þqp 1 þ C2
t 11þqp ðt 1 sÞq1 kPq ðt 2 sÞ Pq ðt 1 sÞkBðXÞ jf ðs; xðsÞÞjds
Z
Z
t1
ðt 1 sÞq1 mðsÞds sup kPq ðt2 sÞ Pq ðt 1 sÞkBðXÞ s2½0;t 1 e
0
h
t1
t 1 e
6
t11þqþ2qp þ 2C 2
Z
i t 11þqp ðt 1 sÞq1 ðt2 sÞqð1þpÞ þ ðt 1 sÞqð1þpÞ mðsÞds
Z
t1 0
t1 t 1 e
ðt 1 sÞð1þqpÞ mðsÞds sup kPq ðt 2 sÞ Pq ðt 1 sÞkBðXÞ s2½0;t 1 e
t 1þqp ðt1 sÞ 1
ð1þqpÞ
mðsÞds:
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L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
By (H0) and limt2 !t1 I1 ¼ 0, it is easy to see I3 tends to zero independently of y 2 BðJÞ as t 2 ! t 1 ; e ! 0. Therefore, jðTyÞðt2 Þ ðTyÞðt1 Þj tends to zero independently of y 2 BðJÞ as t 2 ! t 1 , which means that fTy : y 2 BðJÞg is equicontinuous. h Lemma 4.4. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . Assume that (H1)–(H3) hold. Then the operator T : BðJÞ ! BðJÞ is bounded and continuous provided x0 2 DðAb Þ with b > 1 þ p. Proof. Step I. T maps BðJÞ into BðJÞ. For any y 2 BðJÞ, let xðtÞ ¼ tð1þqpÞ yðtÞ. Then x 2 BrðqÞ ðJ 0 Þ. For t 2 ½0; a, by (H1)–(H3), we have
Z t Z t
jðTyÞðtÞj 6 jPq ðtÞx0 j þ t 1þqp
ðt sÞð1þqpÞ Pq ðt sÞf ðs; xðsÞÞds
6 C 2 jx0 j þ C 2 t 1þqp ðt sÞð1þqpÞ jf ðs; xðsÞÞjds 0 0 ! Z t
6 C 2 jx0 j þ sup t1þqp t2½0;a
ðt sÞð1þqpÞ mðsÞds
6 r:
0
Hence, kTyk 6 r, for any y 2 BðJÞ. Step II. T is continuous in BðJÞ. For any ym ; y 2 BðJÞ; m ¼ 1; 2; . . ., with limm!1 ym ¼ y, we have
lim ym ðtÞ ¼ yðtÞ and
m!1
lim t ð1þqpÞ ym ðtÞ ¼ tð1þqpÞ yðtÞ;
m!1
for t 2 ð0; a:
Then by (H1), we have
f ðt; xm ðtÞÞ ¼ f ðt; t ð1þqpÞ ym ðtÞÞ ! f ðt; t ð1þqpÞ yðtÞÞ ¼ f ðt; xðtÞÞ;
as m ! 1;
where xm ðtÞ ¼ tð1þqpÞ ym ðtÞ and xðtÞ ¼ tð1þqpÞ yðtÞ. On the one hand, using (H2), we get for each t 2 J 0 ,
ðt sÞð1þqpÞ jf ðs; xm ðsÞÞ f ðs; xðsÞÞj 6 ðt sÞð1þqpÞ 2mðsÞ; On the other hand, the function s ! ðt sÞ gence theorem, we get
Z
t
ð1þqpÞ
a:e: in ½0; t:
2mðsÞ is integrable for s 2 ½0; t and t 2 J. By Lebesgue’s dominated conver-
ðt sÞð1þqpÞ jf ðs; xm ðsÞÞ f ðs; xðsÞÞjds ! 0;
as m ! 1:
0
For t 2 ½0; a
jðTym ÞðtÞ ðTyÞðtÞj ¼ jt 1þqp ðTxm ðtÞ TxðtÞÞj
Z t
6 t 1þqp
ðt sÞð1þqpÞ Pq ðt sÞðf ðs; xm ðsÞÞ f ðs; xðsÞÞÞds
0
6 C2t
1þqp
Z
t
ðt sÞð1þqpÞ jf ðs; xm ðsÞÞ f ðs; xðsÞÞjds ! 0;
as m ! 1:
0
Therefore, Tym ! Ty pointwise on J as m ! 1, by which Lemma 4.3 implies that Tym ! Ty uniformly on J as m ! 1 and so T is continuous. h 4.1. The case Q ðtÞðt > 0Þ is compact Suppose that the operator almost sectorial A generates a compact semigroup Q ðtÞðt > 0Þ on X, that is, for any t > 0, the operator Q ðtÞ is compact. Furthermore, Pq ðtÞ is also compact operator for every t > 0. Theorem 4.1. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p=2. Assume that Q ðtÞðt > 0Þ is compact. Furthermore assume that b 0 (H1)–(H3) hold. Then the Cauchy problem (1) has at least one mild solution in BðqÞ r ðJ Þ for every x0 2 DðA Þ with b > 1 þ p. Proof. Since Q ðtÞðt > 0Þ is compact, from Theorem 2.3.2 [38], Q ðtÞðt > 0Þ is equicontinuous, which implies (H0) is satisfied. 0 Then, by Lemmas 4.3 and 4.4, we know that T : BrðqÞ ðJ 0 Þ ! BðqÞ r ðJ Þ is bounded and continuous, and T : BðJÞ ! BðJÞ is bounded, continuous and equicontinuous. Next, we will show that for any t 2 ½0; a, VðtÞ ¼ fðTyÞðtÞ; y 2 BðJÞg is relatively compact in X. Obviously, Vð0Þ is relatively compact in X. Let t 2 ð0; a be fixed. For 8e 2 ð0; tÞ and 8d > 0, define an operator Te;d on BðJÞ by the formula
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L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
ðTe;d yÞðtÞ ¼ t qð1þpÞ Pq ðtÞx0 þ qt1þqp
Z
te
Z
0
¼ tqð1þpÞ Pq ðtÞx0 þ qt 1þqp Q ðeq dÞ
Z
1
9
hðt sÞq1 M q ðhÞQ ððt sÞq hÞf ðs; xðsÞÞdhds
d te
Z
0
1
hðt sÞq1 Mq ðhÞQ ððt sÞq h eq dÞf ðs; xðsÞÞdhds;
d
0 where x 2 BðqÞ Then from the compactness of Pq ðtÞðt > 0Þ and Q ðeq dÞðeq d > 0Þ, we obtain that the set r ðJ Þ. V e;d ðtÞ ¼ fðTe;d yÞðtÞ; y 2 BðJÞg is relatively compact in X for 8e 2 ð0; tÞ and 8d > 0. Moreover, for every y 2 BðJÞ, we have
Z t Z d
jðTyÞðtÞ ðTe;d yÞðtÞj 6
qt 1þqp hðt sÞq1 M q ðhÞQ ððt sÞq hÞf ðs; xðsÞÞdhds
0 0
Z t Z 1
þ
qt1þqp hðt sÞq1 M q ðhÞQ ððt sÞq hÞf ðs; xðsÞÞdhds
te
d
Z 1 ðt sÞð1þqpÞ mðsÞds hp Mq ðhÞdh 0 0 te 0 Z t Z d Z qC Cð1 pÞ 1þqp t ðt sÞð1þqpÞ mðsÞds hp M q ðhÞdh þ 0 t ðt sÞð1þqpÞ mðsÞds 6 qC 0 t 1þqp Cð1 qpÞ 0 0 te
6 qC 0 t 1þqp
! 0;
as
Z
t
ðt sÞð1þqpÞ mðsÞds
Z
d
hp M q ðhÞdh þ qC 0 t1þqp
Z
t
e ! 0; d ! 0:
Therefore, there are relatively compact sets arbitrarily close to the set VðtÞ; t > 0. Hence the set VðtÞ; t > 0 is also relatively compact in X. Therefore, fTy; y 2 BðJÞg is relatively compact by Ascoli–Arzela Theorem. Thus, the continuity of T and relatively compactness of fTy; y 2 BðJÞg imply that T is a completely continuous operator. Schauder fixed point theorem shows that T has a fixed point in y 2 BðJÞ. Let x ðtÞ ¼ tq1 y ðtÞ. Then x is a mild solution of (1). h The condition (H2) can be replaced by the following condition. q 1 p
(H2)0 There exist a constant q1 2 ð0; qÞ and a positive function m 2 L
1
ðJ; Rþ Þ such that
0 jf ðt; xðtÞÞj 6 mðtÞ for all x 2 BðqÞ and a:e: t 2 ð0; a: r ðJ Þ
Corollary 4.1. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p=2. Let all the assumption of Theorem 4.1 be given except (H2). b 0 Assume that (H2)0 holds. Then the Cauchy problem (1) has at least one mild solution in BðqÞ r ðJ Þ for every x0 2 DðA Þ with b > 1 þ p. Proof. In fact, if (H2)0 holds, by using Hölder inequality, for any t 1 ; t 2 2 J 0 and t 1 < t2 , we obtain
Z t1 Z t2
1
ðt 2 sÞð1þqpÞ ðt1 sÞð1þqpÞ mðsÞds þ ðt 2 sÞð1þqpÞ mðsÞds
CðqpÞ 0 t1 Z t1 1þq1 p Z t1 q1 p 1 q 1 p ð1þqpÞ ð1þqpÞ 1þq1 p ððt1 sÞ ðt 2 sÞ Þ ds ðmðsÞÞ 1 ds
qp jðIqp 0þ mÞðt 2 Þ ðI0þ mÞðt 1 Þj ¼
1
6
CðqpÞ þ
1
CðqpÞ
1 CðqpÞ
6
0
Z Z 0
1þq1 p 1þq1 p Z 1 ðt2 sÞð1þqpÞ ds
t1 t1
Z
1 CðqpÞ kmk 1
þ
t2
0
t2
q 1 p 1
ðmðsÞÞ
q1 p ds
t1
1þq1 p ð1þqpÞ ð1þqpÞ ððt 1 sÞ 1þq1 p ðt2 sÞ 1þq1 p Þds kmk 1þq1 p t2 ð1þqpÞ ðt 2 sÞ 1þq1 p ds kmk
t1
q 1 p 1
L
ð6Þ 1 L q1 p
!1þq1 p 1þq1 p ðq1 qÞp ðq1 qÞp ðq1 qÞp 1 þ q1 p 1þq p 1þq p t 1 1 þ ðt 2 t1 Þ 1þq1 p t 2 1 CðqpÞ ðq1 qÞp 1þq1 p kmk q 1p 1 þ q p 1þq1 p ðq1 qÞp 1 L 1 þ ðt 2 t 1 Þ 1þq1 p CðqpÞ ðq1 qÞp 2kmk q 1p 1 þ q p 1þq1 p 1 L 1 6 ðt2 t1 Þðq1 qÞp ! 0; as t2 ! t 1 ; CðqpÞ ðq1 qÞp L
6
q p 1
where
kmk
1 L q1 p
¼
Z
a
q 1 p
ðmðtÞÞ
1
dt
q1 p :
0
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On the other hand,
t1þqp
Z
t
ðt sÞð1þqpÞ mðsÞds 6 t1þqp
0
Z 0
6
1þq1 p Z t q1 p ð1þqpÞ 1 ðt sÞ 1þq1 p ds ðmðsÞÞ q1 p ds
t
1þq1 p
1 þ q1 p ðq1 qÞp
0
t 1þq1 p kmk
L
q 1 p 1
! 0;
as t ! 0:
ð7Þ
0 þ 1þqp qp Thus, (6) and (7) mean that Iqp ðI0þ mÞðtÞ ¼ 0. Hence, (H2) holds. By Theorem 4.1, the Cauchy 0þ m 2 CðJ ; R Þ, and limt!0þ t ðqÞ 0 problem (1) has at least one mild solution in Br ðJ Þ. h
4.2. The case Q ðtÞ is noncompact If Q ðtÞ is noncompact, we give an assumption as follows. (H4) There exists a constant ‘ > 0 such that for any bounded D X,
aðf ðt; DÞÞ 6 ‘aðDÞ; for a:e: t 2 ½0; a:
Theorem 4.2. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . Assume that (H0)–(H4) hold. Then the Cauchy problem (1) has at b 0 least one mild solution in BðqÞ r ðJ Þ for every x0 2 DðA Þ with b > 1 þ p. Proof. By Lemmas 4.3 and 4.4, we have T : BðJÞ ! BðJÞ is bounded, continuous and equicontinuous. Next, we will show that T is compact in a subset of BðJÞ. For each bounded subset B0 BðJÞ, set
T1 ðB0 Þ ¼ TðB0 Þ; Tn ðB0 Þ ¼ T coðTn1 ðB0 ÞÞ ; From Propositions 2.2–2.4, for any
n ¼ 2; 3; . . . : 1
e > 0, there is a sequence fyð1Þ n gn¼1 B0 such that
Z t 1 þe aðT1 ðB0 ðtÞÞÞ ¼ aðTðB0 ðtÞÞÞ 6 2a t1þqp ðt sÞq1 Pq ðt sÞf ðs; fsð1þqpÞ yð1Þ ðsÞg Þds n n¼1 6 4C 2 t 1þqp
0
Z
t
ðt sÞ
0
6 4C 2 ‘t1þqp aðB0 Þ
Z
t
ð1þqpÞ
1
ðt sÞð1þqpÞ sð1þqpÞ ds þ e 6
0
Since
a f ðs; fsð1þqpÞ yð1Þ n ðsÞgn¼1 Þ ds þ e 4C 2 ‘C2 ðqpÞtqp aðB0 Þ þ e: Cð2qpÞ
e > 0 is arbitrary, we have
aðT1 ðB0 ðtÞÞÞ 6
4C 2 ‘C2 ðqpÞt qp aðB0 Þ: Cð2qpÞ
From Propositions 2.2–2.4, for any
1
1 e > 0, there is a sequence fyð2Þ n gn¼1 coðT ðB0 ÞÞ such that
aðT2 ðB0 ðtÞÞÞ ¼ aðTðcoðT1 ðB0 ðtÞÞÞÞÞ
Z t 1 ðt sÞq1 Pq ðt sÞf s; fsð1þqpÞ ynð2Þ ðsÞgn¼1 ds þ e 6 2a t 1þqp
6 4C 2 t
1þqp
Z
0 t
0
6 4C 2 ‘t
1þqp
6 4C 2 ‘t
1þqp
Z
1 ðt sÞð1þqpÞ a f s; fsð1þqpÞ ynð2Þ ðsÞgn¼1 ds þ e t
1 ðt sÞð1þqpÞ a fsð1þqpÞ ynð2Þ ðsÞgn¼1 ds þ e
t
1 ðt sÞð1þqpÞ sð1þqpÞ a fynð2Þ ðsÞgn¼1 ds þ e
0
Z
0
ð4C 2 ‘Þ2 t1þqp C2 ðqpÞtqp 6 aðB0 Þ Cð2qpÞ ¼
Z
t
ðt sÞð1þqpÞ sð1þ2qpÞ ds þ e
0
ð4C 2 ‘Þ2 C3 ðqpÞ 2qp t aðB0 Þ þ e: Cð3qpÞ
2 Nþ , It can be shown, by mathematical induction, that for every n
aðTn ðB0 ðtÞÞÞ 6
ð4C 2 ‘Þn Cnþ1 ðqpÞ nqp t aðB0 Þ: Cððn þ 1ÞqpÞ
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Since
lim
!1 n
ð4C 2 ‘aqp Þn Cnþ1 ðqpÞ ¼ 0; Cððn þ 1ÞqpÞ
^ such that there exists a positive integer n ^
^
ð4C 2 ‘Þn Cnþ1 ðqpÞ n^qp ð4C 2 ‘aqp Þn Cnþ1 ðqpÞ t 6 ¼ k < 1: Cððn^ þ 1ÞqpÞ Cððn^ þ 1ÞqpÞ ^
^
Then
aðTn^ ðB0 ðtÞÞÞ 6 kaðB0 Þ: ^
We know from Proposition 2.1, Tn ðB0 ðtÞÞ is bounded and equicontinuous. Then, from Proposition 2.2, we have
aðTn^ ðB0 ÞÞ ¼ max aðTn^ ðB0 ðtÞÞÞ: t2½0;a
Hence
aðTn^ ðB0 ÞÞ 6 kaðB0 Þ: Let
D0 ¼ BðJÞ;
^
^
D1 ¼ coðTn ðDÞÞ; . . . ; Dn ¼ coðTn ðDn1 ÞÞ;
n ¼ 2; 3; . . .
From [40], we can get (i) D0 D1 D2 . . . Dn1 Dn . . .; (ii) limn!1 aðDn Þ ¼ 0. b ¼ T1 Dn is a nonempty, compact and convex subset in BðJÞ. Then D n¼0 ^ D. ^ Firstly, we show By using the similar method in [40], we will prove TðDÞ
TðDn Þ Dn ;
n ¼ 0; 1; 2; . . .
1
ð8Þ 1
From T ðD0 Þ ¼ TðD0 Þ D0 , we know coðT ðD0 ÞÞ D0 . Therefore
T2 ðD0 Þ ¼ TðcoðT1 ðD0 ÞÞÞ TðD0 Þ ¼ T1 ðD0 Þ; T3 ðD0 Þ ¼ TðcoðT2 ðD0 ÞÞÞ TðcoðT1 ðD0 ÞÞÞ ¼ T2 ðD0 Þ; ^
^
^
^
Tn ðD0 Þ ¼ TðcoðTn1 ðD0 ÞÞÞ TðcoðTn2 ðD0 ÞÞÞ ¼ Tn1 ðD0 Þ: ^
^
^
^
^
Hence, D1 ¼ coðTn ðD0 ÞÞ coðTn1 ðD0 ÞÞ, so TðD1 Þ TðcoðTn1 ðD0 ÞÞÞ ¼ Tn ðD0 Þ coðTn ðD0 ÞÞ ¼ D1 . Employing the same b T1 TðDn Þ T1 Dn ¼ D. b Then Tð DÞ b is compact. method, we can prove TðDn Þ Dn ðn ¼ 0; 1; 2; . . .Þ. By (8), we get Tð DÞ n¼0 n¼0 Hence, Schauder fixed point theorem shows that T has a fixed point y 2 BðJÞ. Let x ðtÞ ¼ t q1 y ðtÞ. Then x is a mild solution of (1). h Corollary 4.2. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . Let all the assumption of Theorem 4.2 be given except (H2). Assume b 0 that (H2)0 holds. Then the Cauchy problem (1) has at least one mild solution in BðqÞ r ðJ Þ for every x0 2 DðA Þ with b > 1 þ p. The proof of Corollary 4.2 is similar to that of Corollary 4.1, it is thus omitted. In the following, we also give the existence and uniqueness result which is based on Banach contraction principle. We will need the following assumption. 0 (H5) There exists a constant k > 0 such that for any x; y 2 BðqÞ r ðJ Þ, and t 2 ð0; a we have
jf ðt; xðtÞÞ f ðt; yðtÞÞj < kkx ykq : Theorem 4.3. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . If assumptions (H2), (H3) and (H5) are satisfied. Then, for every x0 2 DðAb Þ with b > 1 þ p, the Cauchy problem (1) has a unique mild solution in BrðqÞ ðJ 0 Þ provided
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L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
C 2 ak < 1: qp
ð9Þ
Proof. By the proof of Lemma 4.4, we know that T is an operator from BðJÞ into itself. For any y1 ; y2 2 BðJÞ, let x1 ðtÞ ¼ t ð1þqpÞ y1 ðtÞ and x2 ðtÞ ¼ t ð1þqpÞ y2 ðtÞt 2 ð0; a, we have
jðTy1 ÞðtÞ ðTy2 ÞðtÞj ¼ t1þqp jðTx1 ÞðtÞ ðTx2 ÞðtÞj
Z t
¼ t1þqp
ðt sÞq1 Pq ðt sÞ½f ðs; x1 ðsÞÞ f ðs; x2 ðsÞÞds
0 Z t 6 C 2 t1þqp ðt sÞð1þqpÞ jf ðs; x1 ðsÞÞ f ðs; x2 ðsÞÞjds 0
C 2 tk 6 kx1 x2 kq qp C 2 ak ¼ ky1 y2 k; qp which implies
jTy1 Ty2 j 6
C 2 ak ky1 y2 k; qp
which means that T is a contraction according to (9). By applying Banach contraction principle, we know that T has a unique fixed point y 2 BðJÞ. Let x ðtÞ ¼ tq1 y ðtÞ. Then x is a mild solution of (1). h Corollary 4.3. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . Let all the assumption of Theorem 4.3 be given except (H2). Assume that (H2)0 holds. Then the Cauchy problem (1) has a unique mild solution in BrðqÞ ðJ 0 Þ for every x0 2 DðAb Þ with b > 1 þ p. The proof of Corollary 4.3 is similar to that of Corollary 4.1, it is thus omitted.
5. An example Let X be a bounded domain in RN ðN P 0Þ with boundary @ X of class C 4 . Let X ¼ C k ðXÞð0 < k < 1Þ. Set
A ¼ D;
DðAÞ ¼ fu 2 C 2þk ðXÞ : uð0Þ ¼ 0 on @ Xg:
It follows from [36] (Example 2.3) that there exist 1; e > 0, such that k 1
A þ 1 2 H2pe ðC k ðXÞÞ: 2
Consider the following fractional initial-boundary value problem
8 L q > < ð D0þ xÞðt; zÞ ¼ Dxðt; zÞ þ f ðt; xðtÞÞ; uj@X ¼ 0 > : 1q ðI0þ xÞð0; zÞ ¼ x0 ðzÞ; z 2 X
ppt 2 ½0; a; z 2 X ð10Þ
1
in the space X, where q ¼ 12, f ðt; xðtÞÞ ¼ t 3 sin xðtÞ. Then problem (10) can be written abstractly as
8 1 < ðL D20þ xÞðtÞ ¼ AxðtÞ þ t13 sin xðtÞ; 1 : I2 x ð0Þ ¼ x : 0 0þ
ppt 2 ð0; a;
We choose 1
mðtÞ ¼ t3
and r ¼ C 2 jx0 j þ
2 C 2 a3 C 2p C 23 p 2 : C 2 þ 3 ð1Þ
Then, the conditions (H1)–(H3) are satisfied. According to Theorem 4.1, system (10) has a mild solution on Br2 ðð0; aÞ. References [1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. [2] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. [3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
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Zhou, F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal.: RWA 11 (2010) 4465–4475. J. Wang, Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal.: RWA 12 (2011) 262–272. J.R. Wang, M. Fec˘kan, Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ. 8 (2011) 345–361. S. Kumar, N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Differ. Equ. 252 (2012) 6163–6174. R. Wang, D. Chen, T.J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differ. Equ. 252 (2012) 202–235. H. Fan, J. Mu, Initial value problem for fractional evolution equations, Adv. Differ. Equ. 49 (2012). F. Xiao, Nonlocal Cauchy problem for nonautonomous fractional evolution equations, Adv. Differ. Equ. 2011 (2011) 1–17. X.B. Shu, Y.Z. Lai, Y.M. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal.: TMA 74 (2011) 2003–2011. K. Li, J. Peng, J. Jia, Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives, J. Funct. Anal. 263 (2012) 476– 510. M.M. Fall, T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal. 263 (2012) 2205–2227. A.Z.-A.M. Tazali, Local existence theorems for ordinary differential equations of fractional order, Ordinary and Partial Differential Equations, Lecture Notes in Mathematics, 964, Springer, Dundee, 1982. pp. 652–665. J. Banas`, K. Goebel, Measure of Noncompactness in Banach Spaces, Marcel Dekker Inc., New York, 1980. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985. H.-P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal.: TMA 7 (1983) 1351–1371. V. Lakshmikantham, S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, New York, 1969. D.J. Guo, V. Lakshmikantham, X.Z. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic, Dordrecht, 1996. H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal.: TMA 4 (1980) 985–999. D. Bothe, Multivalued perturbation of m-accretive differential inclusions, Israel. J. Math. 108 (1998) 109–138. J.R. Wang, Y. Zhou, M. Fec˘kan, Abstract Cauchy problem for fractional differential equations, Nonlinear Dyn. 71 (2013) 685–700. H. Markus, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, vol. 69, Birkhauser-Verlag, Basel, 2006. F. Periago, B. Straub, A functional calculus for almost sectorial operators and applications to abstract evolution equations, J. Evol. Equ. 2 (2002) 41–68. A.N. Carvalho, T. Dlotko, M.J.D. 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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Fractional Cauchy problems with almost sectorial operators q Lu Zhang, Yong Zhou ⇑ Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, PR China
a r t i c l e
i n f o
a b s t r a c t This paper concerns the abstract Cauchy problem of fractional evolution equations with almost sectorial operators. The suitable mild solutions of evolution equations with Riemann–Liouville derivative and Caputo derivative are introduced respectively. The existence theorems of mild solutions of Cauchy problems are established. The results obtained here improve and generalize some known results. Finally, an example is given for demonstration. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Fractional evolution equations Almost sectorial operator Riemann–Liouville derivative Caputo derivative Measure of noncompactness Cauchy problem
1. Introduction Fractional differential equations have proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic. There has been a significant development in fractional differential equations in recent years, see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3], Lakshmikantham et al. [4], Tarasov [5], Zhou [6] and the references therein. In this paper, we assume that X is a Banach space with the norm j j. Let a 2 Rþ ; J ¼ ½0; a and J 0 ¼ ð0; a. Denote CðJ; XÞ be the Banach space of continuous functions from J into X with the norm kxk ¼ supt2½0;a jxðtÞj, where x 2 CðJ; XÞ, and BðXÞ be the space of all bounded linear operators from X to X with the norm kQ kBðXÞ ¼ supfjQ ðxÞj : jxj ¼ 1g, where Q 2 BðXÞ and x 2 X. Consider the following Cauchy problem of fractional evolution equations with Riemann–Liouville derivative
(
ðL Dq0þ xÞðtÞ ¼ AxðtÞ þ f ðt; xðtÞÞ; ðI1q 0þ xÞð0Þ
almost all t 2 ½0; a;
¼ x0 ;
ð1Þ
where L Dq0þ is Riemann–Liouville derivative of order q, I1q 0þ is Riemann–Liouville integral of order 1 q; 0 < q < 1, A is an almost sectorial operator on a complex Banach space, and f : J X ! X is a given function. The existence of mild solutions for the Cauchy problem of fractional evolution equations has been considered in several recent papers (see [7–25,34]), much less is known about the fractional evolution equations with Riemann–Liouville derivative. As pointed out in [15], the results in papers [7–14] are incorrect since the considered concept of mild solution is not appropriate. In [16], we introduce a concept of mild solution for fractional evolution equations and we study the existence and uniqueness of mild solutions. Our results have been extended and applied to Cauchy problems of many kinds of fractional evolution equations (see papers [17–22], etc.).
q Project supported by National Natural Science Foundation of PR China (11271309), the Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001) and Hunan Provincial Natural Science Foundation of China (12JJ2001). ⇑ Corresponding author. E-mail addresses: [email protected] (L. Zhang), [email protected] (Y. Zhou).
http://dx.doi.org/10.1016/j.amc.2014.07.024 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
In most of the existed articles, Schauder’s fixed point theorem, Krasnoselskii’s fixed point theorem or Darbo’s fixed point theorem, Kuratowski measure of noncompactness are employed to obtain the fixed points of the solution operator of the Cauchy problems under some restrictive conditions. In order to show that the solution operator is compact, a very common approach is to use Arzela–Ascoli’s theorem. However, it is difficult to check the relatively compactness of the solution operator and the equicontinuity of certain family of functions which is given by the solution operator. In this paper, by using the theory of Hausdorff measure of noncompactness, we study the Cauchy problems of fractional evolution equations with almost sectorial operators. In the next Section, we recall some notations and useful concepts from fractional calculus and theory of measure of noncompactness. In Section 3, we introduce some basic propositions of almost sectorial operators. In Section 4, we give a appropriate definition on the mild solution and existence theorems of the problem (1). Finally, an example is given for demonstration. 2. Fractional derivatives and measure of noncompactness In this section, we introduce preliminary facts which are used throughout this paper. Firstly, we recall some basic definitions and properties of the fractional calculus. For more details, see Kilbas et al. [1]. Definition 2.1 [1]. The fractional integral Iq0þ f of order q for a function f 2 AC½0; 1Þ is defined as
ðIq0þ f ÞðtÞ ¼
1 CðqÞ
Z
0
t
f ðsÞ ðt sÞ1q
ds;
t > 0; 0 < q < 1
provided the right side is point-wise defined on ½0; 1Þ, where CðÞ is the gamma function. Definition 2.2 [1]. Riemann–Liouville derivative L Dq0þ f of order q for a function f 2 AC½0; 1Þ can be written as
ðL Dq0þ f ÞðtÞ ¼
1 d Cð1 qÞ dt
Z
t
0
f ðsÞ ds; ðt sÞq
t > 0; 0 < q < 1:
Remark 2.1. (i) If f ðtÞ 2 C 1 ½0; 1Þ, then
ðC Dq0þ f ÞðtÞ ¼
1 Cð1 qÞ
Z 0
t
0
f ðsÞ 0 ds ¼ ðI1q 0þ ½f ðsÞÞðtÞ; ðt sÞq
t > 0; 0 < q < 1:
(ii) If f is an abstract function with values in X, then integrals which appear in Definitions 2.1 and 2.2 are taken in Bochner’s sense.
Lemma 2.1 [26]. (i) Let n; g 2 R such that g > 1. If t > 0, then
In0þ
( nþg t ; sg ðtÞ ¼ Cðnþgþ1Þ Cðg þ 1Þ 0;
if n þ g – n if n þ g ¼ n
ðn 2 Nþ Þ:
(ii) Let n > 0 and g 2 Lðð0; aÞ; XÞ. Define
Gn ðtÞ ¼ In0þ g;
for t 2 ð0; aÞ;
then g ðIg0þ Gn ÞðtÞ ¼ ðInþ 0þ gÞðtÞ;
g > 0; ppt 2 ½0; a;
where ðppt 2 ½0; aÞ is to be read as ‘‘almost all t 2 ½0; a’’. Next, we recall some definitions and properties of measure of noncompactness. The measure of noncompactness a is said to be: (i) Monotone if for all bounded subsets B1 ; B2 of X; B1 # B2 implies aðB1 Þ 6 aðB2 Þ; Please cite this article in press as: L. Zhang, Y. Zhou, Fractional Cauchy problems with almost sectorial operators, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.07.024
L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
3
(ii) Nonsingular if aðflg [ BÞ ¼ aðBÞ for every l 2 X and every nonempty subset B # X; (iii) Regular if aðBÞ ¼ 0 if and only if B is relatively compact in X. One of the most important examples of measure of noncompactness is the Hausdorff measure of noncompactness a defined on each bounded subset B of X by
(
aðBÞ ¼ inf e > 0 : B
m [
) Be ðxj Þ where xj 2 X ;
j¼1
where Be ðxj Þ is a ball of radius 6 e centered at xj ; j ¼ 1; 2; . . . ; m. Without confusion, Kuratowski measure of noncompactness a1 defined on each bounded subset B of X by
(
a1 ðBÞ ¼ inf
)
m [ e > 0 : B Mj
and diam ðM j Þ 6 e ;
j¼1
where the diameter of M j is defined by diamðM j Þ ¼ supfkx yk : x; y 2 M j g; j ¼ 1; 2; . . . ; m. It is well known that the Hausdorff measure of noncompactness a and Kuratowski measure of noncompactness a1 enjoy the above properties (i)–(iii) and other properties (see [27–30]). (iv) aðB1 þ B2 Þ 6 aðB1 Þ þ aðB2 Þ, where B1 þ B2 ¼ fx þ y : x 2 B1 ; y 2 B2 g; (v) aðB1 [ B2 Þ 6 maxfaðB1 Þ; aðB2 Þg; (vi) aðkBÞ 6 jkjaðBÞ for any k 2 R; In particular, the relationship of Hausdorff measure of noncompactness a and Kuratowski measure of noncompactness a1 is given by (vii) aðBÞ 6 a1 ðBÞ 6 2aðBÞ. For any W CðJ; XÞ, we define
Z
t
WðsÞds ¼
0
Z
t
uðsÞds : u 2 W ;
for t 2 J;
0
where WðsÞ ¼ fuðsÞ 2 X : u 2 Wg. Proposition 2.1. If W CðJ; XÞ is bounded and equicontinuous, then coW CðJ; XÞ is also bounded and equicontinuous. Proposition 2.2 [31]. If W CðJ; XÞ is bounded and equicontinuous, then t ! aðWðtÞÞ is continuous on J, and
Z
aðWÞ ¼ maxaðWðtÞÞ; a t2J
t
Z t WðsÞds 6 aðWðsÞÞds;
0
for t 2 J:
0
~ Proposition 2.3 [32]. Let fun g1 n¼1 be a sequence of Bochner integrable functions from J into X with kun ðtÞk 6 mðtÞ for almost all 1 þ ~ 2 L1 ðJ; Rþ Þ, then the function wðtÞ ¼ aðfun ðtÞg1 t 2 J and every n P 1, where m Þ belongs to L ðJ; R Þ and satisfies n¼1
Z
a
t
un ðsÞds : n P 1
0
62
Z
t
wðsÞds: 0
Proposition 2.4 [33]. If W is bounded, then for each
e > 0, there is a sequence fun g1 n¼1 W, such that
aðWÞ 6 2aðfun g1 n¼1 Þ þ e: 3. Almost sectorial operators We firstly introduce some special functions and classes of functions which will be used in the following, for more details, we refer to [35,36]. Let 1 < p < 0, and let S0l with 0 < l < p be the open sector
fz 2 C n f0g : j arg zj < lg and Sl be its closure, that is
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L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
Sl ¼ fz 2 C n f0g : j arg zj 6 lg
[ f0g:
As in [37], we state the concept of almost sectorial operators as follows. Definition 3.1. Let 1 < p < 0 and 0 < x < p2 . By Hpx ðXÞ we denote the family of all linear closed operators A : DðAÞ X ! X which satisfy S (i) rðAÞ Sx ¼ fz 2 C n f0g : j arg zj 6 xg f0g and (ii) for every x < l < p there exists a constant C l such that
kRðz; AÞkBðXÞ 6 C l jzjp ;
for all z 2 C n Sl :
ð2Þ
where Rðz; AÞ ¼ ðzI AÞ1 ; z 2 qðAÞ, which are bounded linear operators the resolvent of A. A linear operator A will be called an almost sectorial operator on X if A 2 Hpx ðXÞ.
Remark 3.1. Let A 2 Hpx ðXÞ. Then the definition implies that 0 2 qðAÞ. We denote the semigroup associated with A by fQ ðtÞgtP0 . For t 2 S0px
QðtÞ ¼ e
tz
1 ðAÞ ¼ 2pi
Z
2
e
tz
Rðz; AÞdz;
Ch
where the integral contour Ch ¼ fRþ eih g lytic semigroup of growth order 1 þ p.
S þ ih fR e g is oriented counter-clockwise and x < h < l < p2 j arg tj, forms an ana-
Remark 3.2. We say that the estimate (2) in Definition 3.1 is ‘‘deficient’’ since p > 1. From [36], note in particular that if A 2 Hpx ðXÞ, then A generates a semigroup Q ðtÞ with a singular behavior at t ¼ 0 in a sense, called semigroup of growth 1 þ p. Moreover, the semigroup Q ðtÞ is analytic in an open sector of the complex plane C, but the strong continuity fails at t ¼ 0 for data which are not sufficiently smooth. Hence, it is impossible to apply to A the general results and techniques on generation of strongly continuous operator semigroup, as it is developed in [38]. Proposition 3.1 [36]. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . Then the following properties remain true. (i) Q ðtÞ is analytic in S0px and 2
dn dtn
Q ðtÞ ¼ ðAÞn Q ðtÞðt 2 S0px Þ; 2
(ii) the functional equation Q ðs þ tÞ ¼ Q ðsÞQ ðtÞ for all s; t 2 S0px holds; 2
(iii) there is a constant C 0 ¼ C 0 ðpÞ > 0 such that kQ ðtÞkBðXÞ 6 C 0 tp1 ðt > 0Þ; (iv) if b > 1 þ p, then DðAb Þ RQ ¼ fx 2 X : limt!0þ Q ðtÞx ¼ xg; R1 (v) Rðk; AÞ ¼ 0 ekt Q ðtÞdt for every k 2 C with Rek > 0. Consider the function of Wright-type (see [39])
Mq ðhÞ ¼
1 X
ðhÞn1 ; ðn 1Þ!Cð1 qnÞ n¼1
h2C
with 0 < q < 1. For 1 < d < 1; k > 0, the following results hold (see [39]). R1 ð1þdÞ (i) 0 hd M q ðhÞdh ¼ CCð1þqdÞ ; R 1 q kh 1 q (ii) 0 hqþ1 e M q hq dh ¼ ek . Define operator families fSq ðtÞgjt2S0p ; fP q ðtÞgjt2S0p
Sq ðtÞ ¼ Pq ðtÞ ¼
Z
1
2
q
M q ðhÞQ ðt hÞdh;
Z0 1
x
for t 2
2
x
by
S0px ; 2
for t 2 S0px :
qhMq ðhÞQðt q hÞdh;
2
0
Lemma 3.1 [20]. For each fixed t 2 S0px ; Sq ðtÞ and P q ðtÞ are linear and bounded operators on X. Moreover, for all t > 0 2
jSq ðtÞj 6 C 1 t qð1þpÞ
and jPq ðtÞj 6 C 2 t qð1þpÞ ;
CðpÞ ð1pÞ where C 1 ¼ C 0 Cð1qð1þpÞÞ ; C 2 ¼ qC 0 CCð1qpÞ :
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Lemma 3.2. For t > 0; Sq ðtÞ and Pq ðtÞ are strongly continuous, which means that, for any x 2 X and 0 < t 0 < t00 6 a, we have
jSq ðt00 Þx Sq ðt 0 Þxj ! 0;
jPq ðt00 Þx Pq ðt 0 Þxj ! 0;
as t 00 ! t0 :
Lemma 3.3. Let b > 1 þ p. For all x 2 DðAb Þ, we have
lim Sq ðtÞx ¼ x and lim P q ðtÞx ¼
t!0þ
t!0þ
x
CðqÞ
:
Proof. For any x 2 X, we have
Sq ðtÞx x ¼
Z
1
M q ðhÞðQðt q hÞx xÞdh
0
and
Pq ðtÞx
x ¼ CðqÞ
Z
1
qhM q ðhÞðQ ðt q hÞx xÞdh:
0
On the other hand, (iv) of Proposition 3.1 it follows that DðAb Þ RQ in view of b > 1 þ p. Therefore, we deduce, using (iii) of Proposition 3.1, that for any x 2 DðAb Þ, there exists a function gðhÞ 2 Lð0; 1Þ depending on M q ðhÞ such that
jMq ðhÞðQ ðtq hÞx xÞj 6 gðhÞ: Hence, by means of the Lebesgue dominated convergence theorem we obtain
Sq ðtÞx x ! 0 and Pq ðtÞx The proof is complete.
x
CðqÞ
! 0 as t ! 0 þ :
h
4. Existence of problem (1) Before presenting the definition of mild solution of problem (1), we firstly prove the following lemmas. Lemma 4.1. The Cauchy problem (1) is equivalent to the integral equation
xðtÞ ¼
x0 q1 1 t þ CðqÞ CðqÞ
Z
t
ðt sÞq1 ½AxðsÞ þ f ðs; xðsÞÞds;
for t 2 ð0; a
ð3Þ
for t > 0
ð4Þ
0
provided that the integral in (3) exists. Lemma 4.2. If
xðtÞ ¼
x0 q1 1 t þ CðqÞ CðqÞ
Z
t
ðt sÞq1 ½AxðsÞ þ f ðs; xðsÞÞds;
0
holds, then we have
xðtÞ ¼ tq1 Pq ðtÞx0 þ
Z
t
ðt sÞq1 Pq ðt sÞf ðs; xðsÞÞds;
for t > 0:
0
Due to Lemma 4.2, we give the following definition of the mild solution of (1). Definition 4.1. By the mild solution of the Cauchy problem (1), we mean that the function x 2 CðJ 0 ; XÞ which satisfies
xðtÞ ¼ tq1 Pq ðtÞx0 þ
Z
t
ðt sÞq1 Pq ðt sÞf ðs; xðsÞÞds;
for t 2 ð0; a:
0
Define
X ðqÞ ðJ 0 Þ ¼ ðqÞ
x 2 CðJ 0 ; XÞ : lim t 1þqp xðtÞ exists and is finite : t!0þ
0
For any x 2 X ðJ Þ, let the norm k kq defined by
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L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
kxkq ¼ sup t1þqp jxðtÞj : t2ð0;a
Then ðX ðqÞ ðJ 0 Þ; k kq Þ is a Banach space. For r > 0, define a closed subset BrðqÞ ðJ 0 Þ X ðqÞ ðJ 0 Þ as follows
n o 0 ðqÞ 0 BðqÞ r ðJ Þ ¼ x 2 X ðJ Þ : kxkq 6 r :
0 ðqÞ 0 Thus, BðqÞ ðJ Þ. r ðJ Þ is a bounded closed and convex subset of X Let BðJÞ be the closed ball of the space CðJ; XÞ with radius r and center at 0, that is
BðJÞ ¼ fy 2 CðJ; XÞ : kyk 6 rg: Thus BðJÞ is a bounded closed and convex subset of CðJ; XÞ. We introduce the following hypotheses: (H0) Q ðtÞðt > 0Þ is equicontinuous, i.e., Q ðtÞ is continuous in the uniform operator topology for t > 0; (H1) for each t 2 J 0 , the function f ðt; Þ : X ! X is continuous and for each x 2 CðJ 0 ; XÞ, the function f ð; xÞ : J 0 ! X is strongly measurable; (H2) there exists a function m 2 LðJ 0 ; Rþ Þ such that qp 0 Dt m
2 CðJ 0 ; Rþ Þ;
lim t1þqp 0 Dqp t mðtÞ ¼ 0
t!0þ
and 0 jf ðt; xÞj 6 mðtÞ for all x 2 BðqÞ and almost all t 2 ½0; a; r ðJ Þ
(H3) there exists a constant r > 0 such that
C2
! Z t ð1þqpÞ 1þqp 6 r: jx0 j þ sup t ðt sÞ mðsÞds t2ð0;a
0
For any x 2 Bqr ðJ 0 Þ, we define an operator T as follows
ðTxÞðtÞ ¼ t q1 Pq ðtÞx0 þ
Z
t
ðt sÞq1 P q ðt sÞf ðs; xðsÞÞds;
for t 2 ð0; a:
0
It is easy to see that limt!0þ t 1þqp ðTxÞðtÞ ¼ 0. For any y 2 BðJÞ, set
xðtÞ ¼ tð1þqpÞ yðtÞ; Then, x 2
BrðqÞ ðJ 0 Þ.
ðTyÞðtÞ ¼
for t 2 ð0; a:
Define T as follows
(
t 1þqp ðTxÞðtÞ; if t 2 ð0; a; 0;
if t ¼ 0:
Before giving the main results, we firstly prove the following lemmas. Lemma 4.3. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . Assume that (H0)–(H3) hold. Then the operator T : BðJÞ ! BðJÞ is equicontinuous provided x0 2 DðAb Þ with b > 1 þ p. Proof. For any y 2 BðJÞ, and t1 ¼ 0; 0 < t2 6 a, we get
Z
qð1þpÞ jðTyÞðt2 Þ ðTyÞð0Þj ¼
t 2 Pq ðt 2 Þx0 þ t 1þqp 2
t2
0
Z
qð1þpÞ
6 t 2 Pq ðt2 Þx0 þ
t21þqp
ðt2 sÞq1 Pq ðt 2 sÞf ðs; xðsÞÞds
t2
0
Z
qð1þpÞ
Pq ðt2 Þx0 þ C 2 t 21þqp 6 t 2
ðt 2 sÞq1 Pq ðt2 sÞf ðs; xðsÞÞds
t2
ðt 2 sÞð1þqpÞ mðsÞds ! 0;
as t 2 ! 0:
0
For 0 < t 1 < t 2 6 a, we have
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L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
7
Z t2
qð1þpÞ
qð1þpÞ jðTyÞðt2 Þ ðTyÞðt 1 Þj 6 t 2 Pq ðt 2 Þx0 t 1 Pq ðt 1 Þx0 þ
t 21þqp ðt 2 sÞq1 Pq ðt 2 sÞf ðs; xðsÞÞds 0
Z t1
qð1þpÞ
qð1þpÞ q1
6 t t1þqp ðt sÞ P ðt sÞf ðs; xðsÞÞds Pq ðt2 Þx0 t 1 P q ðt1 Þx0
1 q 1 2 1
0
Z t2
Z t1
q1
þ
t 1þqp ðt sÞ P ðt sÞf ðs; xðsÞÞds t 1þqp ðt 2 sÞq1 P q ðt2 sÞf ðs; xðsÞÞds þ
2 q 2 2 2
t1 t1
0
Z t1
sÞ Pq ðt 2 sÞf ðs; xðsÞÞdsj þ
t 11þqp ðt 1 sÞq1 Pq ðt2 sÞf ðs; xðsÞÞds 0 0 Z t1
qð1þpÞ
qð1þpÞ t 11þqp ðt 1 sÞq1 P q ðt1 sÞf ðs; xðsÞÞdsj 6 t2 Pq ðt 2 Þx0 t1 Pq ðt 1 Þx0
0
Z t2
Z t1 h
i
q1 q1 q1 1þqp 1þqp
þ
t 1þqp ðt sÞ P ðt sÞf ðs; xðsÞÞds þ t ðt sÞ t ðt sÞ ðt sÞf ðs; xðsÞÞds P 2 q 2 2 1 q 2 2 2 1
Z
t1þqp ðt1 1
q1
t
0
Z 1t1
þ
t 1þqp ðt 1 sÞq1 ½Pq ðt 2 sÞf ðs; xðsÞÞ Pq ðt1 sÞf ðs; xðsÞÞds
¼: I0 þ I1 þ I2 þ I3 ; 1 0
By Lemma 3.2, it is easy to see that limt2 !t1 I0 ¼ 0. Since
Z
I1 6 C 2 t1þqp 2 þ C2
t1
Z
t1
lim
Z
t 2 !t 1
Thus, by
R t1
h
0
qp 0 Dt m
I2 6 C 2
t2
ðt 2 sÞð1þqpÞ mðsÞds t1þqp 1
Z 0
t1
ðt1 sÞð1þqpÞ mðsÞds
i
t1þqp ðt 1 sÞð1þqpÞ t 21þqp ðt 2 sÞð1þqpÞ mðsÞds; 1
t 1þqp ðt 1 sÞð1þqpÞ mðsÞds exists ðs 2 ð0; t1 Þ, then by the Lebesgue dominated convergence theorem, we have 1
0
t1
0
h
0
noting that
Z
ðt 2 sÞð1þqpÞ mðsÞds 6 C 2
t1þqp 2
t2
Z
i t1þqp ðt1 sÞð1þqpÞ t 21þqp ðt 2 sÞð1þqpÞ mðsÞds ¼ 0: 1
2 CðJ 0 ; Rþ Þ, one can deduce that limt2 !t1 I1 ¼ 0. Since
1þqp q1 q1
qð1þpÞ 1þqp mðsÞds;
t2 ðt2 sÞ t 1 ðt 1 sÞ ðt2 sÞ
t1
0
noting that
h i
1þqp q1 q1
qð1þpÞ 1þqp mðsÞ 6 t21þqp ðt 2 sÞð1þqpÞ þ t 1þqp ðt 1 sÞq1 ðt2 sÞqð1þpÞ mðsÞ
t 2 ðt 2 sÞ t1 ðt1 sÞ ðt 2 sÞ 1 h i ðt 1 sÞð1þqpÞ mðsÞ 6 t21þqp ðt 2 sÞð1þqpÞ þ t 1þqp 1 6 2t 11þqp ðt 1 sÞð1þqpÞ mðsÞ and
R t1 0
ð5Þ
t 1þqp ðt 1 sÞð1þqpÞ mðsÞds exists ðs 2 ð0; t1 Þ, then by Lebesgue’s dominated convergence theorem, we have 1
Z 0
t1
1þqp q1 q1
qð1þpÞ 1þqp mðsÞds ! 0 as t 2 ! t 1 ;
t 2 ðt 2 sÞ t 1 ðt 1 sÞ ðt 2 sÞ
then one can deduce that limt2 !t1 I2 ¼ 0. For e > 0 be enough small, we have
I3 6
Z
t 1 e
0
þ
Z
t 11þqp ðt 1 sÞq1 kPq ðt 2 sÞ Pq ðt 1 sÞkBðXÞ jf ðs; xðsÞÞjds
t1
t 1 e
6 t1þqp 1 þ C2
t 11þqp ðt 1 sÞq1 kPq ðt 2 sÞ Pq ðt 1 sÞkBðXÞ jf ðs; xðsÞÞjds
Z
Z
t1
ðt 1 sÞq1 mðsÞds sup kPq ðt2 sÞ Pq ðt 1 sÞkBðXÞ s2½0;t 1 e
0
h
t1
t 1 e
6
t11þqþ2qp þ 2C 2
Z
i t 11þqp ðt 1 sÞq1 ðt2 sÞqð1þpÞ þ ðt 1 sÞqð1þpÞ mðsÞds
Z
t1 0
t1 t 1 e
ðt 1 sÞð1þqpÞ mðsÞds sup kPq ðt 2 sÞ Pq ðt 1 sÞkBðXÞ s2½0;t 1 e
t 1þqp ðt1 sÞ 1
ð1þqpÞ
mðsÞds:
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8
L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
By (H0) and limt2 !t1 I1 ¼ 0, it is easy to see I3 tends to zero independently of y 2 BðJÞ as t 2 ! t 1 ; e ! 0. Therefore, jðTyÞðt2 Þ ðTyÞðt1 Þj tends to zero independently of y 2 BðJÞ as t 2 ! t 1 , which means that fTy : y 2 BðJÞg is equicontinuous. h Lemma 4.4. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . Assume that (H1)–(H3) hold. Then the operator T : BðJÞ ! BðJÞ is bounded and continuous provided x0 2 DðAb Þ with b > 1 þ p. Proof. Step I. T maps BðJÞ into BðJÞ. For any y 2 BðJÞ, let xðtÞ ¼ tð1þqpÞ yðtÞ. Then x 2 BrðqÞ ðJ 0 Þ. For t 2 ½0; a, by (H1)–(H3), we have
Z t Z t
jðTyÞðtÞj 6 jPq ðtÞx0 j þ t 1þqp
ðt sÞð1þqpÞ Pq ðt sÞf ðs; xðsÞÞds
6 C 2 jx0 j þ C 2 t 1þqp ðt sÞð1þqpÞ jf ðs; xðsÞÞjds 0 0 ! Z t
6 C 2 jx0 j þ sup t1þqp t2½0;a
ðt sÞð1þqpÞ mðsÞds
6 r:
0
Hence, kTyk 6 r, for any y 2 BðJÞ. Step II. T is continuous in BðJÞ. For any ym ; y 2 BðJÞ; m ¼ 1; 2; . . ., with limm!1 ym ¼ y, we have
lim ym ðtÞ ¼ yðtÞ and
m!1
lim t ð1þqpÞ ym ðtÞ ¼ tð1þqpÞ yðtÞ;
m!1
for t 2 ð0; a:
Then by (H1), we have
f ðt; xm ðtÞÞ ¼ f ðt; t ð1þqpÞ ym ðtÞÞ ! f ðt; t ð1þqpÞ yðtÞÞ ¼ f ðt; xðtÞÞ;
as m ! 1;
where xm ðtÞ ¼ tð1þqpÞ ym ðtÞ and xðtÞ ¼ tð1þqpÞ yðtÞ. On the one hand, using (H2), we get for each t 2 J 0 ,
ðt sÞð1þqpÞ jf ðs; xm ðsÞÞ f ðs; xðsÞÞj 6 ðt sÞð1þqpÞ 2mðsÞ; On the other hand, the function s ! ðt sÞ gence theorem, we get
Z
t
ð1þqpÞ
a:e: in ½0; t:
2mðsÞ is integrable for s 2 ½0; t and t 2 J. By Lebesgue’s dominated conver-
ðt sÞð1þqpÞ jf ðs; xm ðsÞÞ f ðs; xðsÞÞjds ! 0;
as m ! 1:
0
For t 2 ½0; a
jðTym ÞðtÞ ðTyÞðtÞj ¼ jt 1þqp ðTxm ðtÞ TxðtÞÞj
Z t
6 t 1þqp
ðt sÞð1þqpÞ Pq ðt sÞðf ðs; xm ðsÞÞ f ðs; xðsÞÞÞds
0
6 C2t
1þqp
Z
t
ðt sÞð1þqpÞ jf ðs; xm ðsÞÞ f ðs; xðsÞÞjds ! 0;
as m ! 1:
0
Therefore, Tym ! Ty pointwise on J as m ! 1, by which Lemma 4.3 implies that Tym ! Ty uniformly on J as m ! 1 and so T is continuous. h 4.1. The case Q ðtÞðt > 0Þ is compact Suppose that the operator almost sectorial A generates a compact semigroup Q ðtÞðt > 0Þ on X, that is, for any t > 0, the operator Q ðtÞ is compact. Furthermore, Pq ðtÞ is also compact operator for every t > 0. Theorem 4.1. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p=2. Assume that Q ðtÞðt > 0Þ is compact. Furthermore assume that b 0 (H1)–(H3) hold. Then the Cauchy problem (1) has at least one mild solution in BðqÞ r ðJ Þ for every x0 2 DðA Þ with b > 1 þ p. Proof. Since Q ðtÞðt > 0Þ is compact, from Theorem 2.3.2 [38], Q ðtÞðt > 0Þ is equicontinuous, which implies (H0) is satisfied. 0 Then, by Lemmas 4.3 and 4.4, we know that T : BrðqÞ ðJ 0 Þ ! BðqÞ r ðJ Þ is bounded and continuous, and T : BðJÞ ! BðJÞ is bounded, continuous and equicontinuous. Next, we will show that for any t 2 ½0; a, VðtÞ ¼ fðTyÞðtÞ; y 2 BðJÞg is relatively compact in X. Obviously, Vð0Þ is relatively compact in X. Let t 2 ð0; a be fixed. For 8e 2 ð0; tÞ and 8d > 0, define an operator Te;d on BðJÞ by the formula
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L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
ðTe;d yÞðtÞ ¼ t qð1þpÞ Pq ðtÞx0 þ qt1þqp
Z
te
Z
0
¼ tqð1þpÞ Pq ðtÞx0 þ qt 1þqp Q ðeq dÞ
Z
1
9
hðt sÞq1 M q ðhÞQ ððt sÞq hÞf ðs; xðsÞÞdhds
d te
Z
0
1
hðt sÞq1 Mq ðhÞQ ððt sÞq h eq dÞf ðs; xðsÞÞdhds;
d
0 where x 2 BðqÞ Then from the compactness of Pq ðtÞðt > 0Þ and Q ðeq dÞðeq d > 0Þ, we obtain that the set r ðJ Þ. V e;d ðtÞ ¼ fðTe;d yÞðtÞ; y 2 BðJÞg is relatively compact in X for 8e 2 ð0; tÞ and 8d > 0. Moreover, for every y 2 BðJÞ, we have
Z t Z d
jðTyÞðtÞ ðTe;d yÞðtÞj 6
qt 1þqp hðt sÞq1 M q ðhÞQ ððt sÞq hÞf ðs; xðsÞÞdhds
0 0
Z t Z 1
þ
qt1þqp hðt sÞq1 M q ðhÞQ ððt sÞq hÞf ðs; xðsÞÞdhds
te
d
Z 1 ðt sÞð1þqpÞ mðsÞds hp Mq ðhÞdh 0 0 te 0 Z t Z d Z qC Cð1 pÞ 1þqp t ðt sÞð1þqpÞ mðsÞds hp M q ðhÞdh þ 0 t ðt sÞð1þqpÞ mðsÞds 6 qC 0 t 1þqp Cð1 qpÞ 0 0 te
6 qC 0 t 1þqp
! 0;
as
Z
t
ðt sÞð1þqpÞ mðsÞds
Z
d
hp M q ðhÞdh þ qC 0 t1þqp
Z
t
e ! 0; d ! 0:
Therefore, there are relatively compact sets arbitrarily close to the set VðtÞ; t > 0. Hence the set VðtÞ; t > 0 is also relatively compact in X. Therefore, fTy; y 2 BðJÞg is relatively compact by Ascoli–Arzela Theorem. Thus, the continuity of T and relatively compactness of fTy; y 2 BðJÞg imply that T is a completely continuous operator. Schauder fixed point theorem shows that T has a fixed point in y 2 BðJÞ. Let x ðtÞ ¼ tq1 y ðtÞ. Then x is a mild solution of (1). h The condition (H2) can be replaced by the following condition. q 1 p
(H2)0 There exist a constant q1 2 ð0; qÞ and a positive function m 2 L
1
ðJ; Rþ Þ such that
0 jf ðt; xðtÞÞj 6 mðtÞ for all x 2 BðqÞ and a:e: t 2 ð0; a: r ðJ Þ
Corollary 4.1. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p=2. Let all the assumption of Theorem 4.1 be given except (H2). b 0 Assume that (H2)0 holds. Then the Cauchy problem (1) has at least one mild solution in BðqÞ r ðJ Þ for every x0 2 DðA Þ with b > 1 þ p. Proof. In fact, if (H2)0 holds, by using Hölder inequality, for any t 1 ; t 2 2 J 0 and t 1 < t2 , we obtain
Z t1 Z t2
1
ðt 2 sÞð1þqpÞ ðt1 sÞð1þqpÞ mðsÞds þ ðt 2 sÞð1þqpÞ mðsÞds
CðqpÞ 0 t1 Z t1 1þq1 p Z t1 q1 p 1 q 1 p ð1þqpÞ ð1þqpÞ 1þq1 p ððt1 sÞ ðt 2 sÞ Þ ds ðmðsÞÞ 1 ds
qp jðIqp 0þ mÞðt 2 Þ ðI0þ mÞðt 1 Þj ¼
1
6
CðqpÞ þ
1
CðqpÞ
1 CðqpÞ
6
0
Z Z 0
1þq1 p 1þq1 p Z 1 ðt2 sÞð1þqpÞ ds
t1 t1
Z
1 CðqpÞ kmk 1
þ
t2
0
t2
q 1 p 1
ðmðsÞÞ
q1 p ds
t1
1þq1 p ð1þqpÞ ð1þqpÞ ððt 1 sÞ 1þq1 p ðt2 sÞ 1þq1 p Þds kmk 1þq1 p t2 ð1þqpÞ ðt 2 sÞ 1þq1 p ds kmk
t1
q 1 p 1
L
ð6Þ 1 L q1 p
!1þq1 p 1þq1 p ðq1 qÞp ðq1 qÞp ðq1 qÞp 1 þ q1 p 1þq p 1þq p t 1 1 þ ðt 2 t1 Þ 1þq1 p t 2 1 CðqpÞ ðq1 qÞp 1þq1 p kmk q 1p 1 þ q p 1þq1 p ðq1 qÞp 1 L 1 þ ðt 2 t 1 Þ 1þq1 p CðqpÞ ðq1 qÞp 2kmk q 1p 1 þ q p 1þq1 p 1 L 1 6 ðt2 t1 Þðq1 qÞp ! 0; as t2 ! t 1 ; CðqpÞ ðq1 qÞp L
6
q p 1
where
kmk
1 L q1 p
¼
Z
a
q 1 p
ðmðtÞÞ
1
dt
q1 p :
0
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10
L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
On the other hand,
t1þqp
Z
t
ðt sÞð1þqpÞ mðsÞds 6 t1þqp
0
Z 0
6
1þq1 p Z t q1 p ð1þqpÞ 1 ðt sÞ 1þq1 p ds ðmðsÞÞ q1 p ds
t
1þq1 p
1 þ q1 p ðq1 qÞp
0
t 1þq1 p kmk
L
q 1 p 1
! 0;
as t ! 0:
ð7Þ
0 þ 1þqp qp Thus, (6) and (7) mean that Iqp ðI0þ mÞðtÞ ¼ 0. Hence, (H2) holds. By Theorem 4.1, the Cauchy 0þ m 2 CðJ ; R Þ, and limt!0þ t ðqÞ 0 problem (1) has at least one mild solution in Br ðJ Þ. h
4.2. The case Q ðtÞ is noncompact If Q ðtÞ is noncompact, we give an assumption as follows. (H4) There exists a constant ‘ > 0 such that for any bounded D X,
aðf ðt; DÞÞ 6 ‘aðDÞ; for a:e: t 2 ½0; a:
Theorem 4.2. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . Assume that (H0)–(H4) hold. Then the Cauchy problem (1) has at b 0 least one mild solution in BðqÞ r ðJ Þ for every x0 2 DðA Þ with b > 1 þ p. Proof. By Lemmas 4.3 and 4.4, we have T : BðJÞ ! BðJÞ is bounded, continuous and equicontinuous. Next, we will show that T is compact in a subset of BðJÞ. For each bounded subset B0 BðJÞ, set
T1 ðB0 Þ ¼ TðB0 Þ; Tn ðB0 Þ ¼ T coðTn1 ðB0 ÞÞ ; From Propositions 2.2–2.4, for any
n ¼ 2; 3; . . . : 1
e > 0, there is a sequence fyð1Þ n gn¼1 B0 such that
Z t 1 þe aðT1 ðB0 ðtÞÞÞ ¼ aðTðB0 ðtÞÞÞ 6 2a t1þqp ðt sÞq1 Pq ðt sÞf ðs; fsð1þqpÞ yð1Þ ðsÞg Þds n n¼1 6 4C 2 t 1þqp
0
Z
t
ðt sÞ
0
6 4C 2 ‘t1þqp aðB0 Þ
Z
t
ð1þqpÞ
1
ðt sÞð1þqpÞ sð1þqpÞ ds þ e 6
0
Since
a f ðs; fsð1þqpÞ yð1Þ n ðsÞgn¼1 Þ ds þ e 4C 2 ‘C2 ðqpÞtqp aðB0 Þ þ e: Cð2qpÞ
e > 0 is arbitrary, we have
aðT1 ðB0 ðtÞÞÞ 6
4C 2 ‘C2 ðqpÞt qp aðB0 Þ: Cð2qpÞ
From Propositions 2.2–2.4, for any
1
1 e > 0, there is a sequence fyð2Þ n gn¼1 coðT ðB0 ÞÞ such that
aðT2 ðB0 ðtÞÞÞ ¼ aðTðcoðT1 ðB0 ðtÞÞÞÞÞ
Z t 1 ðt sÞq1 Pq ðt sÞf s; fsð1þqpÞ ynð2Þ ðsÞgn¼1 ds þ e 6 2a t 1þqp
6 4C 2 t
1þqp
Z
0 t
0
6 4C 2 ‘t
1þqp
6 4C 2 ‘t
1þqp
Z
1 ðt sÞð1þqpÞ a f s; fsð1þqpÞ ynð2Þ ðsÞgn¼1 ds þ e t
1 ðt sÞð1þqpÞ a fsð1þqpÞ ynð2Þ ðsÞgn¼1 ds þ e
t
1 ðt sÞð1þqpÞ sð1þqpÞ a fynð2Þ ðsÞgn¼1 ds þ e
0
Z
0
ð4C 2 ‘Þ2 t1þqp C2 ðqpÞtqp 6 aðB0 Þ Cð2qpÞ ¼
Z
t
ðt sÞð1þqpÞ sð1þ2qpÞ ds þ e
0
ð4C 2 ‘Þ2 C3 ðqpÞ 2qp t aðB0 Þ þ e: Cð3qpÞ
2 Nþ , It can be shown, by mathematical induction, that for every n
aðTn ðB0 ðtÞÞÞ 6
ð4C 2 ‘Þn Cnþ1 ðqpÞ nqp t aðB0 Þ: Cððn þ 1ÞqpÞ
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11
Since
lim
!1 n
ð4C 2 ‘aqp Þn Cnþ1 ðqpÞ ¼ 0; Cððn þ 1ÞqpÞ
^ such that there exists a positive integer n ^
^
ð4C 2 ‘Þn Cnþ1 ðqpÞ n^qp ð4C 2 ‘aqp Þn Cnþ1 ðqpÞ t 6 ¼ k < 1: Cððn^ þ 1ÞqpÞ Cððn^ þ 1ÞqpÞ ^
^
Then
aðTn^ ðB0 ðtÞÞÞ 6 kaðB0 Þ: ^
We know from Proposition 2.1, Tn ðB0 ðtÞÞ is bounded and equicontinuous. Then, from Proposition 2.2, we have
aðTn^ ðB0 ÞÞ ¼ max aðTn^ ðB0 ðtÞÞÞ: t2½0;a
Hence
aðTn^ ðB0 ÞÞ 6 kaðB0 Þ: Let
D0 ¼ BðJÞ;
^
^
D1 ¼ coðTn ðDÞÞ; . . . ; Dn ¼ coðTn ðDn1 ÞÞ;
n ¼ 2; 3; . . .
From [40], we can get (i) D0 D1 D2 . . . Dn1 Dn . . .; (ii) limn!1 aðDn Þ ¼ 0. b ¼ T1 Dn is a nonempty, compact and convex subset in BðJÞ. Then D n¼0 ^ D. ^ Firstly, we show By using the similar method in [40], we will prove TðDÞ
TðDn Þ Dn ;
n ¼ 0; 1; 2; . . .
1
ð8Þ 1
From T ðD0 Þ ¼ TðD0 Þ D0 , we know coðT ðD0 ÞÞ D0 . Therefore
T2 ðD0 Þ ¼ TðcoðT1 ðD0 ÞÞÞ TðD0 Þ ¼ T1 ðD0 Þ; T3 ðD0 Þ ¼ TðcoðT2 ðD0 ÞÞÞ TðcoðT1 ðD0 ÞÞÞ ¼ T2 ðD0 Þ; ^
^
^
^
Tn ðD0 Þ ¼ TðcoðTn1 ðD0 ÞÞÞ TðcoðTn2 ðD0 ÞÞÞ ¼ Tn1 ðD0 Þ: ^
^
^
^
^
Hence, D1 ¼ coðTn ðD0 ÞÞ coðTn1 ðD0 ÞÞ, so TðD1 Þ TðcoðTn1 ðD0 ÞÞÞ ¼ Tn ðD0 Þ coðTn ðD0 ÞÞ ¼ D1 . Employing the same b T1 TðDn Þ T1 Dn ¼ D. b Then Tð DÞ b is compact. method, we can prove TðDn Þ Dn ðn ¼ 0; 1; 2; . . .Þ. By (8), we get Tð DÞ n¼0 n¼0 Hence, Schauder fixed point theorem shows that T has a fixed point y 2 BðJÞ. Let x ðtÞ ¼ t q1 y ðtÞ. Then x is a mild solution of (1). h Corollary 4.2. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . Let all the assumption of Theorem 4.2 be given except (H2). Assume b 0 that (H2)0 holds. Then the Cauchy problem (1) has at least one mild solution in BðqÞ r ðJ Þ for every x0 2 DðA Þ with b > 1 þ p. The proof of Corollary 4.2 is similar to that of Corollary 4.1, it is thus omitted. In the following, we also give the existence and uniqueness result which is based on Banach contraction principle. We will need the following assumption. 0 (H5) There exists a constant k > 0 such that for any x; y 2 BðqÞ r ðJ Þ, and t 2 ð0; a we have
jf ðt; xðtÞÞ f ðt; yðtÞÞj < kkx ykq : Theorem 4.3. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . If assumptions (H2), (H3) and (H5) are satisfied. Then, for every x0 2 DðAb Þ with b > 1 þ p, the Cauchy problem (1) has a unique mild solution in BrðqÞ ðJ 0 Þ provided
Please cite this article in press as: L. Zhang, Y. Zhou, Fractional Cauchy problems with almost sectorial operators, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.07.024
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L. Zhang, Y. Zhou / Applied Mathematics and Computation xxx (2014) xxx–xxx
C 2 ak < 1: qp
ð9Þ
Proof. By the proof of Lemma 4.4, we know that T is an operator from BðJÞ into itself. For any y1 ; y2 2 BðJÞ, let x1 ðtÞ ¼ t ð1þqpÞ y1 ðtÞ and x2 ðtÞ ¼ t ð1þqpÞ y2 ðtÞt 2 ð0; a, we have
jðTy1 ÞðtÞ ðTy2 ÞðtÞj ¼ t1þqp jðTx1 ÞðtÞ ðTx2 ÞðtÞj
Z t
¼ t1þqp
ðt sÞq1 Pq ðt sÞ½f ðs; x1 ðsÞÞ f ðs; x2 ðsÞÞds
0 Z t 6 C 2 t1þqp ðt sÞð1þqpÞ jf ðs; x1 ðsÞÞ f ðs; x2 ðsÞÞjds 0
C 2 tk 6 kx1 x2 kq qp C 2 ak ¼ ky1 y2 k; qp which implies
jTy1 Ty2 j 6
C 2 ak ky1 y2 k; qp
which means that T is a contraction according to (9). By applying Banach contraction principle, we know that T has a unique fixed point y 2 BðJÞ. Let x ðtÞ ¼ tq1 y ðtÞ. Then x is a mild solution of (1). h Corollary 4.3. Let A 2 Hpx ðXÞ with 1 < p < 0 and 0 < x < p2 . Let all the assumption of Theorem 4.3 be given except (H2). Assume that (H2)0 holds. Then the Cauchy problem (1) has a unique mild solution in BrðqÞ ðJ 0 Þ for every x0 2 DðAb Þ with b > 1 þ p. The proof of Corollary 4.3 is similar to that of Corollary 4.1, it is thus omitted.
5. An example Let X be a bounded domain in RN ðN P 0Þ with boundary @ X of class C 4 . Let X ¼ C k ðXÞð0 < k < 1Þ. Set
A ¼ D;
DðAÞ ¼ fu 2 C 2þk ðXÞ : uð0Þ ¼ 0 on @ Xg:
It follows from [36] (Example 2.3) that there exist 1; e > 0, such that k 1
A þ 1 2 H2pe ðC k ðXÞÞ: 2
Consider the following fractional initial-boundary value problem
8 L q > < ð D0þ xÞðt; zÞ ¼ Dxðt; zÞ þ f ðt; xðtÞÞ; uj@X ¼ 0 > : 1q ðI0þ xÞð0; zÞ ¼ x0 ðzÞ; z 2 X
ppt 2 ½0; a; z 2 X ð10Þ
1
in the space X, where q ¼ 12, f ðt; xðtÞÞ ¼ t 3 sin xðtÞ. Then problem (10) can be written abstractly as
8 1 < ðL D20þ xÞðtÞ ¼ AxðtÞ þ t13 sin xðtÞ; 1 : I2 x ð0Þ ¼ x : 0 0þ
ppt 2 ð0; a;
We choose 1
mðtÞ ¼ t3
and r ¼ C 2 jx0 j þ
2 C 2 a3 C 2p C 23 p 2 : C 2 þ 3 ð1Þ
Then, the conditions (H1)–(H3) are satisfied. According to Theorem 4.1, system (10) has a mild solution on Br2 ðð0; aÞ. References [1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. [2] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. [3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
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