On some impulsive fractional differential equations in Banach spaces

ON SOME IMPULSIVE

FRACTIONAL DIFFERENTIAL EQUATIONS

IN BANACH SPACES

JinRong Wang, W. Wei, YanLong Yang

Abstract. This paper deals with some impulsive fractional differential equations in Banach spaces. Utilizing the Leray-Schauder fixed point theorem and the impulsive nonlinear singu-lar version of the Gronwall inequality, the existence ofP C_{-mild solutions for some fractional}

differential equations with impulses are obtained under some easily checked conditions. At last, an example is given for demonstration.

Keywords: fractional differential equations with impulses, nonlinear impulsive singular version of the Gronwall inequality,P C-mild solutions, existence.

Mathematics Subject Classification: 45N05, 93C25.

1. INTRODUCTION

During the past decades, impulsive differential equations have attracted much interest since it is much richer than the corresponding theory of differential equations (see for instance [14, 25, 54] and references therein). Recently, impulsive evolution equations and their optimal control problems in infinite dimensional spaces have been investi-gated by many authors including Ahmed, Benchohra, Ntouyas, Liu, Nieto and us (see for instance [1–5, 8, 9, 29], [38, 39, 50–53] and references therein). Specially, we also studied the impulsive periodic system in infinite dimensional spaces (see [43–47]).

On the other hand, the fractional differential equations have recently been proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, economics and science. We can find numerous applica-tions in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. [16, 19–21, 33, 37]. In recent years, there has been a significant development in frac-tional differential equations. One can see the monographs of Kilbas et al. [24], Miller and Ross [31], Podlubny [41], Lakshmikantham et al. [28], and the papers [6, 7, 10, 15–19, 22, 24, 26, 27, 34, 35] and the references therein.

507

However, to our knowledge, the theory for impulsive fractional differential equa-tions in Banach spaces has not yet been sufficiently developed. Very recently, Ben-chohraet al. [10, 12] applied the Banach contraction principle, Schaefer’s fixed point theorem and the nonlinear alternative of the Leray-Schauder type or measure of noncompactness to a class of impulsive fractional differential equations without un-bounded operator. A class of initial value problem for impulsive fractional differential equations with variable times is also considered in [13]. Balachandranet al. [13] using fractional calculus and fixed point theorems for a class of impulsive fractional evolu-tion equaevolu-tions with bounded time-varying linear operator. Mophouet al. [36], Wang et al. [49], apply semigroup theory and fixed point theorems to study the impulsive fractional differential equations with an unbounded operator in Banach spaces.

Motivated by the above work including [34–36, 48, 49], the main purpose of this paper is to consider the following fractional differential equations with impulses

    

Dα

tx(t) =Ax(t) +tnf(t, x(t)), α∈(0,1], n∈Z+, t∈J = [0, b], t6=tk,

x(0) =x0,

∆x(tk) =Ik(x(tk)) =x(t+_k)−x(t−_k), k= 1,2, . . . , δ, 0< t1< t2< . . . < tδ < b, (1.1)

where A: D(A) ⊂ X → X is the generator of a C0-semigroup {T(t), t ≥ 0} on a

Banach space X, Dα

t is the Caputo fractional derivative, f: J×X →X is specified later,x0is an element ofX,Ik: X→X is a nonlinear map which determines the size of the jump attk,0 =t0< t1< t2< . . . < tδ < tδ+1=b,x(t+_k) = limh→0+=x(t_k+h)

andx(t−

k) =x(tk)represents respectively the right and left limits ofx(t)att=tk. In order to obtain the existence of solutions for impulsive fractional differential equations, some authors use Krasnoselskii’s fixed point theorem or contraction map-ping principle. It is obvious that the conditions for Krasnoselskii’s fixed point theorem are not easily verified sometimes and the conditions for the contraction mapping prin-ciple are too strong. Some authors give the prior estimate of the solutions for impulsive fractional differential equations, however, the condition onf is a little strong.

Here, we use the Leray-Schauder fixed point theorem to obtain the existence of

P C-mild solutions for system (1.1) under some easily checked conditions. First, we construct an operatorHfor system (1.1), then use a generalized Ascoli-Arzela theorem (see Theorem 2.5) and overcome some difficulties to show the compactness of H

which is very important. With the help of an impulsive nonlinear singular version of the Gronwall inequality (see Theorem 2.7), the key estimate of the fixed point set {x=σHx, σ ∈[0,1]} can be established successfully. Therefore, the existence of

P C-mild solutions for system (1.1) is shown. Our methods are different from previous work and we give a new way to show the existence of solutions for impulsive fractional differential equations.

2. PRELIMINARIES

Let £_{b(X) be the Banach space of all linear and bounded operators on X. For a}

C0-semigroup {T(t), t ≥ 0} on X, we set M ≡ supt∈JkT(t)k£_b(X). Let C(J, X)

be the Banach space of all X-valued continuous functions from J = [0, b] into X

endowed with the normkxkC= supt∈Jkx(t)k. We also introduce the set of functions

P C(J, X)≡

x:J →X |xis continuous att∈J\{t1, t2, . . . , tδ}, andxis continuous from left and has right hand limits at t∈ {t1, t2, . . . , tδ} . Endowed with the norm

kxkP C = max

sup

t∈J

kx(t+ 0)k,sup

t∈J

kx(t−0)k

,

(P C(J, X),k · kP C)is a Banach space.

Let us recall the following definitions. For more details see [41].

Definition 2.1. A real functionf(t) is said to be in the space Cα, α∈ R if there exists a real numberκ > α, such thatf(t) =tκg(t), whereg∈C[0,∞)and it is said to be in the spaceC_αmifff(m)∈Cα,m∈N.

Definition 2.2. The Riemann-Liouville fractional integral operator of orderα >0

of a functionf ∈Cα, α≥ −1is defined as

I_tαf(t) = 1 Γ(α)

t Z

0

(t−s)α−1_f(s)ds,

whereΓ(·)is the Euler gamma function.

Definition 2.3. If the function f ∈ C−ζ1, ζ ∈N, the fractional derivative of order α >0of a functionf(t)in the Caputo sense is given by

dα_f(t)

dtα =

1 Γ(ζ−α)

t Z

0

(t−s)ζ−α−1_f(ζ)_{(s)ds,ζ−1< α≤ζ.}

Based on [36] (Definition 3.2 and Lemma 3.3), we use the following definition of a P C-mild solution for system (1.1).

Definition 2.4. By a P C-mild solution of the system (1.1) we mean the function

x∈P C(J, X)which satisfies

x(t) =T(t)x0+

1 Γ(α)

X

0<tk<t

tk

Z

tk−1

(tk−s)α−1T(t−s)snf(s, x(s))ds+

+ 1

Γ(α)

t Z

tk

(t−s)α−1

T(t−s)snf(s, x(s))ds+

+ X

0<tk<t

T(t−tk)Ik(x(tk)).

(2.1)

Lemma 2.5 (Generalized Ascoli-Arzela theorem, Theorem 2.1, [50]). SupposeW ⊂

P C(J, X)be a subset. If the following conditions are satisfied:

(1) W is a uniformly bounded subset ofP C(J, X).

(2) W is equicontinuous in(tk, tk+1),k= 0,1,2, . . . , δ, wheret0= 0,tδ+1=b.

(3) W(t)≡ {x(t)|x∈ W, t∈J\{t1, . . . , tδ}},W(tk+ 0)≡ {x(tk+ 0)|x∈ W}and

W(tk−0)≡ {x(tk−0)|x∈ W}are relatively compact subsets ofP C(J, X).

ThenW is a relatively compact subset ofP C(J, X).

Lemma 2.6 (Lemma 2.1, [42]). For all β >0 andϑ >−1,

t Z

0

(t−s)β−1

sϑds=C(β, ϑ)tβ+ϑ,

where

C(β, ϑ) = Γ(β)Γ(ϑ+ 1) Γ(β+ϑ+ 1).

Lemma 2.7 (Impulsive nonlinear singular version of the Gronwall inequality, Theo-rem 3.1, [42]). Letx∈P C([0,∞), X)and satisfy the following inequality

x(t)≤a(t) +b(t)

t Z

0

(t−s)α−1_sγ_F

1(s)xm(s)ds+d(t)

X

0<tk<t

ηkx(tk), t≥0, (2.2)

where a(t), b(t), d(t) and F1(t) are nonnegative continuous functions, ηk ≥ 0 are constants.

(1) If 1

2 ≥α >0,− 1

2 ≥γ >−1, then it holds that fort∈(tk, tk+1],

x(t)≤

(k+ 3)q−1_f

p(t) k Y l=1

(1 + (k+ 2)q−1_)ηq lf(tl)

q1

×

×

1−(m−1)

k X i=1

ti

Z

ti−1

(i+ 2)q−1_m

i−1

Y j=1

(1 + (j+ 2)q−1_)ηq

jf(tj))mF1q(s)fpm(s)ds−

−(m−1)(k+ 3)(q−1)m_× k Y j=1

(1 + (j+ 2)q−1_ηq

jf(tj))m t Z

tk

F₁q(s)f_pm(s)ds

q(1−1m)

as long as the expression between the second brackets is positive, that is, on (0, Tp),

Tp is the sup of all values oft for which

k X i=1

ti

Z

ti−1

(i+ 2)(q−1)m i−1

Y j=1

(1 + (j+ 2)q−1₎

ηq_jf(tj))mF1q(s)fpm(s)ds−

−(m−1)(k+ 3)(q−1)m_× k Y j=1

(1 + (j+ 2)q−1_ηq

jf(tj))m t Z

tk

F₁q(s)f_pm(s)ds <

< 1 m−1;

(2) If 1

2 < α,− 1

2 < γ, then it holds that for t∈(tk, tk+1],

x(t)≤

(k+ 3)f(t)

k Y l=1

(1 + (k+ 2))η2_lf(tl) 12

×

×

1−(m−1)

k X i=1

ti

Z

ti−1

(i+ 2)m

i−1

Y j=1

(1 + (j+ 2))η_j2f(tj))mF12(s)fm(s)ds−

−(m−1)(k+ 3)m_× k Y j=1

(1 + (j+ 2)η2_jf(tj))m t Z

tk

F12(s)fm(s)ds

2(1−1m)

as long as the expression between the second brackets is positive, that is, on (0, T2),

T2 is the sup of all values oft for which

k X

i=1

ti

Z

ti−1

(i+ 2)m

i−1

Y j=1

(1 + (j+ 2))η_j2f(tj))mF12(s)fm(s)ds−

−(m−1)(k+ 3)m_× k Y j=1

(1 + (j+ 2)η2_jf(tj))m t Z

tk

F12(s)fm(s)ds <

where

fp(t) = sup n

aq(t), Cqp(pα−p+ 1, pγ)bq(t)tq(α+γ)−1, dq(t)

o

,

pandqsuch that 1

p+

1

q = 1,

f(t)≡f2(t) = supna2(t), C(2α−2 + 1,2γ)b2(t)t2(α+γ)−1_{, d}2_(t)o_,

ifp=q= 2,

C(pα−p+ 1, pγ) =Γ(pα−p+ 1)Γ(pγ+ 1) Γ(pα−p+ 1 +pγ+ 1).

3. EXISTENCE OF MILD SOLUTIONS

In this section, we will derive the existence result concerning the P C-mild solution for the system (1.1) under some easily checked conditions.

We make the following assumptions.

[HA]: A is the infinitesimal generator of a compact C0-semigroup {T(t), t ≥0}

onX with domain D(A).

[Hf]: (1)f :J×X →X is strongly measurable with respect totonJ and for any

x,y∈X satisfyingkxk,kyk ≤ρthere exists a positive constantLf(ρ)>0such that

kf(t, x)−f(t, y)k ≤Lf(ρ)kx−yk.

(2) There exists a positive constantMf>0 such that

kf(t, x)k ≤Mf(1 +kxkm)for allt∈J, x∈X, some m >1.

[HI]: (1) The nonlinear map Ik: X → X, Ik(X)is a bounded subset of X, k =

1,2, . . . , δ.

(2) There exist constantshk>0, such that

kIk(x)−Ik(y)k ≤hkkx−yk, for all x, y∈X, k= 1,2, . . . , δ.

Proof. Letx0∈X be fixed. Define an operatorH onP C(J, X)which is given by

(Hx)(t) =T(t)x0+ 1 Γ(α)

X

0<tk<t

tk

Z

tk−1

(tk−s)α−1T(t−s)snf(s, x(s))ds+

+ 1

Γ(α)

t Z

tk

(t−s)α−1

T(t−s)snf(s, x(s))ds+

+ X

0<tk<t

T(t−tk)Ik(x(tk)).

(3.1)

Using [HA] and [Hf], one can verify thatH is a continuous mapping from P C(J, X)

to P C(J, X)for x∈P C(J, X). In fact, for 0≤τ < t≤t1, it comes from [HA] and the following inequality

k(Hx)(t)−(Hx)(τ)k ≤ kT(t)x0−T(τ)x0k+

+ 1

Γ(α)

t Z

τ

k(t−s)α−1

T(t−s)snf(s, x(s))kds+

+ 1

Γ(α)

τ Z

0

k(t−s)α−1_{[T(t−s)−T(τ−s)]s}n_{f(s, x(s))kds≤}

≤MkT(t−τ)x0−x0k+

+t

n_Mkfk P C

Γ(α)

t Z

τ

(t−s)α−1_ds+

+τ

n_{MkT(t−τ)−Ikkfk} P C

Γ(α)

τ Z

0

(t−s)α−1 ds≤

≤MkT(t−τ)x0−x0k+

+t

n

1MkfkP C

Γ(α+ 1) (t−τ)

α₊

+

tn

1MkfkP C

Γ(α+ 1) |t

α_−(t−τ)α_|

kT(t−τ)−Ik

(3.2)

thatHx∈C([0, t1], X).

With analogous arguments we can obtainHx∈C([tk, tk+1], X),k= 0,1,2, . . . , δ.

That isHx∈P C(J, X).

Letx1, x2∈P C(J, X)andkx1−x2kP C ≤1, thenkx2kP C≤1 +kx1kP C=ρ. By assumptions [HA], [Hf] and [HI], we obtain

k(Hx1)(t)−(Hx2)(t)k ≤

≤ 1

Γ(α)

X

0<tk<t

tk

Z

tk−1

k(tk−s)α−1T(t−s)sn[f(s, x1(s))−f(s, x2(s))]kds+

+ 1

Γ(α)

t Z

tk

k(t−s)α−1_T(t−s)sn_{[f(s, x1(s))−f(s, x2(s))]kds+}

+ X

0<tk<t

kT(t−tk)[Ik(x1(tk))−Ik(x2(tk))]k ≤

≤ M Lf(ρ)

Γ(α)

X

0<tk<t

tk

Z

tk−1

(tk−s)α−1snkx1(s)−x2(s)kds+

+M Lf(ρ) Γ(α)

t Z

tk

(t−s)α−1_sn_{kx1(s)−x2(s)kds+M} X

0<tk<t

hkkx1(tk)−x2(tk)k ≤

≤

t Z

0

(t−s)α−1_sn_ds !

M Lf(ρ)

Γ(α) +M

X

0<tk<t

hk

kx1−x2kP C.

(3.3)

Using Lemma 2.6, one can deduce that

kHx1−Hx2kP C≤

Γ(α)·Γ(n+ 1) Γ(α+n+ 1) t

α+n

M Lf(ρ)

Γ(α) +M

δ X k=1

hk

kx1−x2kP C ≤

≤Lkx1−x2kP C,

where

L=M bα+nΓ(α)·Γ(n+ 1)

Γ(α+n+ 1)

Lf(ρ)

Γ(α) +

δ X k=1

hk

.

(2) H is a compact operator onP C(J, X).

Let B _{be a bounded subset of P C(J, X), there exists a constant µ > 0 such} that kxkP C ≤ µ for all x ∈ B. Using [HI], there exists a constant N such that

t∈J. Further, HB_{is a bounded subset of P C(J, X). In fact, letx∈}B_{, we have}

k(Hx)(t)k ≤Mkx0k+ M ω Γ(α)

X

0<tk<t

tk

Z

tk−1

(tk−s)α−1snds+

+ M ω Γ(α)

t Z

tk

(t−s)α−1_sn_ds+M X

0<tk<t

N ≤

≤Mkx0k+M N δ+ M ω Γ(α)

t Z

0

(t−s)α−1_sn_ds≤

≤Mkx0k+M N δ+ωMΓ(n+ 1) Γ(α+n+ 1)b

α+n_.

HenceHB_{is bounded.} Define

Π =HB _{and Π(t) ={(Hx)(t)|x∈}B_{)} fort∈J.}

Clearly, Π(0) = {x0} is compact, hence, it is only necessary to check that Π(t) =

{(Hx)(t)|x∈B_{}fort∈(0, b]is also compact. For 0< ε < t≤b, define}

Πε(t)≡(HεB)(t) ={(Hεx)(t)|x∈B} (3.4)

and the operatorHεis defined by

(Hεx)(t) =T(ε)T(t−ε)x0+

+T(ε) 1 Γ(α)

X

0<tk<t

tk

Z

tk−1

(tk−s)α−1T(t−ε−s)snf(s, x(s))ds+

+T(ε) 1 Γ(α)

t−ε Z

tk

(t−s)α−1

T(t−ε−s)snf(s, x(s))ds+

+T(ε) X

0<tk<t

T(t−ε−tk)Ik(x(tk)) =

=T(t)x0+ 1 Γ(α)

X

0<tk<t

tk

Z

tk−1

(tk−s)α−1T(t−s)snf(s, x(s))ds+

+ 1

Γ(α)

t−ε Z

tk

(t−s)α−1

T(t−s)snf(s, x(s))ds+

+ X

0<tk<t

T(t−tk)Ik(x(tk)),

from which implies that Πε(t)is relatively compact for t∈(ε, b] due to{T(t), t≥0} is a compact semigroup.

For interval(0, t1], (3.4) reduces to

Πε(t)≡(HεB)(t) ={(Hεx)(t)|x∈B}.

Combine with (3.1) and (3.5), we can deduce

sup

x∈B

k(Hx)(t)−(Hεx)(t)k ≤

1 Γ(α)

t Z

t−ε

(t−s)α−1_T(t−s)sn_{f(s, x(s))ds}

≤

≤t

n_ωM

Γ(α)

t Z

t−ε

(t−s)α−1_ds≤

≤b

n_{ωM ε}α

Γ(α+ 1).

It shows that the setΠ(t)can be approximated to an arbitrary degree of accuracy by a relatively compact set fort ∈(0, t1]. Hence,Π(t)itself is a relatively compact set fort∈(0, t1].

For interval(t1, t2], define

Π(t1+ 0)≡Π(t1−0) +I1(Π(t1−0)) = Π(t1) +I1(Π(t1)) = ={(Hx)(t1) +I1(x(t1))|x∈B_}.

By assumption [HI], one can verify thatI1(Π(t1))is relatively compact. Hence,Π(t1+ 0)is relatively compact. Then (3.4) reduces to

Πε(t)≡(HεB)(t) =

=

(

(Hx)(t1+ 0) + 1 Γ(α)

t−ε Z

t1

(t−s)α−1_T(t−s)sn_{f(s, x(s))ds|x∈B} )

.

By elementary computation again, we have

sup

x∈B

k(Hx)(t)−(Hεx)(t)k ≤

ωM

Γ(α)

t Z

t−ε

(t−s)α−1_sn_ds≤bnωM εα

Γ(α+ 1).

Hence,Π(t)itself is relatively compact set fort∈(t1, t2].

In general, for any given tk,k= 1,2, . . . , δ, we define thatx(ti+ 0) =xi, and

Π(tk+ 0)≡Π(tk−0) +Ik(Π(tk−0)) =

= Π(tk) +Ik(Π(tk)) =

By assumption [HI] again, Ik(Π(tk) is relatively compact and the associated Πε(t) over the interval(tk, tk+1]is given by

Πε(t)≡(HεB)(t) =

=

(

(Hx)(tk+ 0) +

1 Γ(α)

t−ε Z

tk

(t−s)α−1_T(t−s)sn_{f(s, x(s))ds|x∈B} )

.

Thus, we have

sup

x∈B

k(Hx)(t)−(Hεx)(t)k ≤

bnωM εα

Γ(α+ 1).

Hence,Π(t)itself is a relatively compact set fort∈(tk, tk+1].

Now, we repeat the procedures till the time interval which is expanded. Thus, we can obtain that the setΠ(t)itself is relatively compact fort∈J\ {t1, t2, . . . , tδ} and

Π(tk+ 0)is relatively compact fortk,k= 1,2, . . . , δ.

(3) Πis equicontinuous on the interval(tk, tk+1),k= 1,2, . . . , δ.

For interval(0, t1), we note that fort1> h >0,

k(Hx)(h)−(Hx)(0)k ≤ kT(h)−Ikkx0k+ωM Γ(n+ 1)

Γ(α+n+ 1)h

α+n_,

and fort1≥t+h≥t≥γ≥0,γ < handx∈B_,

(Hx)(t+h)−(Hx)(t) = (T(t+h)−T(t))x0+

+ 1

Γ(α)

t+h Z

t

(t+h−s)α−1_T(t+h−s)sn_{f(s, x(s))ds+}

+ 1

Γ(α)

t Z

t−γ

(t+h−s)α−1_[

T(t+h−s)−T(t−s)]snf(s, x(s))ds+

+ 1

Γ(α)

t Z

t−γ

[(t+h−s)α−1_−(t−s)α−1_]T(t−s)sn_{f(s, x(s))ds+}

+ 1

Γ(α)

t−γ Z

0

(t+h−s)α−1_[

T(t+h−s)−T(t−s)]snf(s, x(s))ds+

+ 1

Γ(α)

t−γ Z

0

hence,

k(Hx)(t+h)−(Hx)(t)k ≤MkT(h)−Ikkx0k+

+ ωM t

n

1

Γ(α+ 1)h

α₊

+ωM tn₁|h

α_−(h+γ)α_|

Γ(α+ 1) kT(h)−Ik+

+ωM tn₁ 1

Γ(α)

t Z

t−γ

|(t+h−s)α−1_−(t−s)α−1_|ds+

+ωM tn₁|(t+h)

α_−(γ+h)α_|

Γ(α+ 1) kT(h)−Ik+

+ωM tn₁ 1

Γ(α)

t−γ Z

0

|(t+h−s)α−1_−(t−s)α−1_|ds.

(3.6)

Since kT(h)−Ik →0,|(t+h−s)α−1_−(t−s)α−1_{| →0 ash→0, thus the right}

hand side of (3.6) can be made as small as desired by choosing hsufficiently small. Hence,Π(t)is equicontinuous in interval(0, t1).

In general, for time interval (tk, tk+1), k = 1,2, . . . , δ, we similarly obtain the

following inequalities

k(Hx)(t+h)−(Hx)(t)k ≤MkT(h)−Ikkxkk+

+ ωM t

n k+1

Γ(α+ 1)h

α₊

+ωM tn_k+1|h

α_−(h+γ)α_|

Γ(α+ 1) kT(h)−Ik+

+ωM tn_k+1 1

Γ(α)

t Z

t−γ

|(t+h−s)α−1_−(t−s)α−1_|ds+

+ωM tn_k+1

|(t+h)α_−(γ+h)α_|

Γ(α+ 1) kT(h)−Ik+

+ωM tn_k+1

1 Γ(α)

t−γ Z

tk

|(t+h−s)α−1_−(t−s)α−1_|ds.

With analogous arguments, one can verify thatΠis also equicontinuous on the interval

(tk, tk+1),k= 1,2, . . . , δ.

Now, we repeat the procedures till the time interval which is expanded. Thus we obtain that the setΠ(t)itself is relatively compact fort∈J\{t1, . . . , tδ}andΠ(tk+0) is relatively compact fortk ∈ {t1, . . . , tδ}.

According to Leray-Schauder fixed point theorem, it suffices to show the follow-ing set

x∈P C(J, X)|x=σHx, σ ∈[0,1]

is a bounded subset ofP C(J, X). In fact, letx∈ {x∈P C(J, X)| x=H(σx), σ ∈

[0,1]}, we have

kx(t)k=kH(σx(t))k ≤ ≤ kT(t)(σx0)k+

+ 1

Γ(α)

X

0<tk<t

tk

Z

tk−1

k(tk−s)α−1T(t−s)snf(s, σx(s))kds+

+ 1

Γ(α)

t Z

tk

k(t−s)α−1

T(t−s)snf(s, σx(s))kds+

+ X

0<tk<t

kT(t−tk)Ik(σx(ti))k ≤

≤σMkx0k+M Mf

1 Γ(α)

t Z

0

(t−s)α−1_sn_{(1 +σkx(s)k}m_)ds+

+M X

0<tk<t

(kIk(0)k+σhkkx(tk)k)≤

≤M

kx0k+Mf

Γ(n+ 1) Γ(α+n+ 1)t

α+n₊ δ X k=1

kIi(0)k

+

+M Mf Γ(α)

t Z

0

(t−s)α−1_sγ_sn−γ_kx(s)km_ds+

+M X

0<tk<t

hkkx(tk)k ≤

≤M

kx0k+Mf

Γ(n+ 1) Γ(α+n+ 1)t

α+n₊ δ X k=1

kIk(0)k

+

+tn−γM Mf

Γ(α)

t Z

0

(t−s)α−1_sγ_kx(s)km_ds+

+M X

0<tk<t

Denote

fp(t) = sup

Mq

kx0k+Mf

Γ(n+ 1) Γ(α+n+ 1)t

α+n₊ δ X k=1

kIk(0)k q

,

Cpq(_pα−_p+ 1_{, pγ})_t(n−γ)q

M Mf

Γ(α)

q

tq(α+γ)−1_{, M}q

,

pandqsuch that 1

p+

1

q = 1,

f(t)≡f2(t) = sup

M2

kx0k+Mf

Γ(n+ 1) Γ(α+n+ 1)t

α+n₊ δ X k=1

kIk(0)k 2

,

C(2α−2 + 1,2γ)t(n−γ)2M Mf

Γ(α)

2

t2(α+γ)−1 , M2

.

(i) If 1₂ ≥ α > 0, −1₂ ≥ γ > −1, by (1) of Lemma 2.7, it holds that for each

(tk, tk+1],

kx(t)k ≤ sup

t∈[0,Tp]

(k+ 3)q−1_f

p(t) k Y l=1

(1 + (k+ 2)q−1_)hq lf(tl)

1q

≡M1∗ k ,

whereTp is the sup of all values oft for which

k X i=1

ti

Z

ti−1

(i+ 2)(q−1)m i−1

Y j=1

(1 + (j+ 2)q−1_)hq

jf(tj))mfpm(s)ds−

−(m−1)(k+ 3)(q−1)m_× k Y j=1

(1 + (j+ 2)q−1_hq

jf(tj))m t Z

tk

f_pm(s)ds < 1 m−1.

(ii) If 1

2 < α <1, − 1

2 < γ, by (2) of Lemma 2.7, it holds that for each (tk, tk+1],

kx(t)k ≤ sup

t∈[0,T2]

(k+ 3)f(t)

k Y l=1

(1 + (k+ 2))h2_lf(tl) 12

≡M2∗ k ,

whereT2 is the sup of all values oft for which

k X i=1

ti

Z

ti−1

((i+ 2)m₎ i−1

Y j=1

(1 + (j+ 2))h2_jf(tj))mfm(s)ds−

−(m−1)(k+ 3)m_× k Y j=1

(1 + (j+ 2)h2_jf(tj))m t Z

tk

SetM1∗_{= max{}

M1∗

1 , M21∗, . . . , Mδ1∗},M2

∗_{= max{}

M2∗

1 , M22∗, . . . , Mδ2∗}. We denote

M∗_{= max{}

M1∗

, M2∗_{}. Then we have}

kxkP C≤M∗ for allx∈ {x∈P C(J, X)|x=σHx, σ∈[0,1]}.

Thus,{x∈P C(J, X)|x=σHx, σ∈[0,1]}is a bounded subset ofP C(J, X). By the Leray-Schauder fixed point theorem, we obtain thatH has a fixed point inP C(J, X). This completes the system (1.1) has at least aP C-mild solution onJ.

At last, an example is given to illustrate our theory. Consider the following im-pulsive fractional differential equations

    

D13

tx(t, y) = ∂

2

∂y2x(t, y) +tx2(t, y) +tsin(t, y), t∈(0,1]\ {

1 2},

∆x(t1, y) =−x(t1, y), t1= 1

2, y∈Ω = (0, π), x(t, y)|y∈∂Ω= 0, t >0, x(0, y) = 0, y∈Ω.

(3.7)

LetX =L2([0, π]). Define

D(A) =

x∈X | ∂x

∂y, ∂2x

∂y2 andx(0) =x(π) = 0

andAx=−∂

2

∂y2xforx∈D(A)

which can determine a compact C0-semigroup {T(t), t ≥ 0} in L2([0, π]) such that

kT(t)k ≤1.

Denote x(·)(y) = x(·, y), sin(·)(y) = sin(·, y), f(·, x(·))(y) = x2_{(·, y) + sin(·, y),} I1(x(t1))(y) =−x(t1, y).Thus, problem (3.7) can be rewritten as

    

Dα

tx(t) =Ax(t) +tnf(t, x(t)), α=13 ∈(0, 1

2], n= 1, t∈(0,1]\ {tk},

∆x(tk) =Ik(x(tk)), tk= 1₂, k= 1,

x(0) = 0.

(3.8)

Obviously, all the assumptions in Theorem 3.1 are satisfied. Our results can be used to solve problem (3.7).

Acknowledgments

JinRong Wang acknowledge support from the National Natural Science Foundation of Guizhou Province (2010, No. 2142) and the Introducing Talents Foundation for the Doctor of Guizhou University (2009, No. 031).

W. Wei acknowledge support from the National Natural Science Foundation of China (No. 10961009).

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JinRong Wang wjr9668@126.com

Guizhou University College of Science

Guiyang, Guizhou 550025, P.R. China

YanLong Yang yylong@163.com

Guizhou University College of Technology

Guiyang, Guizhou 55004, P.R. China

W. Wei

wwei@gzu.edu.cn

Guizhou University College of Science

Guiyang, Guizhou 550025, P.R. China