On a new class of impulsive fractional evolution equations

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R E S E A R C H

Open Access

On a new class of impulsive fractional

evolution equations

Xi Fu

1

_{, Xiaoyou Liu}

2*

_{and Bowen Lu}

1,3

*_{Correspondence:}

[email protected] 2_{School of Mathematics and} Physics, University of South China, Hengyang, Hunan 421001, P.R. China

Full list of author information is available at the end of the article

Abstract

This paper is concerned with the existence ofPC-mild solutions for Cauchy and nonlocal problems of impulsive fractional evolution equations for which the impulses are not instantaneous. By using the theory of operator semigroups, probability density functions, and some suitable ﬁxed point theorems, we establish some existence results for these types of problems.

MSC: 34K30; 35R12; 26A33

Keywords: fractional evolution equations; impulses;PC-mild solutions; Cauchy problems; nonlocal problems

1 Introduction

Recently, much attention has been paid to the study of fractional diﬀerential equations due to the fact that they have been proved to be valuable tools in the mathematical modeling of many phenomena in physics, biology, mechanics,etc.(see [–]).

Impulsive differential equations of integer order have found extensive applications in realistic mathematical modeling of a wide variety of practical situations, such as biolog-ical phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, and frequency modulated systems. For the general theory and relevant developments of impulsive differential equations, please see [–] and the references therein. Usually the impulses of the evolution process described by impulsive differential equations are assumed to be abrupt and instantaneous. That is to say, the perturbations (impulses) start abruptly and the duration of them is negligible in comparison with the duration of the process.

However, in [], the authors introduced a new class of abstract impulsive diﬀerential equations for which the impulses are not instantaneous. Speciﬁcally, they studied the ex-istence of solutions for the following impulsive problem:

⎧ ⎪ ⎨ ⎪ ⎩

u(t) =Au(t) +f(t,u(t)), t∈(si,ti+],i= , , , . . . ,N,

u(t) =gi(t,u(t)), t∈(ti,si],i= , , . . . ,N,

u() =x,

whereA:D(A)⊂X→Xis the generator of aC-semigroup of bounded linear operators

{T(t)}t≥deﬁned on a Banach space (X, · ),x∈X,  =t=s<t≤s<t≤s<· · ·<

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tN ≤sN <tN+=aare pre-ﬁxed numbers,gi∈C((ti,si]×X;X) fori= , , . . . ,Nandf : [,a]×X→Xis a suitable function. The impulses start abruptly at the pointstiand their action continues on the interval [ti,si]. As a motivation for the study of such systems, see [], where an example of the hemodynamical equilibrium of a person was given.

Impulsive diﬀerential equations of fractional order have been studied by some authors, for example [–]. As for the study of impulsive fractional evolution equations, to the best of our knowledge, there are few papers [–] on this topic.

Motivated by [], in this paper we consider a class of impulsive fractional evolution equations of the form

⎧ ⎪ ⎨ ⎪ ⎩

c_Dα_x₍_t_{) =}_Ax₍_t_{) +}_f₍_t_,_x₍_t_)), _t_∈₍_s

i,ti+],i= , , , . . . ,m,

x(t) =Ii(x(ti)) +gi(t,x(t)), t∈(ti,si],i= , , . . . ,m,

x() =x,

()

where c_Dα _{is the Caputo fractional derivative of order} _α _∈_{(, ) with the lower limit} zero,A:D(A)⊂X→Xis the generator of aC-semigroup of bounded linear operators

{T(t)}t≥on a Banach space (X, · ),x∈X,  =t=s<t≤s<t≤s<· · ·<tm≤

sm<tm+=T are ﬁxed numbers,gi∈C((ti,si]×X;X),Ii:X→Xfori= , , . . . ,mand

f : [,T]×X→Xis a nonlinear function.

The impulses in problem () start abruptly at the pointstiand their action continues on the interval [ti,si]. To be precise, the functionxtakes an abrupt impulse attiand follows diﬀerent rules in the two subintervals (ti,si] and (si,ti+] of the interval (ti,ti+]. At the point

si, the functionxis continuous. The termIi(x(ti)) means that the impulses are also related to the value ofx(ti) =x(t–i).

From the results obtained in the papers [–], we know that the deﬁnition of mild solutions for fractional evolution equations is more involved than integer order evolution equations. Moreover, to construct solutions for impulsive fractional diﬀerential equations, we should properly handle the fractional derivative and impulsive conditions due to the memory property of fractional calculus (see [–]).

We remark that if ti=si and the second equation of () takes the form of x(ti) =

Ii(x(ti)) =x(t+i) –x(t–i) withx(ti+) =lim→+x(t_i+),x(t–

i) =lim→–x(t_i+) representing

the right and left limits ofx(t) att=ti, problem () reduces to the case considered in [] (with the ﬁxed impulses).

We also study the nonlocal Cauchy problems for impulsive fractional evolution equa-tions

⎧ ⎪ ⎨ ⎪ ⎩

c_Dα_x₍_t_{) =}_Ax₍_t_{) +}_f₍_t_,_x₍_t_)), _t_∈₍_s

i,ti+],i= , , , . . . ,m,

x() =x+b(x),

()

whereA,f,Ii,giare the same as above,bis a given function; this constitutes a nonlocal Cauchy problem. It is well known that the nonlocal condition has a better eﬀect on the solution and is more precise for physical measurements than the classical initial condition alone.

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the existence results of problems () and (). An example is given in Section  to illustrate the results.

2 Preliminaries

Let us setJ= [,T],J= [,t],J= (t,t], . . . ,Jm–= (tm–,tm],Jm= (tm,tm+] and introduce

the space PC(J,X) :={u:J→X|u∈C(Jk,X),k= , , , . . . ,m, and there exist u(tk+) and

u(t_k–),k= , , . . . ,m, withu(t_k–) =u(tk)}. It is clear thatPC(J,X) is a Banach space with the normuPC=sup{u(t):t∈J}.

Lemma .(Theorem . in []) Suppose W ⊆PC(J,X).If the following conditions are

satisﬁed:

() Wis a uniformly bounded subset ofPC(J,X);

() Wis equicontinuous in(ti,ti+),i= , , , . . . ,m,wheret= ,tm+=T;

() W(t) ={u(t) :u∈W,t∈J\{t,t, . . . ,tm}},W(ti+) ={u(ti+) :u∈W}and

W(t–

i) ={u(ti–) :u∈W},i= , , . . . ,m,are relatively compact subsets ofX.

Then W is a relatively compact subset of PC(J,X).

Let us recall the following well-known deﬁnitions.

Deﬁnition .([]) The Riemann-Liouville fractional integral of orderqwith the lower

limit zero for a functionf is deﬁned as

Iqf(t) = 

(q)

t



(t–s)q–f(s)ds, q> ,

provided the integral exists, where(·) is the gamma function.

Deﬁnition .([]) The Riemann-Liouville derivative of orderqwith the lower limit zero

for a functionf : [,∞)→Rcan be written as

L_Dq_f₍_t_{) =} 

(n–q)

dn

dtn

t



(t–s)n–q–f(s)ds, n–  <q<n,t> .

Deﬁnition .([]) The Caputo derivative of orderqfor a functionf : [,∞)→Rcan

be written as

c_Dq_f₍_t_{) =}L_Dq

f(t) –

n–

k=

tk

k!f

(k)_() _, _n_{–  <}_q_<_n_,_t_{> .}

Remark .

(a) Iff ∈Cn_[,_∞₎_{, then, for}_n_{–  <}_q_<_n,

c_Dq_f₍_t_{) =} 

(n–q)

t



(t–s)n–q–f(n)(s)ds=In–qf(n)(t), t> .

(b) Iff is an abstract function with values inX, then the integrals in Deﬁnitions . and . are taken in Bochner’s sense.

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Deﬁnition .([, ]) A functionx∈C(J,X) is said to be a mild solution of the follow-ing problem:

c_Dα_x₍_t_{) =}_Ax₍_t_{) +}_y₍_t_), _t_∈_(,_T_],

x() =x,

if it satisﬁes the integral equation

x(t) =Pα(t)x+

t



(t–s)α–_Q

α(t–s)y(s)ds.

Here

Pα(t) =

∞



ξα(θ)T

tαθdθ, Qα(t) =α

∞



θ ξα(θ)T

tαθdθ, ()

ξα(θ) = 

αθ

––_α

α

θ–α≥_,

α(θ) = 

π ∞

n=

(–)n–θ–nα–(nα+ )

n! sin(nπ α), θ∈(,∞), ()

andξαis a probability density function deﬁned on (,∞) [], that is,

ξα(θ)≥,θ∈(,∞),

_∞



ξα(θ)dθ= .

It is not diﬃcult to verify that

_∞



θ ξα(θ)dθ= 

( +α). ()

Remark . By applying the Laplace transform and probability density functions, Zhou

and Jiao [, ] introduced the above deﬁnition of mild solutions for fractional evolution equations. For pioneering work on Caputo fractional evolution equations, we refer the readers to [, ].

We make the following assumption onAin the whole paper.

H(A): The operatorAgenerators a strongly continuous semigroup{T(t) :t≥}inX, and there is a constantMA≥such thatsupt∈[,∞)T(t)L(X)≤MA. For anyt> ,T(t)

is compact.

Lemma .(see [, ]) LetH(A)hold,then the operators Pαand Qαhave the following

properties:

() For any ﬁxedt≥,Pα(t)andQα(t)are linear and bounded operators,and for any

x∈X,

Pα(t)x≤MAx, Qα(t)x≤

αMA

( +α)x;

() {Pα(t),t≥}and{Qα(t),t≥}are strongly continuous;

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Finally we recall a ﬁxed point theorem which will be needed in the sequel.

Theorem .(Krasnoselskii ﬁxed point theorem) Let M be a closed, convex,and

non-empty subset of a Banach space X.Let A,B be the operators such that: (a)Ax+By∈M for all x,y∈M, (b)A is compact and continuous, (c)B is a contraction.Then there exists a x∈M such that x=Ax+Bx.

3 The construction of mild solutions

Lety∈PC(J,X). We ﬁrst consider the following fractional impulsive problem:

⎧ ⎪ ⎨ ⎪ ⎩

c_Dα_x₍_t_{) =}_Ax₍_t_{) +}_y₍_t_), _t_∈₍_s

i,ti+],i= , , , . . . ,m,

x() =x.

()

From the property of the Caputo derivative, a general solution of problem () can be writ-ten as

x(t) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x+₍_α₎

t

(t–s)

α–₍_Ax₍_s_{) +}_y₍_s₎₎_ds_, _t_∈_[,_t ),

I(x(t)) +g(t,x(t)), t∈(t,s],

d+₍_α₎

t

(t–s)

α–₍_Ax₍_s_{) +}_y₍_s₎₎_ds_, _t_∈₍_s ,t),

. . . ,

Ii(x(ti)) +gi(t,x(t)), t∈(ti,si],i= , , . . . ,m,

di+(α)

t

(t–s)

α–₍_Ax₍_s_{) +}_y₍_s₎₎_ds_, _t_∈₍_s

i,ti+),

()

wheredi,i= , , . . . ,m, are elements ofX. By () and the functionxis continuous at the pointssi, we have, fori= , , , . . . ,m,

x(t) =diχ[si,ti+)(t) +



(α)

t



(t–s)α–Ax(s) +y(s)ds, t∈[si,ti+), ()

withd=xandχ[si,ti+)(t) is the characteristic function of [si,ti+),i.e.

χ[si,ti+)(t) =

, t∈[si,ti+),

, otherwise.

Now we follow the idea used in the papers [, ] and apply the Laplace transformation for () to obtain

u(λ) = e

–λsi_–_e–λti+

λ di+



λαAu(λ) + + 

λαv(λ),

whereu(λ) =_∞e–λs_x₍_s₎_ds_and_v₍_λ_{) =}∞

 e–

λs_y₍_s₎_ds_,_λ_{> . Then}

u(λ) =λα–λαI–A–e–λsi_d

i–λα–

λαI–A–e–λti+_d

i+

λαI–A–v(λ),

whereIis the identity operator deﬁned onX. Note that the Laplace transform ofα(θ) deﬁned by () is given by

_∞



e–λθα(θ)dθ=e–λ α

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Then by the same computations in [, ] and the properties of Laplace transform (trans-lation theorem and linearity of the inverse Laplace transform), we obtain

x(t) =χ[si,∞)Pα(t–si)di–χ[ti+,∞)Pα(t–ti+)di+

Therefore, a mild solution of problem () is given by

x(t) =

Next, by using the above results, we introduce the following deﬁnition of the mild solu-tion for problem ().

Deﬁnition . A functionx∈PC(J,X) is said to be aPC-mild solution of problem () if

it satisﬁes the following relation:

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where, fori= , , . . . ,m,

di=Ii

x(ti)

+gi

si,x(si)

–

si



(si–s)α–Qα(si–s)f

s,x(s)ds. ()

Remark . For treating the mild solutions for abstract fractional diﬀerential equations,

we can also refer to [].

4 Existence results

This section deals with the existence results for problems () and (). From Deﬁnition ., we deﬁne an operatorS:PC(J,X)→PC(J,X) as

(Sx)(t) =

⎧ ⎪ ⎨ ⎪ ⎩

Pα(t)x+

t

(t–s)

α–_Q

α(t–s)f(s,x(s))ds, t∈[,t],

Ii(x(ti)) +gi(t,x(t)), t∈(ti,si],

Pα(t–si)di+

t

(t–s)

α–_Q

α(t–s)f(s,x(s))ds, t∈[si,ti+]

withdi,i= , , . . . ,m, deﬁned by ().

To prove our ﬁrst existence result we introduce the following assumptions.

H(f ): The functionf ∈C(J×X;X)and there existsLf∈L



τ(J,R+)with_τ ∈(,_α)such that

f(t,x) –f(t,y) ≤Lf(t)x–yfor allx,y∈Xand everyt∈J.

H(I): Fori= , , . . . ,m,Ii∈C(X,X)and there is a constantLI> such thatIi(x) –Ii(y) ≤

LIx–yfor allx,y∈X.

H(g): Fori= , , . . . ,m, the functionsgi∈C([ti,si]×X;X)and there existsLg∈C(J,R+)

such thatgi(t,x) –gi(t,y) ≤Lg(t)x–yfor allx,y∈Xandt∈[ti,si].

Theorem . AssumeH(f), H(I),andH(g)are satisﬁed and

MA

LI+LgC(J)

+ ( +MA)

αMA

(α+ )

 –τ α–τ

–τ

Tα–τLf

Lτ(J)< . ()

Then there exists a unique PC-mild solution of problem().

Proof From the assumptions it is easy to show that the operator S is well deﬁned on

PC(J,X).

Letx,y∈PC(J,X). Fort∈[,t], from Lemma ., we have

(Sx)(t) – (Sy)(t)≤

t



(t–s)α–_Q

α(t–s)

fs,x(s)–fs,y(s)ds

≤ αMA

(α+ )

 –τ α–τ

–τ

t_α–τLf

Lτ([,t])x–yPC.

Similarly, we have, fort∈(ti,si],i= , , . . . ,m,

(Sx)(t) – (Sy)(t)≤Ii

x(ti)

–Ii

y(ti)

+gi

t,x(t)–gi

t,y(t)

≤LI+LgC(J)

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and, fort∈[si,ti+],i= , , . . . ,m,

Then it follows from condition () thatSis a contraction on the spacePC(J,X). Hence by the Banach contraction mapping principle,Shas a unique ﬁxed pointx∈PC(J,X) which is just the uniquePC-mild solution of problem (). The proof is now complete.

The next result is based on the Krasnoselskii ﬁxed point theorem.

H(f ): For anyx∈X, the mapt→f(t,x)is strongly measurable onJ. For a.e.t∈J, the

and there exists a constant r> such that

MA

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Proof We deﬁne two operatorsS,S:PC(J,X)→PC(J,X) as

Letr>  satisfy condition (). We set

M=u∈PC(J,X) :uPC≤r

.

ThenMis a closed, convex, and nonempty subset of the Banach spacePC(J,X).

Next we will show that the operatorsS,Ssatisfy the requirements of Theorem .,i.e.

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Then we get

Sx+SyPC≤MA

ϕI(r) +mgC(J)ϕg(r) +x

+ ( +MA)

αMAϕf(r)

(α+ )

 –τ α–τ

–τ

Tα–τ_m

f Lτ(J).

It follows from () thatSx+Sy∈Mfor allx,y∈M.

Step:Sis a contraction. Letx,y∈PC(J,X). From H(I), H(g), and Lemma ., we have,

fort∈[,t],

(Sx)(t) – (Sy)(t) = 

fort∈(ti,si],i= , , . . . ,m,

(Sx)(t) – (Sy)(t)≤

LI+LgC(J)

x–yPC,

and fort∈[si,ti+],i= , , . . . ,m,

(Sx)(t) – (Sy)(t)≤MA

LI+LgC(J)

x–yPC.

Therefore we deduce that

Sx–SyPC≤MA

LI+LgC(J)

x–yPC.

In view of (), the operatorSis a contraction onPC(J,X).

Step:Sis compact and continuous. Firstly, we will prove thatS is continuous. Let

xn→xinPC(J,X) asn→ ∞. We can assume without any loss of generality thatxnPC≤

Rfor someR>  andn≥. By H(f ), we have

ft,xn(t)

→ft,x(t) a.e.t∈J, ()

ft,xn(t)≤mf(t)ϕf(R) fort∈J,n≥. ()

Since, fort∈[,t],

(Sxn)(t) – (Sx)(t)≤

αMA

(α+ )

t



(t–s)α–fs,xn(s)

–fs,x(s)ds

fort∈(ti,si],i= , , . . . ,m,

(Sxn)(t) – (Sx)(t)= ,

and, fort∈[si,ti+],i= , , . . . ,m,

₍_S__x_n₎₍_t_{) – (}_S__x₎₍_t₎_≤ αMA

(α+ )

t



(t–s)α–fs,xn(s)

–fs,x(s)ds

+MA

αMA

(α+ )

si



(si–s)α–f

s,xn(s)

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Then from (), (), and by means of the Lebesgue dominated convergence theorem, we obtain

Sxn–SxPC→ asn→ ∞.

This means thatSis continuous.

Next we shall show thatS maps bounded set into relatively compact set inPC(J,X).

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Therefore from the property of the probability density functionξαand (), we obtain

(Sx)(t) – (Mh,δx)(t)→ ash→,δ→.

This means that there are relatively compact sets arbitrarily close to the setS(B)(t). Hence

the setS(B)(t) is also relatively compact inX.

Case :ti<t≤si,i= , , . . . ,m. In such a case,

S(B)(t) ={}is compact.

Case :si<t≤ti+,i= , , . . . ,m,

S(B)(t) =

t



(t–s)α–_Q

α(t–s)f

s,x(s)ds

–Pα(t–si)

si



(si–s)α–Qα(si–s)f

s,x(s)ds:x∈B

.

By the same argument as in Case andPα(t–si) is a compact operator (see Lemma .), we knowS(B)(t) is relatively compact.

Therefore it follows from Lemma . thatSis compact and continuous.

As a consequence of Steps -, we know thatS+Ssatisﬁes all conditions of

Krasnosel-skii ﬁxed point theorem (Theorem .). Hence the operatorShas a ﬁxed point inPC(J,X) which is aPC-mild solution of problem (). The proof is complete.

Finally in this section, we extend the results obtained above to nonlocal problems for impulsive fractional evolution equations. Speciﬁcally, we show study the existence and uniqueness of the mild solutions for problem (). Here we only state the existence results for problem () without proofs since these are similar to the ones obtained for problem () above.

Deﬁnition . A functionx∈PC(J,X) is said to be aPC-mild solution of problem () if

it satisﬁes the following relation:

x(t) =

⎧ ⎪ ⎨ ⎪ ⎩

Pα(t)(x+b(x)) +

t

(t–s)

α–_Q

α(t–s)f(s,x(s))ds, t∈[,t],

Ii(x(ti)) +gi(t,x(t)), t∈(ti,si],i= , , . . . ,m,

Pα(t–si)di+

t

(t–s)

α–_Q

α(t–s)f(s,x(s))ds, t∈[si,ti+],

withdi,i= , , . . . ,m, deﬁned by ().

H(b): b:PC(J,X)→Xand there exist a constantLb> andϕb∈C([,∞),R+)

nonde-creasing such that, forx,y∈PC(J,X),

b(x) –b(y)≤Lbx–yPC, b(x)≤ϕb

xPC

.

Theorem . AssumeH(f), H(I), H(g),andH(b)are satisﬁed and

MA

LI+LgC(J)+Lb

+ ( +MA)

αMA

(α+ )

 –τ α–τ

–τ

Tα–τLf

Lτ(J)< .

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Theorem . LetH(f), H(I), H(g), H(Ig),andH(b)hold.Assume that

and there exists a constant r> such that

MA

Then there exists a PC-mild solution of problem().

5 Examples

A simple example is given in this section to illustrate the results.

LetX=L_([,_π_{]). Deﬁne an operator}_A_:_D₍_A₎_⊆_X_→_X_by_Ax₌_x_with_D₍_A_{) =}_{_x_∈

X:x∈X,x() =x(π) = }. It is well known thatA is the inﬁnitesimal generator of a strongly continuous semigroup {T(t) :t≥} inX. Moreover,T(t) is compact fort>  andT(t)L(X)≤e–t≤ =MA,t≥.

Consider the following impulsive problem:

⎧

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Therefore by Theorem ., we deduce that problem () has a uniquePC-mild solution on [, ].

Assumption  Let

ft,u(t,y)=e–|u(t,y)|+ t+sint+ u(t,y)  +u₍_t_,_y₎,

Iu(t,y)= |u(t,y)|

 +|u(t,y)|+ t, g

t,u(t,y)=

arctanu(t,y) + .

Similarly, the functionsf,I, andgcan be rewritten as

ft,x(t)=e–|x(t)|+ t+sint+ x(t)  +x₍_t₎,

Ix(t)= |x(t)|

 +|x(t)|+ t, g

t,x(t)=

arctanx(t) + .

Putmf(t)≡,ϕf(x)≡√π,ϕI(x)≡ _√π,mg(t)≡,ϕg(x)≡(π_ + )√π,LI =_, andLg(t)≡ _, then it is easy to show that H(f ), H(I), H(g), and H(Ig)hold. We have

MA

LI+LgC(J)

=  < .

In such a case, obviously, we can choose a constantr>  such that condition () holds. Therefore it follows from Theorem . that problem () has aPC-mild solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 312000, P.R. China.}2_{School of Mathematics and} Physics, University of South China, Hengyang, Hunan 421001, P.R. China. 3_{School of International Business, Zhejiang} International Studies University, Hangzhou, Zhejiang 310012, P.R. China.

Acknowledgements

The work was supported by Hunan Provincial Natural Science Foundation of China (Grant No. 2015JJ6095), Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ14A010006) and Doctor Priming Fund Project of University of South China (Grant No. 2013XQD16).

Received: 7 May 2015 Accepted: 2 July 2015

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