Liang et al. Fixed Point Theory and Applications (2015) 2015:230 DOI 10.1186/s13663-015-0479-y

0 Fixed Point Theory and Applications

a SpringerOpen Journal

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About periodicity of impulsive evolution equations through fixed point theory

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Jin Liang1*, Ti-Jun Xiao2 and He Yang1

"Correspondence: jinliang@sjtu.edu.cn 1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, P.R. China Full list of author information is available at the end of the article

Abstract

By processing the problem through fixed point theory and propagator theory, we investigate the periodicity of solutions to a class of impulsive evolution equations in Hilbert spaces and establish some existence theorems for periodic solutions. Moreover, the asymptotic stability of periodic solutions is obtained under suitable conditions. As one will see, the concept of an impulsive propagator is introduced for the first time in the paper.

MSC: 47D06; 47H10; 34G10; 34G20; 35Q99; 35K90; 35B40

Keywords: fixed point theorem; periodic solution; propagator; impulsive evolution equation; asymptotic stability; analytic semigroup

1 Introduction

It is well known that fixed point theorems play key roles in obtaining the existence of solutions, positive solution, periodic solutions, and almost periodic solutions to various equations or systems. There are many research publications in this area; for example, see [1-27] and the references therein for interesting results on this issue.

In this paper, by using the fixed point theory and propagator theory, we study the existence and asymptotic stability of periodic solutions for the following impulsive evolution equation in a Hilbert space H:

x'(t)+Ax(t)=f(t, x(t)), t e R+, t = tk, &x\t=tk = yk - akx(tk), k = 1,2,...,

where A : D(A) c H ^ H is a unbounded closed linear operator, f (t, u): R+ x H ^ H is a nonlinear mapping and it is T-periodic in t, 0 < t1 < t2 < ••• < tm < T,

tm+k = tk + T, k > 1,

T > 0 is a fixed number and m e N denotes the number of impulsive points between 0 and T,

ft Spri

Aujt=tk = u(t+)- u(t-),

ringer

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u(t+) and u(t-) represent the right and left limits of u(t) at t = tk, respectively, ak e R, yk e H (k = 1,2,...) satisfy

ak+m = ak, yk+m = yk.

In the last few decades, the theory of impulsive differential equations has been largely developed. For the earlier results, we refer the reader to the monographs of Lakshmikan-tham etal. [13] and Benchohra etal. [3], the papers of Ahmed [28-30] and Liu [20], and the references cited therein. For the recent results, we refer the reader to, e.g., [1,16,17,21, 23, 31] and the references therein.

On the other hand, the existence of periodic solutions or almost periodic solutions of evolution equations has been investigated by many authors (cf., e.g., [2, 4, 5, 9-11, 14, 15, 17-19, 24]). One can see that all these studies are based on the fixed point theory. For example, in [14,15], under the spectral separation condition of selfadjoint operators, Li obtained some existence and uniqueness results for periodic solutions to semilinear evolution equations in Hilbert spaces by using fixed point theorems. Moreover, we can find useful information on the study of almost periodicity of evolution equations with the help of fixed point theory from Cuevas et al. [4, 5], Diagana [9-11], Mophou [24] and references cited there.

Although there have been many papers on periodic solutions of periodic system in finite or infinite dimensional spaces, to our knowledge,impulsive periodic systems in infinite dimensional spaces (with unbounded operators) have been seldom investigated. Recently, Liang etal. (see [17]) studied a class of semilinear impulsive evolution equations with delay in Banach spaces. They proved the existence of T-periodic solutions by using Horn's fixed point theorem.

In this paper, we will use a different method, which is based on fixed point theorems and evolution operators, to study T-periodic solutions of (1.1). First of all, we introduce a new concept of impulsive propagator, corresponding to the linear impulsive evolution equation, and then we introduce a suitable T-periodic solution operator of (1.1). Second, we overcome some difficulties to show the existence and uniqueness of periodic solutions to (1.1) by using Schauder's fixed point theorem. Finally, we present the global asymptotic stability result for (1.1). Particularly, our discussion is made in a framework of Hilbert spaces, which enables us to obtain the existence theorems for strong T-periodic solutions of (1.1).

The rest of this paper is organized as follows. In Section 2, we introduce the impulsive propagator and then prove the existence of T-periodic solutions for the linear impulsive evolution equations. In Section 3, the existence and asymptotic stability theorems for T-periodic solutions to (1.1) are obtained. An example is given in Section 4 to illustrate the applicability of our results.

2 Impulsive propagator and linear impulsive periodic systems

Let (H, (•, •)) be a Hilbert space, A : D(A) c H ^ H be a positive definite selfadjoint operator and the embedding D(A) H be compact. Then the spectrum of A consists of real eigenvalues ¡¡i (i = 1,2,...), and

¡i < ¡2 < ••• < xn < ••• .

By the positive definite property of A, the first eigenvalue ¡1 > 0. It is known from [12, 25] that A generates a compact and exponentially stable analytic semigroup 5(t) (t > 0) in H, and

and A" is defined by

A" = (A-" )-1, D(A") = A-" (H).

Let H" be the Hilbert space (D(A"), <•, •>"), where

<, •>" = (A" •, A" •).

Particularly, we write H0 = H and H1 = D(A). Consider the linear Cauchy problem

x'(t)+Ax(t) = h(t), t > 0,

x(0) = x0.

When x0 e D(A) and h e C1([0, to),H), the linear Cauchy problem (2.1) has a classical solution x e C1([0, to),H) n C([0, to),H1) expressed by

x(t) = S(t)xo + i S(t - s)h(s) ds. (2.2)

When x0 e H and h e ¿1oc([0, to),H), the function x e C([0, to),H) given by (2.2) is a mild solution of the linear Cauchy problem (2.1). If the function x e Wo1([0, to),H) n ¿1oc([0, to), H1), then it is a strong solution of the linear Cauchy problem (2.1) (cf., e.g., [26, 27]).

Let D = {t1, t2,..., tm}c [0, T]. Write

PC([0, T],H) := {x: [0, T] ^ H;x is continuous at t e [0, T] \ D,

||S(t)|| < e-wt, t > 0.

From [12, 25], we also know that for any a > 0, A a is defined by

x is continuous from left and has right hand limits at t e D}

PCr(R+,H) := {x: R+ ^ H;x is continuous at t e R+ \ {t1, t2,...,tk,...},

x is continuous from left and has right hand limits at tk, k = 1,2,...,x(t + T)= x(t) for t e R+}.

It is clear that the restriction of PCt(R+,H) on [0, T] is PC([0, T],H). Set ||x||pc = max] sup ||x(t + 0)||, sup ||x(t — 0)|| |.

Ue[0,T] te[0,T] '

Then PC([0, T],H) (or PCT(R+,H)) is a Banach space endowed with the norm || • ||PC. Definition 2.1 Let V(•, •): A ^ Lb(H) be denoted by

v(t,e)= n (1 — ak)S(t — 0), (2.3)

0 <tk <t

where A := {(t, 0)\0 < 0 < t < to). Then {V(t, 0)\(t, 0) e A} is called an impulse propagator associated with {ak, tk}g=1.

Lemma 2.2 Impulsive propagator {V(t, 0 )\(t, 0) e A) has the following properties: (1) For 0 < 0 < t < T, V(t, 0) e Lb(H), i.e.,

||V(t, 0)|| ^\1 — ak\e—^l(t—0). k=1

Particularly, if 0 < ak <1, k = 1,2,..., m, then

V(t, 0)|| < e—w(t—0) < 1.

(2) For 0 < 0 < r < t < to, r = tk, k = 1,2,..., m,

V(t, 0) = V(t, r) • V(r, 0).

(3) For 0 < 0 < t < to ani N e Z+,

(i) V(t + NT, 0 + NT) = V(t, 0);

(ii) V(t + NT, 0) = V(t, 0)[V(T, 0)]N;

(iii) V(t + T, 0) = V(t,0)V(T, 0).

(4) IfS(t) (t > 0) is a compact semigroup in H, then V (t, 0) is a compact operator for 0 < 0 < t < T.

Proof By (2.3), we see easily that (1), (2), and (4) hold. Now, we prove that (3) also holds. For any 0 < 0 < t < to and N e Z+, by (2.3), we have

V(t + NT, 0 + NT)= Y\ (1 — ak)S(t + NT — 0 — NT)

0 +NT<tk <t+NT

= f](1 — ak )S(t — 0)

0 <tk <t

= V (t, 0)

v (t + nt, e )= n (1- ak)S(t + NT - d)

e <tk <t+NT

= n (1 - ak) n (1 - ak)S(t - e)S(NT)

e <tk <t t<tk <t+NT

= V (t, e) Yi (1- ak)S(NT)

t<tk <t+NT

= v (t, e )[V (T ,0)]N. It implies that (i) and (ii) hold. Moreover, V(t + t,e)= n (1-ak)S(t + t-e)

e <tk <t+T

= n (1- ak) n (1- ak)S(t)S(T - e)

e <tk <t t <tk <t+T

= V(T, e) [] (1- ak)S(t)

0<tk <t

= v (t,0)v (T, e).

This implies that (iii) holds. By (ii) and (iii), we also get

v (t + t , e) = v (t, e )v (t ,0) = v (t,0)v (t , e).

This completes the proof of Lemma 2.2. □

Let h e PCt(R+,H), ak e R, and yk e H, k = 1,2,.... We look at the existence of T-periodic solutions for the linear impulsive evolution equation

x'(t)+Ax(t) = h(t), t e R+, t = tk, ^^

Ax|t=tk = yk -akx(tk), k = 1,2,....

For (2.4), we have the following result.

Lemma 2.3 Let A : -D(A) c H ^ H be a positive definite selfadjoint operator in H and the embedding D(A) H be compact. Ifh e PCT(R+,H), ak e R, and yk e H1, k = 1,2,..., then the linear impulsive evolution equation (2.4) has a unique T-periodic mild solution x := Ah e PCt(R+,H) provided that 0<ak < 1, k = 1,2,...,m, andx e W1,2([0, T],H) n L2([0, T],H1) is a strong solution. Moreover, A: PCT(R+,H) ^ PCT(R+,H) is a compact operator.

Proof Letting h e PC([0, T], H), we first consider the Cauchy problem for linear impulsive evolution equations

x'(t)+Ax(t) = h(t), t e [0, T], t = tk,

Ax\t=tk = yk — akx(tk), k = 1,2,..., m, (2.5)

x(0) = xo.

Let t0 = 0, tm+1 = T, a0 = 0, and y0 = 0. Then, for any t e (tk,tk+1], k = 0,1,2,...,m, the Cauchy problem (2.5) becomes of the following form:

x'(t) + Ax(t) = h(t), t e (tk, tk+1],

x(t+) = (1 — ak)x(tk) +yk. It follows from [12, 27] that the problem (2.6) has a unique mild solution expressed by

x(t) = S(t — tk)[(1 — ak)x(tk) + yk] + / S(t — s)h(s)ds, t e (tk, tk+1].

Iterating successively in the equality above with x(tn), n = k, k — 1,..., 2,1,0, we infer that

x(t) = V(t, 0)x0 + i V(t, 0)h(0) d0 + ^ V(t, t+)yk, t e [0, T]. (2.7)

0<tk <t

In view of the maximal regularity of linear evolution equations with positive definite operators in Hilbert spaces ([25], Chapter II, Theorem 3.3), we see that for x0 e H1 ,the mild solution of the linear Cauchy problem (2.1) satisfies

x e W^,2([0,a],H) nL2([0,a],H1) n C([0,a],H1), (2.8)

where a > 0 is a fixed constant. Since yk e H1, k = 1,2,...,m, we deduce by using (2.8) interval by interval from [0, t1] to (tm, T] that

x e W1;2([0, T],H) nL2([0, T],H1) n PC([0, T],H1),

where x is the mild solution of the Cauchy problem (2.5).

On the other hand, if x e PCT (R+, H) isa T-periodic mild solution of the linear impulsive evolution equation (2.4), then x is a mild solution of the Cauchy problem (2.5), which satisfies

x0 := x(0) = x(T).

By Lemma 2.2,

jyh ln(1 — ak )<^1.

Therefore, I - V(T, 0) has a bounded inverse operator (I - V(T, 0)) 1. Hence, for

xq = (I - V(T,0))

/ v(T, e)h(e) de + J2 v(t, t+)yk

0 k=1 k

the Cauchy problem (2.5) has a unique mild solution x given by (2.7) with

x(T) =x(0) = xQ. Moreover, for any t > 0, we have

x(t) = v(t, o)x(o) + f v(t, e)h(e) de + V v(t, t+)yk

o<tk <t

= v(t, o)x(T) + v(t, e)h(e) de + ^ v(t, t+ )yk

0<tk <t

= V (t,0)

v(t, o)x(o) + f v(T, e)h(e) de + V v(t, t+)yk

0<tk <T

i v(t, e)h(e) de + V v(t, t+>k

0<tk <t

p T p t+T

= v(t,o)v(T,o)x(o) + v(t, o) / v(t, e)h(e) de + / v(t + t, e)h(e) de

+ v(t, o) V v(t, t+)yk + V v(t + T, t+)yk

Q<tk <T T<tk <t+T

v(t + t, e)h(e) de + ^ v(t + t, t+)yk

= x(t+T).

Q<tk <t+T

This implies that the T-periodic extension of x on R+ is the unique T-periodic mild solution of the linear impulsive evolution equation (2.4). Moreover, we can see that x e ^1,2([0, T],H) n L2([0, T],H1) is a strong T-periodic solution of the linear impulsive evolution equation (2.4) and

x(t) = V(t,0)(l - V(T,0))

„ T m

/ v(T, e)h(e) de ^ v(t, t++)yk

0 k=1 k

■j v(t, e)h(e) de + j2 v(t, t+)yk

0<tk <t

:= (Ah)(t), t e [0, T].

0 < a <1, ^ e (o,1-^ |.

Then the solution operator

A: PC([0, T],H) ^ PCA([0, T],Ha)

is a continuous linear operator by [2], Lemma 2.1 and Corollary 2.2. Thus, Aszela-Ascoli's theorem shows that the embedding

PCA([0, T],Ha) ^ PC([0, T],H)

is compact. Thus, A: PCT(R+,H) ^ PCT(R+,H) is a compact linear operator. □

3 Main results

Next we always assume that ak e (0,1) and yk e H1, k = 1,2,..., m. First, we consider the following Cauchy problem:

x'(t)+Ax(t)=f(t, x(t)), t > 0, t = tk,

Ax\t=tk = yk — akx(tk), k = 1,2,..., (3.1)

x(0) = x0 e H.

For the Cauchy problem (3.1), we have the following uniqueness theorem.

Theorem 3.1 Let —A be the infinitesimal generator of a C0-semigroup S(t) (t > 0). Assume that

(HI) There exists a constant C >0 such that

|f (t, u) —f (t, v) || < C||u — v||, t > 0, u, v e H. Then the Cauchy problem (3.1) has a unique strong solution x e Wl1o,2((0, to),H) nL2oc((0, to),H1) n PC([0, to),H),

x(t) = V(t, 0)x0 + i V(t, 0)f (0, x(0)) d0 + ^ V(t, t+)yk, t > 0.

0 0<tk<t

Proof For t e [0, t1], the Cauchy problem (3.1) is in the following form:

x'(t) +Ax(t) = f (t, x(t)), t e [0, ti], x(0) = x0.

By [26], Chapter 6, Theorem 1.2, the Cauchy problem (3.2) has a unique global mild solution x e C([0, t1],H), and

x(t) = S(t)xo + i S(t - s)f (s,x(s)) ds, t e [0, ti]. (3.3)

For t e (ti, t2], the Cauchy problem (3.1) is of the following form:

x'(t) + Ax(t) = f (t,x(t)), t e (ti, £2], x(t+) = (l-fli)x(ti) +yi.

By (3.3), the Cauchy problem (3.4) has a unique mild solution

xi(t) = S(t - ti)[(i - ai)x(ti) + yi\ + / S(t -s)f(s,x(s)) ds

= (i - ai)S(t)xo + / (i - ai)S(t - s)f (s, x(s)) ds Jo

+ / S(t - s)f(s,x(s)) ds + S(t - ti)yi. Jti

Doing this interval by interval, we obtain

x(t) = V(t,0)xo+ i V(t, 0)f (0, x(0)) d0 + ^ V(t, t+)yk, t > 0.

0<tk <t

Let x be the mild solution of Cauchy problem (3.i). Then we can deduce that

x e w£2((0,to),H) ni2oc((°,Hi),

and it is a strong solution of the Cauchy problem (3.i). □

Now, we can consider the existence, uniqueness, and asymptotic stability of T-periodic solutions for the impulsive evolution equation

x'(t)+Ax(t)=f (t, x(t)), t e R+, t = tk,

Ax|t=tk = yk - akx(tk), k = i, 2,...,

where f: R+ x H ^ H is continuous and f (t,x) is T-periodic in t. For the impulsive evolution equation (3.5), we have the following theorem.

Theorem 3.2 Let A : D(A) c H ^ H be a positive definite selfadjoint operator in H and the embeddingD(A) H be compact. Assume that (H2) There exist two constants Mi e (0, ¡xi) and M2 > 0 such that

|f (t, u) || < Mi ||u|| + M2, t e R+.

Then the impulsive evolution equation (3.5) has at least one strong T-periodic solution

x e w£2(R+,H) nL2oc(R+,Hi) n PCt(R+,H).

Proof Set

G(x)(t) :=f (t,x(t)), x e PCt(R+,H), t e R+.

Then G: PCt(R+,H) ^ PCt(R+,H) is continuous and maps bounded sets in PCt(R+,H) into bounded sets. By Lemma 2.3 and the fact that

A: PCr (R+, H) ^ PCr (R+, H)

is a compact operator, we see that the operator

$ := A o G: PCT(R+,H) ^ PCT(R+,H)

is completely continuous. It is clear that T-periodic mild solutions of the impulsive evolution equation (3.5) are equivalent to fixed points of operator $. Choose

M2(l- e-fiT) + ^iEt=i WykII (¡i- Mi)(l- e-nT)

and let

Br := {x e PCt(R+,H): ||x|pc < r}.

Then, for every x e Br and t e [0, T],

(I - V(T,0))

£[v (t ,0)]n

<E n(1- )

«e-«m T <

n=0 k=i

1 - e-fiT

Hence,

|($x)(t)| = || AG(x))(t) ||

V(t,0)(I - V(T,0))-

/ v(t, e)f(e,x(e)) de + J2 v(t, t++)yk Z0 k=i

i |v(t,e)f(e,x(e))|de+ £ |v(t,t++ )y

J0 0<tk <t

/ e-fi(T-e) de (Mir + M2)^V W

(Mir + M2)^ Wyk II

i-e-« T i - e-fit

Mir + M2 I e

-¡¿it

i -e-fiT

i)£ Wyk W

Mir+M2 + L m=i wyk w

i - e-fiT '

This means that W $x||PC < r. So, $(Br) c Br is completely continuous. By Schauder's fixed point theorem, we know that $ has at least one fixed point x in Br. Therefore, x is a

T-periodic mild solution of the impulsive evolution equation (3.5). By Lemma 2.3,

x e w£2(R+,H) nL2oc(R+,Hi) n PCt(R+,H) is also a strong T-periodic solution of the impulsive evolution equation (3.5). □

For the impulsive evolution equation (3.5), we also have the following uniqueness and asymptotic stability theorem.

Theorem 3.3 Let A : D(A) c H ^ H be a positive definite selfadjoint operator in H and the embeddingD(A) ^ H is compact. Assume that (H3) There exists a constant 0 < M < fi1 such that

|f (t, u) —f (t, v) || < M||u - v\\, t e R+, u, v e H.

Then the impulsive evolution equation (3.5) has a unique strong T-periodic solution which belongs to W^c (R+, H) n L2oc(R+, H1) and it is globally asymptotically stable.

Proof Since the condition (H3) ^ (H2) holds, by Theorem 3.2, the impulsive evolution equation (3.5) has at least one strong T-periodic solution which belongs to W^R^,H) n L2oc(R+,H1). Let x1, x2 be strong T-periodic solutions of the impulsive evolution equation (3.5). Then they are fixed points of the operator $. By the definition of we have

|X!(Î)-X2(Î)| = |($X!)(i)-($X2)(i)|

V(t,0)(l - V(T,0))-i i V(T, 0 )f (0, x1(0 )) -f (0, x2(0 ))] d0 Jo

H /t||V (t, 0 )f (0, xi(0 )) -f (0, X2(0 ))]|| d0 Jo

1-e-wT

f e-»l(T-0(0,xi(0))-f (0,X2(0))| d0 Jo

i e^l(t-0)\f (0,xi(0)) -f(0,x2(0))| d0.

Therefore,

llxi - x2 H PC <

Me-^lt M(i- e-^11 ) -+-

llxi - x2 l PC

= -||xi - x2 ^PC.

Hence xi = x2. Thus, the impulsive evolution equation (3.5) has a unique strong T-periodic solution.

Let x e PCT(R+, H) be the unique strong T-periodic solution of the impulsive evolution equation (3.5). By Theorem 3.i, for any x0 e H, the Cauchy problem (3.i) has a unique strong solution

x(t) := x(t;xo) e Wl1o,c2((0,to),h) ni2o^(0,Hi) n [0,to),H).

Hence,

||x(t) -x(t)|| < e"wt||x(0) - x(0)|| + M i e"w(t"s)||x(s)-x(s)|| ds

= e"wt ||x(0) - x(0) || + Me-^lt i e^s ||x(s) - x(s) || ds, t > 0.

Write p(t):= e^lt ||x(t) - x(t) ||. Then, for t e [0, to), by the inequality above, we have p(t) < p(0) + m( p(s) ds.

eMlt ||x(t) -x(t)|| < ||x(0) -x(0) || eMt.

It follows that

Hx(t)-x(t)H < Wx(0)-x(0)||e-(w-M)t ^ 0 (t ^ to).

Therefore, the strong T-periodic solution x of the impulsive evolution equation (3.5) is globally asymptotically stable and it exponentially attracts every solution of the corresponding Cauchy problem. □

4 Impulsive problem for a class of parabolic equations

Consider time 2n -periodic solutions of the impulsive problem for the following parabolic equation:

' dx(z, t) = Ey=l d(Tij(z)ij) - T0(z)x(z, t) +f (z, t,x(z, t)),

z e fi, t e R+ \{fn;k = l,2,...},

2 k (4.l) Ax(z, tk) =yk - akx(z, tk), z e fi, tk = 5 n, k = l, 2,...,

.x(z, t)|ze3fi = 0,

where fi c RN is a bounded domain, whose boundary dfi is sufficiently smooth, f: fi x R+ x R ^ R is continuous and is 2n-periodic in the second variable. We assume that the following conditions are satisfied.

(Al) %ij e Cl+v (fi) (i, j = l, 2,..., N), [Tj(z)]NxN is a positive definite symmetric matrix for z e fi and there exists a constant y >0 such that

J2 Tij(z)ZiZj > YIZ I2, VZ = (Zl, Z2,..., Zn) e Rn,z e fi. i,j=l

(A2) T0 e Cv(fi) for some v e (0, l), T0(z) > 0 on fi.

Furthermore, there exists a function p: [0, to) ^ [0, to) such thatf: fi x R+ x R ^ R satisfies

f (z, t, £ )-f (¿, t', £ ')| < p(n)(|z - z'|v + |t -1'|2 + - £ 'j), for any n > 0 and (z, t, £),(z/, t', £') e fi x R+ xB(R; 0, n), whereB(R;0,n) = {£ e R: If | < n}.

Take H = L2(Q). Define

N d / d x\

Ax :=V d[Tij(z) — - To(z)x, D(A):=H2(fi) nH0(fi). (4.2)

u d zA 9 ZV

Then A is a positive definite selfadjoint operator in H and D(A 2 )= H0 (fi). So, A generates a compact analytic semigroup 5(t) (t > 0) in H which is exponentially stable. Let

ak := —, k =1,2,3,4 k 10

with ak+4 := ak and yk e H0(fi) with yk+4 := yk. It is obvious that

tk+4 = tk + 2tt .

Let /i be the smallest eigenvalue of operator

N d / d x \ g Tij(z) dzJ -To(z)

under the Dirichlet boundary condition x|dQ = 0. Then /i > 0. Under the above assumptions, we have the following existence theorem.

Theorem 4.1 Assume that (Fl) There exist two constants Mi e (0, /i) and M2 > 0 such that

f(z, t, f )| < Mi|f | + M2, z e Q, t e R+, f e R.

Then the impulsive boundary value problem (4.1) has a time 2n -periodic solution x e

C2+y,1+ 2 (ft X R+).

Proof Define

x(-)(z) := x(z, •), f (•,x( ))(z) :=f (z, •,x(z, •)).

Then the impulsive boundary value problem (4.1) can be rewritten in the form of (3.5). By Theorem 3.2, the boundary value problem (4.1) has at least one strong time 2n-periodic solution

x e PC2n(R+,H0(n)) nL2oc(R+,H2(ft)) n Woe (R+,L2(n)).

By the regularization method in [2], Lemma 4.2, we can prove that x e C2+v,1+ 2 (ft x R+) is a classical time 2n-periodic solution of the boundary value problem (4.1). □

Theorem 4.2 Assume that (F2) There exists a constant M e (0, ¡ii) such that

f (z, t, à) -f (z, t, &) | < M\£i - & \ , z e a, t e R+, & e R.

The« the boundary value problem (4.1) has a unique time 2n-periodic solution x e c2+y,i+ 2 (a x R+), which exponentially attracts every solution of the boundary value problem (4.i) with initial value condition x(z, 0) = x0(z) in L2(a).

Proof Clearly, the condition (F2) implies the conditions (Hi) and (H3) hold. By Theorem 3.i, the boundary value problem (4.1) has a unique solution x(z, t; x0) e c2+v,i+ 2 (a x (0, to)), which satisfies the initial value condition x(z, 0) = x0(z). By Theorem 3.3, the boundary value problem (4.1) has a unique time 2n-periodic solution x e PC2n ((0, to), L2(a)) n c2+v,l+ 2(a x R+), which exponentially attracts x(z, t;x0) inL2(a). □

5 Applications

Example 5.1 We consider the following impulsive boundary value problem:

' d x(z t) = VN 32x - x(z t) + w sint |x(z,t)| + 2

dtx(z L)=/-^i,j=l dzd j L) + 2 l+|x(z,t)| + ez ,

z e fi, t e R+ \{fn; k =l,2,...},

2 k (5.l)

Ax(z, tk) =yk - akx(z, tk), z e fi, tk = 2 n, k = l, 2,...,

.x(z, t)|ze3fi = 0,

where fi c RN is a bounded domain whose boundary dfi is sufficiently smooth,

ak := —, k = l, 2,3,4 k l00

with ak+4 := ak, yk e H^(fi) with yk+4 := yk, and ¡l is the smallest eigenvalue of operator £N=i (xij(z)) - T0(z) under the Dirichlet boundary condition x|gfi = 0. Let H = L2(fi), A be the operator as in (4.2), and

f »:= ^ ^ + 7 ■

Then we have

f(z, t,x(z, t)) | <

¡i sin t \x(z, t) \

2 i+\x(z, t)\

\f (t,x(t))|2 <(ja

¡2 |x(z, t)| +2

< ¡2 |x(z, t)| +2

< ¡2\\x\\2 + 2.

It implies that the conditions (Fl) and (H2) hold. Hence the impulsive boundary value problem (5.l) can be rewritten into the abstract form (3.5). Thus, by Theorem 3.2, the

problem (5.1) has at least one strong time 2n-periodic solution

X e PC2n(R+,H0(fi)) nL2oc(R+,H2(ft)) n Wl1o,2(R+,L2(ft)).

By the same argument as in the proof ofTheorem 4.1, we see that x e C2+v,1+ 2 (ft x R+) is a classical time 2n -periodic solution of the boundary value problem (5.1).

Example 5.2 We take a look at the following impulsive problem:

d x(z t) = VN d2x — x(z f) + Hi sint |x(z,t)| dtx(z L)=/-^i,j=i dzidZj x(z L) + 3ez i+|x(z,t)|,

z e ft, t e R+ \{| n ; k = i, 2,...}, Ax(z, tk) =yk - akx(z, tk), z e ft, tk = f n, k = i, 2,..., x(z, t)|ze3Q = 0,

where ft, ak, yk are the same as in (4.1), and ¡x1 is a constant as in (5.1). Let H = L2(ft), A be the operator as in (4.2), and

f(z, t, x(z, t)) :=

/xi sin t |x(z, t) | 3ez i+|x(z, t)| '

f(z, t,xi(z, t)) -f (z, t,x2(z, t)) | <

Hi sin t( |xi(z, t)| |x2(z, t)|

3ez \i + |xi(z, t)| i+|x2(z, t)|

< Hi |xi(z, t) — x2 (z, t)1

\f (t, xi(t)) —f (t, x2(t)) II2 < (Ja

xi(z, t) — x2(z, t)l

< -3 ||xi — xdk.

It implies that the conditions (F2), (Hi), and (H3) hold. Therefore, by Theorem 3.i, the problem (5.2) has a unique solution x(z, t;xo) e C2+y,i+ 2 (ft x (0, to)), which satisfies the initial value condition x(z, 0) = x0(z). Moreover, by Theorems 3.3 and 4.2, the problem (5.2) has a unique time 2n-periodic solution

x e PC2n ((0, to),L2(Q)) n C2+^+ 2 (ft x R+),

which exponentially attracts x(z, t;x0) in L2(ft).

Competing interests

The authors declare that they have no competing interests. Authors' contributions

All the authors contributed equally and significantly in writing this paper. Allauthors read and approved the final manuscript.

Author details

1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, P.R. China. 2School of Mathematical

Sciences, Fudan University, Shanghai, 200433, P.R. China.

Acknowledgements

The authors thank the referees very much for their careful reading and helpful suggestions.

J Liang and TJ Xiao acknowledge support from NSFC (No. 11171210,11271082,11371095). H Yang acknowledges

support from NSF ofGansu province (No. 1308RJZA217) and China Postdoctoral Science Foundation Funded Project (No.

BR0710008).

Received: 6 November 2015 Accepted: 1 December 2015 Published online: 15 December 2015

References

1. Abada, N, Benchohra, M, Hammouche, H: Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J. Differ. Equ. 246,3834-3863 (2009)

2. Amann, H: Periodic solutions of semilinear parabolic equations. In: Cesari, L, Weinberger, R (eds.) Nonlinear Analysis: A Collection of Papers in Honor of Erich H. Rothe, pp. 1-29. Academic Press, New York (1978)

3. Benchohra, M, Henderson, J, Ntouyas, S: In: Impulsive Differential Equations and Inclusions. Comtemp. Math. Appl., vol. 2. Hindawi Publ. Corp., Cairo (2006)

4. Cuevas, C, Sepulveda, A, Soto, H: Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations. Appl. Math. Comput. 218,1735-1745 (2011)

5. Cuevas, C, Pierri, M, Sepulveda, A: Weighted S-asymptotically ^-periodic solutions of a class of fractional differential equations. Adv. Differ. Equ. 2011, Article ID 584874 (2011)

6. De la Sen, M: About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory. Fixed Point Theory Appl. 2011, Article ID 867932 (2011)

7. De la Sen, M: Total stability properties based on fixed point theory for a class of hybrid dynamic systems. Fixed Point Theory Appl. 2009, Article ID 826438 (2009)

8. De la Sen, M: About robust stability of dynamic systems with time delays through fixed point theory. Fixed Point Theory Appl. 2008, Article ID 480187 (2008)

9. Diagana, T: Almost periodic solutions to some second-order nonautonomous differential equations. Proc. Am. Math. Soc. 140,279-289 (2012)

10. Diagana, T: Pseudo-almost periodic solutions for some classes of nonautonomous partial evolution equations. J. Franklin Inst. 348,2082-2098 (2011)

11. Diagana, T: The existence of a weighted mean for almost periodic functions. Nonlinear Anal. 74,4269-4273 (2011)

12. Henry, D: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math., vol. 840. Springer, New York (1981)

13. Lakshmikantham, V, Bainov, D, Simeonov, P: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

14. Li, YX: Existence and uniqueness of periodic solution for a class of semilinear evolution equations. J. Math. Anal. Appl. 349, 226-234 (2009)

15. Li, YX: Existence and asymptotic stability of periodic solution for evolution equations with delays. J. Funct. Anal. 261, 1309-1324 (2011)

16. Liang, J, Liu, JH, Xiao,TJ: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. Math. Comput. Model. 49, 798-804 (2009)

17. Liang, J, Liu, JH, Xiao, TJ: Periodic solutions of delay impulsive differential equations. Nonlinear Anal. 74,6835-6842 (2011)

18. Liu, JH: Periodic solutions of infinite delay evolution equations. J. Math. Anal. Appl. 247,644-727 (2000)

19. Liu, JH: Bounded and periodic solutions of infinite delay evolution equations. J. Math. Anal. Appl. 286, 705-712 (2003)

20. Liu, XZ: Impulsive stabilization and applications to population growth models. Rocky Mt. J. Math. 25,381-395 (1995)

21. Machado, JA, Ravichandran, C, Rivero, M, Trujillo, JJ: Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions. Fixed Point Theory Appl. 2013,66 (2013)

22. Mophou, GM, N'Guerekata, GM: Existence of Antiperiodic Solutions to Semilinear Evolution Equations in Intermediate Banach Spaces. In: Advances in Interdisciplinary Mathematical Research, Springer Proc. Math. Stat., vol. 37, pp. 133-139. Springer, New York (2013)

23. Mophou, GM: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal. 72,1604-1615 (2010)

24. Mophou, GM: Almost automorphic solutions of some semilinear fractional differential equations. Int. J. Evol. Equ. 5(1), 109-115 (2010)

25. Pazy, A: Semigroup of Linear Operator and Applications to Partial Differential Equations. Springer, Berlin (1983)

26. Temam, R: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York (1997)

27. Xiao, TJ, Liang, J: The Cauchy Problem for Higher Order Abstract Differential Equations. Lecture Notes in Math., vol. 1701. Springer, Berlin (1998)

28. Ahmed, N: Optimal impulsive control for impulsive systems in Banach spaces. Int. J. Differ. Equ. 1, 37-52 (2000)

29. Ahmed, N: Some remarks on the dynamics of impulsive systems in Banach spaces. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 8,261-274 (2001)

30. Ahmed, N: Existence of optimal controls for a general class of impulsive systems on Banach spaces. SIAM J. Control Optim. 42,669-685 (2003)

31. Barreira, L, Valls, C: Lyapunov regularity of impulsive differential equations. J. Differ. Equ. 249,1596-1619 (2010)