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RESEARCH Open Access

Existence and stability of almost periodicsolutions for impulsive differential equationsJunwei Liu* and Chuanyi Zhang

* Correspondence:[email protected] of Mathematics,Harbin Institute of Technology,Harbin 150001, PR China

Abstract

In this article, by using Schauder’s fixed point theorem, we study the existence ofalmost periodic solutions for abstract impulsive differential equations. In addition,sufficient conditions for their asymptotic stability are obtained by means ofgeneralized Gronwall-Bellman inequality.

Keywords: abstract impulsive differential equations, Schauder’s fixed point theorem,asymptotic stability, generalized Gronwall-Bellman inequality

1 IntroductionWhen a dynamical system processes aftereffect and state changing by jumps, this sys-

tem is called an impulsive differential system. Because of the needs of modern technol-

ogy, such as simulation in physics, biology, populations dynamics, control theory,

industrial robotics, etc., the study of impulsive differential equations attracts more and

more researchers’ interest, see [1-8]. There are many articles [9-16] about existence of

solutions, periodic solutions and stability for impulsive differential equations. But, only

a few articles [17,18] have studied the existence of almost periodic solutions to abstract

impulsive differential equations in Banach space.

By means of fractional powers of operators and Contraction mapping principle, Sta-

mov and Alzabut in [17] studied the existence and uniqueness of exponentially stable

almost periodic solutions for the abstract impulsive differential equation as follows:

x(t) + Ax = F(t, x) +∞∑

k=−∞Gk(x)δ(t − τk), (1:1)

where Gk(⋅) are continuous impulsive operators.

In [18], by using Contraction mapping principle, Henriquez et al. studied the exis-

tence of almost periodic solutions to the following impulsive differential equation:{u′(t) = A(t)u(t) + F(u(t)) + f (t), t ∈ R, t �= ti, i ∈ Z,

�u(ti) = Ii(u(ti)).(1:2)

To the best of authors’ knowledge, most of the previous research on existence of

almost periodic solutions for impulsive differential equations was based on Contraction

mapping principle. Lipschitz condition is necessary. Motivated by the studies in

[8,12,13,19-21], in this article we use Schauder’s fixed point theorem and generalized

Liu and Zhang Advances in Difference Equations 2012, 2012:34http://www.advancesindifferenceequations.com/content/2012/1/34

© 2012 Liu and Zhang; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

mailto:[email protected]

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Gronwall-Bellman inequality to investigate the existence and asymptotic stability of

almost periodic solutions to the following impulsive differential equation:{u′(t) = A(t)u(t) + f (t, u(t)), t ∈ R, t �= ti, i ∈ Z,

�u(ti) = Ii(u(ti)).(1:3)

where A(t) : X ® X are closed linear operators on a Banach space (X, || ⋅ ||), f, Ii,and ti satisfy suitable conditions that will be established later and the symbol Δξ (t)

represents the jump of the function ξ at t, which is defined by Δξ (t) = ξ (t+) − ξ(t−).

2 PreliminariesIn this section, we give some notations, definitions, and preliminary facts about impul-

sive differential equations. Throughout this article, we denote by (X, || ⋅ ||) a Banach

space and by R the set of real numbers. The symbol L(X) stands for the Banach space

formed by all bounded linear operators form X into X endowed with the uniform

operator topology. For a linear operator A, r(A) is its resolvent set and for l Î r(A), R(l,A) = (lI − A)−1 denotes its resolvent operator. Let T be the set consisting of all real

sequences {ti}iÎZ such that a = inf iÎZ(ti+1 - ti)> 0 and limi®−∞ ti = −∞, limi®−∞ ti = ∞.

For {ti}i∈Z ∈ T, let PC(R, X) be the space formed by all piecewise continuous func-

tions j : R ® X such that j(⋅) is continuous at t for any t ∉ {ti}iÎZ and φ(ti) = φ(t−i )for all i Î Z; let PC(R × X, X) be the space formed by all piecewise continuous func-

tions j : R × X ® X such that for any xÎ X, j(⋅, x) is continuous at t for any t ∉ {ti}

iÎZ and φ(ti, x) = φ(t−i , x) for all i Î Z and for any t Î R, j(t, ⋅) is continuous at x Î X.

For a function f Î PC (R, X) (respectively PC(R × X, X)) and τ Î R, the translation of

f by τ is the function Rτ f(t) = f(t + τ) (respectively, Rτ f(t, x) = f(t + τ, x)).

A number τ Î R is called an ε-translation number of the function f Î PC(R, X) (PC(

R × X, X )) if∥∥Rτ f (t) − f (t)∥∥ < ε, (

∥∥Rτ f (f , x) − f (t, x)∥∥ < ε, x ∈ K, for every compact K ⊆ X)

for all t Î R which satisfies |t - ti| >ε. Denote T(f , ε) by the set of all ε-translation

numbers of f.

Definition 2.1 (D1) A function j Î PC(R, X)(PC(R × X, X)) is said to be piecewise

almost periodic if the following conditions are fulfilled:

(1){tji = ti+j − ti

}, j Î Z, are equipotentially almost periodic.

(2) For any ε > 0 (every compact subset K of X), there exists a positive number δ =

δ (ε)(δ = δ(ε, K)) such that if the points t′ and t″ belong to a same continuity and |t

′ - t″| <δ, then ||j (t′) - j(t″)|| <ε (||j(t′, x) - j(t″, x)|| <ε, for any x Î K).

(3) For every ε > 0 (every compact subset K of X), T(φ, ε) (T(φ, ε, K)) is a rela-

tively dense set in R.

We denote by APT(R, X)(APT(R × X, X)) the space of all piecewise almost periodic

functions. Obviously, the space APT(R, X)(APT(R × X, X)) endowed with the norm of

the uniform convergence || ⋅ || is a Banach space.

(D2) A set B ⊆ APT(R, X) is called a uniformly piecewise almost periodic family if it is

uniformly bounded and if given ε > 0, then T(B, ε) = ∩f∈BT(f , ε) is relatively dense in R.


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Obviously, if f Î APT(R × X, X) and for each compact set K ⊆ X, f(t, x) : R × X ® X

is uniformly continuous in x Î K uniformly in t Î R, then {f(⋅, x) : x Î K} is a uni-

formly piecewise almost periodic family

Lemma 2.2 [18] Let j Î APT(R, X), then the range of j, R(j), is a relatively compact

subset of X.

Lemma 2.3 If a set B ⊆ APT(R, X) is relatively compact, then ∪fÎB R ( f ) is a rela-

tively compact set in X.

Proof Since B is a relatively compact set, for any ε > 0, there exists a finite number of

functions f1, f2, ..., fN Î B such that for any f Î B, there exists a number i Î {1, 2, ...,

N}, then

∥∥f − fi∥∥ ≤ ε

2.

By Lemma 2.2, R(f1), R(f2), ..., R(fN) are relatively compact sets in X, so is ∪1≤ i≤ NR ( fi ),

that is, ∪1≤ i≤ NR ( fi ) can be embedded in a finite number of balls of radiusε

2. Let us

denote by x1, x2, ..., xn the centers of the balls of radiusε

2which cover the set ∪1≤ i≤ NR (fi ).

For any t Î R, f Î B, let xi be the center of the ball of radiusε

2which contains fi(t), we have

∥∥f (t) − xi∥∥ ≤ ∥∥f (t) − fi(t)

∥∥ +∥∥fi(t) − xi

∥∥ ≤ ε

2+

ε

2= ε.

This show that UfÎB R(f) is covered for any ε > 0 by a finite number of balls of radius

ε. This proof is complete.

For the operators {A(t) : t Î R}, we make the following assumption (H) :

(H1) {A(t) : t Î R} are linear operators on a Banach space X and there are con-

stants K ≥ 0, r Î R and φ ∈(π

2,π

)such that

∥∥R(λ,A(t))∥∥ ≤ K

1 + |λ − r|for l Î r(A(t)) ∩ ∑(j, r) and t Î R, where ∑(j, r) = {r} ∪ {l : | arg(l - r)| ≤ j}.

(H2) There are constants L ≥ 0 and µ, ν Î (0, 1] with µ + ν > 1 such that

|λ|ν∥∥∥Ar(t)R

(λ,Ar(t)

)(Ar(t)

−1 − (Ar(s)−1)

∥∥∥ ≤ L|t − s|μ

for Ar(t) = A(t) -r, t, s Î R and | arg l| <j.

(H3) R (l, A(⋅)) Î AP (L(X)).

(H4) The evolution family {T(t, s) : t, s Î R, t ≥ s} generated by {A(t) : t Î R} is

exponentially stable, i.e., there exists numbers M > 0, w > 0, such that ||T(t, s)|| ≤

Me-w(t - s), t ≥ s, t, s Î R.

(H5) For fixed t, s Î R, t ≥ s, the operator T(t, s) : X ® X is compact, that is, T(t, s)

is continuous and maps a bounded set into a relatively compact set.

(H6) For each x Î X, T(t+h, t)x ® x as h ® 0+ uniformly for t Î R, i.e., {T(t, s) : t

≥ s} is uniformly continuous.


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Operators {A(t) : t Î R} fulfilling (H1) and (H2) are called sectorial, see [22-25].

Lemma 2.4 [25] Assume that the conditions (H1)-(H4) hold. Then for each ε > 0

and δ > 0, there is a relatively dense set Ωε,δ such that

∥∥T(t + τ , s + τ ) − T(t, s)∥∥ ≤ εe−

w2 (t−s), t − s > δ, t, s ∈ R, τ ∈ ∈,δ.

This property can be abbreviated by writing T Î AP(L(X)).

Lemma 2.5 [18] Assume that f Î APT(R, X), T Î AP(L(X)), the sequence {xi : i Î Z}

is almost periodic, and{tji}, j Î Z, are equipotentially almost periodic. Then for each ε

> 0 there exist relatively dense sets ε,f ,xi,T of R and Qε,f ,xi,T of Z satisfying:

(i) ||f(t + τ) - f(t)|| <ε for all t Î R, |t - ti| >ε, τ ∈ ε,f ,xi,T and i Î Z.

(ii)∥∥T (t + τ , s + τ) − T(t, s)

∥∥ < εe−w2 (t−s) for all t, s Î R, |t - s| > 0, |t - ti| >ε, |s - ti|

>ε, τ ∈ ε,f ,xi,T and i Î Z; w > 0 is the same as in (H4).

(iii) ||xi + q - xi|| <ε for all q ∈ Qε,f ,xi,T and i Î Z.

(iv)∣∣∣tqi − τ

∣∣∣ < ε for all q ∈ Qε,f ,xi,T , τ ∈ ε,f ,xi,T and i Î Z.

Definition 2.6 A bounded function u(t) : R ® X is called a mild solution of (1.3) if

for any t Î R, s <ti <t <ti+1,

u(t) = T(t, σ )u(σ ) +

t∫σ

T(t, s)f (s, u(s))ds +∑

σ<ti<t

T(t, ti)Ii(u(ti)

).

Under the assumption (H), ||T(t, s)|| ≤ Me-w(t - s) for all t >s. Let s ® - ∞, then ||T

(t, s) || ® 0 and so, the above formula can be replaced by

u(t) =

t∫−∞

T(t, s)f(s, u(s)

)ds +

∑ti<t

T(t, ti)Ii(u(ti)

).

In fact, for t >s,

t∫σ

T(t, s)f(s, u(s)

)ds +

∑σ<ti<t

T(t, ti)Ii(u(ti)

)

=

t∫−∞

T(t, s)f(s, u(s)

)ds −

σ∫−∞

T(t, s)f(s, u(s)

)ds

+∑

−∞<ti<t

T(t, ti)Ii(u(ti)

) −∑

−∞<ti<σ

T(t, ti)Ii(u(ti)

)

=

t∫−∞

T(t, s)f(s, u(s)

)ds +

∑−∞<ti<t

T(t, ti)Ii(u(ti)

)

− T(t, σ )

⎡⎣ σ∫

−∞T(σ , s)f

(s, u(s)

)ds −

∑−∞<ti<σ

T(σ , ti)Ii(u(ti)

)⎤⎦= u(t) − T(t, σ )u(σ ),


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so that,

u(t) = T(t, σ )u(σ ) +

t∫σ

T(t, s)f(s, u(s)

)ds +

∑σ<ti<t

T(t, ti)Ii(u(ti)

).

The following lemma is a criterion of relative compactness for subsets in APT(R, X),

its proof is similar to the case in AP(R, X) (e.g., see [26]).

Lemma 2.7 Let W ⊂ APT(R,X). Then W is relatively compact in APT(R, X) if and

only if the following properties hold true:

(i) W is equicontinuous, which means that for any ε > 0, there exists δ(ε) > 0 such

that for any f ∈ W one has || f (t) - f(s) || <ε if |t - s| <δ(ε) and t and s in a same

continuity.

(ii) W is equi-almost-periodic, which means that for each ε > 0 there exists l(ε) > 0

such that any interval (a, a + l) ⊂ R contains a number τ that is an ε- almost per-

iod for all f ∈ W.

(iii) For fixed t Î R, the set {f (t) : f ∈ W} is relatively compact in the space X.

Definition 2.8 [27] The solution x(t, s, j) of (1.3) is said to be stable, if for any s ÎR, ε > 0, there is a number δ = δ(s, ε) > 0, such that for any other solution y(t, s, ψ) of(1.3) satisfying ||� - ψ|| <δ then ||x(t, s, �)-y(t, s, ψ)|| <ε.x(t, s, �) is said to be asymptotically stable if it is stable and if there is a constant b >

0 such that ||� - ψ|| <b, then limt ® ∞ ||x(t, s, �) - y(t, s, ψ)|| = 0.

Lemma 2.9 ([1], generalized Gronwall-Bellman inequality) Let a nonnegative func-

tion u(t) Î PC(R, X) satisfy for t ≥ t0 the inequality

u(t) ≤ C +

t∫t0

υ(τ )u(τ )dτ +∑

t0<τi<t

βiu(τi),

where C ≥ 0, bi ≥ 0, v(τ) > 0, and τ ′i s are discontinuity points of first type of the func-

tion u(t). Then the following estimate holds for the function u(t),

u(t) ≤ C∏

t0<τi<t

(1 + βi)e∫ tt0

υ(τ)dτ .

3 Main resultsIn order to get the almost periodic solutions of (1.3), in addition to the previous

assumptions, we require the following additional assumptions:

(A1) ∀L > 0, CL = suptÎR, ||x|| ≤ L ||f(t, x)|| <∞, C = supiÎZ ||Ii|| < ∞. Moreover,

there exists a number L0 > 0, such that M(CL0

w+

C1 − e−wα

)≤ L0 and

CL0 +Mw

+C +M

1 − e−wα≤ 1

2.

(A2) For each compact set K ⊆ X, f(t, x) Î APT(R × X, X) is uniformly continuous

in x Î K uniformly in t Î R; Ii(x) is almost periodic in i Î Z uniformly in x Î K

and is a uniformly continuous function defined on the set K ⊆ X for all i Î Z.


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(A3) Let {xn} ⊆ APT(R, X) be uniformly bounded in R and uniformly convergent in

each compact set of R, then {f(⋅, xn(⋅))} is relatively compact in PC(R, X).

(A4) There exists a compact set K0 ⊆ X, for any r > 0,

Br = {u ∈ APT(R,X) : ‖u‖ ≤ L0,∥∥u(t + τ ) − u(t)

∥∥ ≤ r, |t − ti| > r, i ∈ Z, τ ∈ r,f ,Ii ,K0,T},

such that R(x) ⊆ K0 for all x Î Br, where the choice of r,f ,Ii ,K0,T is similar to that of

Lemma 2.5.

Theorem 3.1 Suppose the conditions (H), (A1)-(A4) hold, then (1.3) has a mild pie-

cewise almost periodic solution.

Proof Define the operator Γ on APT(R, X) by

�u(t) =

t∫−∞

T(t, s)f(s, u(s)

)ds +

∑ti<t

T(t, ti)Ii(u(ti)

).

Now we use Schauder’s fixed point theorem to prove that Γ has a fixed point in

APT(R, X).

For any n Î N, let

B 1n= {u ∈ APT(R,X) : ‖u‖ ≤ L0,

∥∥u(t + τ ) − u(t)∥∥ ≤ 1

n, |t − ti| >

1n, i ∈ Z, τ ∈ 1

n ,f ,Ii,K0,T},

and

B =⋂n∈N

B 1n.

By (A4), for any n ∈ N, {u(t) : u ∈ B 1n, t ∈ R} ⊆ K0. Note that

{u(t) : u ∈ B, t ∈ R} ⊆ {u(t) : u ∈ B 1n, t ∈ R}, so {u(t): u Î B, t Î R} ⊆ K0. By (A2), {f(⋅,

x) : x Î K0} and {Ii(x) : x Î K0} are uniformly piecewise almost periodic families. Then

by Lemma 2.5, there exists a relatively dense set 1n ,f ,Ii,K0,T of R. As the proof of [[28],

Theorem 2.6] for the case of AP(R, X), one shows that for each n Î N, the set B 1nis

relatively compact and so is B. Obviously, B is a nonempty closed convex set. For any

u Î B,

∥∥�u(t)∥∥ =

∥∥∥∥∥∥t∫

−∞T(t, s)f

(s, u(s)

)ds +

∑ti<t

T(t, ti)Ii(u(ti)

)∥∥∥∥∥∥≤

t∫−∞

Me−w(t−s)∥∥f (

s, u(s))∥∥ ds + ∑

ti<t

Me−w(t−ti)∥∥Ii (u(ti))∥∥

≤ MwCL0 +

∑ti<t

Me−w(t−ti)C.

In order to estimate the second term on the right-hand side of the above formula, we

assume tj ≤ t <tj+1, j Î Z, so

t − ti = (t − tj) + (t j − ti) ≥ (j − i)α,


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and,

∑ti<t

Me−w(t−ti)C ≤∑

−∞<i≤j

Me−w(j−i)αC =∑

0≤k=j−i<∞Me−wkαC =

M

1 − e−wαC.

Thus

∥∥�u(t)∥∥ ≤ M

wCL0 +

M

1 − e−wαC ≤ L0.

By (A2), for any n Î N, there exists a number 0 < δ ≤ 1n, if ||x - y|| ≤ δ then

∥∥f (t, x) − f (t, y)∥∥ ≤ 1

nand

∥∥Ii(x) − Ii(y)∥∥ ≤ 1

nfor all i Î Z.

For any u ∈ B ⊂ B 1n(∀n Î N) and τ ∈ 1

n ,f ,Ii,K0,T,

∥∥u(t + τ ) − u(t)∥∥ ≤ 1

n(= δ)

(|t − ti| >

1n

).

So, for |t − ti| > 1n,∥∥f (

t + τ , u(t + τ )) − f

(t, u(t)

)∥∥≤ ∥∥f (

t + τ , u(t + τ )) − f

(t, u(t + τ )

)∥∥ +∥∥f (

t, u(t + τ )) − f

(t, u(t)

)∥∥≤ 1

n+1n=2n.

(3:1)

Similarly, for q Î Z with∣∣ti+q − ti − τ

∣∣ <1n, we have

∥∥Ii+q (u(ti+q)

) − Ii(u(ti)

)∥∥ ≤ 2n. (3:2)

Then,

�u(t + τ ) − �u(t) =

t+τ∫−∞

T (t + τ , s) f(s, u(s)

)ds +

∑ti<t+τ

T (t + τ , ti) Ii(u(ti)

)

−t∫

−∞T(t, s)f

(s, u(s)

)ds −

∑ti<t

T(t, ti)Ii(u(ti)

)

=

t∫−∞

T(t + τ , s + τ )f(s + τ , u(s + τ )

)ds

+∑ti<t

T (t + τ , ti + τ ) Ii+q(u(ti+q)

)

−t∫

−∞T(t, s)f

(s, u(s)

)ds −

∑ti<t

T(t, ti)Ii(u(ti)

)

=

t∫−∞

[T(t + τ , s + τ )f

(s + τ , u(s + τ )

) − T(t, s)f(s, u(s)

)]ds

+∑ti<t

[T(t + τ , ti + τ )Ii+q

(u(ti+q)

) − T(t, ti)Ii(u(ti)

)].


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By Lemmas 2.4, 2.5, Equations (3.1) and (3.2), we have∥∥∥∥∥∥t∫

−∞

[T(t + τ , s + τ )f (s + τ , u(s + τ )) − T(t, s)f

(s, u(s)

)]ds

∥∥∥∥∥∥=

∥∥∥∥∥∥t∫

−∞T(t + τ , s + τ )

[f(s + τ , u(s + τ )

) − f(s, u(s)

)]ds

+

t∫−∞

[T(t + τ , s + τ ) − T(t, s)

]f(s, u(s)

)ds

∥∥∥∥∥∥≤

t∫−∞

∥∥T(t + τ , s + τ )∥∥ ∥∥f (

s + τ , u(s + τ )) − f

(s, u(s)

)∥∥ ds

+

t∫−∞

∥∥T(t + τ , s + τ ) − T(t, s)∥∥ ∥∥f (

s, u(s))∥∥ ds

≤t∫

−∞

2nMe−w(t−s)ds +

t∫−∞

1ne−

w(t−s)2 CL0ds

=2nw

M +2nw

CL0,

and ∥∥∥∥∥∑ti<t

[T(t + τ , ti + τ )Ii+q

(u(ti+q)

) − T(t, ti)Ii(u(ti)

)]∥∥∥∥∥=

∥∥∥∥∥∑ti<t

T(t + τ , ti + τ )[Ii+q

(u(ti+q)

) − Ii(u(ti)

)]

+∑ti<t

[T(t + τ , ti + τ ) − T(t, ti)

]Ii

(u(ti)

)∥∥∥∥∥≤

∑ti<t

∥∥T(t + τ , ti + τ )∥∥ ∥∥Ii+q(u(ti+q)) − Ii(u(ti))

∥∥+

∑ti<t

∥∥T(t + τ , ti + τ ) − T(t, ti)∥∥ ∥∥Ii (u(ti))∥∥

≤∑ti<t

Me−w(t−ti) 2n+

∑ti<t

1ne−

w(t−ti)2 C

≤ 2n(1 − e−wα)

M +2

n(1 − e−wα)C.

Therefore, for any n Î N, there exists a relatively dense set 1n ,f ,Ii,K0,T of R, for∥∥�u(t + τ ) − �u(t)

∥∥ ≤ 2M + 2CL0

nw+

2M + 2Cn(1 − e−wα)

≤ 1n.,

∥∥�u(t + τ ) − �u(t)∥∥ ≤ 2M + 2CL0

nw+

2M + 2Cn(1 − e−wα)

≤ 1n.

That is, �u ∈ B 1n. Since n is arbitrary, �u ∈ ∩n∈NB 1

n= B, that is ΓB ⊆ B.


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We will show Γ is compact. To this end, first we show Γ is continuous. In fact, By

(A2), for the above1n, δ > 0 for each compact set K ⊆ X, x1, x2 Î K and ||x1 - x2|| ≤ δ,

we have

∥∥f (t, x1) − f (t, x2)∥∥ ≤ 1

n

∥∥Ii(x1) − Ii(x2)∥∥ ≤ 1

n,

for all t Î R, i Î N. Let x, y Î B, ||x - y|| ≤ δ. Since R(x) ⊆ K0 and R(y) ⊆ K0,

∥∥f (t, x(t)

) − f(t, y(t)

)∥∥ ≤ 1n

∥∥Ii (x(t)) − Ii(y(t)

)∥∥ ≤ 1n.

So

∥∥�x(t) − �y(t)∥∥ =

∥∥∥∥∥∥t∫

−∞T(t, s)f

(s, x(s)

)ds +

∑ti<t

T(t, ti)Ii(x(ti)

)

−t∫

−∞T(t, s)f

(s, y(s)

)ds −

∑ti<t

Ii(y(ti)

)∥∥∥∥∥∥≤

t∫−∞

∥∥T(t, s)∥∥∥∥f (s, x(s)

) − f(s, y(s)

)∥∥ds+

∑ti<t

∥∥T(t, ti)∥∥∥∥Ii (x(ti)) − Ii(y(ti)

)∥∥

≤t∫

−∞Me−w(t−s)

∥∥f (s, x(s)

) − f(s, y(s)

)∥∥ ds+

∑ti<t

Me−w(t−ti)∥∥Ii (x(ti)) − Ii

(y(ti)

)∥∥≤ M

wn+

Mn(1 − e−wα)

→ 0 if n → ∞.

Next, we show the following three statements:

(S1) {Γu(t) : u Î B} is a relatively compact subset of X for each t Î R.

(S2) {Γu : u Î B} is equicontinuous.

(S3) {Γu : u Î B} is equi-almost-periodic.

(S3) is obvious because B is.

For any ε > 0, let

�εu(t) =

t−ε∫−∞

T(t, s)f(s, u(s)

)ds +

∑ti<t−ε

T(t, ti)Ii(u(ti)

)

= T(t, t − ε)

⎡⎣ t−ε∫

−∞T(t − ε, s)f

(s, u(s)

)ds +

∑ti<t−ε

T (t − ε, ti) Ii(u(ti)

)⎤⎦= T(t, t − ε)�u(t − ε).

Since {Γu(t - ε) : u Î B} is uniformly bounded in X and T(t, t - ε) is compact, so {Γεu

(t) : uÎB} is relatively compact in X. Moreover,


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∥∥�u(t) − �εu(t)∥∥ =

∥∥∥∥∥∥t∫

t−ε

T(t, s)f(s, x(s)

)ds +

∑t−ε<ti<t

T(t, ti)Ii(u(ti)

)∥∥∥∥∥∥≤

t∫t−ε

∥∥T(t, s)∥∥∥∥f (s, x(s)

)∥∥ ds + ∑t−ε<ti<t

∥∥T(t, ti)∥∥∥∥Ii (u(ti))∥∥

≤t∫

t−ε

Me−w(t−s)∥∥f (

s, x(s))∥∥ ds + ∑

t−ε<ti<t

Me−w(t−ti)∥∥Ii (u(ti))∥∥

≤ εMwCL0 + ε

MαC.

So, {Γu(t) : u Î B} is a relatively compact subset of X for each t Î R. This show (S1).

It remains to prove (S2). Let t″ <t′, t″, t′ Î R, u Î B,

�u(t′) − �u(t′′) =t′∫

−∞T(t′, s)f

(s, u(s)

)ds +

∑ti<t′

T(t′, ti)Ii(u(ti)

)

−t′′∫

−∞T(t′′, s)f

(s, u(s)

)ds −

∑ti<t′′

T(t′′, ti)Ii(u(ti)

)

=

t′′∫−∞

[T(t′, s) − T(t′′, s)

]f(s, u(s)

)ds

+

t′∫t′′

T(t′, s)f(s, u(s)

)ds

+∑ti<t′′

[T(t′, ti) − T(t′′, ti)

]Ii

(u(ti)

)+

∑t′′<ti<t′

T(t′, ti)Ii(u(ti)

).

Moreover,

t′′∫−∞

[T(t′, s) − T(t′′, s)

]f(s, u(s)

)ds

=

∞∫0

[T(t′, t′′ − s) − T(t′′, t′′ − s)

]f(t′′ − s, u(t′′ − s)

)ds

=

∞∫0

[T(t′, t′′)T(t′′, t′′ − s) − T(t′′, t′′ − s)

]f(t′′ − s, u(t′′ − s)

)ds

=

∞∫0

[T(t′, t′′) − I

]T(t′′, t′′ − s)f

(t′′ − s, u(t′′ − s)

)ds.

By (H6), for the given ε > 0, there exists δ(ε) <ε

4M(CL0 +Cα) such that if t′, t″

belongs to a same continuity and 0 <t′ - t″ <δ, then


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∥∥T(t′, t′′) − I∥∥ < min

{wε

4MCL0,(1 − e−wα)ε

4MC

}.

So, ∥∥∥∥∥∥t′′∫

−∞

[T(t′, s) − T(t′′, s)

]f(s, u(s)

)ds

∥∥∥∥∥∥≤

∞∫0

∥∥T(t′, t′′) − I∥∥ ∥∥T(t′′, t′′ − s)

∥∥∥∥f (t′′ − s, u(t′′ − s)

)∥∥ ds

≤∞∫0

wε

4MCL0Me−wsCL0ds <

ε

4,

and,∥∥∥∥∥∥t′∫

t′′

T(t′, s)f(s, u(s)

)ds

∥∥∥∥∥∥ ≤t′∫

t′′

∥∥T(t′, s)∥∥ ∥∥f (s, u(s)

)ds

∥∥ < δMCL0 <ε

4.

Similarly,∥∥∥∥∥∥∑ti<t′′

[T(t′, ti) − T(t′′, ti)

]Ii

(u(ti)

)∥∥∥∥∥∥=

∥∥∥∥∥∥∑ti<t′′

[T(t′, t′′)T(t′′, ti) − T(t′′, ti)

]Ii

(u(ti)

)∥∥∥∥∥∥=

∥∥∥∥∥∥∑ti<t′′

[T(t′, t′′) − I

]T(t′′, ti)Ii

(u(ti)

)∥∥∥∥∥∥≤

∑ti<t′′

∥∥T(t′, t′′) − I∥∥ ∥∥T(t′′, ti)∥∥ ∥∥Ii (u(ti))∥∥

≤∑ti<t′′

(1 − e−wα)ε4MC

Me−w(t′′−ti)C ≤ ε

4,

and,∥∥∥∥∥∥∑

t′′<ti<t′T(t′, ti)Ii

(u(ti)

)∥∥∥∥∥∥ ≤∑

t′′<ti<t′

∥∥T(t′, ti)∥∥ ∥∥Ii (u(ti))∥∥ <δ

αMC <

ε

4.

Thus, for u Î B, when t′, t″ Î R belongs to a same continuity and 0 <t′- t″ <δ,∥∥�u(t′) − �u(t′′)∥∥ ≤ ε.

That is, {Γu: u Î B} is equicontinuous and (S2) holds.

Now the conditions of Lemma 2.7 are met, so Γ is compact. By Schauder’s point

fixed theorem, Γ has a fixed point u Î B. That is, (1.3) has a mild piecewise almost

periodic solution u(t).

The proof is complete.


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Remark 3.2 As we pointed out in Introduction, to get the almost periodic solutions

of impulsive differential equations, most people use the Contraction theorem. To be

different, we use Schauder’s fixed point theorem. This makes proofs difficult. In this

case, we impose a stronger condition (A4). To investigate this problem without (A4)

and give a easy example will be our next study.

Remark 3.3 Note that the uniform continuity is weaker than the Lipschitz continuity

So if the conditions of uniformly continuity of f and Ii in (A2) is replaced by∥∥f (t, x) − f (t, y)∥∥ +

∥∥Ii(x) − Ii(y)∥∥ ≤ η

∥∥x − y∥∥ (3:3)

for all x, y Î X, h Î (0,1), we can also get the mild piecewise almost periodic solu-

tion to (1.3) by Theorem 3.1.

In the end, we will use generalized Gronwall-Bellman inequality (Lemma 2.9) to

show the asymptotic stability of piecewise almost periodic solutions to (1.3) under the

Lipschitz condition.

Theorem 3.4 Assume the conditions of Theorem 3.1 are fulfilled except that (A2) is

replaced by (3.3). Assume further that Mh - w < 0, then (1.3) has an asymptotically

stable almost periodic solution.

Proof By Remark 3.3, (1.3) has a mild piecewise almost periodic solution x(t),

x(t) = T(t, σ )x(σ ) +

t∫σ

T(t, s)f(s, x(s)

)ds +

∑σ<ti<t

T(t, ti)Ii(u(ti)

).

Let u(t) = u(t, s, �) and v(t) = v(t, s, ψ) be two solutions of Equation (1.3), then

u(t) = T(t, σ )ϕ +

t∫σ

T(t, s)f(s, u(s)

)ds +

∑σ<ti<t

T(t, ti)Ii(u(ti)

),

υ(t) = T(t, σ )ψ +

t∫σ

T(t, s)f(s,υ(s)

)ds +

∑σ<ti<t

T(t, ti)Ii(υ(ti)

).

So

∥∥u(t) − υ(t)∥∥ =

∥∥∥∥∥∥T(t, σ )ϕ − T(t, σ )ψ +

t∫σ

T(t, s)[f(s, u(s)

) − f(s,υ(s)

)]ds

+∑

σ<ti<t

T(t, ti)[Ii

(u(ti)

) − Ii(υ(ti)

)]∥∥∥∥∥≤ ∥∥T(t, σ )∥∥ ‖ϕ − ψ‖ +

t∫σ

∥∥T(t, s)∥∥ ∥∥f (s, u(s)

) − f(s,υ(s)

)∥∥ ds+

∑σ<ti<t

∥∥T(t, ti)∥∥∥∥Ii (u(ti)) − Ii(υ(ti)

)∥∥

≤ Me−w(t−σ ) ‖ϕ − ψ‖ +

t∫σ

Mηe−w(t−s)∥∥u(s) − υ(s)

∥∥ ds+

∑σ<ti<t

Mηe−w(t−ti)∥∥u(ti) − υ(ti)

∥∥.


of 14

Then,

ewt∥∥u(t) − υ(t)

∥∥ ≤ Mewσ ‖ϕ − ψ‖ +

t∫σ

Mηews∥∥u(s) − υ(s)

∥∥ ds+

∑σ<ti<t

Mηewti∥∥u(ti) − υ(ti)

∥∥ .Let ϒ(t) = ewt||u(t) - v(t)||, then

ϒ(t) ≤ Mϒ(σ ) +

t∫σ

Mηϒ(s)ds +∑

σ<ti<t

Mηϒ(ti).

By Lemma 2.9, we have

ϒ(t) ≤ Mϒ(σ )∏

σ<ti<t

(1 +Mη)e∫

tσMηds = Mϒ(σ )

∏σ<ti<t

(1 +Mη)eMη(t−σ ),

that is,∥∥u(t) − υ(t)∥∥ ≤ M ‖ϕ − ψ‖

∐σ<ti<t

(1 +Mη)e(Mη−w)(t−σ ).

This completes the proof.

4 ConclusionThe aim of this article was to give sufficient conditions for existence and asymptotic

stability of almost periodic solutions for abstract impulsive differential equations. By

taking into consideration the relative compactness for subsets in APT(R, X), we get

almost periodic solutions to impulsive differential equation. In addition, we use gener-

alized Gronwall-Bellman inequality to show the above solution is asymptotically stable.

AcknowledgementsThis research was supported by the National Natural Science Foundation of China (No. 11071048). The authors wouldlike to thank the referees for carefully reading the manuscript, giving valuable comments and suggestions on thetopic of this article.

Authors’ contributionsJL carried out the main part of this manuscript. CZ participated discussion and corrected the main theorem. Allauthors read and approved the final manuscript.

Competing interestsThe authors declare that they have no competing interests.

Received: 12 November 2011 Accepted: 20 March 2012 Published: 20 March 2012

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doi:10.1186/1687-1847-2012-34Cite this article as: Liu and Zhang: Existence and stability of almost periodic solutions for impulsive differentialequations. Advances in Difference Equations 2012 2012:34.

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