

Articulo 1.dvi


CUBO A Mathematical Journal
Vol.12, No¯ 02, (1–17). June 2010

Existence Results for Semilinear Differential Evolution
Equations with Impulses and Delay

Nadjet Abada

Département de Mathématiques,

Université Mentouri Constantine,

email: n65abada@yahoo.fr

Mouffak Benchohra 1

Laboratoire de Mathématiques,

Université de Sidi Bel Abbès,

BP 89, 22000 Sidi Bel Abbès, Algérie.

email: benchohra@univ-sba.dz

and

Hadda Hammouche

Département de Mathematiques,

Université Kasdi Merbah Ouargla,

email: h.hammouche@yahoo.fr

ABSTRACT

In this paper, we establish sufficient conditions for the existence of mild and extremal

solutions for some densely defined impulsive functional differential equations in separable

Banach spaces with local and nonlocal conditions. We shall rely for the existence of mild

solutions on a fixed point theorem due to Burton and Kirk for the sum of completely con-

tinuous and contraction operators, and for the existence of extremal solutions on Dhage’s

fixed point theorem.

1Corresponding author


2 N. Abada , M. Benchohra and H. Hammouche CUBO
12, 2 (2010)

RESUMEN

En este art́ıculo establecemos condiciones suficientes para la existencia de soluciones suaves

y extremas para algunas ecuaciones diferenciales funcionales impulsivas densamente definidas

en espacios de Banach separables con condiciones locales y no locales. Para la existencia

de soluciones suaves usaremos un teorema de punto fijo debido a Burton y Kirk para la

suma de un operador completamente continuo y otro contractivo; para la existencia de

soluciones extremas usaremos el teorema de punto fijo de Dhage.

Key words and phrases: Densely defined operator, impulsive functional differential equations, fixed

point, semigroups, mild solutions, extremal mild solutions, nonlocal condition, separable Banach space.

AMS (MOS) Subj. Class.: 34A37, 34K30, 34K35, 34K45.

1 Introduction

In this paper, we establish sufficient conditions for the existence of mild and extremal mild solutions

of first order impulsive functional equations in a separable Banach space (E. |.|) of the form:

y′(t) − Ay(t) = f(t,yt), a.e. t ∈ J = [0,b] , t 6= tk, k = 1, . . . ,m (1)

∆y|t=tk = Ik(y(t
−

k
)), k = 1, . . . ,m (2)

y(t) = φ(t), t ∈ [−r, 0] , (3)

where f : J × D → E is a given function, D = {ψ : [−r, 0] → E,ψ is continuous every where except

for a finite number of points s at which ψ (s−) ,ψ (s+) exist and ψ (s−) = ψ (s)}, φ ∈ D, 0 < r <

∞, 0 = t0 < t1 < ... < tm < tm+1 = b, Ik ∈ C (E,E) , k = 1, 2, . . . ,m, A : D(A) ⊂ E → E is

the infinitesimal generator of a C0-semigroup T (t), t ≥ 0, and E a real separable Banach space with

norm |.|.

We denote by yt the element of D defined by

yt(θ) = y(t + θ), θ ∈ [−r, 0].

Here yt(·) represents the history of the state from t − r, up to the present time t.

Functional differential equations arise in many areas of applied mathematics and such equations

have received much attention in recent years. A good guide to the literature for functional differential

equations is the books by Hale [18] and Hale and Verduyn Lunel [19] and Kolmanovskii and Myshkis

[23] and the references therein.

Impulsive differential equations have become important in recent years as mathematical models of

phenomena in both the physical and social sciences. There has a significant development in impulsive

theory especially in the area of impulsive differential equations with fixed moments; see for instance

the monoghraphs by Bainov and Simeonov [5], Benchohra et al [6], Lakshmikantham et al [24], and

Samoilenko and Perestyuk [29]. In the case where the impulses are absent (i.e Ik = 0,k = 1, . . . ,m) and

F is a single or multivalued map and A is a densely defined linear operator generating a C0-semigroup


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Impulsive Semilinear Functional Differential Equations 3

of bounded linear operators the problem (1)–(3) has been investigated on compact intervals in, for

instance, the monographs by Ahmed [1], Hu and Papageorgiou [21], Kamenskii et al [22] and Wu [30]

and the papers of Benchohra and Ntouyas [7, 8, 9].

During the last decades problems of the form (1)–(3) have received much attention. Some ex-

istence results were given in the monograph Ahmed [2] and Benchohra et al [6] and the papers by

Ahmed [3, 4], Cardinali and Rubbioni [15], Liu [25] and Rogovchenko [27, 28], and the references

therein.

The plan of this paper is as follows: in Section 2, we will recall briefly some basic definitions

and preliminary facts which will be used throughout the following sections. In Section 3, we prove

existence of mild solutions for problem (1)–(3). Our approach will be based for the existence of mild

solutions, on a fixed point theorem of Burton and Kirk [10] for the sum of a contraction map and a

completely continuous map. In Section 4, we shall prove the existence of extremal solutions of the

problem (1)–(3), and our approach here is based on the concept of upper and lower solutions combined

with a fixed point theorem on ordered Banach spaces established recently by Dhage [16]. In Section 5

we study the impulsive functional differential equations with non-local initial conditions, of the form

y′(t) − Ay(t) = f(t,yt), a.e. t ∈ J = [0,b] , t 6= tk, k = 1, . . . ,m (4)

∆y|t=tk = Ik(y(t
−

k
)), k = 1, . . . ,m (5)

y(t) + ht (y) = φ(t), t ∈ [−r, 0] , (6)

where ht : PC([−r,b],E) → E is given function (see Section 2 for the definition of PC([−r,b],E)).

The non-local condition can be applied in physics with better effect than the classical initial condition

y (0) = y0. For example, ht (y) may be given by

ht (y) =

p∑

i=1

ciy(ti + t), t ∈ [−r, 0]

where ci, i = 1, ...,p, are given constants and 0 < t1 < ... < tp ≤ b. At time t = 0, we have

h0 (y) =

p∑

i=1

ciy (ti)

Non-local conditions were initiated by Byszewski [11] (see also [12, 13, 14]) in which we refer for

motivation and other references. Finally, Section 6 is devoted to an example illustrating the abstract

theory considered in the previous sections.

2 Preliminaries

In this section, we introduce notations and definitions, preliminaries facts which are used throughout

this paper. For ψ ∈ D the norm of ψ is defined by

‖ψ‖D = sup{|ψ(θ)| : θ ∈ [−r, 0]},

and B(E) denotes the Banach space of bounded linear operators from E into E, with norm

‖N‖B(E) = sup{|N(y)| : |y| = 1}.


4 N. Abada , M. Benchohra and H. Hammouche CUBO
12, 2 (2010)

L1(J,E) denotes the Banach space of measurable functions y : J −→ E which are Bochner integrable

normed by

‖y‖L1 =

∫ b

0

|y(t)|dt.

It is well known that the operator A generates a (C0) semigroup if A satisfies:

(i) D(A) = E.

(ii) The Hille-Yosida condition, that is, there exists M ≥ 0 and ω ∈ R such that (ω,∞) ⊂ ρ(A) and

sup{(λ − ω)n|(λI − A)−n| : λ, n ∈ N} ≤ M,

where ρ(A) is the resolvent set of A and I is the identity operator in E.

For more details on semigroup theory we refer the interested reader to the books of Ahmed [1]

and Pazy [26].

In order to define a mild solution of problems (1)–(3) and (4)–(6), we shall consider the space

PC ([−r,b] ,E) ={y : [−r,b] → E : y (t) is continuous everywhere except for some tk at which y
(
t
−

k

)

and y
(
t
+
k

)
, k = 1, 2, . . . ,m exist and y

(
t
−

k

)
= y (tk)}.

Obviously, for any t ∈ [0,b] and y ∈ PC ([−r,b] ,E), we have yt ∈ D and PC ([−r,b] ,E) is a

Banach space with the norm

‖y‖ = sup {|y (t)| : t ∈ [−r,b]} .

Definition 2.1. A map f : J × D → E is said to be Carathéodory if

(i) the function t 7−→ f(t,x) is measurable for each x ∈ D;

(ii) the function x 7−→ f(t,x) is continuous for almost all t ∈ J.

3 Existence of Mild Solutions

In this section, we give our main existence result for problem (1)–(3). Before stating and proving this

result, we give the definition of the mild solution.

Definition 3.1. A function y ∈ PC ([−r,b] ,E) is said to be a mild solution of problem (1)–(3) if

y(t) = φ(t), t ∈ [−r, 0], and y is a solution of impulsive integral equation

y(t) = T (t)φ(0) +

∫ t

0

T (t − s)f(t,ys)ds +
∑

0<tk <t

T (t − tk)Ik(y(t
−

k
)), t ∈ J.

Our main result in this section is based upon the following fixed point theorem due to Burton

and Kirk [10].

Theorem 3.1. Let X be a Banach space, and A, B : X → X two operators satisfying:

(i) A is a contraction, and


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Impulsive Semilinear Functional Differential Equations 5

(ii) B is completely continuous.

Then either

(a) the operator equation y = A(y) + B(y) has a solution, or

(b) the set E =
{
u ∈ X : λA

(
u

λ

)
+ λB(u) = u, λ ∈ (0, 1)

}
is unbounded.

Let us introduce the following hypotheses:

(H1) A : D(A) ⊂ E → E is the infinitesimal generator of a C0-semigroup {T (t)}, t ∈ J which is

compact for t > 0 in the Banach space E. Let M = sup{‖T (t)‖B(E) : t ∈ J};

(H2) There exist constants dk > 0, k = 1, ...,m with M

m∑

k=1

dk < 1 such that

|Ik (y) − Ik (x)| ≤ dk |y − x| for each y,x ∈ E

(H3) The function f : J × D → E is Carathéodory;

(H4) There exists a function p ∈ L1(J, R+) and a continuous nondecreasing function ψ : [0,∞) →

(0,∞) such that

|f(t,x)| ≤ p(t)ψ(‖x‖D), a.e. t ∈ J, for all x ∈ D

with ∫
∞

Do

ds

ψ (s)
> D1 ‖p‖L1 ,

where

D0 =

M(‖φ‖ +
m∑

k=1

|Ik(0)|)

1 − M
m∑

k=1

dk

, D1 =
M

1 − M
m∑

k=1

dk

.

Theorem 3.2. Assume that (H1)-(H4) hold. Then the problem (1)–(3) has at least one mild solution

on [−r,b].

Proof. Transform the problem (1)-(3) into a fixed point problem. Consider the two operators:

A,B : PC ([−r,b] ,E) → PC ([−r,b] ,E)

defined by

A(y) (t) :=






0, if t ∈ [−r, 0];

∑

0<tk <t

T (t − tk) Ik
(
y
(
t
−

k

))
, if t ∈ J,

and


6 N. Abada , M. Benchohra and H. Hammouche CUBO
12, 2 (2010)

B(y) (t) :=






φ(t), if t ∈ [−r, 0];

T (t)φ(0) +

∫ t

0

T (t − s)f (s,ys) ds, if t ∈ J.

Then the problem of finding the solution of problem (1)–(3) is reduced to finding the solution of

the operator equation A (y) (t) +B (y) (t) = y (t) , t ∈ [−r,b]. We shall show that the operators A and

B satisfy all the conditions of Theorem 3.1. For better readability, we break the proof into a sequence

of steps.

Step 1: B is continuous.

Let {yn} be a sequence such that yn → y in PC([−r,b],E). Then for t ∈ J

|B(yn)(t) − B(y)(t)| =

∣∣∣∣
∫ t

0

T (t − s)[f(s,yns ) − f(s,ys)]ds

∣∣∣∣

≤ M

∫ b

0

|f(s,yns ) − f(s,ys)|ds.

Since f(s, ·) is continuous for a.e. s ∈ J, we have by the Lebesgue dominated convergence theorem

|B(yn)(t) − B(y)(t)| → 0 as n → ∞.

Thus B is continuous.

Step 2: B maps bounded sets into bounded sets in PC([−r,b],E).

It is enough to show that for any q > 0 there exists a positive constant l such that for each

y ∈ Bq = {y ∈ PC([−r,b],E) : ‖y‖ ≤ q} we have ‖B (y)‖ ≤ l. So choose y ∈ Bq, then we have for

each t ∈ J,

|B(y)(t)| =

∣∣∣∣T (t)φ(0) +
∫ t

0

T (t − s)f(s,ys)ds

∣∣∣∣

≤ M|φ(0)| + Mψ(q)

∫ b

0

p(s) ds;

Then we have

‖B(y)‖ ≤ M‖φ‖ + Mψ(q)‖p‖L1 := l.

Step 3: B maps bounded sets into equicontinuous sets of PC([−r,b],E).

We consider Bq as in step 2 and let τ1,τ2 ∈ J\ {t1, ..., tm} , τ1 < τ2.Thus if ǫ > 0 and ǫ ≤ τ1 < τ2
we have


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Impulsive Semilinear Functional Differential Equations 7

|B(y)(τ2) − B(y)(τ1)| ≤ |T (τ2)φ(0) − T (τ1)φ(0)|

+ψ(q)

∫ τ1−ǫ

0

‖T (τ2 − s) − T (τ1 − s)‖B(E)p(s)ds

+ψ(q)

∫ τ1

τ1−ǫ

‖T (τ2 − s) − T (τ1 − s)‖B(E)p(s)ds

+ψ(q)

∫ τ2

τ1

‖T (τ2 − s)‖B(E)p(s)ds.

As τ1 → τ2 and ǫ become sufficiently small, the right-hand side of the above inequality tends to

zero, since T (t) is a strongly continuous operator and the compactness of T (t) for t > 0 implies the

continuity in the uniform operator topology ([1, 26]). This proves the equicontinuity for the case

where t 6= ti,k = 1, 2, . . . ,m + 1. It remains to examine the equicontinuity at t = ti.

First we prove equicontinuity at t = t−i . Fix δ1 > 0 such that {tk : k 6= i} ∩ [ti − δ1, ti + δ1] = ∅.

For 0 < h < δ1 we have

|B(y)(ti − h) − B(y)(ti)| ≤ | (T (ti − h) − T (ti)) φ(0)|

+

∫ ti−h

0

| (T (ti − h − s) − T (ti − s)) f(s,ys)|ds

+ψ(q)M

∫ ti

ti−h

p(s)ds;

which tends to zero as h → 0.

Define

B̂0(y)(t) = B(y)(t), t ∈ [0, t1]

and

B̂i(y)(t) =

{
B(y)(t), if t ∈ (ti, ti+1]

B(y)(t+i ), if t = ti.

Next we prove equicontinuity at t = t+i . Fix δ2 > 0 such that {tk : k 6= i} ∩ [ti − δ2, ti + δ2] = ∅.

For 0 < h < δ2 we have

|B̂(y)(ti + h) − B̂(y)(ti)| ≤ | (T (ti + h) − T (ti)) φ(0)|

+

∫ ti

0

| (T (ti + h − s) − T (ti − s)) f(s,ys)|ds

+ψ(q)M

∫ ti+h

ti

p(s)ds.


8 N. Abada , M. Benchohra and H. Hammouche CUBO
12, 2 (2010)

The right-hand side tends to zero as h → 0. The equicontnuity for the cases τ1 < τ2 ≤ 0 and

τ1 ≤ 0 ≤ τ2 follows from the uniform continuity of φ on the interval [−r, 0].

As consequence of Steps 1 to 3 together with Arzelá-Ascoli theorem it suffices to show that B

maps B into a precompact set in E.

Let 0 < t < b be fixed and let ǫ be a real number satisfying 0 < ǫ < t. For y ∈ Bq we define

Bǫ(y)(t) = T (t)φ(0) + T (ǫ)

∫ t−ǫ

0

T (t − s − ǫ)f(s,ys)ds.

Since T (t) is a compact operator, the set

Yǫ(t) = {Bǫ(y)(t) : y ∈ Bq}

is precompact in E for every ǫ, 0 < ǫ < t. Moreover, for every y ∈ Bq we have

|B(y)(t) − Bǫ(y)(t)| ≤ ψ(q)

∫ t

t−ǫ

‖T (t − s)‖B(E)p(s)ds

≤ ψ(q)M

∫ t

t−ǫ

p(s)ds.

Therefore, there are precompact sets arbitrarily close to the set Yǫ(t) = {Bǫ(y)(t) : y ∈ Bq}. Hence

the set Y (t) = {B(y)(t) : y ∈ Bq} is precompact in E. Hence the operator B : PC ([−r,b] ,E) →

PC ([−r,b] ,E) is completely continuous.

Step 4: A is a contraction

Let x,y ∈ PC([−r,b],E). Then for t ∈ J

|A(y)(t) − A(x)(t)| =

∣∣∣∣∣
∑

0<tk<t

T (t − tk)
(
Ik
(
y
(
t
−

k

))
− Ik

(
x
(
t
−

k

)))
∣∣∣∣∣

≤ M
∑

0<tk <t

∣∣Ik
(
y
(
t
−

k

))
− Ik

(
x
(
t
−

k

))∣∣

≤ M
m∑

k=1

dk
∣∣y
(
t
−

k

)
− x

(
t
−

k

)∣∣

≤ M
m∑

k=1

dk ‖y − x‖ .

Then

‖A(y) − A(x)‖ ≤ M
m∑

k=1

dk ‖y − x‖ ,

which is a contraction, since M
m∑

k=1

dk < 1 .

Step 5: A priori bounds.


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Impulsive Semilinear Functional Differential Equations 9

Now it remains to show that the set

E =
{
y ∈ PC([−r,b],E) : y = λB(y) + λA

(
y

λ

)
for some 0 < λ < 1

}

is bounded.

Let y ∈ E, then y = λB(y) + λA
(
y

λ

)
for some 0 < λ < 1. Thus, for each t ∈ J,

y(t) = λT (t)φ(0) + λ

t∫

0

T (t − s)f(s,ys)ds + λ
∑

0<tk<t

T (t − tk) Ik
(
y

λ

(
t
−

k

))
.

This implies by (H2) and (H4) that, for each t ∈ J, we have

|y(t)| ≤ λM|φ(0)| + λM

t∫

0

p(s)ψ(‖ys‖)ds + λM
m∑

k=1

∣∣∣Ik
(
y

λ

(
t
−

k

))∣∣∣

≤ λM‖φ‖ + λM

t∫

0

p(s)ψ(‖ys‖)ds

+λM

m∑

k=1

∣∣∣Ik
(
y

λ

(
t
−

k

))
− Ik(0)

∣∣∣ + λM
m∑

k=1

|Ik (0)|

≤ λM

(
‖φ‖ +

m∑

k=1

|Ik(0)|

)
+ λM

t∫

0

p(s)ψ(‖ys‖)ds

+λM

m∑

k=1

dk

∣∣∣
y

λ

(
t
−

k

)∣∣∣

≤ M

(
‖φ‖ +

m∑

k=1

|Ik(0)|

)
+ M




t∫

0

p(s)ψ(‖ys‖)ds +
m∑

k=1

dk
∣∣y
(
t
−

k

)∣∣



 .

Now we consider the function µ defined by

µ(t) = sup{|y(s)| : −r ≤ s ≤ t}, 0 ≤ t ≤ b.

Then ‖ys‖ ≤ µ(t) for all t ∈ J and there is a point t
∗ ∈ [−r,t] such that µ(t) = |y(t∗)|. If

t∗ ∈ [0,b], by the previous inequality we have for t ∈ [0,b] (note t∗ ≤ t)

µ(t) ≤ M

(
‖φ‖ +

m∑

k=1

|Ik(0)|

)
+ M

t∫

0

p(s)ψ(µ(s))ds + M

m∑

k=1

dkµ(t).

Then (
1 − M

m∑

k=1

dk

)
µ(t) ≤ M

(
‖φ‖ +

m∑

k=1

|Ik(0)|

)
+ M

t∫

0

p(s)ψ(µ(s))ds.


10 N. Abada , M. Benchohra and H. Hammouche CUBO
12, 2 (2010)

Thus we have

µ(t) ≤ D0 + D1

t∫

0

p(s)ψ(µ(s))ds.

Let us take the right hand-side of above inequality as v(t). Then we have

µ(t) ≤ v(t) for all t ∈ J,

v(0) = D0,

and

v′(t) = D1p(t)ψ(µ(t)), a.e. t ∈ J.

Using the nondecreasing character of ψ we get

v′(t) ≤ D1p(t)ψ(v(t)), a.e. t ∈ J.

That is
v′(t)

ψ(v(t))
≤ D1p (t) , a.e. t ∈ J.

Integrating from 0 to t we get

∫ t

0

v′(s)

ψ(v(s))
ds ≤ D1

∫ t

0

p (s) ds.

By a change of variable we get

∫ v(t)

v(0)

du

ψ(u)
≤ D1

∫ b

0

p (s) ds = D1‖p‖L1 <

∫
∞

D0

du

ψ(u)
.

Hence there exist a constant N such that

µ(t) ≤ v(t) ≤ N for all t ∈ J.

Now from the definition of µ it follows that

‖y‖ = sup
t∈[−r,b]

|y(t)| ≤ µ(b) ≤ N, for all y ∈ E.

This shows that the set E is bounded. As a consequence of Theorem 3.1 we deduce that A + B

has a fixed point which is a mild solution of problem (1)–(3).

4 Existence of Extremal Mild Solutions

In this section we shall prove the existence of maximal and minimal solutions of problem (1)–(3) under

suitable monotonicity conditions on the functions involved in it.

Definition 4.1. A nonempty closed subset C of a Banach space (X,‖ · ‖) is said to be a cone if


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Impulsive Semilinear Functional Differential Equations 11

(i) C + C ⊂ C,

(ii) λC ⊂ C

(iii) {−C}
⋂
{C} = {0}

A cone C is called normal if the norm ‖·‖ is semi-monotone on C, i.e., there exists a constant N > 0

such that ‖x‖ ≤ N‖y‖, whenever x ≤ y. We equip the space X = C(J,E) with the order relation ≤

induced by a regular cone C in E, that is for all y,y ∈ X : y ≤ y if and only if y(t) −y(t) ∈ C, ∀t ∈ J.

In what follows will assume that the cone C is normal. Cones and their properties are detailed in

[17, 20]. Let a,b ∈ X be such that a ≤ b. Then, by an order interval [a,b] we mean a set of points in

X given by

[a,b] = {x ∈ X | a ≤ x ≤ b}.

Definition 4.2. Let X be an ordered Banach space. A mapping T : X → X is called isotone

increasing if T (x) ≤ T (y) for any x,y ∈ X with x < y. Similarly, T is called isotone decreasing if

T (x) ≥ T (y) whenever x < y.

Definition 4.3. We say that x ∈ X is the least fixed point of G in X if x = Gx and x ≤ y whenever

y ∈ X and y = Gy. The greatest fixed point of G in X is defined similarly by reversing the inequality.

If both least and greatest fixed point of G in X exist, we call them extremal fixed point of G in X.

Very recently Dhage has proved the following.

Theorem 4.1. [16]. Let [a,b] be an order interval in a Banach space X and let B1,B2 : [a,b] → X

be two functions satisfying:

(a) B1 is a contraction,

(b) B2 is completely continuous,

(c) B1 and B2 are strictly monotone increasing, and

(d) B1(x) + B2(x) ∈ [a,b], ∀x ∈ [a,b].

Further if the cone C in X is normal, then the equation x = B1(x) + B2(x) has a least fixed

point x∗ and a greatest fixed point x
∗ ∈ [a,b]. Moreover x∗ = lim

n→∞
xn and x

∗ = lim
n→∞

yn, where

{xn} and {yn} are the sequences in [a,b] defined by

xn+1 = B1(xn) + B2(xn), x0 = a and yn+1 = B1(yn) + B2(yn), y0 = b.

We need the following definitions in the sequel.

Definition 4.4. We say that a function v ∈ PC([−r,b],E) is a lower mild solution of problem (1)–(3)

if v(t) = φ(t), t ∈ [−r, 0], and

v(t) ≤ T (t)φ(0) +

∫ t

0

T (t − s)f (s,vs) ds +
∑

0<tk<t

T (t − tk)Ik(v(t
−

k
)), t ∈ J, t 6= tk

and v(t+
k

) − v(t−
k

) ≤ Ik(v(tk)), t = tk, k = 1, . . . ,m. Similarly an upper mild solution w of problem

(1)–(3) is defined by reversing the order.


12 N. Abada , M. Benchohra and H. Hammouche CUBO
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Definition 4.5. A solution xM of problem (1)–(3) is said to be maximal if for any other solution x

of problem (1)-(3) on J, we have that x(t) ≤ xM (t) for each t ∈ J.

Similarly a minimal solution of problem (1)–(3) is defined by reversing the order of the inequalities.

Definition 4.6. A function f(t,x) is called strictly monotone increasing in x almost everywhere for

t ∈ J, if f(t,x) ≤ f(t,y) a.e. t ∈ J for all x,y ∈ D with x < y. Similarly f(t,x) is called strictly

monotone decreasing in x almost everywhere for t ∈ J, if f(t,x) ≥ f(t,y) a.e. t ∈ J for all x,y ∈ D

with x < y.

We consider the following assumptions in the sequel.

(H5) The function f(t,y) is strictly monotone increasing in y for almost each t ∈ J.

(H6) T (t) is preserving the order, that is T (t)v ≥ 0 whenever v ≥ 0.

(H7) The function Ik, k = 1, ...m are continuous and nondecreasing.

(H8) The problem (1)–(3) has a lower mild solution v and an upper mild solution w with v ≤ w.

Theorem 4.2. Assume that assumptions (H1)-(H8) hold. Then problem (1)–(3) has minimal and

maximal solutions on [−r,b].

Proof. It can be shown, as in the proof of Theorem 3.2 that B is completely continuous and A is

a contraction on [v,w]. We shall show that A and B are isotone increasing on [v,w]. Let y,y ∈ [a,b]

be such that y ≤ y, y 6= y. Then by (H5), (H6), we have for each t ∈ J

B(y) (t) = T (t)φ(0) +

∫ t

0

T (t − s)f (s,ys) ds

≤ T (t)φ(0) +

∫ t

0

T (t − s)f (s,ys) ds

= B(y) (t) .

and by (H7), we have for each t ∈ J

A(y) (t) =
∑

0<tk<t

T (t − tk) Ik
(
y
(
t
−

k

))

≤
∑

0<tk<t

T (t − tk) Ik
(
y
(
t
−

k

))

= A(y) (t) .

Therefore A and B are isotone increasing on [v,w]. Finally, let x ∈ [v,w] be any element. By (H8)

we deduce that

v ≤ A(v) + B(v) ≤ A(x) + B(x) ≤ A(w) + B(w) ≤ w,

which shows that A(x) + B(x) ∈ [v,w] for all x ∈ [v,w]. Thus, A and B satisfy all conditions of

Theorem 4.1 , hence problem (1)–(3) has maximal and minimal solutions on [−r,b]. This completes

the proof. 2


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Impulsive Semilinear Functional Differential Equations 13

5 Impulsive Functional Differential Equations with Non-local

Conditions

In this section we shall prove existence results for problem (4)–(6). Nonlocal conditions were initiated

by Byszewski [11] when he proved the existence and uniqueness of mild and classical solutions of non-

local Cauchy problems. As remarked by Byszewski [12], the nonlocal condition can be more useful

than the standard initial condition to describe some physical phenomena.

Definition 5.1. A function y ∈ PC ([−r,b] ,E) is said to be a mild solution of problem (4)–(6) if

y(t) = φ(t) − ht (y) , t ∈ [−r, 0], and

y(t) = T (t) (φ(0) − h0 (y)) +

∫ t

0

T (t − s)f (s,ys) ds +
∑

0<tk<t

T (t − tk) Ik
(
y
(
t
−

k

))
, t ∈ J.

Theorem 5.1. Assume that hypotheses (H1)-(H3) hold and moreover

(A1) The function h is continuous with respect to t, and there exists a constant α > 0 such that

|ht(u)| ≤ α, u ∈ PC([−r,b],E)

and for each k > 0 the set

{φ(0) − h0(y), y ∈ PC([−r,b],E),‖y‖ ≤ k}

is precompact in E

(A2) There exists a function p ∈ L1(J, R+) and a continuous nondecreasing function ψ : [0,∞) →

(0,∞) such that

|f(t,x)| ≤ p(t)ψ(‖x‖D), a.e. t ∈ J, for all x ∈ D

with ∫
∞

D̃0

ds

ψ (s)
> D1 ‖p‖L1 ,

and

D̃0 =

M[‖φ‖D + α +
m∑

k=1

|Ik(0)|]

1 − M
m∑

k=1

dk

.

Then the problem 4)–(6) has at least one mild solution on [−r,b].

Proof. Transform the problem (4)–(6) into a fixed point problem. Consider the two operators :

B1 : PC ([−r,b] ,E) → PC ([−r,b] ,E) defined by

B1(y)(t) =






φ(t) − ht(y), if t ∈ [−r, 0];

T (t) (φ(0) − h0(y)) +

∫ t

0

T (t − s)f (s,ys) ds, if t ∈ J,


14 N. Abada , M. Benchohra and H. Hammouche CUBO
12, 2 (2010)

and

A1(y)(t) =






0, if t ∈ [−r, 0];

∑

0<tk<t

T (t − tk)Ik(y(t
−

k
)), if t ∈ J.

Then the problem of finding the solution of problem (4)–(6) is reduced to finding the solution of the

operator equation A1 (y) (t) +B2 (y) (t) = y (t) , t ∈ [−r,b]. As in Section 3, we can show that the

operators A1 and B1 satisfy all conditions of Theorem 3.1.

6 An Example

As an application of our results we consider the following impulsive partial functional differential

equation of the form

∂
∂t
z(t,x) = ∂

2

∂x2
z(t,x)

+Q(t,z(t − r,x)), x ∈ [0,π], t ∈ [0,b]\{t1, t2, . . . , tm}.
(7)

z(t+
k
,x) − z(t−

k
,x) = bkz(t

−

k
,x), x ∈ [0,π], k = 1, . . . ,m (8)

z(t, 0) = z(t,π) = 0, t ∈ [0,b] (9)

z(t,x) = φ(t,x), t ∈ [−r, 0], x ∈ [0,π], (10)

where r > 0, bk > 0, k = 1, . . . ,m, φ ∈ D = {ψ : [−r, 0] × [0,π] → IR; ψ is continuous everywhere

except for a countable number of points at which ψ(s−),ψ(s+) exist with ψ(s−) = ψ(s)}, 0 = t0 <

t1 < t2 < ... < tm < tm+1 = b, z(t
+
k
) = lim

(h,x)→(0+,x)
z(tk + h,x),z(t

−

k
) = lim

(h,x)→(0−,x)
z(tk + h,x) and

Q : [0,b] × IR → IR is a given function.

Let

y(t)(x) = z(t,x), t ∈ J, x ∈ [0,π],

Ik(y(t
−

k
))(x) = bkz(t

−

k
,x), x ∈ [0,π], k = 1, . . . ,m

F(t,φ)(x) = Q(t,φ(θ,x)), θ ∈ [−r, 0], x ∈ [0,π],

φ(θ)(x) = φ(θ,x), θ ∈ [−r, 0], x ∈ [0,π].

Take E = L2[0,π] and define A : D(A) ⊂ E → E by Aw = w′′ with domain

D(A) = {w ∈ E,w,w′ are absolutely continuous, w′′ ∈ E,w(0) = w(π) = 0}.

Then

Aw =

∞∑

n=1

n2(w,wn)wn, w ∈ D(A)

where ( , ) is the inner product in L2 and wn(s) =
√

2
π
sinns, n = 1, 2, . . . is the orthogonal set

of eigenvectors in A. It is well known (see [26]) that A is the infinitesimal generator of an analytic

semigroup T (t), t ∈ [0,b] in E and is given by

T (t)w =
∞∑

n=1

exp(−n2t)(w,wn)wn, w ∈ E.


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Impulsive Semilinear Functional Differential Equations 15

Since the analytic semigroup T (t) is compact, there exists a constant M ≥ 1 such that

‖T (t)‖B(E) ≤ M.

Also assume that there exists an integrable function σ : [0,b] → IR+ such that

|Q(t,w(t − r,x))| ≤ σ(t)Ω(|w|)

where Ω : [0,∞) → (0,∞) is continuous and nondecreasing with

∫
∞

1

ds

s + Ω(s)
= +∞.

Assume that there exists a function l̃ ∈ L1([0,b], IR+) such that

|Q(t,w) − Q(t,w)| ≤ l̃(t)|w − w|, t ∈ [0,b], w,w ∈ IR.

We can show that problem (1)-(3) is an abstract formulation of problem (7)-(10). Since all the

conditions of Theorem 3.2 are satisfied, the problem (7)-(10) has a solution z on [−r,b] × [0,π].

Acknowledgement. This work was completed when the second author was visiting the ICTP

in Trieste as a Regular Associate. It is a pleasure for him to express gratitude for its financial support

and the warm hospitality.

Received: December 2008. Revised: January 2009.

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