BeNgSe1\(09\).dvi


CUBO A Mathematical Journal

Vol.12, No¯ 03, (35–48). October 2010

Measure of Noncompactness and Nondensely

Defined Semilinear Functional Differential

Equations with Fractional Order

MOUFFAK BENCHOHRA1

Laboratoire de Mathématiques, Université de Sidi Bel-Abbès,

B.P. 89, 22000, Sidi Bel-Abbès, Algérie

email: benchohra@univ-sba.dz

GASTON M. N’GUÉRÉKATA

Department of Mathematics, Morgan State University,

1700 E. Cold Spring Lane, Baltimore M.D. 21252, USA

email: Gaston.N’Guerekata@morgan.edu

AND

DJAMILA SEBA

Département de Mathématiques, Université de Boumerdès,

Avenue de l’indépendance, 35000 Boumerdès, Algérie

email: djam_seba@yahoo.fr

ABSTRACT

This paper is devoted to study the existence of integral solutions for a nondensely defined
semilinear functional differential equations involving the Riemann-Liouville derivative in

1Corresponding author



36 M. Benchohra, Gaston M. N’Guérékata & D. Seba CUBO
12, 3 (2010)

a Banach space. The arguments are based upon Mönch’s fixed point theorem and the
technique of measures of noncompactness.

RESUMEN

Este artículo es dedicado al estudio de existencia de soluciones integrales para ecuaciones
diferenciales funcionales semilineales envolviendo la derivada de Riemann-Liouville en un
espacio de Banach. Los argumentos se basan en un teorema de punto fijo de Mönch y la
técnica de no compacidad.

Key words and phrases: Partial differential equations, fractional derivative, fractional inte-

gral, fixed point, semigroups, integral solutions, finite delay, measure of noncompactness, fixed

point, Banach space.

Math. Subj. Class.: 34G20, 34G25, 26A33, 26A42.

1 Introduction

The theory of functional differential equations has emerged as an important branch of non-

linear analysis. It is worthwhile mentioning that several important problems of the theory

of ordinary and delay differential equations lead to investigations of functional differential

equations of various types, see the books of Hale and Verduyn Lunel [22], Kolmanovskii and

Myshkis [26], Wu [42], and the references therein. On the other hand the theory of fractional

differential equations is also intensively studied and finds numerous applications in describ-

ing real world problems (see for instance the monographs of Lakshmikantham et al. [27],

Kilbas et al. [25], Miller and Ross [31], Podlubny [39], Samko et al. [40], and the papers of

Agarwal et al. [1], Benchohra et al. [11, 12], Chang and Nieto [14], Diethelm et al. [16], Furati

and Tatar [17, 18], Gaul et al. [19], Glockle and Nonnenmacher [20], Lakshmikantham and

Devi [28], Mainardi [29], Metzler et al. [30], N’Guérékata et al [33, 34, 35], and the references

therein). Jaradat et al. [23], studied the existence and uniqueness of mild solutions for a class

of initial value problem for a semilinear integrodifferential equation involving the Caputo’s

fractional derivative.

In this paper we will examine the following semilinear functional differential equation

of fractional order

D
r
y(t) = A y(t) + f (t, yt ), t ∈ J = [0, b], r > 0 (1)

y(t) = φ(t), t ∈ [−ρ, 0], (2)

where D r is the standard Riemann-Liouville fractional derivative, f : J × C([−ρ, 0], E) → E

is a given function, A : D(A) ⊂ E → E is a nondensely defined closed linear operator on E.



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Semilinear Fractional Differential Equations 37

φ : [−ρ, 0] → E a given continuous function with φ(0) = 0 and (E,| · |) a Banach space. For any

function y defined on [−ρ, b] and any t ∈ J we denote by yt the element of C([−ρ, 0], E) defined

by

yt(θ) = y(t +θ), θ ∈ [−ρ, 0].

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

Let us mention that the functional differential equation of the type (1) was investigated,

in the case when A generates a C0−semigroup, in a lot of papers and developed with the help

of various tools of fixed-point theory see, for instance Belmekki et al. [8, 9, 10].

The principal goal of this paper is to extend such results to the case when the operator A

is nondensely defined and satisfies the Hille-Yosida condition, and to initiate the application

of the technique of measures of noncompactness to investigate the problem of the existence of

integral solutions for (1)–(2). Especially that technique combined with an appropriate fixed

point theorem has proved to be a very useful tool in the study of the existence of solutions for

several types of integral and differential equations; see for example Alvàrez [3], Banas̀ et al.

[5, 6, 7], Benchohra et al. [13], Guo et al. [21], Mönch [32], Mönch and Von Harten [37], and

Szufla [41].

2 Preliminaries

In this section we collect some definitions, notations and results needed in the sequel. At first,

we recall the definition of Riemann-Liouville fractional primitive and fractional derivative.

Denote by C(J, E) the Banach space of continuous functions J → E, with the usual supre-

mum norm

‖y‖∞ = sup{|y(t)|, t ∈ J}.

For ψ ∈ C([−ρ, 0], E) the norm of ψ is defined by

‖ψ‖C = sup{|ψ(θ)|, θ ∈ [−ρ, 0]}.

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

‖N‖B(E) = su p{|N( y)| : |y| = 1}.

Let L1(J, E) be the Banach space of measurable functions y : J → E which are Bochner inte-

grable, equipped with the norm

‖y‖L1 =

ˆ

J

|y(t)|dt.

Let L∞(J, E) be the Banach space of measurable functions y : J → E which are bounded,

equipped with the norm

‖y‖L∞ = inf{c > 0 : ‖y(t)‖ ≤ c, a.e. t ∈ J}.



38 M. Benchohra, Gaston M. N’Guérékata & D. Seba CUBO
12, 3 (2010)

For a given set V of functions v : [−ρ, b] → E, let us denote by

V (t) = {v(t) : v ∈ V }, t ∈ [−ρ, b]

and

V (J) = {v(t) : v ∈ V , t ∈ [−ρ, b]}.

Definition 2.1. ([25, 39]). The Riemann-Liouville fractional primitive of order r > 0 of a

function h : (0, b] → E is defined by

I
r
0 h(t) =

1

Γ(r)

ˆ t

0
(t − s)r−1 h(s)ds,

provided the right side is pointwise defined on (0, b], and where Γ is the gamma function.

Definition 2.2. ([25, 39]). The Riemann-Liouville fractional derivative of order r ∈ (0, 1] of a

continuous function h : (0, b] → E is defined by

dr h(t)

dtr
=

1

Γ(1 − r)

d

dt

ˆ

t

0
(t − s)−r h(s)ds

=
d

dt
I

1−r
0 h(t).

Definition 2.3. A map f : J × C([−ρ, 0], E) → E is said to be Carathéodory if

(i) t 7−→ f (t, u) is measurable for each u ∈ C([−ρ, 0], E);

(ii) u 7−→ F(t, u) is continuous for almost each t ∈ J.

For completeness we gather some definitions and basic facts of integrated semigroups,

and operators satisfying Hille-Yosida condition.

Definition 2.4. [4]. Let E be a Banach space. An integrated semigroup is a family of operators

(S(t))t≥0 of bounded linear operators S(t) on E with the following properties:

(i) S(0) = 0;

(ii) t → S(t) is strongly continuous;

(iii) S(s)S(t) =

ˆ s

0
(S(t +τ) − S(τ))dτ, for all t, s ≥ 0.

Definition 2.5. [24]. An operator A is called a generator of an integrated semigroup if there

exists ω ∈ R such that (ω,∞) ⊂ ρ0(A) (ρ0(A), is the resolvent set of A) and there exists a strongly

continuous exponentially bounded family (S(t))t≥0 of bounded operators such that S(0) = 0 and

R(λ, A) := (λI − A)−1 = λ

ˆ

∞

0
e
−λt

S(t)dt exists for all λ with λ > ω.



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Semilinear Fractional Differential Equations 39

Proposition 2.1. [4]. Let A be the generator of an integrated semigroup (S(t))t≥0. Then for

all x ∈ E and t ≥ 0,

ˆ t

0
S(s)xds ∈ D(A) and S(t)x = A

ˆ t

0
S(s)xds + tx.

Definition 2.6. We say that the linear operator A satisfies the Hille-Yosida condition if there

exists M ≥ 0 and ω ∈ R such that (ω,∞) ⊂ ρ0(A) and

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

Definition 2.7. [24].

(i) An integrated semigroup (S(t))t≥0 is called locally Lipschitz continuous if, for all τ > 0,

there exists a constant L such that

|S(t) − S(s)| ≤ L|t − s|, t, s ∈ [0,τ].

(ii) An integrated semigroup (S(t))t≥0 is called non degenerate if S(t)x = 0, for all t ≥ 0,

implies that x = 0.

Theorem 2.1. [24]. The following assertions are equivalent:

(i) A is the generator of a non degenerate, locally Lipschitz continuous integrated semi-

group;

(ii) A satisfies the Hille-Yosida condition.

If A is the generator of an integrated semigroup (S(t))t≥0 which is locally Lipschitz,

then from [4], S(·)x is continuously differentiable if and only if x ∈ D(A) and (S′(t))t≥0 is a

C0−semigroup on D(A).

Let (S(t))t≥0 be the integrated semigroup generated by A. We note that, if A satisfies

the Hille-Yosida condition, then ‖S′(t)‖B(E) ≤ M e
ωt, t ≥ 0, where M and ω are the constants

considered in the Hille-Yosida condition.

Now let us recall some fundamental facts of the notion of Kuratowski measure of non-

compactness.

Definition 2.8. ([6]) Let E be a Banach space and ΩE the bounded subsets of E. The Kura-

towski measure of noncompactness is the map α : ΩE → [0,∞] defined by

α(B) = inf{ǫ > 0 : B ⊆ ∪n
i=1Bi and d iam(Bi ) ≤ ǫ}; here B ∈ ΩE .



40 M. Benchohra, Gaston M. N’Guérékata & D. Seba CUBO
12, 3 (2010)

Properties: The Kuratowski measure of noncompactness satisfies the following properties

(for more details see [6]).

(a) α(B) = 0 ⇔ B is compact (B is relatively compact).

(b) α(B) = α(B).

(c) A ⊂ B ⇒ α(A) ≤ α(B).

(d) α(A + B) ≤ α(A) +α(B)

(e) α(cB) = |c|α(B); c ∈ IR.

(f) α(convB) = α(B).

Theorem 2.2. ([2, 32]) Let D be a bounded, closed and convex subset of a Banach space such

that 0 ∈ D, and let N be a continuous mapping of D into itself. If the implication

V = conv N(V ) or V = N(V ) ∪ {0} ⇒ α(V ) = 0

holds for every subset V of D, then N has a fixed point.

Lemma 2.1. ([41]) Let D be a bounded, closed and convex subset of the Banach space C(J, E),

G a continuous function on J × J and f a function from J × C([−ρ, 0], E) → E which satisfies

the Carathéodory conditions and there exists p ∈ L1(J, IR+) such that for each t ∈ J and each

bounded set B ⊂ C([−ρ, 0], E) we have

lim
k→0+

α( f (Jt,k × B)) ≤ p(t)α(B); here Jt,k = [t − k, t] ∩ J.

If V is an equicontinuous subset of D, then

α

({
ˆ

J

G(s, t) f (s, ys )ds : y ∈ V
})

≤

ˆ

J

‖G(t, s)‖p(s)α(V (s))ds.

3 Main Results

We start with the following principal assumption and the definition of integral solutions to

the problem (1)-(2).

(H1) A satisfies the Hille-Yosida condition.

Definition 3.1. We say that a continuous function y : [−ρ, b] → E is an integral solution of

problem (1)-(2) if



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Semilinear Fractional Differential Equations 41

(i)

ˆ

t

0
(t − s)r−1 y(s)ds ∈ D(A) for t ∈ J,

(ii) y(t) = φ(t), t ∈ [−ρ, 0], and

(iii) y(t) =
1

Γ(r)
A

ˆ t

0
(t − s)r−1 y(s)ds +

1

Γ(r)

ˆ t

0
(t − s)r−1 f (s, ys )ds, t ∈ J.

From the definition it follows that y(t) ∈ D(A),∀ t ≥ 0. Moreover, y satisfies the following

variation of constants formula:

y(t) =
1

Γ(r)

d

dt

ˆ t

0
S(t − s)(t − s)r−1 f (s, ys )ds, t ≥ 0. (3)

Let Bλ = λR(λ, A), then for all x ∈ D(A), Bλx 7→ x as λ 7→ ∞.

We notice also that, if y satisfies (3), then

y(t) = lim
λ→∞

1

Γ(r)

ˆ t

0
S
′(t − s)(t − s)r−1 Bλ f (s, ys )ds, t ≥ 0.

Without lost of generality, we will assume that w > 0.

Let us list some conditions on the functions involved in the problem (1)-(2).

(H2) The operator S′(t) is compact in D(A) whenever t > 0 and

‖S
′(t)‖B(E) ≤ M e

ωt, t ∈ J.

(H3) f : J × C([−ρ, 0], E) → E is of Carathéodory.

(H4) There exists a function p ∈ L∞(J, IR+) such that

|f (t, u)| ≤ p(t)(‖u‖C + 1), for a.e. t ∈ J, and each u ∈ C([−ρ, 0], E).

(H5) For almost each t ∈ J and each bounded set B ⊂ C([−ρ, 0], E) we have

lim
h→0+

α( f (Jt,h × B)) ≤ p(t)α(B); here Jt,h = [t − h, t] ∩ J.

(H6) Assume
Mbr p∗ eωb

Γ(r + 1)
< 1.

Let p∗ = ‖p‖L∞ . Our main result reads as follows

Theorem 3.1. Assume that assumptions (H1) − (H6) hold. Then the the problem (1)-(2) has

at least one integral solution.



42 M. Benchohra, Gaston M. N’Guérékata & D. Seba CUBO
12, 3 (2010)

Proof. We shall reduce the existence of solutions of (1)-(2) to a fixed point problem.

Consider the operator N : C([−ρ, b], E) → C([−ρ, b], E) defined by

N( y)(t) =











φ(t), t ∈ [−ρ, 0],
1

Γ(r)

d

dt

ˆ t

0
(t − s)r−1 S(t − s) f (s, ys )ds, t ∈ [0, b].

Let r0 > 0 be such that

r0 ≥
M p∗br eωb

Γ(r + 1) − Mbr p∗ eωb
,

and consider the set

Dr0 = { y ∈ C([−ρ, b], E) : ‖y‖∞ ≤ r0}.

Clearly, the subset Dr0 is closed, bounded and convex. We shall show that N satisfies the

assumptions of Theorem 2.2. The proof will be given in three steps.

Step 1: N is continuous.

Let us consider a sequence { yn} such that yn → y in C([−ρ, b], E). Then for each t ∈ J

|N( yn)(t) − N( y)(t)| =

∣

∣

∣

∣

1

Γ(r)

d

dt

ˆ t

0
(t − s)r−1 S(t − s)[ f (s, yns ) − f (s, ys )]ds

∣

∣

∣

∣

≤
M eωt

Γ(r)

ˆ

t

0
e
−ωs(t − s)r−1|f (s, yns ) − f (s, ys )|ds

≤
M eωb

Γ(r)

ˆ t

0
(t − s)r−1|f (s, yns ) − f (s, ys )|ds.

Let µ > 0 be such that

‖yn‖∞ ≤ µ, ‖y‖∞ ≤ µ.

By (H4) we have

|(t − s)r−1 [ f (s, yns ) − f (s, ys )]| ≤ 2 p
∗(µ+ 1)(t − s)r−1 ∈ L1(J, IR+).

Since f is a Carathéodory function, the Lebesgue dominated convergence theorem im-

plies that

‖N( yn) − N( y)‖∞ → 0 as n → ∞.

Step 2: N maps Dr0 into itself.

For each y ∈ Dr0 , by (H4) and (H6) we have for each t ∈ J

|N( y)(t)| =

∣

∣

∣

∣

1

Γ(r)

d

dt

ˆ t

0
(t − s)r−1 S(t − s) f (s, ys )ds

∣

∣

∣

∣



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Semilinear Fractional Differential Equations 43

≤
Mbr p∗ eωb(r0 + 1)

Γ(r + 1)
≤ r0

Step 3: N(Dr0 ) is bounded and equicontinuous.

By Step 2, it is obvious that N(Dr0 ) ⊂ Dr0 is bounded.

For the equicontinuity of N(Dr0 ). Let τ1, τ2 ∈ J, τ1 < τ2, thus if ǫ > 0 and ǫ ≤ τ1 ≤ τ2 we

have for any y ∈ Dr0 ;

|N( y)(τ2) − N( y)(τ1)| =

∣

∣

∣

∣

lim
λ 7→∞

1

Γ(r)

ˆ τ2

0
(τ2 − s)

r−1
S
′(τ2 − s)Bλ f (s, ys )ds

− lim
λ 7→∞

1

Γ(r)

ˆ τ1

0
(τ1 − s)

r−1
S
′(τ1 − s)Bλ f (s, ys )ds

∣

∣

∣

∣

≤ M p
∗(r0 + 1)

(

1

Γ(r)

ˆ τ1−ǫ

0
[(τ2 − s)

r−1
− (τ1 − s)

r−1]ds

+ ‖S
′(τ2 −τ1 +ǫ) − S

′(ǫ)‖B(E)

{

1

Γ(r)

ˆ τ1−ǫ

0
(τ2 − s)

r−1
ds

}

+
1

Γ(r)

ˆ τ1

τ1−ǫ

[(τ2 − s)
r−1

− (τ1 − s)
r−1]ds

+ ‖S
′(τ2 −τ1) − I‖B(E)

{

1

Γ(r)

ˆ τ1

τ1−ǫ

(τ2 − s)
r−1

ds

}

+
1

Γ(r)

ˆ τ2

τ1

(τ2 − s)
r−1

ds

)

.

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

since S′(t) is a strongly continuous operator and the compactness of S′(t) for t > 0 implies the

continuity in the uniform operator topology (see [38]).

Now let V be a subset of Dr0 such that V ⊂ conv(N(V ) ∪ {0}).

V is bounded and equicontinuous and therefore the function v → v(t) = α(V (t)) is continuous

on [−ρ, b]. By (H5), Lemmas 2.1 and the properties of the measure α we have for each t ∈

[−ρ, b]

v(t) ≤ α(N(V )(t) ∪ {0})

≤ α(N(V )(t))

≤ lim
λ 7→∞

1

Γ(r)

ˆ t

0
(t − s)r−1 S′(t − s)Bλ p(s)α(V (s))ds



44 M. Benchohra, Gaston M. N’Guérékata & D. Seba CUBO
12, 3 (2010)

≤
M p∗ eωt

Γ(r)

ˆ

t

0
(t − s)r−1 v(s)ds

≤ ‖v‖∞
Mbr p∗ eωb

Γ(r + 1)
.

This means that

‖v‖∞

(

1 −
Mbr p∗ eωb

Γ(r + 1)

)

≤ 0.

By (H6) it follows that ‖v‖∞ = 0, that is v(t) = 0 for each t ∈ [−ρ, b], and then V (t) is relatively

compact in E. In view of the Ascoli-Arzelà theorem, V is relatively compact in Dr0 . Applying

now Theorem 2.2 we conclude that N has a fixed point which is an integral solution for the

problem (1)-(2). �

4 An Example

As an application of our results we consider the following fractional time partial functional

differential equation of the form

∂α

∂tα
z(t, x) =

∂2

∂x2
z(t, x) + Q(t, z(t − r, x)), x ∈ [0,π], t ∈ [0, 1], α ∈ (0, 1], (4)

z(t, 0) = z(t,π) = 0, t ∈ [0, 1] (5)

z(t, x) = ϕ(t, x), t ∈ [−r, 0], x ∈ [0,π], (6)

where r > 0, ϕ : [−r, 0] × [0,π] → IR is continuous and Q : [0, 1] × IR → IR is a given function.

Let

y(t)(x) = z(t, x), t ∈ J, x ∈ [0,π],

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

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

We choose E = C([0,π]; IR) endowed with the uniform topology and consider the operator

A : D(A) ⊂ E → E defined by:

D(A) = { y ∈ C2([0,π], IR) : y(0) = y(π) = 0} A y = y′′.

It is well known (see [15]) that the operator A Satisfies the Hille-Yosida condition with

(0,+∞) ⊂ ρ(A), ‖(λI − A)−1‖ ≤
1

λ
for λ > 0, and

D(A) = { y ∈ E; y(0) = y(π) = 0} 6= E.



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Semilinear Fractional Differential Equations 45

It follows that A generates an integrated semigroup (S(t))t≥0 and ‖S
′(t)‖ ≤ e−µt for t ≥ 0 and

for some constant µ > 0. We can show that problem (1)-(2) is an abstract formulation of

problem (4)-(6).

Assume that the function Q satisfies the following conditions

(i) The function Q : J × IR → IR is of Carathéodory.

(ii) |Q(t, z)| ≤
1

et+2
(|z|+ 1) for each (t, z) ∈ J × IR.

It is clear that conditions (H1)-(H4) are satisfied. We shall show that (H6) holds with

p(t) =
1

et+2
, t ∈ [0, 1],

M = 1, b = 1, p∗ =
1

e2
.

Indeed, we have
Mbr p∗ eωb

Γ(r + 1)
≤

1

e2Γ(r + 1)
< 1, for each r ∈ (0, 1].

Hence, Theorem 3.1 implies that problem (4)-(6) has an integral solution z on [−r, 1] ×

[0,π].

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