FRACTIONAL INTEGRO-DIFFERENTIAL INCLUSIONS WITH STATE-DEPENDENT DELAY

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doi:10.7151/dmdico.1160

FRACTIONAL INTEGRO-DIFFERENTIAL INCLUSIONS WITH STATE-DEPENDENT DELAY

Khalida Aissani

Laboratory of Mathematics, University of Sidi Bel-Abb`es P.O. Box 89, Sidi Bel-Abb`es 22000, Algeria

Mouffak Benchohra

¹

Laboratory of Mathematics, University of Sidi Bel-Abb`es P.O. Box 89, Sidi Bel-Abb`es 22000, Algeria

and

Department of Mathematics, King Abdulaziz University P.O. Box 80203, Jeddah 21589, Saudi Arabia

e-mail: benchohra@univ-sba.dz

and Khalil Ezzinbi

Departement of Mathematics, Facult´e of Sciences Semlalia, B.P. 2390, Marrakech, Morocco

e-mail: ezzinbi@ucam.ac.ma

Abstract

In this paper, we establish sufficient conditions for the existence of mild solutions for fractional integro-differential inclusions with state-dependent delay. The techniques rely on fractional calculus, multivalued mapping on a bounded set and Bohnenblust-Karlin’s fixed point theorem. Finally, we present an example to illustrate the theory.

Keywords: fractional integro-differential inclusions, Caputo fractional derivative, mild solution, multivalued map, Bohnenblust-Karlin’s fixed point, state-dependent delay.

2010 Mathematics Subject Classification:26A33, 34A08, 34A60, 34G20, 34G25, 34K09, 34K37.

1Corresponding author.

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1. Introduction

Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, mechanics, chemistry, engineer- ing, control, etc. (see [33, 39, 41]). There has been a significant development in ordinary and partial fractional differential equations in recent years; see the monographs of Abbas et al. [1, 2], Baleanu et al. [10], Diethelm [20], Kilbas et al. [35], and the papers [4, 5, 12, 23, 37, 7].

Functional differential equations arise in many areas of applied mathemat- ics, this type of equation has received much attention in recent years. On the other hand, functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason the study of this type of equations has received great attention in the last few years, see for instance [6, 22, 28, 29, 27, 30, 32, 31, 40] and the references therein. In [13, 14], the authors provide sufficient conditions for the existence of solutions for frac- tional integro-differential equations with state-dependent delay. Very recently, Aissani and Benchohra [8] investigated the existence of mild solutions for a class of fractional integro-differential equations with state-dependent delay.

Functional differential inclusions with fractional order are first considered by El Sayed and Ibrahim [24]. Recently, Aissani and Benchohra [9] have investi- gated the existence of solutions of impulsive fractional differential inclusions with infinite delay involving Caputo’s fractional derivative. In [11] Benchohra et al.

proved the existence and controllability results for fractional integro-differential inclusions with state-dependent delay in Fr´echet spaces. Cernea [16, 17] estab- lished some Filippov type existence theorems for solutions of fractional semilinear differential inclusions involving Caputo’s fractional derivative in Banach spaces.

Motivated by the papers cited above, in this paper, we consider the existence of a class of fractional integro-differential inclusions with state-dependent delay described by the form

(1) D

_t^q

x(t) ∈ Ax(t) + Z

t

0

a(t, s)F (s, x

_ρ(s,x_s₎

, x(s))ds, t ∈ J = [0, T ],

x

₀

= φ ∈ B, t ∈ (−∞, 0],

where D

_t^q

is the Caputo fractional derivative of order 0 < q < 1, A generates

a compact and uniformly bounded linear semigroup {S(t)}

_t≥0

on the separable

Banach space X, F : J × B × X −→ P(X) is a multivalued map (P(X) is the

family of all nonempty subsets of X). ρ : J × B → (−∞, T ] and a : D → R (D =

{(t, s) ∈ J × J : t ≥ s}) are appropriated functions. We denote by x

t

the element

of B defined by x

_t

(θ) = x(t + θ), for θ ∈ (−∞, 0]. Here x

_t

represents the history

of the state up to the present time t. We assume that the histories x

t

belong to

some abstract phase space B, to be specified later.

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2. Preliminaries

Let (X, k · k) be a real Banach space. C = C(J, X) be the space of all X-valued continuous functions on J. L(X) is the Banach space of all linear and bounded operators on X. L

¹

(J, X) the space of X−valued Bochner integrable functions on J with the norm

kyk

_L1

= Z

T

0

ky(t)kdt.

L

^∞

(J, R) is the Banach space of essentially bounded functions, normed by kyk

_L^∞

= inf{d > 0 : |y(t)| ≤ d, a.e. t ∈ J}.

Denote by P

_cl

(X) = {Y ∈ P (X) : Y closed}, P

_b

(X) = {Y ∈ P (X) : Y bounded}, P

_cp

(X) = {Y ∈ P (X) : Y compact}, P

cp,c

(X) = {Y ∈ P (X) : Y compact, convex}.

A multivalued map G : X → P (X) is convex (closed) valued if G(x) is convex (closed) for all x ∈ X. G is bounded on bounded sets if G(B) = ∪

x∈B

G(x) is bounded in X for all B ∈ P

_b

(X) (i.e., sup

x∈B

{sup{kyk : y ∈ G(x)}} < ∞).

G is called upper semi-continuous (u.s.c.) on X if for each x

₀

∈ X the set G(x

₀

) is a nonempty, closed subset of X, and if for each open set U of X containing G(x

₀

), there exists an open neighborhood V of x

₀

such that G(V ) ⊆ U.

G is said to be completely continuous if G(B) is relatively compact for every B ∈ P

b

(X). If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e., x

n

−→ x

∗

, y

n

−→ y

∗

, y

n

∈ G(x

n

) imply y

∗

∈ G(x

∗

)).

For more details on multivalued maps see the books of Deimling [19], and G´ orniewicz [25].

Definition 2.1. The multivalued map F : J × B × X −→ P(X) is said to be Carath´eodory if

(i) t 7−→ F (t, x, y) is measurable for each (x, y) ∈ B × X;

(ii) (x, y) 7−→ F (t, x, y) is upper semicontinuous for almost all t ∈ J.

We need some basic definitions and properties of the fractional calculus theory which are used further in this paper.

Definition 2.2. Let α > 0 and f : R

+

→ X be in L

¹

(R

+

, X). Then the Riemann- Liouville integral is given by:

I

_t^α

f(t) = 1 Γ(α)

Z

t 0

f(s)

(t − s)

^1−α

ds,

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where Γ(·) is the Euler gamma function.

For more details on the Riemann-Liouville fractional derivative, we refer the reader to [18].

Definition 2.3 [39]. The Caputo derivative of order α for a function f : [0, +∞) → R can be written as

D

_t^α

f (t) = 1 Γ(n − α)

Z

t 0

f

^(n)(s)

(t − s)

^α+1−n

ds = I

^n−α

f

⁽ⁿ⁾

(t), t > 0, n − 1 ≤ α < n.

If 0 < α ≤ 1, then

D

_t^α

f (t) = 1 Γ(1 − α)

Z

t 0

f

^′

(s) (t − s)

^α

ds.

Obviously, the Caputo derivative of a constant is equal to zero.

In this paper, we will employ an axiomatic definition for the phase space B which is similar to those introduced by Hale and Kato [26]. Specifically, B is a linear space of functions mapping (−∞, 0] into X and endowed with a seminorm k · k

B

, satisfying the following axioms:

(A1) If x : (−∞, T ] −→ X is continuous on J and x

₀

∈ B, then x

t

∈ B and x

t

is continuous in t ∈ J and

(2) kx(t)k ≤ Ckx

t

k

B

,

where C ≥ 0 is a constant.

(A2) There exist a continuous function C

1

(t) > 0 and a locally bounded function C

₂

(t) ≥ 0 in t ≥ 0 such that

(3) kx

t

k

B

≤ C

1

(t) sup

s∈[0,t]

kx(s)k + C

2

(t)kx

0

k

B

, for t ∈ [0, T ] and x as in (A1).

(A3) The space B is complete.

Remark 2.4. Condition (2) in (A1) is equivalent to kφ(0)k ≤ Ckφk

B

, for all φ ∈ B.

Now we state the following lemmas which are necessary to establish our main result.

Let S

F,x

be a set defined by

S

_F,x

= {v ∈ L

¹

(J, X) : v(t) ∈ F (t, x

_ρ(t,x_t₎

, x(t)) a.e. t ∈ J}.

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Lemma 2.5 [36]. Let X be a Banach space. Let F : J × B × X −→ P

cp,c

(X) be an L

¹

-Carath´ eodory multivalued map and let Ψ be a linear continuous mapping from L

¹

(J, X) to C(J, X). Then the operator

Ψ ◦ S

F

: C(J, X) −→ P

cp,c

(C(J, X)),

x 7−→ (Ψ ◦ S

F

)(x) := Ψ(S

F,x

) is a closed graph operator in C (J, X) × C(J, X).

The next result is known as the Bohnenblust-Karlin fixed point theorem.

Lemma 2.6 (Bohnenblust-Karlin [15]). Let X be a Banach space and D ∈ P

_cl,c

(X). Suppose that the operator G : D → P

_cl,c

(D) is upper semicontinuous and the set G(D) is relatively compact in X. Then G has a fixed point in D.

Let B be a set defined by

B = {x : (−∞, T ] → X such that x|

J

∈ C(J, X), x

₀

∈ B} .

3. Main results

In this section, our aim is to discuss the existence of mild solutions to the problem (1). We give first the definition of the mild solution of our problem.

Definition 3.1. A function x ∈ B is said to be a mild solution of (1) if there exists v(·) ∈ L

¹

(J, X), such that v(t) ∈ F (t, x

_ρ(t,x_t₎

, x(t)) a.e. t ∈ [0, T ], and x satisfies

(4) x(t) =



 



 



φ(t), t ∈ (−∞, 0];

−Q(t)φ(0) + Z

t

0

Z

s 0

R(t − s)a(s, τ )v(τ )dτ ds, t ∈ J, where

Q(t) = Z

∞

0

ξ

_q

(σ)S(t

^q

σ)dσ, R(t) = q Z

∞

0

σt

^q−1

ξ

_q

(σ)S(t

^q

σ)dσ and for σ ∈ (0, ∞),

ξ

_q

(σ) = 1

q σ

⁻¹⁻¹^q

̟

_q

(σ

⁻¹^q

) ≥ 0,

̟

_q

(σ) = 1 π

∞

X

n=1

(−1)

ⁿ⁻¹

σ

^−qn−1

Γ(nq + 1)

n! sin(nπq).

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Here, ξ

q

is a probability density function defined on (0, ∞) ([38]), that is ξ

q

(σ) ≥ 0, σ ∈ (0, ∞) and

Z

∞ 0

ξ

q

(σ)dσ = 1.

It is not difficult to verify that Z

∞

0

σξ

_q

(σ)dσ = 1 Γ(1 + q) .

Remark 3.2. Note that {S(t)}

_t≥0

is a uniformly bounded semigroup, i.e., there exists a constant M > 0 such that kS(t)k ≤ M for all t ∈ [0, T ].

Remark 3.3. Note that

(5) kR(t)k ≤ C

q,M

t

^q−1

, t > 0, where C

q,M

= qM

Γ(1 + q) . Set

R(ρ

⁻

) = {ρ(s, ϕ) : (s, ϕ) ∈ J × B, ρ(s, ϕ) ≤ 0}.

We always assume that ρ : J × B → (−∞, T ] is continuous. Additionally, we introduce following hypothesis:

(H

_ϕ

) The function t → ϕ

_t

is continuous from R(ρ

⁻

) into B and there exists a continuous and bounded function L

^φ

: R(ρ

⁻

) → (0, ∞) such that

kφ

_t

k

B

≤ L

^φ

(t)kφk

B

for every t ∈ R(ρ

⁻

).

Remark 3.4. The condition (H

ϕ

), is frequently verified by functions continuous and bounded. For more details, see for instance [34].

Remark 3.5. In the rest of this section, C

₁^∗

and C

₂^∗

are the constants C

₁^∗

= sup

s∈J

C

₁

(s) and C

₂^∗

= sup

s∈J

C

₂

(s).

Lemma 3.6 [32]. If x : (−∞, T ] → X is a function such that x

₀

= φ, then kx

s

k

B

≤ (C

₂^∗

+ L

^φ

)kφk

B

+ C

₁^∗

sup{|y(θ)|; θ ∈ [0, max{0, s}]}, s ∈ R(ρ

⁻

) ∪ J, where L

^φ

= sup

t∈R(ρ⁻)

L

^φ

(t).

We assume the following.

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(H1) The multivalued map F : J × B × X −→ P

cp,cv

(X) is Carath´eodory.

(H2) There exists a function µ ∈ L

¹

(J, R

⁺

) and a continuous nondecreasing function ψ : R

⁺

→ (0, +∞) such that

kF (t, v, w)k ≤ µ(t)ψ (kvk

B

+ kwk

X

) , (t, v, w) ∈ J × B × X.

(H3) For each t ∈ J, a(t, s) is measurable on [0, t] and a(t) = esssup{|a(t, s)|, 0 ≤ s ≤ t} is bounded on J. The map t → a

_t

is continuous from J to L

^∞

(J, R), where, a

t

(s) = a(t, s).

Set a = sup

t∈J

a(t).

Theorem 3.7. Suppose that (H

ϕ

), (H1)–(H3) hold. Then the problem (1) has a mild solution on (−∞, T ].

Proof. We transform the problem (1) into a fixed-point problem. Consider the multivalued operator N : B −→ P(B) defined by N (h) = {h ∈ B} with

h(t) =



 

 

φ(t), t ∈ (−∞, 0];

−Q(t)φ(0) + Z

t

0

Z

s 0

R(t − s)a(s, τ )v(τ )dτ ds, v ∈ S

_F,x

, t ∈ J.

Clearly, fixed points of the operator N are mild solutions of the problem (1).

For φ ∈ B, we will define the function y(.) : (−∞, T ] → X by

y(t) =







φ(t), if t ∈ (−∞, 0];

−Q(t)φ(0), if t ∈ J.

Then y

₀

= φ. For each z ∈ C(J, X) with z(0) = 0, we denote by z the function defined by

z(t) =







0, if t ∈ (−∞, 0];

z(t), if t ∈ J.

If x(·) verifies (4), we can decompose it as x(t) = y(t) + z(t), for t ∈ J, which implies x

t

= y

t

+ z

t

, for every t ∈ J and the function z(t) satisfies

z(t) = Z

t

0

Z

s 0

R(t − s)a(s, τ )v(τ )dτ ds,

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where

v ∈ S

_F,y+z

= {v ∈ L

¹

(J, X) : v(t) ∈ F (t, y

_ρ(t,y_t_+z_t₎

+ z

_ρ(t,y_t_+z_t₎

, y(t) + z(t)) for a.e. t ∈ J}.

Let

Z

₀

= {z ∈ B : z

0

= 0}.

For any z ∈ Z

₀

, we have kzk

Z0

= sup

t∈J

kz(t)k + kz

0

k

B

= sup

t∈J

kz(t)k.

Thus (Z

₀

, k.k

Z₀

) is a Banach space. We define the operator P : Z

₀

−→ P(Z

₀

) by:

P (z) = {h ∈ Z

₀

} with h(t) =

Z

t 0

Z

s 0

R(t − s)a(s, τ )v(τ )dτ ds, v(s) ∈ S

_F,y+z

, t ∈ J.

Obviously the operator N has a fixed point if and only if the operator P has one, so it turns to prove that P has a fixed point. We shall show that the operator P satisfies all conditions of Lemma 2.6. For better readability, we break the proof into some steps. Choose

r ≥ aC

q,M

T

^q

q ψ

(C

₁^∗

+ 1)r + (C

₂^∗

+ L

^φ

+ C

₁^∗

M C )kφk

B

Z

T 0

µ(τ )dτ, and consider the set

B

_r

= {z ∈ Z

0

: kzk

Z0

≤ r}.

It is clear that B

r

is a closed, convex, bounded set in Z

0

.

Step 1. P(z) is convex for each z ∈ B

r

. Indeed, if h

₁

and h

₂

belong to P (z), then there exist v

₁

, v

₂

∈ S

_F,yρ(τ,yτ +zτ )+zρ(τ,yτ +zτ )

such that, for t ∈ J, we have

h

i

(t) = −Q(t)φ(0) + Z

t

0

Z

s 0

R(t − s)a(s, τ )v

i

(τ )dτ ds, i = 1, 2.

Let 0 ≤ d ≤ 1. Then, for each t ∈ J, we have dh

₁

(t) + (1 − d)h

₂

(t)

= −Q(t)φ(0) + Z

t

0

Z

s 0

R(t − s)a(s, τ ) [dv

₁

(τ ) + (1 − d)v

₂

(τ )] dτ ds.

Since S

F,yρ(τ,yτ +zτ )+zρ(τ,yτ +zτ )

is convex (because F has convex values), we have

dh

₁

+ (1 − d)h

₂

∈ P (z).

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Step 2. P (B

r

) ⊂ B

r

. Let h ∈ P (z) and z ∈ B

r

. For each t ∈ [0, T ], we have kh(t)k ≤

Z

t 0

Z

s 0

kR(t − s)a(s, τ )v(τ )kdτ ds

≤ a C

q,M

Z

t 0

Z

s 0

(t − s)

^q−1

µ(τ )ψ [ky

τ

+ z

τ

k + ky(τ ) + z(τ )k] dτ ds

≤ a C

q,M

Z

t 0

Z

s 0

(t − s)

^q−1

µ(τ )ψ h

C

₁^∗

r +

C

₂^∗

+ L

^φ

+ C

₁^∗

M C + r i

dτ ds

≤ T

^q

a C

q,M

q ψ h

(C

₁^∗

+ 1) r +

C

₂^∗

+ L

^φ

+ C

₁^∗

M C i Z

T 0

µ(τ )dτ ≤ r, which proves that P (B

_r

) ⊂ B

_r

.

Step 3. We will prove that P (B

r

) is equicontinuous. Let τ

₁

, τ

₂

∈ [0, T ], with τ

₁

> τ

₂

. We have

kh(τ

₂

) − h(τ

₁

)k ≤ Q

₁

+ Q

₂

, where

Q

₁

=

Z

τ2

0

Z

s 0

[R(τ

₁

− s) − R(τ

₂

− s)]a(s, τ )v(τ )dτ ds Q

₂

=

Z

τ₁ τ2

Z

s 0

kR(τ

₁

− s)kka(s, τ )kkv(τ )kdτ ds.

In view of (5), we have Q

₁

≤ a

Z

τ₂ 0

Z

s 0

kR(τ

₁

− s) − R(τ

₂

− s)kkv(τ )kdτ ds

≤ aψ h

(C

₁^∗

+ 1) r +

C

₂^∗

+ L

^φ

+ C

₁^∗

M C i kµk

_L¹

Z

τ2

0

kR(τ

₁

−s) −R(τ

₂

−s)kds

≤ aψ h

(C

₁^∗

+ 1) r +

C

₂^∗

+ L

^φ

+ C

₁^∗

M C i kµk

_L1

× [q Z

τ2

0

Z

∞ 0

σk[(τ

1

− s)

^q−1

− (τ

2

− s)

^q−1

]ξ

q

(σ)S((τ

1

− s)

^q

σ)kdσds + q

Z

τ₂ 0

Z

∞ 0

σ(τ

₂

− s)

^q−1

ξ

q

(σ)kS((τ

₁

− s)

^q

σ) − S((τ

₂

− s)

^q

σ)kdσds]

≤ aψ h

(C

₁^∗

+ 1) r +

C

₂^∗

+ L

^φ

+ C

₁^∗

M C i kµk

_L¹

× [C

q,M

Z

τ₂ 0

(τ

₁

− s)

^q−1

− (τ

₂

− s)

^q−1

ds + q

Z

τ2

0

Z

∞ 0

σ(τ

2

− s)

^q−1

ξ

_q

(σ)kS((τ

1

− s)

^q

σ) − S((τ

2

− s)

^q

σ)kdσds].

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Clearly, the first term on the right-hand side of the above inequality tends to zero as τ

₂

→ τ

₁

. From the continuity of S(t) in the uniform operator topology for t > 0, the second term on the right-hand side of the above inequality tends to zero as τ

₂

→ τ

₁

.

For Q

₂

, we obtain Q

₂

≤ aψ h

(C

₁^∗

+ 1) r +

C

₂^∗

+ L

^φ

+ C

₁^∗

M C i kµk

_L¹

Z

τ1

τ2

kR(τ

1

− s)kds

≤ aC

q,M

ψ h

(C

₁^∗

+ 1) r +

C

₂^∗

+ L

^φ

+ C

₁^∗

M C i kµk

_L¹

Z

τ1

τ₂

(τ

₁

− s)

^q−1

ds.

As τ

₂

→ τ

₁

, Q

₂

tends to zero. So P (B

r

) is equicontinuous.

Step 4. (P B

_r

)(t) is relatively compact for each t ∈ J, where (P B

r

)(t) = {h(t) : h ∈ P (B

r

)}.

Let 0 < t ≤ T be fixed and let ε be a real number satisfying 0 < ε < t. For arbitrary δ > 0, we define

h

_ε,δ

(t) = q Z

t−ε

0

(t − s)

^q−1

Z

∞

δ

σξ

_q

(σ)S((t − s)

^q

σ) Z

s

0

a(s, τ )v(τ )dτ dσds

= qS(ε

^q

δ) Z

t−ε

0

(t − s)

^q−1

Z

∞

δ

σξ

_q

(σ)S((t − s)

^q

σ −ε

^q

δ) Z

s

0

a(s, τ )v(τ )dτ dσds, where v ∈ S

F,yρ(τ,yτ +zτ )+zρ(τ,yτ +zτ )

. Since S(t) is a compact operator, the set

H

_ε,δ

= {h

_ε,δ

(t) : h ∈ P (B

_r

)}

is relatively compact. Moreover, for every h ∈ P (B

r

) we have kh(t) − h

ε,δ

(t)k

≤ q Z

t−ε

0

(t − s)

^q−1

Z

δ

0

σξ

_q

(σ)kS((t − s)

^q

σ)k Z

s

0

ka(s, τ )kkv(τ )kdτ dσds + q

Z

t t−ε

(t − s)

^q−1

Z

∞

0

σξ

_q

(σ)kS((t − s)

^q

σ)k Z

s

0

ka(s, τ )kkv(τ )kdτ dσds

≤ T

^q

M aψ h

(C

₁^∗

+ 1) r +

C

₂^∗

+ L

^φ

+ C

₁^∗

M C i kµk

_L¹

Z

δ 0

σξ

_q

(σ)dσ + ε

^q

M a

Γ(1 + q) ψ h

(C

₁^∗

+ 1) r +

C

₂^∗

+ L

^φ

+ C

₁^∗

M C

i

kµk

_L1

.

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Therefore, (P B

r

)(t) is relatively compact.

As a consequence of Step 2 to 4 together with Arzel´ a-Ascoli theorem we conclude that the multivalued operator P is completely continuous.

Step 5. P has a closed graph. Let z

n

→ z

∗

, h

n

∈ P (z

n

), and h

n

→ h

∗

. We need to show that h

∗

∈ P (z

∗

). Now, h

n

∈ P (z

n

) implies there exists v

n

∈ S

_F,ynρ(τ,ynτ +znτ )+znρ(τ,ynτ +znτ )

such that for each t ∈ J,

h

_n

(t) = Z

t

0

Z

s 0

R(t − s)a(s, τ )v

_n

(τ )dτ ds, t ∈ J.

We have to show that there exists v

∗

∈ S

_F,y_∗ρ(τ,y∗τ +z∗τ )+z∗ρ(τ,y∗τ +z∗τ )

such that for each t ∈ J,

h

∗

(t) = Z

t

0

Z

s 0

R(t − s)a(s, τ )v

∗

(τ )dτ ds, t ∈ J.

Consider the linear and continuous operator Υ : L

¹

(J, X) −→ C(J, X) defined by

(Υv)(t) = Z

t

0

Z

s 0

R(t − s)a(s, τ )v(s)dτ ds.

From the definition of Υ, we know that

h

_n

(t) ∈ Υ(S

F,ynρ(τ,ynτ +znτ )+znρ(τ,ynτ +znτ )

).

Since z

n

→ z

∗

and ΥoS

F

is a closed graph operator by Lemma 2.5 then there exists v

∗

∈ S

_F,y_∗ρ(τ,y∗τ +z∗τ )+z∗ρ(τ,y∗τ +z∗τ )

such that

h

∗

(t) = Z

t

0

Z

s 0

R(t − s)a(s, τ )v

∗

(s)dτ ds.

Hence h

∗

∈ P (z

∗

).

As a consequence of Lemma 2.6, we deduce that P has a fixed point z on the interval (−∞, T ], so x = y + z is a fixed point of the operator N which is the mild solution of problem (1).

4. An example

To apply our abstract result, we consider the following fractional partial func-

tional integrodifferential inclusion with state dependent delay of the form

(12)

(6)

∂

_t^q

∂t

^q

v(t, ζ) ∈ ∂

²

∂ζ

²

v(t, ζ) + Z

t

0

(t − s) Z

s

−∞

η(s, τ − s, ζ)

× G(τ, v(τ − σ(v(τ, 0)), ζ))dτ ds, t ∈ [0, T ], ζ ∈ [0, π], v(t, 0) = v(t, π) = 0, t ∈ [0, T ],

v(θ, ζ) = v

₀

(θ, ζ), θ(−∞, 0], ζ ∈ [0, π], where 0 < q < 1, σ ∈ C(R, [0, ∞)) and G : [0, T ] × B → P (R) is a u.s.c. multival- ued map with compact convex values.

Set X = L

²

([0, π]) and define A by

D(A) = {u ∈ X : u

^′′

∈ X, u(0) = u(π) = 0}, Au = u

^′′

.

It is well known that A is the infinitesimal generator of an analytic semigroup (S(t))

_t≥0

on X. For the phase space, we choose B = B

γ

defined by

B

γ

:=

φ ∈ C((−∞, 0], X) : lim

θ→−∞

e

^γθ

φ(θ) exists in X

endowed with the norm

kφk = sup{e

^γθ

|φ(θ)| : θ ≤ 0}.

Notice that the phase space B

_γ

satisfies axioms (A1)–(A3).

For t ∈ [0, T ], ζ ∈ [0, π] and ϕ ∈ B

γ

we set x(t)(ζ) = v(t, ζ),

φ(θ)(ζ) = v

0

(θ, ζ), θ ∈ (−∞, 0], a(t, s) = t − s,

F (t, ϕ, x(t))(ζ) = Z

0

−∞

η(t, θ, ζ)G(t, ϕ(0, ζ))dθ, ρ(t, ϕ) = t − σ(ϕ(0, 0)).

Then problem (6) can be rewritten in the abstract form (1). An application of Theorem 3.7 yields the following result.

Theorem 4.1. Let ϕ ∈ B

γ

be such that (H

ϕ

) holds, and let t → ϕ

t

be continuous

on R(ρ

⁻

). Then the system (6) admits a mild solution on (−∞, T ].

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Received 3 March 2014

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