Controllability of fractional order integro-differential inclusions with infinite delay

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Controllability of fractional order integro-differential inclusions with infinite delay

Khalida Aissani

¹

, Mouffak Benchohra

^B^{1, 2}

and Mohamed Abdalla Darwish

^{3, 4}

1Laboratory of Mathematics, University of Sidi Bel-Abbès, PO Box 89, 22000, Sidi Bel-Abbès, Algeria

2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589 Saudi Arabia

3Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, Jeddah, Saudi Arabia

4Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt

Received 14 July 2014, appeared 14 November 2014 Communicated by Michal Feˇckan

Abstract. This paper concerns for controllability of fractional order integro-differential inclusions with infinite delay in Banach spaces. A theorem about the existence of mild solutions to the controllability of fractional order integro-differential inclusions is obtained based on Dhage fixed point theorem. An example is given to illustrate the existence result.

Keywords: controllability, Caputo fractional derivative, integro-differential inclusions, fixed point, semigroup, infinite delay.

2010 Mathematics Subject Classification: 26A33, 34A08, 34A60, 34B15, 34G25, 34H05, 34K09.

1 Introduction

Fractional differential equations have recently been proved to be valuable tools in the mod- eling of many phenomena in various fields of engineering, physics, economics and science.

We can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. [26,27,35,37]. In recent years, there has been a significant development in fractional differential equations. One can see the monographs of Abbas et al.[1,2], Kilbas et al.[31], Lakshmikanthamet al.[32], Miller and Ross [38], Podlubny [40], Zhou [46], and the papers [3–5,10,17–20,24,34,41,42] and the references therein.

On the other hand, the most important qualitative behavior of a dynamical system is controllability. It is well known that the issue of controllability plays an important role in control theory and engineering [7,8,12,15] because they have close connections to pole as- signment, structural decomposition, quadratic optimal control and observer design etc. In

BCorresponding author. Email: benchohra@univ-sba.dz

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recent years, the problem of controllability for various kinds of fractional differential and integro-differential equations have been discussed in [6,16,44].

El-Sayed and Ibrahim initiated the study of fractional differential inclusions in [25]. Re- cently several qualitative results for fractional differential inclusion were obtained in [11,14, 39]. Recently, Benchohraet al. [9] studied the existence and controllability results for fractional order integro-differential inclusions with state-dependent delay in Fréchet spaces. Wang and Zhou [45] investigated the existence and controllability results for fractional semilinear differential inclusions.

Motivated by the papers cited above, in this paper, we consider the controllability results for fractional order integro-differential inclusions with infinite delay described by the form

D_t^qx(t)∈ Ax(t) +Bu(t) +

Z _t

0 a(t,s)F(s,xs,x(s))ds, t ∈ J = [0,T],

x0=φ∈ B, t∈ (−_{∞, 0}],

(1.1)

where D_t^qis the Caputo fractional derivative of order 0 < q< 1, Agenerates a compact and uniformly bounded linear semigroupS(·)onX,F: J× B ×X−→ P(X)is a multivalued map (P(X)is the family of all nonempty subsets of X), a: D → _R (D = {(t,s) ∈ [0,T]×[0,T] : t≥s}),φ∈ BwhereB is called phase space to be defined in Section 2. Bis a bounded linear operator fromXintoX, the controlu∈ L²(J;X), the Banach space of admissible controls. For any functionxdefined on(−_∞,T]and anyt ∈ J, we denote byx_t the element ofBdefined by

x_t(θ) =x(t+θ), θ ∈ (−_{∞, 0}]. Herext represents the history of the state up to the present timet.

Our results are based on the Dhage fixed point theorem and the semigroup theory. To our knowledge, very few results are available for controllability for fractional integro-differential inclusions. So the present results complement this literature.

The paper is organized as follows. In Section 2 some preliminary results are introduced.

The main result is presented in Section 3, and an example illustrating the abstract theory is presented in Section 4.

2 Preliminaries

Let(X,k · k)be a real Banach space.

C=C(J,X)be the space of allX-valued continuous functions on J. L(X)be the Banach space of all linear and bounded operators onX.

L¹(J,X)the space ofX-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}_.

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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_∈_BG(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 x₀ ∈ X the set G(x₀) is a nonempty, closed subset of X, and if for each open setU of X containing G(x₀), there exists an open neighborhoodVof x₀ such thatG(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, thenG 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 [22], Górniewicz [28] and Hu and Papageorgiou [30] .

Definition 2.1. The multivalued mapF: J× B ×X−→ P(X)is said to be an Carathéodory 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 L¹(_R₊_,X). Then the Riemann–Liouville integral is given by:

I_t^αf(t) = ¹ Γ(α)

Z _t

0

f(s) (t−s)¹⁻^α ^ds, whereΓ(·)is the Euler gamma function.

For more details on the Riemann–Liouville fractional derivative, we refer the reader to [21].

Definition 2.3([40]). The Caputo derivative of order αfor a function f: [0,+_∞)→_R can be written as

D^α_t f(t) = ¹ Γ(n−α)

Z _t

0

f⁽ⁿ⁾⁽^s⁾

(t−s)^α⁺¹⁻ⁿ^ds= Iⁿ⁻^αf⁽ⁿ⁾(t), t >0, n−1≤α<n.

If 0<α≤1, then

D_t^αf(t) = ¹ Γ(1−α)

Z _t

0

f⁰(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 spaceBwhich is similar to those introduced by Hale and Kato [29]. Specifically,B will be a linear space of functions mapping(−_{∞, 0}]intoXendowed with a seminormk · k_B, and satisfies the following axioms:

(A1) Ifx: (−_∞,T ]−→Xis continuous on J andx₀ ∈ B, thenx_t ∈ Band x_t is continuous int ∈ J and

kx(t)k ≤Ckx_tk_B_, _(2.1)

whereC≥0 is a constant.

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(A2) There exist a continuous functionC₁(t)>0 and a locally bounded functionC₂(t)≥0 int≥0 such that

kx_tk_B ≤ C₁(t) sup

s∈[0,t]

kx(s)k+C₂(t)kx₀k_B, (2.2) fort ∈[0,T]andxas in(A1).

(A3) The spaceB is complete.

Remark 2.4. Condition (2.1) in(A1)is equivalent to kφ(0)k ≤Ckφk_B, for all φ∈ B. LetS_F,xbe a set defined by

S_F,x= {v∈ L¹(J,X):v(t)∈ F(t,x_t,x(t))a.e.t∈ J}.

Lemma 2.5([33]). Let X be a Banach space. Let F: J× B ×X−→Pcp,c(X)be an L¹-Carathéodory multivalued map and letΨbe a linear continuous mapping from L¹(J,X)to C(J,X), then the operator

Ψ◦S_F: C(J,X)−→Pcp,c(C(J,X)),

x7−→(_Ψ◦SF)(x):= _Ψ(SF,x) is a closed graph operator in C(J,X)×C(J,X).

Proposition 2.6([13, Proposition III.4]). IfΓ1andΓ2are compact valued measurable multifunctions, then the multifunction t→_Γ₁(t)∩_Γ₂(t)is measurable. If(_Γ_n)is a sequence of compact valued measurable multifunctions, then t → ∩_Γ_n(t)is measurable, and if ∪_Γ_n(t)is compact, then t→ ∪_Γ_n(t) is measurable.

Definition 2.7. A multivalued operatorN: X→P_cl(X)is called a) γ-Lipschitz if and only if there existsγ>0 such that

H_d(N(x),N(y))≤γd(x,y), for each x,y∈ X, b) a contraction if and only if it isγ-Lipschitz withγ<1.

Theorem 2.8 (Dhage theorem [23]). Let E be a Banach space, A: E → P_cl,cv,bd(E) and B: E → Pcp,cv(E),two multivalued operators satisfying:

1. A is a contraction, and 2. B is completely continuous.

Then either

(i) the operator inclusion u∈ Au+Bu has a solution , or

(ii) the setE = {u∈E,u∈ λA(u) +λB(u), 0≤λ≤1}is unbounded.

LetΩbe a set defined by

Ω= ⁿx: (−_∞,T]→ Xsuch thatx|₍₋_∞,0_] ∈ B, x|_J ∈C(J,X)^o.

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3 Main results

In this section, we state and prove the controllability results for the system (1.1). Now we define the mild solution for our problem.

Definition 3.1. A function x ∈ _Ω is said to be a mild solution of (1.1) if there exists v(·)∈ L¹(J,X), such thatv(t)∈ F(t,x_t,x(t))a.e.t∈ [0,T], andx satisfies

x(t) =











φ(t), t ∈ (−_∞, 0];

−Q(t)φ(₀) +Rt

0R(t−s)Bu(s)ds +Rt

0

Rs

0 R(t−s)a(s,τ)v(τ)dτds, t ∈ J,

(3.1)

where

Q(t) =

Z _∞

0 ξ_q(σ)S(t^qσ)dσ, R(t) =q Z _∞

0 σt^q⁻¹ξ_q(σ)S(t^qσ)dσ and for σ∈(0,∞),

ξ_q(σ) = ¹ qσ⁻¹⁻

1qv_q(σ⁻

1q)≥0,

v_q(σ) = ¹ π

∑

∞ n=1

(−1)ⁿ⁻¹σ⁻^qn⁻¹Γ(nq+1)

n! sin(nπq). Here,ξ_qis a probability density function defined on(0,∞)[36], 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 σξq(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 thatkS(t)k ≤ Mfor all t∈[0,T]. Remark 3.3. Note that

kR(t)k ≤C_q,Mt^q⁻¹, t >0, (3.2) whereC_q,M = _Γ₍^qM

1+q).

Definition 3.4. The problem (1.1) is said to be controllable on the interval Jif for every initial functionφ∈ Bandx₁ ∈ Xthere exists a controlu∈ L²(J,X)such that the mild solution x(·) of (1.1) satisfies x(T) =x₁.

We impose the following assumptions:

(H1) The multifunction F: J× B ×X−→P_cp,cv(X)is Carathéodory.

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(H2) There exists a functionµ∈L¹(J,R⁺)and a continuous nondecreasing functionψ: R⁺ → (0,+_∞)such that

kF(t,x,y)k=sup{kvk:v ∈F(t,x,y)}

≤µ(t)ψ(kxk_B+kyk_X), (t,x,y)∈ J× B ×X, with

ω₂ Z _T

0

µ(_s)_ds<

Z ₊_∞

v(0)

du

ψ(u)^, ^(3.3)

where

ω₂= β₁

M₁M₂a²C_q,M² T^2q

q² +aC_q,MT^q q

, v(0) =ω₁= β₂+β₁M₁M₂aC_q,MT^q

q

hkx₁k+Mkφk_Bⁱ, and

β₁=C₁^∗+1, β₂ =C₂^∗kφk_B. (H3) There exists a functionk ∈ L¹(J,R+)such that

H_d(F(t,x₁,y₁)_,F(t,x₂,y₂))≤ k(t) [kx₁−x₂k_B+ky₁−y₂k_X].

(H4) For each t ∈ J, a(t,s) is measurable on [_0,t]_and a(t) = _{ess sup}{|a(t,s)|_{, 0} ≤ s ≤ t}_is bounded on J. The map t→atis continuous from J toL^∞(J,R), here,at(s) =a(t,s). (H5) The linear operatorW: L²(J,X)→ Xdefined by

Wu =

Z _T

0 R(T−s)Bu(s)ds.

has an inverse operatorW⁻¹, which takes values inL²(J,X)_{/ ker}W and there exist two positive constantsM₁ andM₂such that

kBk_L₍_X₎≤ M₁, kW⁻¹k_L₍_X₎ ≤ M2. (3.4) Theorem 3.5. Assume that the hypotheses(_H1)_–(_H5)hold. Then the problem(1.1)is controllable on the interval(−_∞,T]provided that

M₁ M₂ a² C²_q,MT^2q

q² (C₁^∗+₁)kkk_L1 <_1. _(3.5) Proof. We transform the problem (1.1) into a fixed-point problem. Consider the multivalued operatorN: Ω−→ P(_Ω)defined by N(h) ={h∈ _Ω}with

h(t) =











φ(t), t ∈ (−_{∞, 0}];

−Q(t)φ(0) +Rt

0R(t−s)Bu(s)ds +Rt

0

Rs

0 R(t−s)a(s,τ)v(τ)dτds, t ∈ J.

Using hypothesis(H5)for an arbitrary functionx(·)define the control u(t) =W⁻¹h

x₁+Q(t)φ(0)−

Z _T

0

Z _s

0 R(T−s)a(s,τ)v(τ)dτdsi

(t). (3.6)

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Obviously, fixed points of the operator N are mild solutions of the problem (1.1). Forφ ∈ B, we will define the functiony(·): (−_∞,T]−→X by

y(t) = (

φ(t), t∈ (−_{∞, 0}];

−Q(t)φ(0), t∈ J.

Theny₀= φ. For each functionz∈C(J,X)withz(0) =0, we denote byzthe function defined by

z(t) =

(0, t∈ (−_{∞, 0}]; z(t), t∈ J.

If x(·) verifies (3.1), 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 functionz(t)satisfies

z(t) =

Z _t

0 R(t−s)Bu_y+z(s)ds+

Z _t

0

Z _s

0 R(t−s)a(s,τ)v(τ)dτds, where

v∈SF,y+z =ⁿv∈ L¹(J,X): v(t)∈F(t,yt+zt,y(t) +z(t))for a.e.t ∈ Jo . Let

Z₀={z∈ _Ω:z₀=0}. For anyz ∈Z₀, we have

kzk_Z₀ =sup

t∈J

kz(t)k+kz₀k_B =sup

t∈J

kz(t)k.

Thus (Z₀,k · k_Z₀) is a Banach space. We define the operator P: Z₀ −→ P(Z₀) by P(z) = {h∈Z₀}with

h(t) =

Z _t

0

R(t−s)Bu_y+z(s)ds+

Z _t

0

Z _s

0

R(t−s)a(s,τ)v(τ)dτds, v(s)∈ S_F,y+z, t∈ J. Obviously the operator N having a fixed point is equivalent to P having one, so it turns to prove that Phas a fixed point. Letr >0 and consider the set

B_r={z ∈Z₀:kzk_Z₀ ≤r}_. We need the following lemma.

Lemma 3.6. Set

C^∗_i =_sup

t∈J

C_i(t) (i=_{1, 2})_. _(3.7)

Then for any z ∈B_rwe have

kyt+ztk_B ≤C^∗₂kφk_B+C₁^∗r, and

ku(s)k ≤M₂

kx₁k+Mkφk_B+aC_q,M Z _T

0

Z _τ

0

(t−τ)^q⁻¹kv(ι)kdιdτ

. (3.8)

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Proof. Using (2.2), (3.4), (3.6) and (3.7), we obtain ky_t+z_tk_B ≤ ky_tk_B+kz_tk_B

≤C₁(t) sup

0≤τ≤t

ky(τ)k+C₂(t)ky₀k_B+C₁(t) sup

0≤τ≤t

kz(τ)k+C₂(t)kz₀k_B

≤C2(t)kφk_B+C₁(t) sup

0≤τ≤t

kz(τ)k

≤C^∗₂kφk_B+C₁^∗r.

Also, we get

ku(s)k ≤ kW⁻¹k

kx₁k+kQ(t)φ(0)k+

Z _T

0

Z _τ

0

kR(t−τ)kka(τ,ι)kkv(ι)kdιds

≤ M₂h

kx₁k+Mkφk_B+aC_q,M Z _T

0

Z _τ

0

(t−τ)^q⁻¹kv(ι)kdιdτi . The lemma is proved.

Now, we define the following multivalued operators P₁,P₂: Z₀ −→ P(Z₀)as P₁(z) =

h∈Z₀ :h(t) =

Z _t

0

R(t−s)Bu_y+z(s)ds, t∈ J

and

P₂(z) =

h ∈Z₀:h(t) =

Z _t

0

Z _s

0 R(t−s)a(s,τ)v(τ)dτds, v(s)∈S_F,y+z, t∈ J

.

It is clear that P = P₁+P₂. The problem of finding solutions of (1.1) is reduced to finding solutions of the operator inclusionz∈ P₁(z) +P2(z). We shall show that the operatorsP₁and P₂satisfy all conditions of the Theorem2.8. The proof will be given in several steps.

Step 1: P₁ is a contraction.

Letz,z^∗ ∈Z0andh∈ P₁(z). Then, there existsv(t)∈F(t,yt+zt, y(t) +z(t))such that h(t) =

Z _t

0 R(t−s)Buy+z(s)ds, t∈ J. From(H3), it follows that

H_d(F(t,yt+zt,y(t) +z(t)),F(t,yt+z^∗_t,y(t) +z^∗(t)))

≤ k(t) [kz_t−z^∗_tk_B+kz(t)−z^∗(t)k_X]. Hence there isω∈ F(t,y_t+z^∗_t,y(t) +z^∗(t))such that

|v(t)−ω| ≤k(t) [kz_t−z^∗_tk_B+kz(t)−z^∗(t)k_X]. ConsiderU: J → P(E)given by

u(t) ={ω∈ E:|v(t)−ω| ≤k(t) [kz_t−z^∗_tk_B+kz(t)−z^∗(t)k_X].

Since the multivalued operator V(t) = U(t)∩F(t,y_t+z^∗_t,y(t) +z^∗(t)) is measurable (see Proposition 2.6), there exists a function v∗(t), which is a measurable selection for V. So, v∗(t)∈ F(t,yt+z^∗_t,y(t) +z^∗(t)), and using(A2), for eacht ∈ J, we obtain

kv(t)−v^∗(t)k ≤k(t) [kzt−z^∗_tk_B+kz(t)−z^∗(t)k_X]

≤k(t)[C₁^∗kz(t)−z^∗(t)k+kz(t)−z^∗(t)k]

≤k(t)(C₁^∗+1)kz(t)−z^∗(t)k

≤k(t)(C₁^∗+1)kz(t)−z^∗(t)k.

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Let us define for eacht∈ J

h^∗(t) =

Z _t

0 R(t−s)Bu_y+z^∗(s)ds.

Then we have

kh(t)−h^∗(t)k ≤

Z _t

0

kR(t−s)kkBu_y+z(s)−Bu_y₊_z^∗(s)kds

≤ M₁ a C_q,M Z _t

0

(t−s)^q⁻¹kuy+z(s)−u_y+z^∗(s)kds

≤ M₁ a C_q,M Z _t

0

(t−s)^q⁻¹W⁻¹h

x₁+Q(t)φ(0)

−

Z _T

0

Z _τ

0

R(t−τ)a(_τ,ι)v(ι)kdιdτi

−W⁻¹h

x₁+Q(t)φ(0)−

Z _T

0

Z _τ

0 R(t−τ)a(τ,ι)v^∗(ι)kdιdτi ds

≤ M₁ M₂ a² C²_q,M Z _t

0

(t−s)^q⁻¹

Z _T

0

Z _τ

0

(t−τ)^q⁻¹kv(ι)−v^∗(ι)kdιdτds

≤ M₁ M₂ a² C²_q,M Z _t

0

(t−s)^q⁻¹

×

Z _T

0

Z _τ

0

(t−τ)^q⁻¹k(ι)(C₁^∗+1)kz(ι)−z^∗(ι)kdιdτds

≤ M₁ M₂ a² C²_q,MT^2q

q² (C₁^∗+1)kkk_L1kz−z^∗k.

By an analogous relation, obtained by interchanging the roles ofzandz^∗, it follows that H_d(P₁(z),P₁(z^∗))≤ M₁ M2 a²C²_q,MT^2q

q² (C^∗₁+1)kkk_L1kz−z^∗k. By (3.5), the mapping P₁ is a contraction.

Step 2: P₂has compact, convex values, and it is completely continuous. This will be given in several claims.

Claim 1: P₂is convex for eachz ∈Z₀.

Indeed, if h₁ andh₂ belong to P₂, then there exist v₁,v₂ ∈ S_F,y+z such that, for t ∈ J, we have

h_i(t) =

Z _t

0

Z _s

0 R(t−s)a(s,τ)v_i(τ)dτds, i=1, 2.

Letd ∈[0, 1]. Then for eacht∈ J, we have dh₁(t) + (1−d)h2(t) =

Z _t

0

Z _s

0 R(t−s)a(s,τ) [dv₁(τ) + (1−d)v2(τ)] dτds.

SinceS_F,y+z is convex (because Fhas convex values), we have dh₁+ (1−d)h₂ ∈ P₂. Claim 2: P₂maps bounded sets into bounded sets in Z₀.

Indeed, it is enough to show that for anyr >0, there exists a positive constant`such that for eachz∈ B_r ={z∈ Z₀: kzk_Z₀ ≤r}, we havekP₂(z)k_Z₀ ≤ `. Then for each h∈ P₂(z), there exists v∈S_F,y+z such that

h(t) =

Z _t

0

Z _s

0 R(t−s)a(s,τ)v(τ)dτds.

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Using (H2) and Lemma3.6we have for eacht ∈ J, kh(t)k ≤

Z _t

0

Z _s

0

kR(t−s)a(s,τ)v(τ)kdτds

≤a C_q,M Z _t

0

Z _s

0

(t−s)^q⁻¹[µ(τ)ψ(ky_τ+z_τk+ky(τ) +z(τ)k)] dτds

≤a C_q,M Z _t

0

Z _s

0

(t−s)^q⁻¹[µ(τ)ψ(C^∗₂kφk_B+C₁^∗r+r)]dτds

≤a Cq,M

Z _t

0

Z _s

0

(t−s)^q⁻¹[µ(τ)ψ(C^∗₂kφk_B+ (C₁^∗+1)r)] dτds

≤ ^T

qa C_q,M

q ψ(C₂^∗kφk_B+ (C₁^∗+1)r)

Z _T

0 µ(τ)dτ

≤`. HenceP2(_B_r)is bounded.

Claim 3: P₂ maps bounded sets into equicontinuous sets ofZ₀. Leth∈ P₂(z)forz∈ Z₀and let τ₁,τ₂ ∈[0,T], withτ₁ <τ₂, we have

kh(τ₂)−h(τ₁)k ≤

Z _τ₂

0

Z _s

0

[R(τ₁−s)−R(τ₂−s)]a(s,τ)v(τ)dτds +

Z _τ₁

τ2

Z _s

0

kR(τ₁−s)kka(s,τ)kkv(τ)kdτds

≤ I₁+I2, where

I₁=

Z _τ₂

0

Z _s

0

[R(τ₁−s)−R(τ₂−s)]a(s,τ)v(τ)dτds I2=

Z _τ₁

τ₂

Z _s

0

kR(τ₁−s)kka(_s,τ)kkv(τ)kdτds.

For I₁, using (3.2) and (H2), we have I₁ ≤a

Z _τ₂

0

Z _s

0

kR(τ₁−s)−R(τ₂−s)kkv(τ)kdτds

≤aψ(C^∗₂kφk_B+ (C^∗₁+1)r)kµk_L1

Z _τ₂

0

kR(τ₁−s)−R(τ₂−s)kds

≤aψ(C^∗₂kφk_B+ (C^∗₁+1)r)kµk_L1

×[q Z _τ₂

0

Z _∞

0 σk[(τ₁−s)^q⁻¹−(τ₂−s)^q⁻¹]ξ_q(σ)S((τ₁−s)^qσ)kdσds +q

Z _τ₂

0

Z _∞

0 σ(τ₂−s)^q⁻¹ξ_q(σ)kS((τ₁−s)^qσ)−S((τ₂−s)^qσ)kdσds]

≤aψ(C^∗₂kφk_B+ (C^∗₁+1)r)kµk_L1 ×[C_q,M Z _τ₂

0

(τ₁−s)^q⁻¹−(τ₂−s)^q⁻¹ ds +q

Z _τ₂

0

Z _∞

0

σ(τ₂−s)^q⁻¹ξ_q(σ)kS((τ₁−s)^qσ)−S((τ₂−s)^qσ)kdσds].

Clearly, the first term on the right-hand side of the above inequality tends to zero asτ₂→ τ₁. From the continuity ofS(t) in the uniform operator topology for t > 0, the second term on the right-hand side of the above inequality tends to zero asτ₂ →τ₁.

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In view of (3.2), we have

I₂ ≤aψ(C₂^∗kφk_B+ (C₁^∗+1)r)kµk_L1

Z _τ₁

τ2

kR(τ₁−s)kds

≤aC_q,Mψ(C₂^∗kφk_B+ (C₁^∗+1)r)kµk_L1

Z _τ₁

τ₂

(τ₁−s)^q⁻¹ds.

Asτ₂→τ₁, I₂ tends to zero.

SoP₂(B_r)is equicontinuous.

Claim 4: (P₂B_r)(t)is relatively compact for eacht ∈ J, where (P₂B_r)(t) ={h(t):h∈ P₂(B_r)}.

Let 0< t≤ Tbe fixed and let εbe a real number satisfying 0<ε< t. For arbitraryδ >0, we define

h_ε,δ(t) =q Z _t₋_ε

0

(t−s)^q⁻¹

Z _∞

δ

σξq(σ)S((t−s)^qσ)

Z _s

0 a(s,τ)v(τ)dτdσds

=qS(ε^qδ)

Z _t₋_ε

0

(t−s)^q⁻¹

Z _∞

δ

σξ_q(σ)S((t−s)^qσ−ε^qδ)

Z _s

0 a(s,τ)v(τ)dτdσds, wherev ∈S_F,y+z. SinceS(t)is a compact operator, the set

H_ε,δ ={h_ε,δ(t):h ∈P₂(B_r)}

is relatively compact. Moreover, 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)^qσ)k

Z _s

0

ka(s,τ)kkv(τ)kdτdσds +q

Z _t

t−ε

(t−s)^q⁻¹

Z _∞

0

σξ_q(σ)kS((t−s)^qσ)k

Z _s

0

ka(s,τ)kkv(τ)kdτdσds

≤T^qMaψ(C₂^∗kφk_B+ (C^∗₁+1)r)kµk_L1

Z _δ

0 σξq(σ)dσ + ^ε

qMa

Γ(1+q)^ψ(C₂^∗kφk_B+ (C₁^∗+1)r)kµk_L1. Therefore, (P₂B_r)(t)is relatively compact.

As a consequence of Claim 2 to 4 together with the Arzelà–Ascoli theorem we can conclude that P₂is completely continuous.

Claim 5: P₂has a closed graph.

Letz_n→z∗,h_n∈ P₂(z_n), andh_n →h∗. We shall show thath∗ ∈P₂(z∗).h_n∈ P₂(z_n)means that there existsv_n∈S_F,y_n+zn such that

h_n(t) =

Z _t

0

Z _s

0 R(t−s)a(s,τ)v_n(τ)dτds, t∈ J. We have to prove that there existsv∗ ∈ S_F,y_∗+z∗ such that

h∗(t) =

Z _t

0

Z _s

0 R(t−s)a(s,τ)v∗(τ)dτds, t ∈ J.

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Consider the linear and continuous operatorΥ: L¹(J,X)−→C(J,X)defined by (_Υv)(t) =

Z _t

0

Z _s

0 R(t−s)a(s,τ)v(s)dτds.

From Lemma2.5it follows thatΥ◦S_Fis a closed graph operator and from the definition ofΥ one has

hn(t)∈ _Υ(S_F,y_n+zn). Aszn→z∗ andhn→h∗, there is av∗ ∈S_F,y_∗+z∗ such that

h∗(t) =

Z _t

0

Z _s

0 R(t−s)a(s,τ)v(s)dτds.

Hence the multivalued operatorP₂ is upper semi-continuous.

Claim 6: A priori bounds.

Now it remains to show that the set

E = {z∈Z0: z∈λP₁(z) +λP2(z), for some 0<λ<1} is bounded.

Letz∈ E be any element, then there existsv∈SF,y+z such that z(t) =λ

Z _t

0 R(t−s)Bu(s)ds+λ Z _t

0

Z _s

0 R(t−s)a(s,τ)v(s)dτds for some 0< λ<1.

Thus, by (3.8), (H2) and Lemma3.6, for each t∈ J we have kz(t)k ≤

Z _t

0

kR(t−s)kkBu(s)kds+

Z _t

0

Z _s

0

kR(t−s)kka(s,τ)kkv(s)kdτds

≤ M₁M2aCq,M

Z _t

0

(t−s)^q⁻¹^hkx₁k+Mkφk_B +aC_q,M

Z _T

0

Z _τ

0

(t−τ)^q⁻¹kv(ι)kdιdτi

ds+aC_q,M Z _t

0

(t−s)^q⁻¹

Z _s

0

kv(τ)kdτds

≤ M₁M2aCq,MT^q q

hkx₁k+Mkφk_Bⁱ +M₁M₂a²C_q,M²

Z _t

0

(t−s)^q⁻¹

Z _T

0

Z _τ

0

(t−τ)^q⁻¹kv(ι)kdιdτds +aC_q,M

Z _t

0

(t−s)^q⁻¹

Z _s

0

kv(τ)kdτds

≤ M₁M₂aC_q,MT^q q

hkx₁k+Mkφk_Bⁱ+M₁M₂a²C_q,M²

×

Z _t

0

(t−s)^q⁻¹

Z _T

0

Z _τ

0

(t−τ)^q⁻¹[µ(ι)ψ(kyι+zιk+ky(ι) +z(ι)k)]dιdτds +aC_q,M

Z _t

0

(t−s)^q⁻¹

Z _s

0

[µ(τ)ψ(ky_τ+z_τk+ky(τ) +z(τ)k)]dτds

hkx₁k+Mkφk_Bⁱ+M₁M₂a²C_q,M²

×

Z _t

0

(t−s)^q⁻¹

Z _T

0

Z _τ

0

(t−τ)^q⁻¹[µ(ι)ψ(C^∗₂kφk_B+ (C₁^∗+1)kz(ι)k)]dιdτds +aCq,M

Z _t

0

(t−s)^q⁻¹

Z _s

0

[µ(τ)ψ(C₂^∗kφk_B+ (C₁^∗+1)kz(τ)k)]dτds

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≤ M₁M2aCq,M

T^q q

hkx₁k+Mkφk_Bⁱ +M₁M₂a²C_q,M² T^2q

q² Z _t

0

[µ(s)ψ(C₂^∗kφk_B+ (C^∗₁+₁)kz(s)k)]ds +aC_q,MT^q

q Z _t

0

[µ(s)ψ(C^∗₂kφk_B+ (C^∗₁+1)kz(s)k)]ds

hkx₁k+Mkφk_Bⁱ +

M₁M₂a²C²_q,MT^2q

q² +aC_q,MT^q q

_Z _t

0

[µ(s)ψ(β₂+β₁kz(s)k)]ds Then

β₂+β₁kz(_t)k ≤β₂+β₁M1M2aCq,M

T^q q

h

kx1k+_Mkφk_Bⁱ +β₁

M₁M₂a²C²_q,MT^2q

q² +aC_q,MT^q q

_Z _t

0

[µ(s)ψ(β₂+β₁kz(s)k)]ds

≤ω₁+ω₂ Z _t

0

[µ(s)ψ(β₂+β₁kz(s)k)]ds.

Let

m(t):=sup{β₂+β₁kz(s)k : 0≤ s≤t}, t∈ J.

By the previous inequality, we have

m(t)≤ω₁+ω₂ Z _t

0

[µ(s)ψ(m(s))]ds.

Let us take the right-hand side of the above inequality asv(t). Then we have m(t)≤v(t) for allt∈ J,

with

v(₀) =ω₁, and

v⁰(t) =ω₂µ(t)ψ(m(t)), a.e.t∈ J.

Using the nondecreasing character ofψwe get

v⁰(t)≤ω₂µ(t)ψ(v(t)), a.e.t∈ J.

Integrating from 0 totwe get Z _t

0

v⁰(s)

ψ(v(s))^ds≤ ω₂ Z _t

0 µ(s)ds.

By a change of variable we get

Z _v₍_t₎

v(0)

du

ψ(u) ≤ω2

Z _t

0 µ(s)ds.

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Using the condition (3.3), this implies that for eacht∈ J, we have Z _v₍_t₎

v(0)

du

ψ(u) ≤ω₂ Z _t

0

µ(_s)_ds≤ω₂ Z _T

0

µ(_s)_ds<

Z ₊_∞

v(0)

du ψ(u)^.

Thus, for everyt ∈ J, there exists a constantΛsuch thatv(t)≤_Λand hencem(t)≤ _{Λ. Since} kzk_Z₀ ≤m(t), we havekzk_Z₀ ≤_Λ.

This shows that the setE is bounded. As a consequence of Theorem 2.8 we deduce that P₁+P₂ has a fixed point z defined on the interval (−_∞,T]which is the solution of problem (1.1). This completes the proof.

4 An example

Consider the following integro-differential equation with fractional derivative of the form

∂^q

∂t^qv(t,ζ)∈ ∂²

∂ζ²v(t,ζ) +µ(t,ζ) +

Z _t

0

(t−s)²

Z ₀

−_∞G(t,v(t+θ,ζ))η(t,θ,ζ)dθds

, t∈ [0, 1], ζ ∈[0,π]; (4.1)

v(t, 0) =v(t,π) =0, t∈ [0, 1];

v(_θ,ζ) = ϕ(_θ,ζ)_, θ ∈(−_{∞, 0}]_, ζ ∈[_0,π]_, where 0< q< 1,µ: [0, 1]×[0,π]→[0,π], and G: [0, 1]×_R → P(_R)is an u.s.c. multivalued map with compact convex values.

SetX =L²([0,π])and define Aby

D(A) ={u∈X: u⁰⁰ ∈X,u(0) =u(π) =0}, Au =u⁰⁰.

It is well known that Ais the infinitesimal generator of an analytic semigroup (S(t))_t_≥₀ on X [43]. Furthermore, A has a discrete spectrum with eigenvalues of the form −n²,n ∈ _{N, and} the corresponding normalized eigenfunctions are given by

u_n(x) = r2

πsin(nx). In addition,{u_n :n∈_N}is an orthogonal basis forX,

S(t)u=

∑

∞ n=1

e⁻ⁿ²^t(u,u_n)u_n, for all u∈Xand every t≥0.

From these expressions it follows that(S(t))_t_≥₀is uniformly bounded compact semigroup.

For the phase space, we chooseB= B_γ defined by B_γ :=

φ∈ C((−_{∞, 0}],X): lim

θ→−_∞e^γθφ(θ)exists in X

endowed with the norm

kφk=_sup{e^γθ|φ(θ)|_:θ≤₀}_.

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Notice that the phase spaceB_γ satisfies axioms(A1)–(A3).

Fort ∈[0, 1],ζ ∈ [0,π]and ϕ∈ B_γ, we set x(t)(ζ) =v(t,ζ),

a(t,s) = (t−s)², F(t,ϕ,x(t))(ζ) =

Z ₀

−_∞G(t,ϕ(θ)(ζ))η(t,θ,ζ)dθ, Bu(t)(ζ) =µ(t,ζ).

With the above choices, we see that the system (4.1) is the abstract formulation of (1.1). Assume that the operatorW: L²([0, 1],X)→ Xdefined by

Wu(·) =

Z ₁

0 R(1−s)µ(s,·)ds, has a bounded invertible operatorW⁻¹ in L²([0, 1],X)/ kerW.

Thus all the conditions of Theorem3.5 are satisfied. Hence, system (4.1) is controllable on (−_∞,T].

Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz Univer- sity, under grant No. (38-130-35-HiCi). The second and third authors, therefore, acknowledge technical and financial support of KAU.

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