OF FRACTIONAL DIFFERENTIAL INCLUSIONS INVOLVING CAPUTO’S FRACTIONAL DERIVATIVE
AURELIAN CERNEA
Using Bressan-Colombo results, concerning the existence of continuous selections of lower semicontinuous multifunctions with decomposable values, we prove a continuous version of Filippov’s theorem for a fractional differential inclusion in- volving Caputo’s fractional derivative. This result allows to obtain a continuous selection of the solution set of the problem considered.
AMS 2010 Subject Classification: 34A60, 26A33, 26A42.
Key words: differential inclusion, fractional derivative, decomposable set.
1. INTRODUCTION
Differential equations with fractional order have recently proved to be strong tools in the modeling of many physical phenomena. As a consequence there was an intensive development of the theory of differential equations of fractional order ([2, 17, 18, 20] etc.). The study of fractional differential inclusions was initiated by El-Sayed and Ibrahim ([13]). Very recently several qualitative results for fractional differential inclusions were obtained in [1, 3, 10, 15, 16, 18] etc.. Applied problems require definitions of fractional derivative allowing the utilization of physically interpretable initial conditions. Caputo’s fractional derivative, originally introduced in [5] and afterwards adopted in the theory of linear visco elasticity, satisfies this demand. For a consistent bibliography on this topic, historical remarks and examples we refer to [1, 16].
In this paper we study fractional differential inclusions of the form (1.1) Dcαx(t)∈F(t, x(t)) a.e. ([0, T]), x(0) =x0, x0(0) =x1, whereα ∈(1,2],Dαc is the Caputo fractional derivative,F : [0, T]×R→ P(R) is a set-valued map and x0, x1∈R,x0, x1 6= 0.
The aim of this paper is to prove the existence of solutions continuously depending on a parameter for problem (1.1). Our result may be interpreted as a continuous variant of the celebrated Filippov’s theorem ([14]) for problem (1.1). In addition, as usual at a Filippov type existence theorem, our result
REV. ROUMAINE MATH. PURES APPL.,55(2010),2, 121–129
provides an estimate between the family of starting “quasi” solutions and the family of solutions of the fractional differential inclusion. This result allows to obtain a continuous selection of the solution set of problem (1.1).
The key tool in the proof of our theorem is a result of Bressan and Colombo [4] concerning the existence of continuous selections of lower semi- continuous multifunctions with decomposable values. Similar results for other classes of second-order differential inclusions are obtained in [8, 9]. In [8] it is considered a Sturm-Liouville type differential inclusion, while in [9] it is studied a second-order differential inclusion defined by the infinitesimal gene- rator of a strongly continuous cosine family of operators on a separable Banach space. Even if the hypotheses on the set-valued map are the same, the three results can not be obtained one from another thanks to the particular form of the integral representation of the solution.
As an application of our main result we point out the possibility to obtain sufficient conditions for local controllability for fractional differential inclusions. The very nature of this problem does not allows an easy and immediately result, so this application is subject of other studies ([11]).
The paper is organized as follows: in Section 2 we present the notations, definitions and the preliminary results to be used in the sequel and in Section 3 we prove our main result.
2. PRELIMINARIES
LetT > 0,I := [0, T] and denote by L(I) theσ-algebra of all Lebesgue measurable subsets of I. Let X be a real separable Banach space with the norm | · |. Denote by P(X) the family of all nonempty subsets of X and by B(X) the family of all Borel subsets of X. If A ⊂ I then χA(·) :I → {0,1}
denotes the characteristic function of A. For any subsetA⊂X we denote by cl(A) the closure ofA.
The distance between a pointx ∈ X and a subset A ⊂X is defined as usual by d(x, A) = inf{|x−a|;a ∈ A}. We recall that Pompeiu-Hausdorff distance between the closed subsets A, B ⊂ X is defined by dH(A, B) = max{d∗(A, B),d∗(B, A)}, d∗(A, B) = sup{d(a, B);a∈A}.
As usual, we denote byC(I, X) the Banach space of all continuous func- tions x(·) : I → X endowed with the norm |x(·)|C = supt∈I|x(t)| and by L1(I, X) the Banach space of all (Bochner) integrable functions x(·) :I →X endowed with the norm |x(·)|1=RT
0 |x(t)|dt.
We recall first several preliminary results we shall use in the sequel.
Lemma 2.1([22]). Let u :I → X be measurable and let G:I → P(X) be a measurable closed-valued multifunction.
Then for every measurable function r :I → (0,∞), there exists a mea- surable selection g : I → X of G(·) (i.e., such that g(t) ∈ G(t) a.e. (I)) such that
|u(t)−g(t)|<d(u(t), G(t)) +r(t) a.e.(I).
Definition 2.2. A subset D⊂ L1(I, X) is said to be decomposableif for any u(·), v(·) ∈D and any subset A ∈ L(I) one has uχA+vχB ∈D, where B =I\A.
We denote by D(I, X) the family of all decomposable closed subsets of L1(I, X).
Next, (S,d) is a separable metric space; we recall that a multifunction G(·) : S → P(X) is said to be lower semicontinuous (l.s.c.) if for any closed subset C⊂X, the subset {s∈S;G(s)⊂C} is closed.
Lemma 2.3 ([4]). Let F∗(·,·) :I×S→ P(X) be a closed-valuedL(I)⊗ B(S)-measurable multifunction such that F∗(t, .) is l.s.c. for any t∈I.
Then the multifunction G(·) :S→ D(I, X) defined by G(s) ={v∈L1(I, X); v(t)∈F∗(t, s) a.e. (I)}
is l.s.c. with nonempty closed values if and only if there exists a continuous mapping p(·) :S→L1(I, X) such that
d(0, F∗(t, s))≤p(s)(t) a.e.(I), ∀s∈S.
Lemma 2.4([4]). Let G(·) :S → D(I, X) be a l.s.c. multifunction with closed decomposable values and let φ(·) :S → L1(I, X), ψ(·) : S → L1(I,R) be continuous such that the multifunction H(·) :S→ D(I, X) defined by
H(s) = cl{v(·)∈G(s);|v(t)−φ(s)(t)|< ψ(s)(t)a.e.(I)}
has nonempty values.
ThenH(·)has a continuous selection, i.e., there exists a continuous map- ping h(·) :S→L1(I, X) such that
h(s)∈H(s), ∀s∈S.
Definition2.5 ([17]). a)The fractional integral of orderα >0 of a Lebes- gue integrable function f(·) : (0,∞)→R is defined by
Iαf(t) = Z t
0
(t−s)α−1
Γ(α) f(s)ds,
provided the right-hand side is pointwise defined on (0,∞) and Γ(·) is the (Euler’s) Gamma function defined by Γ(α) =R∞
0 tα−1e−tdt.
b) The Caputo fractional derivative of order α > 0 of a function f(·) : [0,∞)→Ris defined by
Dαcf(t) = 1 Γ(n−α)
Z t
0
(t−s)−α+n−1f(n)(s)ds,
where n= [α] + 1. It is assumed implicitly thatf(·) is ntimes differentiable whose nth derivative is absolutely continuous.
We recall (e.g., [17]) that ifα >0 andf(·)∈C(I,R) orf(·)∈L∞(I,R) then (DαcIαf)(t)≡f(t).
Definition 2.6 ([16]). A function x(·) ∈ C(I,R) is called a solution of problem (1.1) if there exists a function v(·)∈L1(I,R) withv(t)∈F(t, x(t)), a.e. (I) such thatDαcx(t) =v(t), a.e. (I) andx(0) =x0, x0(0) =x1.
We shall use the following notation for the solution sets of (1.1).
S(x0, x1) ={x(·); x(·) is a solution of (1.1)}.
3. THE MAIN RESULTS
In order to establish our continuous version of Filippov theorem for prob- lem (1.1) we need the following hypotheses.
Hypothesis 3.1. i)F(·,·) :I×R→ P(R) has nonempty closed values and is L(I)⊗ B(R) measurable.
ii)There existsL(·)∈L1(I,(0,∞))such that, for almost allt∈I, F(t,·) is L(t)-Lipschitz in the sense that
dH(F(t, x), F(t, y))≤L(t)|x−y|, ∀x, y∈R.
Hypothesis 3.2. (i) S is a separable metric space, a(·), b(·) : S → R and c(·) :S →(0,∞) are continuous mappings.
(ii)There exists the continuous mappingsg(·), p(·) :S →L1(I,R),y(·) : S →C(I,R) such that
(Dy(s))αc(t) =g(s)(t) a.e. t∈I, ∀s∈S, d(g(s)(t), F(t, y(s)(t))≤p(s)(t) a.e. t∈I, ∀s∈S.
As we already mentioned in Definition 2.5, in Hypothesis 3.2n= [α] + 1 and it is assumed implicitly that y(s)(·) is n times differentiable whose nth derivative is absolutely continuous for any s∈S.
We use next the notation ξ(s) = 1
1− |IαL|(|a(s)−y(s)(0)|+T|b(s)−(y(s))0(0)|+c(s) +|Iαp(s)|), s∈S, where |IαL|:= supt∈I|IαL(t)|and |Iαp(s)|:= supt∈I|Iαp(s)(t)|.
Theorem 3.3. Assume that Hypotheses 3.1and3.2 are satisfied.
If |IαL|< 1 then there exist a continuous mapping x(·) : S → C(I,R) such that for any s∈S, x(s)(·) is a solution of problem
Dcαz(t)∈F(t, z(t)), z(0) =a(s), z0(0) =b(s) such that
|x(s)(t)−y(s)(t)| ≤ξ(s), ∀(t, s)∈I×S.
Proof. We make the notationd(s) =|a(s)−y(s)(0)|+T|b(s)−(y(s))0(0)|, pn(s) :=|IαL|n−1(d(s) +|Iαp(s)|),n≥1, andx0(s)(t) =y(s)(t), ∀s∈S.
We consider the multifunctionsG0(·), H0(·) defined, respectively, by G0(s) ={v∈L1(I,R); v(t)∈F(t, y(s)(t)) a.e. (I)},
H0(s) = cl
v∈G0(s); |v(t)−g(s)(t)|< p(s) +Γ(α+ 1) Tα c(s)
. Since d(g(s)(t), F(t, y(s)(t))≤p(s)(t) < p(s)(t) + Γ(α+1)Tα c(s), according with Lemma 2.1, the set H0(s) is not empty.
SetF0∗(t, s) =F(t, y(s)(t)) and note that
d(0, F0∗(t, s))≤ |g(s)(t)|+p(s)(t) =p∗(s)(t) and p∗(·) :S→L1(I,R) is continuous.
Applying now Lemmas 2.3 and 2.4 we obtain the existence of a continuous selection f0 of H0, i.e., such that
f0(s)(t)∈F(t, y(s)(t)) a.e.(I), ∀s∈S,
|f0(s)(t)−g(s)(t)| ≤p(s)(t) + Γ(α+ 1)
Tα c(s), ∀s∈S, t∈I.
We define x1(s)(t) =a(s) +tb(s) +Rt 0
(t−u)α−1
Γ(α) f0(s)(u)du and using the fact that Γ(α+ 1) =αΓ(α) one has
|x1(s)(t)−x0(s)(t)| ≤ |a(s)−y(s)(0)|+T|b(s)−(y(s))0(0)|+
+ Z t
0
(t−u)α−1
Γ(α) |f0(s)(u)−g(s)(u)|du≤ |a(s)−y(s)(0)|+T|b(s)−(y(s))0(0)|+
+ Z t
0
(t−u)α−1
Γ(α) (p(s)(u) + Γ(α+ 1)
Tα c(s))du≤ |a(s)−y(s)(0)|+T|b(s)−
−(y(s))0(0)|+Iα(p(s)(·))(t) +Γ(α+ 1) Tα c(s)
Z t
0
(t−u)α−1 Γ(α) du≤
≤d(s) +Iα(p(s)(·))(t)≤d(s) +|Iαp(s)|=p1(s).
We shall construct, using the same idea as in [12], two sequences of approximations fn(·) :S → L1(I,R), xn(·) :S → C(I,R) with the following properties:
a)fn(·) :S →L1(I,R),xn(·) :S →C(I,R) are continuous.
b) fn(s)(t)∈F(t, xn(s)(t)), a.e. (I), s∈S.
c)|fn(s)(t)−fn−1(s)(t)| ≤L(t)pn(s), a.e. (I), s∈S.
d) xn+1(s)(t) =a(s) +tb(s) +Rt 0
(t−u)α−1
Γ(α) fn(s)(u)du,∀t∈I,s∈S.
Suppose we have already constructed fi(·), xi(·) satisfying a)–c) and define xn+1(·) as in d). From c) and d) one has
|xn+1(s)(t)−xn(s)(t)| ≤ Z t
0
(t−u)α−1
Γ(α) |fn(s)(u)−fn−1(s)(u)|du≤ (3.1)
≤ Z t
0
(t−u)α−1
Γ(α) L(u)pn(s)du≤ |IαL(t)|pn(s)≤ |IαL|pn(s) =pn+1(s).
On the other hand,
d(fn(s)(t), F(t, xn+1(s)(t))≤L(t)|xn+1(s)(t)−xn(s)(t)|< L(t)pn+1(s).
Consider for anys∈S the multifunctions
Gn+1(s) ={v ∈L1(I,R); v(t)∈F(t, xn+1(s)(t)) a.e.(I)}, Hn+1(s) = cl{v∈Gn+1(s);|v(t)−fn(s)(t)|< L(t)pn+1(s) a.e.(I)}.
To prove that Hn+1(s) is nonempty we note first that the real function t → rn(s)(t) = (|IαL|n−1 − |IαL|n)(d(s) +|Iαp(s)|)L(t) is measurable and strictly positive for any s. One has
d(fn(s)(t), F(t, xn+1(s)(t))≤L(t)|xn+1(s)(t)−xn(s)(t)| −rn(s)(t)≤
≤L(t)pn+1(s)
and therefore according to Lemma 2.1 there exists v(·)∈ L1(I,R) such that v(t)∈F(t, xn(s)(t)) a.e. (I) and
|v(t)−fn(s)(t)|<d(fn(s)(t), F(t, xn(s)(t)) +rn(s)(t) and hence Hn+1(s) is not empty.
SetFn+1∗ (t, s) =F(t, xn+1(s)(t)) and note that we may write d(0, Fn+1∗ (t, s))≤ |fn(s)(t)|+L(t)|xn+1(s)(t)−xn(s)(t)| ≤
≤ |fn(s)(t)|+L(t)pn+1(s) =p∗n+1(s)(t) a.e.(I) and p∗n+1(·) :S →L1(I,R) is continuous.
By Lemmas 2.3 and 2.4 there exists a continuous map fn+1(·) : S → L1(I,R) such that
fn+1(s)(t)∈F(t, xn+1(s)(t)) a.e.(I), ∀s∈S,
|fn+1(s)(t)−fn(s)(t)| ≤L(t)pn+1(s) a.e.(I), ∀s∈S.
From (3.1) c) and d) we obtain
|xn+1(s)(·)−xn(s)(·)|C ≤ |IαL|pn(s) = (3.2)
=pn+1(s) =|IαL|n(d(s) +|Iαp(s)|),
(3.3) |fn+1(s)(·)−fn(s)(·)|1 ≤ |L(·)|1pn(s) =|L(·)|1|IαL|n(d(s) +|Iαp(s)|).
Therefore, fn(s)(·), xn(s)(·) are Cauchy sequences in the Banach space L1(I,R) and C(I,R), respectively. Let f(·) : S → L1(I,R), x(·) : S → C(I,R) be their limits. The functions→d(s) +|Iαp(s)|is continuous, hence locally bounded. Therefore, (3.3) implies that for every s0 ∈ S the sequence fn(s0)(·) satisfies the Cauchy condition uniformly with respect to s0 on some neighborhood of s. Hence,s→f(s)(·) is continuous from S intoL1(I,R).
From (3.2), as before, xn(s)(·) is Cauchy in C(I,R) locally uniformly with respect to s. So, s→x(s)(·) is continuous fromS intoC(I,R). On the other hand, since xn(s)(·) converges uniformly to x(s)(·) and
d(fn(s)(t), F(t, x(s)(t))≤L(t)|xn(s)(t)−x(s)(t)| a.e.(I), ∀s∈S passing to the limit along a subsequence of fn(·) converging pointwise tof(·) we obtain
f(s)(t)∈F(t, x(s)(t)) a.e.(I), ∀s∈S.
One may write
Z t
0
(t−u)α−1
Γ(α) fn(s)(u)du− Z t
0
(t−u)α−1
Γ(α) f(s)(u)du
≤
≤ Z t
0
(t−u)α−1
Γ(α) |fn(s)(u)−f(s)(u)|du≤
≤ Z t
0
(t−u)α−1
Γ(α) L(u)|xn+1(s)(·)−xn(s)(·)|Cdu≤
≤ |IαL| · |xn+1(s)(·)−xn(s)(·)|C. Therefore, one may pass to the limit in d) and we get
x(s)(t) =a(s) +tb(s) + Z t
0
(t−u)α−1
Γ(α) f(s)(u)du.
By adding inequalities (3.1) for all n≥1 we obtain (3.4) |xn+1(s)(t)−y(s)(t)| ≤
n
X
l=1
pl(s)≤ξ(s).
By passing to the limit in (3.4) we obtain the conclusion of the theorem.
Remark3.4. We note that in [10] it is provided a similar result to Theo- rem 3.3, but the main result in [10] is obtained for fractional differential inclu- sions defined by Riemann-Liouville fractional derivative and in the case when
α ∈ (0,1). Since in the approach in [10] there are no initial conditions for problem considered, that result does not allow to deduce continuous selections of solution sets as we can see in what follows for problem (1.1).
Hypothesis 3.5. Hypothesis 3.1 is satisfied, |IαL|<1 and there exists q0(·)∈L1(I,R+) such that d(0, F(t,0))≤q0(t) a.e. (I).
Corollary 3.6. Assume that Hypothesis3.5 are satisfied.
Then there exists a function x(·,·) :I ×R2 →R such that a)x(·,(ξ, η))∈ S(ξ, η), ∀(ξ, η)∈R2.
b) (ξ, η)→x(·,(ξ, η))is continuous from R2 into C(I,R).
Proof. We take S =R2,a(ξ, η) =ξ, b(ξ, η) =η ∀(ξ, η)∈R2,c(·) :X× X → (0,∞) an arbitrary continuous function, g(·) = 0, y(·) = 0, q(ξ, η)(t) = q0(t) ∀(ξ, η) ∈ R2, t ∈ I and we apply Theorem 3.3 in order to obtain the conclusion of the theorem.
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Received 15 September 2009 University of Bucharest
Faculty of Mathematics and Informatics Str. Academiei 14
010014 Bucharest, Romania
