# Existence Results of Nondensely Defined Fractional Evolution Differential Inclusions

Authors: Zufeng Zhang; Bin Liu
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

This paper deals with the existence results of integral solutions for nondensely defined fractional evolution differential inclusions. Our approach is based on integrated semigroup theory and a fixed point theorem for condensing map due to Martelli. An example is also given to illustrate our results.


---

## Body

## 1. Introduction

In the past decades, the theory of fractional differential equations and inclusions has become an important area of investigation because of its wide applicability in many branches of physics, economics, and technical sciences [1–10].Our aim in this paper is to study the existence of the integral solutions for the fractional semilinear differential inclusions, of the form(1.1)Dqx(t)∈Ax(t)+F(t,x(t)),t∈(0,b],x(0)=x0,
where Dq is the Caputo fractional derivative of order 0<q<1, b>0. A:D(A)⊂X→X is a nondensely closed linear operator on X, X is a real Banach space with the norm |·|. F:[0,∞)×X→𝒫(X) is a nonempty, bounded, closed, and convex multivalued map, and 𝒫(X) denotes the family of all nonempty subsets of X.It is well known that one important way to introduce the concepts of mild solutions for fractional evolution equations is based on some probability densities and Laplace transform. This method was initialed by El-Borai [11] and developed by Zhou and Jiao [12]. Since then, many interesting existence results of mild solutions for fractional evolution equations appeared [13–16]. We will point out that the unbounded operators A in their papers were assumed to be densely defined and generate a strongly continuous semigroup.However, as indicated in [17], we sometimes need to deal with nondensely defined operators and there are extensive work on this subject when equations involve the integral-order derivative, see monograph [18–23] and references therein. Very recently, Wang and Zhou [24] considered problem (1.1) in the case when A is densely defined and generates a strongly continuous semigroup. As far as we know, there are few papers dealing with semilinear fractional differential systems with nondense domain. Motivated by this, we discuss the integral solution to problem (1.1) by using probability densities and integral semigroup. We turn the integral solutions of problem (1.1) to a new formula something like the mild solutions. This new formula of integral solutions is firstly introduced even in fractional evolution equations. Thus, our work can be seen as a supplement to work [24] and a contribution to this emerging field of fractional differential equations with nondense domain.This paper will be organized as follows. In Section2, we recall some basic definitions and preliminary facts for integrated semigroup, fractional calculus, and multivalued map which will be used later. Section 3 is devoted to the existence results of integral solutions for problem (1.1). We will present in Section 4 an example which illustrates our main theorem.


## 2. Preliminaries

In this section, we introduce notations, definitions, and preliminary results which are used in the rest of the paper.We denote byC([0,b],X) the Banach space of all continuous functions from [0,b] into X with the norm(2.1)‖y‖=sup⁡{|y(t)|:t∈[0,b]}.B(X) denotes the Banach space of bounded linear operators from X into X, with the norm(2.2)‖N‖=sup⁡{|N(y)|:|y|=1},
where N∈B(X) and y∈X.Assume thatJ⊂R and 1≤p≤∞. For a measurable function m:J→R, define the norm(2.3)‖m‖LpJ={(∫J|m(t)|pdt)1/p,1≤p<∞,inf⁡μ(J¯)=0{sup⁡t∈J-J¯|m(t)|},p=∞,
where μ(J¯) is the Lebesgue measure on J¯. Let Lp(J,R) be the Banach space of all Lebesgue measurable functions m:J→R with ∥·∥LpJ<∞.Lemma 2.1 (HöLder inequality).
Assume thatr,p≥1 and (1/r)+(1/p)=1. If l∈Lr(J,R), m∈Lp(J,R), then for 1≤p≤∞, lm∈L1(J,R) and
(2.4)‖lm‖L1J≤‖l‖LrJ‖m‖LpJ.Lemma 2.2 (Bochner theorem).
A measurable functionH:[0,b]→X is Bochner’s integrable if |H| is Lebesgue integrable.Definition 2.3 (see [25]).
LetX be a Banach space; an integrated semigroup is a family of operators (S(t))t≥0 of bounded linear operators S(t) on X with the following properties:(i)
S(0)=0;(ii)
t→S(t) is strongly continuous;(iii)
S(s)S(t)=∫0s(S(t+r)-S(r))dr for all t,s≥0.Definition 2.4 (see [26]).
An operatorA is called a generator of an integrated semigroup, if there exists ω∈R such that (ω,+∞)⊂ρ(A) and there exists a strongly continuous exponentially bounded family (S(t))t≥0 of linear bounded operators such that S(0)=0 and (λI-A)-1=λ∫0∞e-λtS(t)dt for all λ>ω.Proposition 2.5 (see [25]).
LetA be the generator of an integrated semigroup (S(t))t≥0. Then for all x∈X and t≥0,
(2.5)∫0tS(s)xds∈D(A),S(t)x=A∫0tS(s)xds+tx.Definition 2.6 (see [26]).
We say that linear operatorA satisfies the Hille-Yosida condition if there exist M≥0 and ω∈R such that (ω,+∞)⊂ρ(A) and
(2.6)sup⁡{(λ-ω)n‖R(λ,A)n‖,n∈N,λ>ω}≤M.Here and hereafter, we assume thatA satisfies the Hille-Yosida condition. Let us introduce the part A0 of A in D(A)¯:A0=A on D(A0)={x∈D(A);Ax∈D(A)¯}. Let (S(t))t≥0 be the integrated semigroup generated by A. We note that (S′(t))t≥0 is a C0-semigroup on D(A)¯ generated by A0 and ∥S′(t)∥≤Meωt, t≥0, where M and ω are the constants considered in the Hille-Yosida condition ([19, 27]).LetBλ=λR(λ,A):=λ(λI-A)-1; then for all x∈D(A)¯, Bλx→x as λ→∞. Also from the Hille-Yosida condition it is easy to see that limλ→∞|Bλx|≤M|x|.For more properties on integral semigroup theory the interested readers may refer to [18, 27].Definition 2.7 (see [3]).
The Riemann-Liouville fractional integral of orderα∈R+ of a function f:R+→X is defined by
(2.7)I0αf(t)=1Γ(α)∫0t(t-s)α-1f(s)ds,t>0,
provided the right-hand side is pointwise defined on R+, where Γ is the gamma function.Remark 2.8.
According to [3], I0qI0β=I0q+β holds for all q, β≥0.Definition 2.9 (see [3]).
The Caputo fractional derivative of order0<α<1 of a function f∈C1([0,∞),X) is defined by
(2.8)Dαf(t)=1Γ(1-α)∫0t(t-s)-αf′(s)ds,t>0.
We will remark that integrals which appear in Definitions 2.7 and 2.9 are taken in Bochner’s sense.Lemma 2.10 (see [28]).
Supposeβ>0, a(t) is a nonnegative, function locally integrable on 0≤t<T and g(t) is a nonnegative, nondecreasing continuous function defined on 0≤t<T, g(t)≤M (constant), and suppose u(t) is nonnegative and locally integrable on 0≤t<T with
(2.9)u(t)≤a(t)+g(t)∫0t(t-s)β-1u(s)ds
on this interval. Then
(2.10)u(t)≤a(t)+∫0t[∑n=1∞(g(t)Γ(β))nΓ(nβ)(t-s)nβ-1a(s)]ds,0≤t<T.Corollary 2.11 (see [28]).
Under the hypothesis of Lemma2.10, let a(t) be a nondecreasing function on [0,T). Then
(2.11)u(t)≤a(t)Eβ(g(t)Γ(β)tβ),
where Eβ is the Mittag-Leffler function defined by Eβ(z)=∑k=0∞(zk/Γ(kβ+1)).We also introduce some basic definitions and results of multivalued maps. See [29] for more details.Let(X,d) be a metric space; 𝒫(X) denotes the family for all nonempty subsets of X. We use the following notations:(2.12)Pcl⁡(X)={Y∈P(X):Yclosed},Pb(X)={Y∈P(X):Ybounded},Pc(X)={Y∈P(X):Yconvex},Pcp(X)={Y∈P(X):Ycompact}.A multivalued mapF:X→𝒫(X) is convex (closed) valued if F(x) is convex (closed) for all x∈X and F is bounded on bounded sets if F(B)=Ux∈BF(x) is bounded in X for all B∈Pb(X), that is, supx∈B{sup{|y|:y∈F(x)}}<∞. F is called upper semicontinuous (u.s.c. for short) on X if for each x0∈X the set F(x0) is nonempty, closed subset of X, and for each open set 𝒰 of X containing F(x0), there exists an open neighborhood 𝒱 of x0 such that F(𝒱)⊂𝒰. F is said to be completely continuous if F(B) is relatively compact for every B∈Pb(X).If the multivalued mapF is completely continuous with nonempty compact valued, the F is u.s.c. if and only if F has closed graph, that is, xn→x*, yn→y*, yn∈F(xn) imply y*∈F(x*).Definition 2.12 (see [30]).
An upper semicontinuous mapG:X→X is said to be condensing if for any bounded subset V⊂X with α(V)≠0, one has α(G(V))<α(V), where α denotes the Kuratowski measure of noncompactness.We remark that a completely continuous multivalued map is the easiest example of a condensing map.Theorem 2.13 (see [30]).
LetJ be a compact interval and X a Banach space. Let F:J×C(J,X)→Pb,cl,c(X), (t,u)↦F(t,u) be measurable with respect to t for each u∈X, upper semicontinuous with respect to u for each t∈J. Moreover, for each fixed u∈C(J,X) the set
(2.13)NF,u={f∈L1(J,X):f(t)∈F(t,u)fora.e.t∈J}
is nonempty. Also let 𝒯 be a linear continuous mapping from L1(J,X) to C(J,X); then the operator
(2.14)T∘NF:C(J,X)⟶Pb,cl⁡,c(C(J,X)),u⟶(T∘NF)(u)=T(NF,u)
is a closed graph operator in C(J,X)×C(J,X).Theorem 2.14 . (Martelli, [31]).
LetX be a Banach space and Φ:X→Pb,cl,c(X) a condensing map. If the set
(2.15)U={x∈X:δx∈Φxforsomeδ>1}
is bounded, then Φ has a fixed point.


## 3. Existence of Integral Solutions

In this section we will establish the existence results for problem (1.1). Let us consider the following problem:(3.1)Dqx(t)=Ax(t)+f(t,x(t)),t∈(0,b],x(0)=x0,
where f:[0,∞)×X→X is a given function and A is the same as that in problem (1.1).Definition 3.1.
One says that a continuous functionx:[0,b]→X is an integral solution of problem (3.1) if(i)
(1/Γ(q))∫0t(t-s)q-1x(s)ds∈D(A) for t∈[0,b],(ii)
x(t)=x0+(1/Γ(q))A∫0t(t-s)q-1x(s)ds+(1/Γ(q))∫0t(t-s)q-1f(s,x(s))ds, t∈[0,b].Lemma 3.2.
Ifx is an integral solution of (1.1), then for all t∈[0,b], x(t)∈D(A)¯. In particular, x(0)=x0∈D(A)¯.Proof.
By Remark2.8 and I0qx(t)∈D(A), for each t∈(0,b], we get that I01x(t)=I01-qI0qx(t)∈D(A). From I01x(t)=∫0tx(s)ds∈D(A) we have (1/h)∫tt+hx(s)ds∈D(A) for h>0, t+h∈(0,b]. Hence, we deduce that
(3.2)x(t)=lim⁡h→01h∫tt+hx(s)ds∈D(A)¯.
The proof is completed.Lemma 3.3 (see [32]).
LetΨq(θ)=(1/π)∑n=1∞(-1)n-1θ-qn-1(Γ(nq+1)/n!)sin(nπq), θ∈R+; then Ψq(θ) is a one-sided stable probability density function and its Laplace transform is given by
(3.3)∫0∞e-pθΨq(θ)dθ=e-pq,q∈(0,1),p>0.Lemma 3.4.
The integral solution in Definition3.1 is given by
(3.4)x(t)=∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s,x(s))dθds,
where hq(θ)=(1/q)θ-1-(1/q)Ψq(θ-1/q) is the probability density function defined on R+.Proof.
From the definition, we have(3.5)x(t)=x0+1Γ(q)A∫0t(t-s)q-1x(s)ds+1Γ(q)∫0t(t-s)q-1f(s,x(s))ds,t∈[0,b].
Consider the Laplace transform
(3.6)v(p)=∫0∞e-ptBλx(t)dt,w(p)=∫0∞e-ptBλf(t,x(t))dt,p>0.
Note that for each t>0, Bλx(t),Bλf(t,x(t))∈D(A), then we have v(p),w(p)∈D(A)¯. Applying (3.6) to (3.5) yields
(3.7)v(p)=1pBλx0+1pqAv(p)+1pqw(p)=pq-1(pqI-A)-1Bλx0+(pqI-A)-1w(p)=pq-1∫0∞e-pqsS′(s)Bλx0ds+∫0∞e-pqsS′(s)w(p)ds,
where I is the identity operator defined on X.
From (3.3), we get
(3.8)pq-1∫0∞e-pqsS′(s)Bλx0ds=∫0∞pq-1e-(pt)qS′(tq)Bλx0qtq-1dt=∫0∞-1pddt(e-(pt)q)S′(tq)Bλx0dt=∬0∞[θΨq(θ)e-ptθS′(tq)Bλx0]dθdt=∬0∞[Ψq(θ)e-psS′((sθ)q)Bλx0]dθds=∫0∞e-pt[∫0∞Ψq(θ)S′((tθ)q)Bλx0dθ]dt,(3.9)∫0∞e-pqsS′(s)w(p)ds=∬0∞e-pqse-ptS′(s)Bλf(t,x(t))dtds=∬0∞qsq-1e-(ps)qe-ptS′(sq)Bλf(t,x(t))dtds=∬0∞∫0∞qΨq(θ)e-psθe-ptS′(sq)Bλf(t,x(t))dθdtds=∬0∞∫0∞qΨq(θ)e-p(s+t)sq-1θqS′((sθ)q)Bλf(t,x(t))dθdtds=∫0∞e-psq∫0s∫0∞Ψq(θ)(s-t)q-1θqS′((s-t)qθq)Bλf(t,x(t))dθdtds=∫0∞e-ptq∫0t∫0∞Ψq(θ)(t-s)q-1θqS′((t-s)qθq)Bλf(s,x(s))dθdsdt.
According to (3.7), (3.8), and (3.9), we have
(3.10)v(p)=∫0∞e-pt[∫0∞Ψq(θ)S′((tθ)q)Bλx0dθ]dt+∫0∞e-ptq∫0t∫0∞Ψq(θ)(t-s)q-1θqS′((t-s)qθq)Bλf(s,x(s))dθdsdt.
Inverting the last Laplace transform, we obtain
(3.11)Bλx(t)=∫0∞Ψq(θ)S′((tθ)q)Bλx0dθ+q∫0t∫0∞Ψq(θ)(t-s)q-1θqS′((t-s)qθq)Bλf(s,x(s))dθds=∫0∞hq(θ)S′(tqθ)Bλx0dθ+q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s,x(s))dθds.
In view of limλ→∞Bλx=x for x∈D(A)¯ and Lemma 3.2, we have
(3.12)x(t)=∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s,x(s))dθds.
The proof is completed.Remark 3.5.
According to [32], one can easily check that
(3.13)∫0∞θhq(θ)dθ=∫0∞1θqΨq(θ)dθ=1Γ(1+q).
Based on the Lemma 3.4, we will define the concept of integral solution of (1.1) as follows.Definition 3.6.
One says that a continuous functionx:[0,b]→X is an integral solution of problem (1.1) if(i)
(1/Γ(q))∫0t(t-s)q-1x(s)ds∈D(A) for t∈[0,b],(ii)
x(0)=x0 and there exists f∈L1([0,b],X) such that f(t)∈F(t,x(t)) for a.e. t∈[0,b] and(3.14)x(t)=∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds,t∈[0,b].
We are now in a position to state and prove our main results of the existence of solutions for problem (1.1).
Let us list the following hypotheses:
(H1)
A satisfies the Hille-Yosida condition;(H2)
the operatorS′(t) is compact in D(A)¯ whenever t>0 and satisfies supt∈[0,∞]∥S′(t)∥=M0<∞, where M0 is a constant;(H3)
F:[0,b]×X→Pb,cl,c(X), for each x∈X, F(·,x) is measurable and for each t∈[0,b], F(t,·) is upper semicontinuous; for each fixed x∈X, the set NF,x = {f∈L1([0,b],X):f(t)∈F(t,x),fora.e.t∈[0,b]} is not empty;(H4)
for eachx∈X, there exist m∈L1/q1([0,b],R+) and n∈C([0,b],R+) such that
(3.15)sup⁡{|f(t)|:f(t)∈F(t,x)}≤m(t)+n(t)|x|fora.e.t∈[0,b],
where q1∈[0,q).Theorem 3.7.
Assume that hypotheses (H1)–(H4) hold; then problem (1.1) has an integral solution x∈C([0,b],D(A)¯).Proof.
DenoteC0=C([0,b],D(A)¯), which is a closed subset of C([0,b],X). Obviously, C0 with the same norm in C([0,b],X) is also a Banach space. Transform the problem (1.1) into a fixed point problem. Consider the multivalued operator Φ:C0→𝒫(C0) defined by
(3.16)Φx={∫0th∈C0:h(t)=∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds},
where f∈NF,x={f∈L1([0,b],X):f(t)∈F(t,x(t)),fora.e.t∈[0,b]}. Obviously, the fixed points of the operator Φ are integral solutions of problem (1.1). Now we will show that Φ satisfies all conditions of Theorem 2.14. The proof would be divided into the following steps.Step 1 (Φ(x) is convex for each x∈C0).
Indeed, ifh1 and h2 belong to Φx, then there exist f1, f2∈NF,x such that for each t∈[0,b], we have
(3.17)hi(t)=∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλfi(s)dθds,i=1,2.
Let0≤k≤1; then for each t∈[0,b], we have
(3.18)(kh1+(1-k)h2)(t)=∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλ×(kf1(s)+(1-k)f2(s))dθds.
Since NF,x is convex, we have kh1+(1-k)h2∈Φx.Step 2 (Φ maps bounded sets into bounded sets in C0).
Indeed, it is enough to show that there exists a positive constantl such that for each h∈Φx, x∈ℬr={x∈C0,∥x∥≤r} one has ∥h∥≤l.
Leth∈Φx; then there exists f∈NF,x such that for t∈[0,b], we have
(3.19)h(t)=∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds.
From (H2) and the fact that ∥Bλ∥≤M, for t∈[0,b] we have
(3.20)|h(t)|≤|∫0∞hq(θ)S′(tqθ)x0dθ|+|lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds|≤M0|x0|+MM0∫0t∫0∞θhq(θ)|(t-s)q-1f(s)|dθds≤M0|x0|+qMM0Γ(1+q)∫0t|(t-s)q-1f(s)|ds.
From Lemma 2.1 and (H4), for t∈[0,b] we have
(3.21)∫0t|(t-s)q-1f(s)|ds≤(∫0t(t-s)(q-1)/(1-q1)ds)1-q1‖m‖L1/q1[0,t]+n¯r∫0t(t-s)q-1ds≤M1(1+a)1-q1b(1+a)(1-q1)+n¯rbqq,
where a=(q-1)/(1-q1)∈(-1,0), M1=∥m∥L1/q1[0,b], n¯=sup{n(t),t∈[0,b]}.
Then from (3.20) and (3.21), we get that
(3.22)‖h‖≤M0|x0|+MM0Γ(1+q)(qM1(1+a)1-q1b(1+a)(1-q1)+n¯rbq):=l.Step 3 (Φ maps bounded sets into equicontinuous sets of C0).
Lett1, t2∈[0,b], t1<t2, and ℬr={x∈C0,∥x∥≤r} be a bounded set of C0. For each x∈ℬr and h∈Φx, there exists f∈NF,x such that
(3.23)h(t)=∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds.
Then,
(3.24)|h(t2)-h(t1)|=|∫0∞hq(θ)S′(t2qθ)x0dθ-∫0∞hq(θ)S′(t1qθ)x0dθ+lim⁡λ→∞q∫0t2∫0∞θ(t2-s)q-1hq(θ)S′((t2-s)qθ)Bλf(s)dθds-lim⁡λ→∞q∫0t1∫0∞θ(t1-s)q-1hq(θ)S′((t1-s)qθ)Bλf(s)dθds|≤∫0∞hq(θ)‖S′(t2qθ)-S′(t1qθ)‖|x0|dθ+|lim⁡λ→∞q∫t1t2∫0∞θ(t2-s)q-1hq(θ)S′((t2-s)qθ)Bλf(s)dθds|+|lim⁡λ→∞q∫0t1∫0∞θ(t2-s)q-1hq(θ)S′((t2-s)qθ)Bλf(s)dθds-lim⁡λ→∞q∫0t1∫0∞θ(t1-s)q-1hq(θ)S′((t2-s)qθ)Bλf(s)dθds|+|lim⁡λ→∞q∫0t1∫0∞θ(t1-s)q-1hq(θ)S′((t2-s)qθ)Bλf(s)dθds-lim⁡λ→∞q∫0t1∫0∞θ(t1-s)q-1hq(θ)S′((t1-s)qθ)Bλf(s)dθds|=∫0∞hq(θ)‖S′(t2qθ)-S′(t1qθ)‖|x0|dθ+|lim⁡λ→∞q∫t1t2∫0∞θ(t2-s)q-1hq(θ)S′((t2-s)qθ)Bλf(s)dθds|+|lim⁡λ→∞q∫0t1∫0∞θ[(t2-s)q-1-(t1-s)q]hq(θ)S′((t2-s)qθ)Bλf(s)dθds|+|lim⁡λ→∞q∫0t1∫0∞θ(t1-s)q-1hq(θ)[(S′(t2-s)qθ)-S′((t1-s)qθ)]Bλf(s)dθds|=I1+q(I2+I3+I4),
where
(3.25)I1=∫0∞hq(θ)‖S′(t2qθ)-S′(t1qθ)‖|x0|dθ,I2=|lim⁡λ→∞∫t1t2∫0∞θ(t2-s)q-1hq(θ)S′((t2-s)qθ)Bλf(s)dθds|,I3=|lim⁡λ→∞∫0t1∫0∞θ[(t2-s)q-1-(t1-s)q-1]hq(θ)S′((t2-s)qθ)Bλf(s)dθds|,I4=|lim⁡λ→∞∫0t1∫0∞θ(t1-s)q-1hq(θ)[S′((t2-s)qθ)-S′((t1-s)qθ)]Bλf(s)dθds|.
By using analogous argument performed in (3.20) and (3.21), we can conclude that
(3.26)I2≤MM0Γ(1+q)(M1(1+a)1-q1(t2-t1)(1+a)(1-q1)+n¯r(t2-t1)qq),I3≤MM0Γ(1+q)[(∫0t1((t1-s)q-1-(t2-s)q-1)1/(1-q1)ds)1-q1‖m‖L1/q1[0,t1]+n¯r∫0t1((t1-s)q-1-(t2-s)q-1)ds(∫0t1((t1-s)q-1-(t2-s)q-1)11-q1ds)1-q1]≤MM0Γ(1+q)[M1(∫0t1((t1-s)a-(t2-s)a)ds)1-q1+n¯r((t2-t1)qq-t2qq+t1qq)(∫0t1((t1-s)a-(t2-s)a)ds)1-q1]=MM0Γ(1+q)(M1(1+a)1-q1(t11+a-t21+a+(t2-t1)1+a)1-q1+n¯rq[(t2-t1)q-t2q+t1q]M1(1+a)1-q1)≤MM0Γ(1+q)(M1(1+a)1-q1(t2-t1)(1+a)(1-q1)+n¯rq(t2-t1)q).
Hence limt2→t1I2=0 and limt2→t1I3=0.
On the other hand, (H2) implies thatS′(t) for t>0 is continuous in the uniform operator topology; then from the Lebesgue dominated convergence theorem, we get limt2→t1I1=0 and
(3.27)lim⁡t2→t1I4≤lim⁡t2→t1M∫0t1∫0∞θ(t1-s)q-1hq(θ)‖S′((t2-s)qθ)-S′((t1-s)qθ)‖|f(s)|dθds≤M∫0t1∫0∞θ(t1-s)q-1hq(θ)lim⁡t2→t1‖S′((t2-s)qθ)-S′((t1-s)qθ)‖|f(s)|dθds=0.
Consequently, |h(t2)-h(t1)|→0 independently of x∈ℬr as t2→t1, which means that Φ(ℬr) is equicontinuous.Step 4 (For eacht∈[0,b], V(t)={(Φx)(t),x∈ℬr} is relatively compact in X).
Obviously,V(0)={x0} is relatively compact in X. Let 0<t<b be fixed. For x∈ℬr and h∈Φx, there exists f∈NF,x such that
(3.28)h(t)=∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds.
For arbitraryϵ∈(0,t) and δ>0, define an operator Fϵ,δ on ℬr by
(3.29)(Fϵ,δx)(t)=∫δ∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t-ϵ∫δ∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds=∫δ∞hq(θ)S′(tqδ)S′((tqθ)-tqδ)x0dθ+lim⁡λ→∞q∫0t-ϵ∫δ∞θ(t-s)q-1hq(θ)S′(ϵqδ)S′((t-s)qθ-ϵqδ)Bλf(s)dθds=S′(tqδ)∫δ∞hq(θ)S′((tqθ)-tqδ)x0dθ+S′(ϵqδ)lim⁡λ→∞q∫0t-ϵ∫δ∞θ(t-s)q-1hq(θ)S′((t-s)qθ-ϵqδ)Bλf(s)dθds.
Then from the compactness of S′(t), t>0, we get that the set Vϵ,δ(t)={(Fϵ,δx)(t),x∈ℬr} is relatively compact in X for each ϵ∈(0,t) and δ>0. Moreover, for every x∈ℬr, we have
(3.30)|(Φx)(t)-(Fϵ,δx)(t)|=|∫0δhq(θ)S′(tqθ)x0dθ|+|lim⁡λ→∞q∫0t∫0δθ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds+lim⁡λ→∞q∫0t∫δ∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds-lim⁡λ→∞q∫0t-ϵ∫δ∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds|≤M0|x0|∫0δhq(θ)dθ+q|lim⁡λ→∞∫0t∫0δθ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds|+q|lim⁡λ→∞∫t-ϵt∫δ∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds|≤M0|x0|∫0δhq(θ)dθ+qMM0∫0t(t-s)q-1|f(s)|ds∫0δθhq(θ)dθ+qMM0∫t-ϵt(t-s)q-1|f(s)|ds∫0∞θhq(θ)dθ.
In view of (3.21), we have
(3.31)|(Φx)(t)-(Fϵ,δx)(t)|≤M0|x0|∫0δhq(θ)dθ+qMM0∫0δθhq(θ)dθ(M1(1+a)1-q1b(1+a)(1-q1)+n¯rbqq)+qMM0Γ(1+q)[(∫t-ϵt(t-s)(q-1)/(1-q1)ds)1-q1‖m‖L1/q1[t-ϵ,t]+n¯r∫t-ϵt(t-s)q-1ds]≤M0|x0|∫0δhq(θ)dθ+qMM0∫0δθhq(θ)dθ(M1(1+a)1-q1b(1+a)(1-q1)+n¯rbqq)+qMM0Γ(1+q)(M1(1+a)1-q1ϵ(1+a)(1-q1)+n¯rϵqq).
From limδ→0∫0δhq(θ)dθ=0 and limδ→0∫0δθhq(θ)dθ=0, we get that there are relatively compact sets arbitrarily close to the set V(t), t>0. Hence the set V(t), t>0, is also relatively compact in X.Step 5 (Φ has a closed graph).
Letxn→x*, hn∈Φxn, and hn→h* as n→∞; we will prove that h*∈Φx*. hn∈Φxn means that there exists fn∈NF,xn such that
(3.32)hn(t)=∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλfn(s)dθds.
We must prove that there existsf*∈NF,x* such that
(3.33)h*(t)=∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf*(s)dθds.
Consider the linear continuous operator 𝒯:L1([0,b],X)→C([0,b],X) defined by
(3.34)(Tf)(t)=lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds.
We can easily see that 𝒯 is continuous. On the other hand,
(3.35)|(hn(t)-∫0∞hq(θ)S′(tqθ)x0dθ)-(h*(t)-∫0∞hq(θ)S′(tqθ)x0dθ)|≤‖hn-h*‖⟶0,asn⟶0.
From Theorem 2.13, it follows that 𝒯∘NF is a closed graph operator. Moreover, we have that
(3.36)hn-∫0∞hq(θ)S′(tqθ)x0dθ∈T(NF,xn).
Since xn→x*, it follows from Theorem 2.13 that there exists f*∈NF,x* such that
(3.37)h*(t)-∫0∞hq(θ)S′(tqθ)x0dθ=lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf*(s)dθds.
Thus,
(3.38)h*(t)=∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞q∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf*(s)dθds.
This implies that h*∈Φx*.
ThereforeΦ is a completely continuous multivalued map, u.s.c. with convex closed values. In order to prove that Φ has a fixed point, we need one more step.Step 6 (The setU={x∈C0:δx∈Φx,forsomeδ>1} is bounded).
Letx∈U; then δx∈Φx for some δ>1. Thus there exists f∈NF,x such that for t∈[0,b],
(3.39)x(t)=1δ∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞qδ∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds.
From (H4), for each t∈[0,b] we have
(3.40)|x(t)|=|1δ∫0∞hq(θ)S′(tqθ)x0dθ+lim⁡λ→∞qδ∫0t∫0∞θ(t-s)q-1hq(θ)S′((t-s)qθ)Bλf(s)dθds|≤M0|x0|+qMM0Γ(1+q)∫0t(t-s)q-1|f(s)|ds≤M0|x0|+qMM0Γ(1+q)∫0t(t-s)q-1m(s)ds+qn¯MM0Γ(1+q)∫0t(t-s)q-1|x(s)|ds≤M0|x0|+qMM0M1Γ(1+q)(1+a)1-q1b(1+a)(1-q1)+qn¯MM0Γ(1+q)∫0t(t-s)q-1|x(s)|ds≤a¯+b¯∫0t(t-s)q-1|x(s)|ds,
where a¯=M0|x0|+(qMM0M1/Γ(1+q)(1+a)(1-q1))b(1+a)(1-q1), b¯=qn¯MM0/Γ(1+q).
Then from Corollary2.11, we have
(3.41)|x(t)|≤a¯Eq(b¯Γ(q)tq).
Therefore, we obtain that
(3.42)‖x‖≤a¯Eq(b¯Γ(q)bq).
This shows that U is bounded.
As a consequence of Theorem2.14, we conclude that Φ has a fixed point which is the integral solution of problem (1.1). This completes the proof.


## 4. An Example

As an application of our results we consider the following fractional differential inclusions of the form(4.1)Dqu(t,z)∈∂2∂z2u(t,z)+G(t,u(t,z)),z∈[0,π],t∈(0,b],u(t,0)=u(t,π)=0,t∈[0,b],u(0,z)=u0,z∈[0,π],
where b>0, G:[0,b]×X→𝒫(X) satisfies semi-continuous assumptions (H3) and (H4).ConsiderX=C([0,π];R) endowed with the supnorm and the operator A:D(A)⊂X→X defined by(4.2)D(A)={u∈C2([0,π];R):u(t,0)=u(t,π)=0},Au=∂2∂z2u(t,z).Now, we haveD(A)¯={u∈X:u(t,0)=u(t,π)=0}≠X. As we know from [17] that A satisfies the Hille-Yosida condition with (0,+∞)⊆ρ(A) and λ>0, ∥R(λ,A)∥≤1/λ. Hence, operator A satisfies (H1), (H2), and M=M0=1.Then the system (4.1) can be reformulated as(4.3)Dqx(t)∈Ax(t)+F(t,x(t)),t∈[0,b],x(0)=u0,
where x(t)(z)=u(t,z), F(t,x(t))(z)=G(t,u(t,z)).If we assume thatF satisfies (H3) and (H4), then all conditions of Theorem 3.7 are satisfied and we deduce (4.1) has at least one integral solution.



---

*Source: 316850-2012-06-19.xml*