# On Higher-Order Sequential Fractional Differential Inclusions with Nonlocal Three-Point Boundary Conditions

Authors: Bashir Ahmad; Sotiris K. Ntouyas
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study a nonlinear three-point boundary value problem of sequential fractional differential inclusions of orderξ+1 with n-1<ξ≤n, n≥2. Some new existence results for convex as well as nonconvex multivalued maps are obtained by using standard fixed point theorems. The paper concludes with an example.


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## Body

## 1. Introduction

The topic of fractional differential equations has attracted a great attention in the recent years. It is mainly due to the intensive development of the theory and applications of fractional calculus. In fact, the tools of fractional calculus have considerably improved the modeling of several real world phenomena in physics, chemistry, bioengineering, etc. The systematic development of theory, methods, and applications of fractional differential equations can be found in [1–6]. For some recent results on fractional differential equations and inclusions, see [7–23] and the references cited therein.In this paper, we study the following boundary value problem:(1)cDξ(D+λ)x(t)∈F(t,x(t)),aaiiaa0<t<1,n-1<ξ≤n,x(0)=0,x′(0)=0,x′′(0)=0,…,x(n-1)(0)=0,x(1)=αx(σ),
where cD is the Caputo fractional derivative, D is the ordinary derivative, F:[0,1]×R→P(R) is a multivalued map, P(R) is the family of all subsets of R, 0<σ<1, λ is a positive real number, and α is a real number.The present work is motivated by a recent paper of the authors [14], where the problem (1) was considered for a single-valued case. The existence of solutions for the given multivalued problem is discussed for three cases: (a) convex-valued maps; (b) not necessarily convex-valued maps; (c) nonconvex-valued maps. To establish the existence results, we make use of nonlinear alternative for Kakutani maps, nonlinear alternative of Leray-Schauder type for single-valued maps, selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values, and a fixed point theorem for contractive multivalued maps due to Covitz and Nadler. The tools employed in this paper are standard; however, their exposition in the framework of the problem at hand is new.The paper is organized as follows: in Section2 we recall some preliminary facts that we used in the sequel. Section 3 contains the main results and an example. In Section 4, we summarize the work obtained in this paper and discuss some special cases.


## 2. Preliminaries

Let us recall some basic definitions of fractional calculus [2, 4, 6].Definition 1.
For(n-1)-times absolutely continuous function g:[0,∞)→R, the Caputo derivative of fractional order q is defined as
(2)cDqg(t)=1Γ(n-q)∫0t‍(t-s)n-q-1g(n)(s)ds,aaaaaaaaaaaaaan-1<q<n,n=[q]+1,
where [q] denotes the integer part of the real number q.Definition 2.
The Riemann-Liouville fractional integral of orderq is defined as
(3)Iqg(t)=1Γ(q)∫0t‍g(s)(t-s)1-qds,q>0,
provided the integral exists.Definition 3.
A functionx∈ACn-1([0,1],R) is called a solution of problem (1) if there exists a function v∈L1([0,1],R) with v(t)∈F(t,x(t)), a.e. [0,1], such that  cDξ(D+λ)x(t)=v(t), a.e. [0,1], and x(0)=0, x′(0)=0, x′′(0)=0,…,x(n-1)(0)=0, and x(1)=αx(σ).For the forthcoming analysis, we define(4)P(t)=Po(t)=tn-1λ-(n-1)tn-2λ2+(n-1)(n-2)tn-3λ3-⋯-(n-1)!tλn-1+(n-1)!λn(1-e-λt),aaaaaaaaaaaaaaaaaaaaaaaaaanisodd,(5)P(t)=Pe(t)=tn-1λ-(n-1)tn-2λ2+(n-1)(n-2)tn-3λ3-⋯+(n-1)!tλn-1-(n-1)!λn(1-e-λt),aaaaaaaaaaaaaaaaaaaaaaaaaniseven.
Furthermore, we assume the nonresonance condition, that is, for P=Po and P=Pe, we choose α such that
(6)P(1)-αP(σ)≠0,for0<σ<1.Lemma 4 (see [14]).
Assume that the nonresonance condition (6) holds. Given y∈C([0,1],R), the unique solution of the problem
(7)cDξ(D+λ)x(t)=y(t),0<t<1,x(0)=0,x′(0)=0,x′′(0)=0,…,x(n-1)(0)=0,x(1)=αx(σ)
is given by
(8)x(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)y(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)y(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)y(u)du)ds],
where P(t)=Po(t) and P(t)=Pe(t) are given by (4) and (5), respectively.


## 3. Existence Results

We begin this section with some preliminary material on multivalued maps [24, 25] that we need in the sequel.LetC([0,1]) denote a Banach space of continuous functions from [0,1] into R with the norm ∥x∥=sup⁡t∈[0,1]|x(t)|. Let L1([0,1],R) be the Banach space of measurable functions x:[0,1]→R which are Lebesgue integrable and normed by ∥x∥L1=∫01‍|x(t)|dt.Let(X,∥·∥) denote a normed space. Then we define
(9)Pcl⁡(X)={Z∈P(X):Zisclosed},Pb(X)={Z∈P(X):Zisbounded},Pcp(X)={Z∈P(X):Ziscompact},Pcp,c(X)={Z∈P(X):Ziscompactandconvex}.Definition 5.
A multivalued mapG:X→P(X) is convex- (closed-) valued if G(x) is convex (closed) for all x∈X.Definition 6.
The mapG is bounded on bounded sets if G(B)=∪x∈BG(x) is bounded in X for all B∈Pb(X) (i.e., sup⁡x∈B{sup⁡{|y|:y∈G(x)}}<∞).Definition 7.
G is called upper semicontinuous (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 set N of X containing G(x0), there exists an open neighborhood N0 of x0 such that G(N0)⊆N.Definition 8.
G is said to be completely continuous if G(B) is relatively compact for every B∈Pb(X).If the multivalued mapG is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph; that is, xn→x*, yn→y*, and yn∈G(xn) imply that y*∈G(x*). G has a fixed point if there is x∈X such that x∈G(x). The fixed point set of the multivalued operator G will be denoted by Fix⁡G.Definition 9.
A multivalued mapG:[0;1]→Pcl⁡(R) is said to be measurable if for every y∈R, the function
(10)t⟼d(y,G(t))=inf⁡{|y-z|:z∈G(t)}
is measurable.


### 3.1. The Carathéodory Case

Definition 10.
A multivalued mapF:[0,1]×R→P(R) is said to be Carathéodory if (i)
t↦F(t,x) is measurable for each x∈R;(ii)
x↦F(t,x) is upper semicontinuous for almost all t∈[0,1].Further a Carathéodory function F is called L1-Carathéodory if (iii)
for eachρ>0, there exists φρ∈L1([0,1],R+) such that
(11)∥F(t,x)∥=sup⁡{|v|:v∈F(t,x)}≤φρ(t)for all∥x∥≤ρ and for a.e.t∈[0,1].For eachy∈C([0,1],R), define the set of selections of F by
(12)SF,y:={v∈L1([0,1],R):v(t)∈F(t,y(t))fora.e.t∈[0,1]v∈L1([0,1],R):v(t)∈F(t,y(t))}.For the forthcoming analysis, we need the following lemmas.Lemma 11 (nonlinear alternative for Kakutanimaps [26]).
LetE be a Banach space, C a closed convex subset of E, U an open subset of C, and 0∈U. Suppose that F:U¯→Pcp,c(C) is an upper semicontinuous compact map. Then either(i)
F has a fixed point in U¯, or(ii)
there is au∈∂U and λ∈(0,1) with u∈λF(u).Lemma 12 (see [27]).
LetX be a Banach space. Let F:[0,1]×R→Pcp,c(X) be an L1-Carathéodory multivalued map and let Θ be a linear continuous mapping from L1([0,1],X) to C([0,1],X). Then the operator
(13)Θ∘SF:C([0,1],X)⟶Pcp,c(C([0,1],X)),x⟼(Θ∘SF)(x)=Θ(SF,x)
is a closed graph operator in C([0,1],X)×C([0,1],X).Now we are in a position to prove the existence of the solutions for the boundary value problem (1) when the right-hand side is convex-valued.Theorem 13.
Assume that the nonresonance condition (6) holds. In addition, we suppose that(H1)
F:[0,1]×R→P(R) is Carathéodory and has nonempty compact and convex values;(H2)
there exist a continuous nondecreasing functionψ:[0,∞)→(0,∞) and a function p∈L1([0,1],R+) such that
(14)∥F(t,x)∥P:=sup⁡{|y|:y∈F(t,x)}≤p(t)ψ(∥x∥)foreach(t,x)∈[0,1]×R;(H3)
there exists a constantM>0 such that
(15)M(ψ(M)Γ(ξ){+αP1∫0σe-λ(σ-s)p(s)ds})-1(1+P1)∫01e-λ(1-s)p(s)ds+αP1∫0σe-λ(σ-s)p(s)ds})-1>1,whereP1=max⁡t∈[0,1]|P(t)/(P(1)-αP(σ))| (P(t) is defined in (4) and (5)). Then the boundary value problem (1) has at least one solution on [0,1].Proof.
Define the operatorΩF:C([0,1],R)→P(C([0,1],R)) by
(16)ΩF(x)={-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds]}h∈C([0,1],R):h(t)={∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds]}},
for v∈SF,x. We will show that ΩF satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that ΩFis convex for eachx∈C([0,1],R). This step is obvious since SF,x is convex (F has convex values), and therefore we omit the proof.
In the second step, we show thatΩFmaps bounded sets (balls) into bounded sets inC([0,1],R). For a positive number r, let Br={x∈C([0,1],R):∥x∥≤r} be a bounded ball in C([0,1],R). Then, for each h∈ΩF(x),x∈Br, there exists v∈SF,x such that
(17)h(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds].
Then for t∈[0,1], we have
(18)|h(t)|≤|∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds]|≤∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)p(s)ψ(∥x∥)du)ds+|P(t)P(1)-αP(σ)|×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)p(s)ψ(∥x∥)du)ds+∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)p(s)ψ(∥x∥)du)ds]≤ψ(∥x∥)Γ(ξ){∫01‍e-λ(1-s)p(s)ds+αP1∫0σ‍e-λ(σ-s)p(s)ds+P1∫01‍e-λ(1-s)p(s)ds}=ψ(∥x∥)Γ(ξ){(1+P1)∫01‍e-λ(1-s)p(s)ds+αP1∫0σ‍e-λ(σ-s)p(s)ds}.
Consequently,
(19)∥h∥≤ψ(r)Γ(ξ){(1+P1)∫01e-λ(1-s)p(s)ds+αP1∫0σe-λ(σ-s)p(s)ds(1+P1)∫01‍e-λ(1-s)p(s)ds}.
Now we show thatΩF maps bounded sets into equicontinuous sets of C([0,1],R). Let t1,t2∈[0,1] and x∈Br. For each h∈ΩF(x), we obtain
(20)|h(t1)-h(t2)|=|∫0t1‍e-λ(t1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫0t2‍e-λ(t2-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+[P(t1)-P(t2)]P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds]|≤∫0t1‍e-λ(t1-s)(∫0s‍(s-u)ξ-1Γ(ξ)ψ(r)p(u)du)ds-∫0t2‍e-λ(t2-s)(∫0s‍(s-u)ξ-1Γ(ξ)ψ(r)p(u)du)ds+|P(t1)-P(t2)P(1)-αP(σ)|×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)ψ(r)p(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)ψ(r)p(u)du)ds].
Obviously the right-hand side of the above inequality tends to zero independently ofx∈Br as t2-t1→0. As ΩF satisfies the above three assumptions, therefore it follows from the Ascoli-Arzelá theorem that ΩF:C([0,1],R)→P(C([0,1],R)) is completely continuous.
In our next step, we show thatΩFhas a closed graph. Let xn→x*, hn∈ΩF(xn), and hn→h*. Then we need to show that h*∈ΩF(x*). Associated with hn∈ΩF(xn), there exists vn∈SF,xn such that, for each t∈[0,1],
(21)hn(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)vn(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)vn(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)vn(u)du)ds].
Thus, it suffices to show that there exists v*∈SF,x* such that, for each t∈[0,1],
(22)h*(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v*(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v*(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v*(u)du)ds].
Let us consider the linear operator Θ:L1([0,1],R)→C([0,1],R) given by
(23)f⟼Θ(v)(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds].
Observe that
(24)∥hn(t)-h*(t)∥=∥∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)(vn(u)-v*(u))du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)(vn(u)-v*(u))du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)(vn(u)-v*(u))du)ds]∥⟶0,
as n→∞.
Thus, it follows from Lemma12 that Θ∘SF is a closed graph operator. Further, we have hn(t)∈Θ(SF,xn). Since xn→x*, therefore, we have
(25)h*(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v*(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v*(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v*(u)du)ds],
for some v*∈SF,x*.
Finally, we show that there exists an open setU⊆C([0,1],R) with x∉ΩF(x) for any λ∈(0,1) and all x∈∂U. Let λ∈(0,1) and x∈λΩF(x). Then there exists v∈L1([0,1],R) with v∈SF,x such that, for t∈[0,1], we have
(26)x(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds].
Using the computations of the second step above we have
(27)|x(t)|≤ψ(∥x∥)Γ(ξ)×{(1+P1)∫01e-λ(1-s)p(s)ds+αP1∫0σe-λ(σ-s)p(s)ds}.
Consequently, we have
(28)∥x∥(ψ(∥x∥)Γ(ξ){(1+P1)∫01e-λ(1-s)p(s)ds+αP1∫0σe-λ(σ-s)p(s)ds})-1≤1.
In view of (H3), there exists M such that ∥x∥≠M. Let us set
(29)U={x∈C([0,1],R):∥x∥<M}.
Note that the operator ΩF:U¯→P(C([0,1],R)) is upper semicontinuous and completely continuous. From the choice of U, there is no x∈∂U such that x∈λΩF(x) for some λ∈(0,1). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 11), we deduce that ΩF has a fixed point x∈U¯ which is a solution of the problem (1). This completes the proof.Remark 14.
The condition(H3) in the statement of Theorem 13 may be replaced with the following one.
(
H
3
)
′ There exists a constant M>0 such that
(30)M(ψ(M)/Γ(ξ)){(1+(1+α)P1)∥p∥L1}>1,
where P1 is the same as defined in (H3).


### 3.2.  The Lower Semicontinuous Case

As a next result, we study the case whenF is not necessarily convex-valued. Our strategy to deal with this problem is based on the nonlinear alternative of Leray-Schauder type together with the selection theorem of Bressan and Colombo [28] for lower semicontinuous maps with decomposable values.LetX be a nonempty closed subset of a Banach space E and let G:X→P(E) be a multivalued operator with nonempty closed values. G is lower semicontinuous (l.s.c.) if the set {y∈X:G(y)∩B≠∅} is open for any open set B in E. Let A be a subset of [0,1]×R. A is L⊗B measurable if A belongs to the σ-algebra generated by all sets of the form J×D, where J is Lebesgue measurable in [0,1] and D is Borel measurable in R. A subset A of L1([0,1],R) is decomposable if, for all u,v∈A and measurable J⊂[0,1]=J, the function uχJ+vχJ-J∈A, where χJ stands for the characteristic function of J.Definition 15.
LetY be a separable metric space and let N:Y→P(L1([0,1],R)) be a multivalued operator. We say N has a property (BC) if N is lower semicontinuous (l.s.c.) and has nonempty closed and decomposable values.LetF:[0,1]×R→P(R) be a multivalued map with nonempty compact values. Define a multivalued operator F:C([0,1]×R)→P(L1([0,1],R)) associated with F as
(31)F(x)={w∈L1([0,1],R):w(t)∈F(t,x(t))fora.e.t∈[0,1]w∈L1([0,1],R):w(t)},
which is called the Nemytskii operator associated with F.Definition 16.
LetF:[0,1]×R→P(R) be a multivalued function with nonempty compact values. We say F is of lower semicontinuous type (l.s.c. type) if its associated Nemytskii operator F is lower semicontinuous and has nonempty closed and decomposable values.Lemma 17 (see [29]).
LetY be a separable metric space and let N:Y→P(L1([0,1],R)) be a multivalued operator satisfying the property (BC). Then N has a continuous selection; that is, there exists a continuous function (single-valued) g:Y→L1([0,1],R) such that g(x)∈N(x) for every x∈Y.Theorem 18.
Assume that(H2), (H3), and the following condition hold:(H4)
F:[0,1]×R→P(R) is a nonempty compact-valued multivalued map such that
(a)
(t,x)↦F(t,x) is L⊗B measurable,(b)
x↦F(t,x) is lower semicontinuous for each t∈[0,1].Further the nonresonance condition (6) holds. Then the boundary value problem (1) has at least one solution on [0,1].Proof.
It follows from(H2) and (H4) that F is of l.s.c. type. Then from Lemma 17, there exists a continuous function f:AC1([0,1],R)→L1([0,1],R) such that f(x)∈F(x) for all x∈C([0,1],R).
Consider the problem(32)cDξ(D+λ)x(t)=f(x(t)),0<t<1,x(0)=0,x′(0)=0,x′′(0)=0,…,x(n-1)(0)=0,x(1)=αx(σ).
Observe that ifx∈AC1([0,1],R) is a solution of (32), then x is a solution to the problem (1). In order to transform the problem (32) into a fixed point problem, we define the operator ΩF¯ as
(33)ΩF¯x(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)f(x(u))du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)f(x(u))du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)f(x(u))du)ds].
It can easily be shown that ΩF¯ is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 13. So we omit it. This completes the proof.


### 3.3. The Lipschitz Case

Now we prove the existence of solutions for the problem (1) with a nonconvex-valued right-hand side by applying a fixed point theorem for multivalued map due to Covitz and Nadler [30].Let(X,d) be a metric space induced from the normed space (X,∥·∥). Consider Hd:P(X)×P(X)→R∪{∞} given by
(34)Hd(A,B)=max⁡{sup⁡a∈Ad(a,B),sup⁡b∈Bd(A,b)},
where d(A,b)=inf⁡a∈Ad(a;b) and d(a,B)=inf⁡b∈Bd(a;b). Then (Pb,cl⁡(X),Hd) is a metric space and (Pcl⁡(X),Hd) is a generalized metric space (see [31]).Definition 19.
A multivalued operatorN:X→Pcl⁡(X) is called(a)
γ-Lipschitz if and only if there exists γ>0 such that
(35)Hd(N(x),N(y))≤γd(x,y)foreachx,y∈X;(b)
a contraction if and only if it isγ-Lipschitz with γ<1.Lemma 20 (see [30]).
Let(X,d) be a complete metric space. If N:X→Pcl(X) is a contraction, then Fix⁡N≠∅.Theorem 21.
Assume that the nonresonance condition (6) holds. In addition, suppose that the following conditions hold: (H5)
F:[0,1]×R→Pcp(R) is such that F(·,x):[0,1]→Pcp(R) is measurable for each x∈R;(H6)
Hd(F(t,x),F(t,x¯))≤m(t)|x-x¯| for almost all t∈[0,1] and x,x¯∈R with m∈L1([0,1],R+) and d(0,F(t,0))≤m(t) for almost all t∈[0,1].Then the boundary value problem (1) has at least one solution on [0,1] if
(36)1Γ(ξ){(1+P1)∫01e-λ(1-s)m(s)ds+αP1∫0σe-λ(σ-s)m(s)ds}<1.Proof.
Observe that the setSF,x is nonempty for each x∈C([0,1],R) by the assumption (H5), so F has a measurable selection (see Theorem III.6 [32]). Now we show that the operator ΩF, defined in the beginning of proof of Theorem 13, satisfies the assumptions of Lemma 20. To show that ΩF(x)∈Pcl⁡((C[0,1],R)) for each x∈C([0,1],R), let {un}n≥0∈ΩF(x) be such that un→u(n→∞) in C([0,1],R). Then u∈C([0,1],R) and there exists vn∈SF,xn such that, for each t∈[0,1],
(37)un(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)vn(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)vn(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)vn(u)du)ds].
AsF has compact values, we pass onto a subsequence (if necessary) to obtain that vn converges to v in L1([0,1],R). Thus, v∈SF,x and, for each t∈[0,1], we have
(38)vn(t)⟶v(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds].
Hence,u∈Ω(x).
Next we show that there existsδ<1 such that
(39)Hd(ΩF(x),ΩF(x¯))≤δ∥x-x¯∥foreachx,x¯∈AC1([0,1],R).
Let x,x¯∈AC1([0,1],R) and h1∈ΩF(x). Then there exists v1(t)∈F(t,x(t)) such that, for each t∈[0,1],
(40)h1(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v1(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v1(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v1(u)du)ds].
By(H6), we have
(41)Hd(F(t,x),F(t,x¯))≤m(t)|x(t)-x¯(t)|.
So, there exists w∈F(t,x¯(t)) such that
(42)|v1(t)-w|≤m(t)|x(t)-x¯(t)|,t∈[0,1].
DefineU:[0,1]→P(R) by
(43)U(t)={w∈R:|v1(t)-w|≤m(t)|x(t)-x¯(t)|}.
Since the multivalued operator U(t)∩F(t,x¯(t)) is measurable (Proposition III.4 [32]), there exists a function v2(t) which is a measurable selection for U. So v2(t)∈F(t,x¯(t)) and for each t∈[0,1], we have |v1(t)-v2(t)|≤m(t)|x(t)-x¯(t)|.
For eacht∈[0,1], let us define
(44)h2(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v2(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v2(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v2(u)du)ds].
Thus,(45)|h1(t)-h2(t)|≤∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)|v1(u)-v2(u)|du)ds+|P(t)P(1)-αP(σ)|×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)|v1(u)-v2(u)|du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)|v1(u)-v2(u)|du)ds].
Hence,(46)∥h1-h2∥≤1Γ(ξ){(1+P1)∫01e-λ(1-s)m(s)ds+αP1∫0σe-λ(σ-s)m(s)ds}×∥x-x¯∥.
Analogously, interchanging the roles ofx and x¯, we obtain
(47)Hd(ΩF(x),ΩF(x¯))≤δ∥x-x¯∥≤1Γ(ξ){(1+P1)∫01‍e-λ(1-s)m(s)ds+αP1∫0σ‍e-λ(σ-s)m(s)ds}∥x-x¯∥.
SinceΩF is a contraction, it follows from Lemma 20 that ΩF has a fixed point x which is a solution of (1). This completes the proof.Remark 22.
An alternative to the condition (36) in the statement of Theorem 21 may be the following one:
(48)1Γ(ξ){(1+(1+α)P1)∥m∥L1}<1.Example 23.
Consider the problem(49)cD7/2(D+2)x(t)∈F(t,x(t)),0≤t≤1,x(0)=0,x′(0)=0,x′′(0)=0,x′′′(0)=0,x(1)=x(12).Here,ξ=7/2,  n=4,  λ=2,  α=1,  σ=1/2, and F:[0,1]×R→P(R) is a multivalued map given by
(50)x⟶F(t,x)=[|x|5|x|5+3+t3+t2+4,|x|3|x|3+1+t+2].
For f∈F, we have
(51)|f|≤max⁡(|x|5|x|5+3+t3+t2+4,|x|3|x|3+1+t+2)≤9,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaax∈R.
Thus,
(52)∥F(t,x)∥P:=sup⁡{|y|:y∈F(t,x)}≤7=p(t)ψ(∥x∥),x∈R,
with p(t)=1,ψ(∥x∥)=7. In this case
(53)P(t)=Pe(t)=t32-3t(t-1)4-3(1-e-2t)8,P1≈6.214821.
By the condition (H3), that is,
(54)M(ψ(M)Γ(ξ){(1+P1)∫01e-λ(1-s)p(s)ds+αP1∫0σe-λ(σ-s)p(s)ds})-1>1,
we find that M>M1 with M1≈10.707326. Therefore, it follows from Theorem 13 that problem (49) has at least one solution.


## 3.1. The Carathéodory Case

Definition 10.
A multivalued mapF:[0,1]×R→P(R) is said to be Carathéodory if (i)
t↦F(t,x) is measurable for each x∈R;(ii)
x↦F(t,x) is upper semicontinuous for almost all t∈[0,1].Further a Carathéodory function F is called L1-Carathéodory if (iii)
for eachρ>0, there exists φρ∈L1([0,1],R+) such that
(11)∥F(t,x)∥=sup⁡{|v|:v∈F(t,x)}≤φρ(t)for all∥x∥≤ρ and for a.e.t∈[0,1].For eachy∈C([0,1],R), define the set of selections of F by
(12)SF,y:={v∈L1([0,1],R):v(t)∈F(t,y(t))fora.e.t∈[0,1]v∈L1([0,1],R):v(t)∈F(t,y(t))}.For the forthcoming analysis, we need the following lemmas.Lemma 11 (nonlinear alternative for Kakutanimaps [26]).
LetE be a Banach space, C a closed convex subset of E, U an open subset of C, and 0∈U. Suppose that F:U¯→Pcp,c(C) is an upper semicontinuous compact map. Then either(i)
F has a fixed point in U¯, or(ii)
there is au∈∂U and λ∈(0,1) with u∈λF(u).Lemma 12 (see [27]).
LetX be a Banach space. Let F:[0,1]×R→Pcp,c(X) be an L1-Carathéodory multivalued map and let Θ be a linear continuous mapping from L1([0,1],X) to C([0,1],X). Then the operator
(13)Θ∘SF:C([0,1],X)⟶Pcp,c(C([0,1],X)),x⟼(Θ∘SF)(x)=Θ(SF,x)
is a closed graph operator in C([0,1],X)×C([0,1],X).Now we are in a position to prove the existence of the solutions for the boundary value problem (1) when the right-hand side is convex-valued.Theorem 13.
Assume that the nonresonance condition (6) holds. In addition, we suppose that(H1)
F:[0,1]×R→P(R) is Carathéodory and has nonempty compact and convex values;(H2)
there exist a continuous nondecreasing functionψ:[0,∞)→(0,∞) and a function p∈L1([0,1],R+) such that
(14)∥F(t,x)∥P:=sup⁡{|y|:y∈F(t,x)}≤p(t)ψ(∥x∥)foreach(t,x)∈[0,1]×R;(H3)
there exists a constantM>0 such that
(15)M(ψ(M)Γ(ξ){+αP1∫0σe-λ(σ-s)p(s)ds})-1(1+P1)∫01e-λ(1-s)p(s)ds+αP1∫0σe-λ(σ-s)p(s)ds})-1>1,whereP1=max⁡t∈[0,1]|P(t)/(P(1)-αP(σ))| (P(t) is defined in (4) and (5)). Then the boundary value problem (1) has at least one solution on [0,1].Proof.
Define the operatorΩF:C([0,1],R)→P(C([0,1],R)) by
(16)ΩF(x)={-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds]}h∈C([0,1],R):h(t)={∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds]}},
for v∈SF,x. We will show that ΩF satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that ΩFis convex for eachx∈C([0,1],R). This step is obvious since SF,x is convex (F has convex values), and therefore we omit the proof.
In the second step, we show thatΩFmaps bounded sets (balls) into bounded sets inC([0,1],R). For a positive number r, let Br={x∈C([0,1],R):∥x∥≤r} be a bounded ball in C([0,1],R). Then, for each h∈ΩF(x),x∈Br, there exists v∈SF,x such that
(17)h(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds].
Then for t∈[0,1], we have
(18)|h(t)|≤|∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds]|≤∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)p(s)ψ(∥x∥)du)ds+|P(t)P(1)-αP(σ)|×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)p(s)ψ(∥x∥)du)ds+∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)p(s)ψ(∥x∥)du)ds]≤ψ(∥x∥)Γ(ξ){∫01‍e-λ(1-s)p(s)ds+αP1∫0σ‍e-λ(σ-s)p(s)ds+P1∫01‍e-λ(1-s)p(s)ds}=ψ(∥x∥)Γ(ξ){(1+P1)∫01‍e-λ(1-s)p(s)ds+αP1∫0σ‍e-λ(σ-s)p(s)ds}.
Consequently,
(19)∥h∥≤ψ(r)Γ(ξ){(1+P1)∫01e-λ(1-s)p(s)ds+αP1∫0σe-λ(σ-s)p(s)ds(1+P1)∫01‍e-λ(1-s)p(s)ds}.
Now we show thatΩF maps bounded sets into equicontinuous sets of C([0,1],R). Let t1,t2∈[0,1] and x∈Br. For each h∈ΩF(x), we obtain
(20)|h(t1)-h(t2)|=|∫0t1‍e-λ(t1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫0t2‍e-λ(t2-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+[P(t1)-P(t2)]P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds]|≤∫0t1‍e-λ(t1-s)(∫0s‍(s-u)ξ-1Γ(ξ)ψ(r)p(u)du)ds-∫0t2‍e-λ(t2-s)(∫0s‍(s-u)ξ-1Γ(ξ)ψ(r)p(u)du)ds+|P(t1)-P(t2)P(1)-αP(σ)|×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)ψ(r)p(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)ψ(r)p(u)du)ds].
Obviously the right-hand side of the above inequality tends to zero independently ofx∈Br as t2-t1→0. As ΩF satisfies the above three assumptions, therefore it follows from the Ascoli-Arzelá theorem that ΩF:C([0,1],R)→P(C([0,1],R)) is completely continuous.
In our next step, we show thatΩFhas a closed graph. Let xn→x*, hn∈ΩF(xn), and hn→h*. Then we need to show that h*∈ΩF(x*). Associated with hn∈ΩF(xn), there exists vn∈SF,xn such that, for each t∈[0,1],
(21)hn(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)vn(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)vn(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)vn(u)du)ds].
Thus, it suffices to show that there exists v*∈SF,x* such that, for each t∈[0,1],
(22)h*(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v*(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v*(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v*(u)du)ds].
Let us consider the linear operator Θ:L1([0,1],R)→C([0,1],R) given by
(23)f⟼Θ(v)(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds].
Observe that
(24)∥hn(t)-h*(t)∥=∥∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)(vn(u)-v*(u))du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)(vn(u)-v*(u))du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)(vn(u)-v*(u))du)ds]∥⟶0,
as n→∞.
Thus, it follows from Lemma12 that Θ∘SF is a closed graph operator. Further, we have hn(t)∈Θ(SF,xn). Since xn→x*, therefore, we have
(25)h*(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v*(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v*(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v*(u)du)ds],
for some v*∈SF,x*.
Finally, we show that there exists an open setU⊆C([0,1],R) with x∉ΩF(x) for any λ∈(0,1) and all x∈∂U. Let λ∈(0,1) and x∈λΩF(x). Then there exists v∈L1([0,1],R) with v∈SF,x such that, for t∈[0,1], we have
(26)x(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds].
Using the computations of the second step above we have
(27)|x(t)|≤ψ(∥x∥)Γ(ξ)×{(1+P1)∫01e-λ(1-s)p(s)ds+αP1∫0σe-λ(σ-s)p(s)ds}.
Consequently, we have
(28)∥x∥(ψ(∥x∥)Γ(ξ){(1+P1)∫01e-λ(1-s)p(s)ds+αP1∫0σe-λ(σ-s)p(s)ds})-1≤1.
In view of (H3), there exists M such that ∥x∥≠M. Let us set
(29)U={x∈C([0,1],R):∥x∥<M}.
Note that the operator ΩF:U¯→P(C([0,1],R)) is upper semicontinuous and completely continuous. From the choice of U, there is no x∈∂U such that x∈λΩF(x) for some λ∈(0,1). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 11), we deduce that ΩF has a fixed point x∈U¯ which is a solution of the problem (1). This completes the proof.Remark 14.
The condition(H3) in the statement of Theorem 13 may be replaced with the following one.
(
H
3
)
′ There exists a constant M>0 such that
(30)M(ψ(M)/Γ(ξ)){(1+(1+α)P1)∥p∥L1}>1,
where P1 is the same as defined in (H3).


## 3.2.  The Lower Semicontinuous Case

As a next result, we study the case whenF is not necessarily convex-valued. Our strategy to deal with this problem is based on the nonlinear alternative of Leray-Schauder type together with the selection theorem of Bressan and Colombo [28] for lower semicontinuous maps with decomposable values.LetX be a nonempty closed subset of a Banach space E and let G:X→P(E) be a multivalued operator with nonempty closed values. G is lower semicontinuous (l.s.c.) if the set {y∈X:G(y)∩B≠∅} is open for any open set B in E. Let A be a subset of [0,1]×R. A is L⊗B measurable if A belongs to the σ-algebra generated by all sets of the form J×D, where J is Lebesgue measurable in [0,1] and D is Borel measurable in R. A subset A of L1([0,1],R) is decomposable if, for all u,v∈A and measurable J⊂[0,1]=J, the function uχJ+vχJ-J∈A, where χJ stands for the characteristic function of J.Definition 15.
LetY be a separable metric space and let N:Y→P(L1([0,1],R)) be a multivalued operator. We say N has a property (BC) if N is lower semicontinuous (l.s.c.) and has nonempty closed and decomposable values.LetF:[0,1]×R→P(R) be a multivalued map with nonempty compact values. Define a multivalued operator F:C([0,1]×R)→P(L1([0,1],R)) associated with F as
(31)F(x)={w∈L1([0,1],R):w(t)∈F(t,x(t))fora.e.t∈[0,1]w∈L1([0,1],R):w(t)},
which is called the Nemytskii operator associated with F.Definition 16.
LetF:[0,1]×R→P(R) be a multivalued function with nonempty compact values. We say F is of lower semicontinuous type (l.s.c. type) if its associated Nemytskii operator F is lower semicontinuous and has nonempty closed and decomposable values.Lemma 17 (see [29]).
LetY be a separable metric space and let N:Y→P(L1([0,1],R)) be a multivalued operator satisfying the property (BC). Then N has a continuous selection; that is, there exists a continuous function (single-valued) g:Y→L1([0,1],R) such that g(x)∈N(x) for every x∈Y.Theorem 18.
Assume that(H2), (H3), and the following condition hold:(H4)
F:[0,1]×R→P(R) is a nonempty compact-valued multivalued map such that
(a)
(t,x)↦F(t,x) is L⊗B measurable,(b)
x↦F(t,x) is lower semicontinuous for each t∈[0,1].Further the nonresonance condition (6) holds. Then the boundary value problem (1) has at least one solution on [0,1].Proof.
It follows from(H2) and (H4) that F is of l.s.c. type. Then from Lemma 17, there exists a continuous function f:AC1([0,1],R)→L1([0,1],R) such that f(x)∈F(x) for all x∈C([0,1],R).
Consider the problem(32)cDξ(D+λ)x(t)=f(x(t)),0<t<1,x(0)=0,x′(0)=0,x′′(0)=0,…,x(n-1)(0)=0,x(1)=αx(σ).
Observe that ifx∈AC1([0,1],R) is a solution of (32), then x is a solution to the problem (1). In order to transform the problem (32) into a fixed point problem, we define the operator ΩF¯ as
(33)ΩF¯x(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)f(x(u))du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)f(x(u))du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)f(x(u))du)ds].
It can easily be shown that ΩF¯ is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 13. So we omit it. This completes the proof.


## 3.3. The Lipschitz Case

Now we prove the existence of solutions for the problem (1) with a nonconvex-valued right-hand side by applying a fixed point theorem for multivalued map due to Covitz and Nadler [30].Let(X,d) be a metric space induced from the normed space (X,∥·∥). Consider Hd:P(X)×P(X)→R∪{∞} given by
(34)Hd(A,B)=max⁡{sup⁡a∈Ad(a,B),sup⁡b∈Bd(A,b)},
where d(A,b)=inf⁡a∈Ad(a;b) and d(a,B)=inf⁡b∈Bd(a;b). Then (Pb,cl⁡(X),Hd) is a metric space and (Pcl⁡(X),Hd) is a generalized metric space (see [31]).Definition 19.
A multivalued operatorN:X→Pcl⁡(X) is called(a)
γ-Lipschitz if and only if there exists γ>0 such that
(35)Hd(N(x),N(y))≤γd(x,y)foreachx,y∈X;(b)
a contraction if and only if it isγ-Lipschitz with γ<1.Lemma 20 (see [30]).
Let(X,d) be a complete metric space. If N:X→Pcl(X) is a contraction, then Fix⁡N≠∅.Theorem 21.
Assume that the nonresonance condition (6) holds. In addition, suppose that the following conditions hold: (H5)
F:[0,1]×R→Pcp(R) is such that F(·,x):[0,1]→Pcp(R) is measurable for each x∈R;(H6)
Hd(F(t,x),F(t,x¯))≤m(t)|x-x¯| for almost all t∈[0,1] and x,x¯∈R with m∈L1([0,1],R+) and d(0,F(t,0))≤m(t) for almost all t∈[0,1].Then the boundary value problem (1) has at least one solution on [0,1] if
(36)1Γ(ξ){(1+P1)∫01e-λ(1-s)m(s)ds+αP1∫0σe-λ(σ-s)m(s)ds}<1.Proof.
Observe that the setSF,x is nonempty for each x∈C([0,1],R) by the assumption (H5), so F has a measurable selection (see Theorem III.6 [32]). Now we show that the operator ΩF, defined in the beginning of proof of Theorem 13, satisfies the assumptions of Lemma 20. To show that ΩF(x)∈Pcl⁡((C[0,1],R)) for each x∈C([0,1],R), let {un}n≥0∈ΩF(x) be such that un→u(n→∞) in C([0,1],R). Then u∈C([0,1],R) and there exists vn∈SF,xn such that, for each t∈[0,1],
(37)un(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)vn(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)vn(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)vn(u)du)ds].
AsF has compact values, we pass onto a subsequence (if necessary) to obtain that vn converges to v in L1([0,1],R). Thus, v∈SF,x and, for each t∈[0,1], we have
(38)vn(t)⟶v(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v(u)du)ds].
Hence,u∈Ω(x).
Next we show that there existsδ<1 such that
(39)Hd(ΩF(x),ΩF(x¯))≤δ∥x-x¯∥foreachx,x¯∈AC1([0,1],R).
Let x,x¯∈AC1([0,1],R) and h1∈ΩF(x). Then there exists v1(t)∈F(t,x(t)) such that, for each t∈[0,1],
(40)h1(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v1(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v1(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v1(u)du)ds].
By(H6), we have
(41)Hd(F(t,x),F(t,x¯))≤m(t)|x(t)-x¯(t)|.
So, there exists w∈F(t,x¯(t)) such that
(42)|v1(t)-w|≤m(t)|x(t)-x¯(t)|,t∈[0,1].
DefineU:[0,1]→P(R) by
(43)U(t)={w∈R:|v1(t)-w|≤m(t)|x(t)-x¯(t)|}.
Since the multivalued operator U(t)∩F(t,x¯(t)) is measurable (Proposition III.4 [32]), there exists a function v2(t) which is a measurable selection for U. So v2(t)∈F(t,x¯(t)) and for each t∈[0,1], we have |v1(t)-v2(t)|≤m(t)|x(t)-x¯(t)|.
For eacht∈[0,1], let us define
(44)h2(t)=∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)v2(u)du)ds+P(t)P(1)-αP(σ)×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)v2(u)du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)v2(u)du)ds].
Thus,(45)|h1(t)-h2(t)|≤∫0t‍e-λ(t-s)(∫0s‍(s-u)ξ-1Γ(ξ)|v1(u)-v2(u)|du)ds+|P(t)P(1)-αP(σ)|×[α∫0σ‍e-λ(σ-s)(∫0s‍(s-u)ξ-1Γ(ξ)|v1(u)-v2(u)|du)ds-∫01‍e-λ(1-s)(∫0s‍(s-u)ξ-1Γ(ξ)|v1(u)-v2(u)|du)ds].
Hence,(46)∥h1-h2∥≤1Γ(ξ){(1+P1)∫01e-λ(1-s)m(s)ds+αP1∫0σe-λ(σ-s)m(s)ds}×∥x-x¯∥.
Analogously, interchanging the roles ofx and x¯, we obtain
(47)Hd(ΩF(x),ΩF(x¯))≤δ∥x-x¯∥≤1Γ(ξ){(1+P1)∫01‍e-λ(1-s)m(s)ds+αP1∫0σ‍e-λ(σ-s)m(s)ds}∥x-x¯∥.
SinceΩF is a contraction, it follows from Lemma 20 that ΩF has a fixed point x which is a solution of (1). This completes the proof.Remark 22.
An alternative to the condition (36) in the statement of Theorem 21 may be the following one:
(48)1Γ(ξ){(1+(1+α)P1)∥m∥L1}<1.Example 23.
Consider the problem(49)cD7/2(D+2)x(t)∈F(t,x(t)),0≤t≤1,x(0)=0,x′(0)=0,x′′(0)=0,x′′′(0)=0,x(1)=x(12).Here,ξ=7/2,  n=4,  λ=2,  α=1,  σ=1/2, and F:[0,1]×R→P(R) is a multivalued map given by
(50)x⟶F(t,x)=[|x|5|x|5+3+t3+t2+4,|x|3|x|3+1+t+2].
For f∈F, we have
(51)|f|≤max⁡(|x|5|x|5+3+t3+t2+4,|x|3|x|3+1+t+2)≤9,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaax∈R.
Thus,
(52)∥F(t,x)∥P:=sup⁡{|y|:y∈F(t,x)}≤7=p(t)ψ(∥x∥),x∈R,
with p(t)=1,ψ(∥x∥)=7. In this case
(53)P(t)=Pe(t)=t32-3t(t-1)4-3(1-e-2t)8,P1≈6.214821.
By the condition (H3), that is,
(54)M(ψ(M)Γ(ξ){(1+P1)∫01e-λ(1-s)p(s)ds+αP1∫0σe-λ(σ-s)p(s)ds})-1>1,
we find that M>M1 with M1≈10.707326. Therefore, it follows from Theorem 13 that problem (49) has at least one solution.


## 4. Conclusions

In this paper, we have solved a three-point boundary value problem of Caputo-type sequential fractional differential inclusions of an arbitrary orderξ∈(n-1,n). The existence of solutions for the given problem with the convex-valued map is obtained by means of nonlinear alternative for Kakutani maps, while the existence result for not necessarily convex-valued map is established by combining nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with decomposable values. The nonconvex-valued case relies on a fixed point theorem for contractive multivalued maps due to Covitz and Nadler. Some new existence results follow by fixing the parameters involved in the given problem. For instance, by taking α=0, our results correspond to a two-point Caputo-type multivalued problem of an arbitrary order ξ∈(n-1,n), while the results for sequential differential inclusions of order (n+1) can be obtained by fixing ξ=n in the results of this paper.



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