# Some Results for Nonlinear (n+1)-Term Fractional Integrodifferential Inclusions with Multipoint Boundary Conditions

Authors: Huacheng Zhou; Chunhai Kou; Wen Song
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

We are concerned with the nonlinear(n+1)-term fractional integrodifferential inclusions L(D)u∈F(t,u,Tu,Su), a.e.t∈[0,1], where L(D)=Dα-bnDβn-bn-1Dβn-1-⋯-b1Dβ1, 0<β1<β2<⋯<βn-1<βn<1<α<2, b1,b2,…,bn are constant coefficients, and (Tu)(t)=∫01k(t,s)u(s)ds, (Su)(t)=∫0tl(t,s)u(s)ds, subject to the nonlocal conditions u(0)=0, u(1)=∑i=1mγiu(ηi). The existence results are obtained by using two fixed-point theorems due to Bohnenblust-Karlin and Covitz-Nadler, respectively. Our results partly generalize and improve the known ones.


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## Body

## 1. Introduction

Fractional differential equations (FDEs) have received increasing interest for the last three decades. It is benefited by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science, engineering economics, and other fields, see for instance [1–12] and references therein.Babakhani and Gejji [7] considered the existence of positive solutions for the nonlinear fractional differential equations(1.1)L(D)u=f(t,u),u(0)=0,0<t<1,
where L(D)=Dsn-an-1Dsn-1-⋯-a1Ds1, 0<s1<s2<⋯<sn<1, aj>0, j=1,2,…,n-1, and Dsj is the standard Riemann-Liouville fractional derivative. Some existence results of positive solutions are obtained by using some fixed-point theorems on a cone.Stojanović [11] considered the existenceuniqueness of solutions for a nonlinear n-term fractional differential equation(1.2)b0Dβ0u(t)+∑i=1m-1biDβiu(t)+∑i=mn-1biDαiu(t)+bnDαnu(t)=f(t,u(t)),t∈(0,1),
where 0<β1<β2<⋯<βm-1<1<αm<αm+1<⋯<αn<2, with initial data u(0)=f(0), ut(0)=g(0).On the other hand, realistic problems arising from economics, optimal control, and so on can be modeled as differential inclusions. Recently, El-Sayed and Ibrahim [13] initiated the study of fractional differential equations inclusions. Differential inclusions have been widely investigated by many authors, see [14–30] and references therein.Very recently, in the survey paper [16], Agarwal et al. establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problems for fractional differential equations and inclusions involving the Caputo fractional derivative.Ouahab [26] studied the following boundary value problem of fractional differential inclusions:(1.3)-Dαy(t)∈F(t,y(t)),a.e.t∈J=[0,1],1<α≤2,y(0)=y(1)=0,
where Dα is the standard Riemann-Liouville fractional derivative, F:J×ℝ→𝒫(ℝ) is a multivalued map with compact values.Chang and Nieto [27] studied boundary value problem of fractional differential inclusions(1.4)0CDδy(t)∈F(t,y(t)),t∈J:=[0,1],δ∈(1,2),y(0)=α,y(1)=β,α,β≠0,
where 0CDδy(t) is the Caputo's derivative, F:J×ℝ→𝒫(ℝ).However, to the best of our knowledge, the existence of solutions for fractional integro-differential inclusions with multipoint boundary conditions has not been paid much attention. Our goal is to fill this gap in literature.In the present work, we consider more general fractional integro-differential inclusions(1.5)L(D)u∈F(t,u,Tu,Su),a.e.t∈[0,1]
with multipoint boundary conditions(1.6)u(0)=0,u(1)=∑i=1mγiu(ηi),
where L(D)=Dα-bnDβn-bn-1Dβn-1-⋯-b1Dβ1, 0<β1<β2<⋯<βn-1<βn<1<α<2, Dα,Dβj, j=1,2,…,n, is the standard Riemann-Liouville fractional derivative, 0<η1<η2⋯<ηm<1, 0≤∑i=1mγiηiα-1<1, (Tu)(t)=∫01k(t,s)u(s)ds,  (Su)(t)=∫0tl(t,s)u(s)ds, k,l are two continuous functions on [0,1]×[0,1], and F:[0,1]×ℝ×ℝ×ℝ→𝒫(ℝ) is a given multivalued function (𝒫(ℝ) is the family of all nonempty subsets of ℝ). We shall consider both the cases of convex and nonconvex valued right hand side and establish some sufficient conditions which admit that the integro-differential inclusions problem has at least one solution. These results obtained by applying two fixed point theorems due to Bohnenblust-Karlin and Covitz-Nadler are complement of previously known results.The organization of this paper is as follows. In Section2, we present some necessary definitions and preliminaries which are used throughout this paper. Main results and their proofs are given in Section 3.


## 2. Preliminaries and Several Lemmas

In this section, we recall some basic definitions and notations and give several lemmas which are useful in our discussion.Definition 2.1.
The Riemann-Liouville fractional integral of orderα>0 of a function f:ℝ+→ℝ is given by
(2.1)Iαf(t)=1Γ(α)∫0t(t-s)α-1f(s)dt,
provided the right side is pointwise defined on ℝ+.Definition 2.2.
The Riemann-Liouville fractional derivative of functionf:ℝ+→ℝ is given by
(2.2)Dαf(t)=DnIn-αf(t),
where n=[α]+1, Dn=dn/dtn, provided the right side is pointwise defined on ℝ+.Lemma 2.3.
Letα>0, then
(2.3)IαDαx(t)=x(t)+c1tα-1+c2tα-2+⋯+cntα-n,
for some ci∈ℝ, i=0,1,2,…,n-1,  n=-[-α].Remark 2.4.
If the fractional derivativeDαx(t) is integrable, then
(2.4)IαDβx(t)=Iα-βx(t)-[I1-βx(t)]t=0tα-1Γ(α),0<β≤α<1.
If x is continuous on [0,1], then [I1-βx(t)]t=0=0, and (2.4) reduces to
(2.5)IαDβx(t)=Iα-βx(t),0<β≤α<1.The reader is referred to [4, 8, 9] for more details on fractional integrals and fractional derivatives.Lemma 2.5.
Letg∈L([0,1],ℝ), 0<β1<β2<⋯<βn-1<βn<1<α<2, and 0<∑i=1mγiηiα-1<1. Then u(t) is a solution of the BVP
(2.6)(Dα-bnDβn-bn-1Dβn-1-⋯-b1Dβ1)u(t)=g(t),a.e.t∈[0,1],(2.7)u(0)=0,u(1)=∑i=1mγiu(ηi).
if and only if u(t) satisfies the integral equation
(2.8)u(t)=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1u(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1u(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1u(s)ds+1Γ(α)∫0t(t-s)α-1g(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1g(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1g(s)ds.Proof.
In view of Lemma2.3 and Remark 2.4, (2.6) is equivalent to the integral equation
(2.9)u(t)=c1tα-1+c2tα-2+∑i=1nbiIα-βiu(t)+Iαg(t),
for some c1,c2∈ℝ, that is,
(2.10)u(t)=c1tα-1+c2tα-2+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1u(s)ds+1Γ(α)∫0t(t-s)α-1g(s)ds.
The boundary condition u(0)=0 implies c2=0. Thus,
(2.11)u(t)=c1tα-1+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1u(s)ds+1Γ(α)∫0t(t-s)α-1g(s)ds.
In view of the boundary condition u(1)=∑i=1mγiu(ηi), we conclude that
(2.12)c1=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1u(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1u(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1g(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1g(s)ds.
Therefore, the solution u(t) of (2.6) and (2.7) satisfies (2.8).
Conversely, ifu(t) is a solution of (2.8), it is easy to verify that u(t) satisfies (2.6) and (2.7). The proof is complete.Now, we recall some facts from multivalued analysis.Let(X,d) be a metric space, and 𝒫(X)={Y⊂X:Y≠∅}, 𝒫bd(X)={Y∈𝒫(X):Ybounded}, 𝒫cl(X)={Y∈𝒫(X):Yclosed}, 𝒫cv(X)={Y∈𝒫(X):Yconvex}, 𝒫cp(X)={Y∈𝒫(X):Ycompact}, 𝒫cv,cp(X)=𝒫cv(X)∩𝒫cp(X), and so forth.ConsiderHd:𝒫(X)×𝒫(X)→ℝ+∪{∞}, given by(2.13)Hd(A,B)=max⁡{sup⁡x∈Ad(x,B),sup⁡y∈Bd(A,y)},
where d(x,B)=infy∈Bd(x,y), d(A,y)=infx∈Ad(x,y). Then (𝒫bd,cl(X),Hd) is a metric space and (𝒫cl(X),Hd) is a generalized metric space (see [31]).A multivalued mapN:[0,1]→𝒫cl(X) is said to be measurable if for each u∈X, the function Y:[0,1]→ℝ, defined by(2.14)Y(t)=d(u,N(t))=inf⁡{d(u,z):z∈N(t)},
is measurable.A multivalued mapF:X→𝒫(X) is convex (closed) valued if F(u) is convex (closed) for all u∈X. F is bounded on bounded sets if F(B)=∪u∈BF(u) is bounded in X for all B∈𝒫(X). That is, supu∈B{sup{|y|:y∈F(u)}}<∞. F is called upper semicontinuous (u.s.c for short) on X if for each u0∈X, the set F(u0) is nonempty closed subset of X, and if for each open set 𝒰 of X containing F(u0), there exists an open neighborhood 𝒱 of u0 such that F(𝒱)⊂𝒰. G is said to be completely continuous if F(B) is relatively compact for every B∈𝒫bd(X). If the multivalued map F is completely continuous with nonempty compact valued, then G is u.s.c. if and only if F has closed graph, that is, xn→x*, yn→y*, yn∈G(x*) imply y*∈G(x*).More details on multivalued maps can be found in the books of Deimling [3], Górniewicz [32], Hu and Papageorgiou [33], and Tolstonogov [34].Definition 2.6.
The multivalued mapF:[0,1]×ℝ×ℝ×ℝ→𝒫(ℝ) is L1-Carathéodory if (i)
t↦F(t,u,v,w) is measurable for each (u,v,w)∈ℝ×ℝ×ℝ;(ii)
(u,v,w)↦F(t,u,v,w) is upper semicontinuous for almost all t∈[0,1];(iii)
for eachr>0, there exists φr∈L1([0,1],ℝ+) such that for all |u|,|v|,|w|≤r and for almost all t∈[0,1],(2.15)‖F(t,u,v,w)‖P=sup⁡{|f|:f∈F(t,u,v,w)}≤φr(t).For anyu∈C([0,1],ℝ), we define the set(2.16)SF,u={f∈L1([0,1],R):f(t)∈F(t,u(t),(Tu)(t),(Su)(t))fora.e.t∈[0,1]},
which is known as the set of selection functions.Lemma 2.7 (see [35]).
LetX be a Banach space. Let F:[0,1]×X→𝒫cp,cv(X) be an L1-Carathéodory multivalued map with SF,u≠∅, and let Γ be a linear continuous mapping from L1([0,1],X) into C([0,1],X). Then the operator
(2.17)Γ∘SF:C([0,1],X)⟶Pcp,cv(C([0,1],X)),u⟼(Γ∘SF)(y):=Γ(SF,u)
is a closed graph operator in C([0,1],X)×C([0,1],X).The following Bohnenblust-Karlin fixed-point lemma and the fixed-point theorem for contractive multivalued operators given by Covitz and Nadler are of great importance in the proofs of our main results. The proofs of these results can be found in Bohnenblust and Karlin [30] and in Covitz and Nadler [36].Lemma 2.8 (see [30]).
LetX be a Banach space, D be a nonempty subset of X, which is bounded, closed, and convex. Suppose G:D→𝒫(X) is u.s.c. with closed, convex values, and such that G(D)⊂D and G(D) is relatively compact. Then G has a fixed point.Lemma 2.9 (see [36]).
Let(X,d) be a complete metric space. If N is a contraction operator, then FixN≠∅.
For convenience, let us list some conditions.(H1)
∑i=1m∑j=1n(γibjηiα-βj/(1-∑i=1mγiηiα-1)Γ(α-βj+1))+∑i=1n(bi/(1-∑i=1mγiηiα-1)Γ(α-βi+1))+∑i=1n(bi/Γ(α-βi+1))<1;(H2)
F:[0,1]×ℝ×ℝ×ℝ→𝒫cp,cv(ℝ) is Carethéodory multivalued map.Lemma 2.10.
Assume that hypothesis (H1) is satisfied. For anyg∈C([0,1],ℝ), the integral equation
(2.18)u(t)=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1u(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1u(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1u(s)ds+g(t)
has a unique solution in C([0,1],ℝ).Proof.
We define the operatorA as follows:
(2.19)(Au)(t):=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1u(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1u(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1u(s)ds+g(t)
Obviously, A is a map from C([0,1],ℝ) into C([0,1],ℝ). Also, we have
(2.20)|(Au1)(t)-(Au2)(t)|≤∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1|u1(s)-u2(s)|ds+∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1|u1(s)-u2(s)|ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1|u1(s)-u2(s)|ds≤(∑i=1m∑j=1nγibjηiα-βj(1-∑i=1mγiηiα-1)Γ(α-βj+1)+∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi+1)+∑i=1nbiΓ(α-βi+1)∑i=1m∑j=1n)‖u1-u2‖.
Therefore, ∥Au1-Au2∥≤λ∥u1-u2∥, where λ=∑i=1m∑j=1n(γibjηiα-βj/(1-∑i=1mγiηiα-1)Γ(α-βj+1))+∑i=1n(bi/(1-∑i=1mγiηiα-1)Γ(α-βi+1))+∑i=1n(bi/Γ(α-βi+1)). By (H1) and Banach contraction principle, the conclusion of lemma is true.Definition 2.11.
A functionu∈C[0,1] is said to be a solution of the BVP (1.5) and (1.6), if there exists a function f∈L1([0,1],ℝ) such that f(t)∈F(t,u(t),(Tu)(t),(Su)(t)) a.e. on [0,1] and
(2.21)u(t)=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1u(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1u(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1u(s)ds+1Γ(α)∫0t(t-s)α-1f(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1f(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1f(s)ds.


## 3. Main Results

In this section, we present our main results and prove them. Firstly, under convexity condition on the multivalued right-hand side, we are to establish the existence theorem of solutions for fractional differential inclusions (1.5) and (1.6), by employing the Bohnenblust-Karlin fixed-point theorem. Then, under nonconvexity condition on the multivalued right-hand side, the existence theorem of solutions are gotten, by employing the Covitz-Nadler fixed-point theorem.Theorem 3.1.
Assume that hypotheses (H1) and (H2) are satisfied. Then BVP (1.5) and (1.6) has at least one solution in C([0,1],ℝ), provided that(H3)
τ/Γ(α)+∑i=1m(γiμi/(1-∑i=1mγiηiα-1)Γ(α))+(ω/(1-∑i=1mγiηiα-1)Γ(α))<1-λ,
where(3.1)τ=lim⁡r→∞inf⁡∫01max⁡{φr(s),φk̃r(s),φl̃r(s)}rds,μi=lim⁡r→∞inf⁡∫0ηi(ηi-s)α-1max⁡{φr(s),φk̃r(s),φl̃r(s)}rds,ω=lim⁡r→∞inf⁡∫01(1-s)α-1max⁡{φr(s),φk̃r(s),φl̃r(s)}rds,k̃=sup⁡t∈[0,1]∫01k(t,s)ds,l̃=sup⁡t∈[0,1]∫0tl(t,s)ds,
and λ is defined in the proof of Lemma 2.10.Proof.
To transform the problem into a fixed-point problem, we consider the multivalued operator,N:C([0,1],ℝ)→𝒫(C([0,1],ℝ)), where for any u∈C([0,1],ℝ),N(u) is defined by
(3.2){1(1-∑i=1mγiηiα-1)Γ(α)h∈C([0,1],R):h(t)=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1h(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1h(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1h(s)ds+1Γ(α)∫0t(t-s)α-1f(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1f(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1f(s)ds,f∈SF,u}.
For anyf∈L1([0,1],ℝ), we have ∫0t(t-s)α-1f(s)ds, ∫0ηi(ηi-s)α-1tα-1f(s)ds, ∫01(1-s)α-1tα-1f(s)ds∈C([0,1],ℝ). In view of Lemma 2.10, N is well defined. Moreover, it follows from the convexity of SF,u (because F has convex values) that N(u) is convex for each u∈C([0,1],ℝ). Clearly, the fixed points of N are solutions of (1.5) and (1.6).
We shall show thatN has a fixed point in three steps.
Step 1.
we claim that there exists ar*>0, such that N(Br*)⊂Br*, where Br*={u∈C([0,1],ℝ):∥u∥≤r*}.
In fact, if it is not true, then for anyr>0, there exists a function ur∈Br, hr∈N(ur) but hr∉Br, that is ∥hr∥>r and for some fr∈SF,u,
(3.3)hr(t)=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1hr(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1hr(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1hr(s)ds+1Γ(α)∫0t(t-s)α-1fr(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1fr(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1fr(s)ds.
On the other hand, from (H2), we obtain(3.4)|hr(t)|≤λ‖hr‖+1Γ(α)∫0t(t-s)α-1fr(s)ds+1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1fr(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1fr(s)ds≤λ‖hr‖+1Γ(α)∫01(|t-s|)α-1φmax⁡{r,k̃r,l̃r}(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1φmax⁡{r,k̃r,l̃r}(s)ds+1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1φmax⁡{r,k̃r,l̃r}(s)ds.
Thus,
(3.5)r<‖hr‖≤1(1-λ)Γ(α)∫01(|t-s|)α-1φmax⁡{r,k̃r,l̃r}(s)ds+∑i=1mγi(1-λ)(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1φmax⁡{r,k̃r,l̃r}(s)ds+1(1-λ)(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1φmax⁡{r,k̃r,l̃r}(s)ds.
Dividing both sides byr and taking the lower limit as r→+∞, we get
(3.6)1≤τ(1-λ)Γ(α)+∑i=1mγiμi(1-λ)(1-∑i=1mγiηiα-1)Γ(α)+ω(1-λ)(1-∑i=1mγiηiα-1)Γ(α).
This is a contraction to (H3). Hence there exists ar*>0 such that N(Br*)⊂Br*.Step 2.
N(Br*) is equicontinuous.
Lett1,t2∈[0,1], t1<t2, u∈Br*, and h∈N(u). Then there exists f∈SF,u such that for each t∈[0,1], we have
(3.7)h(t)=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1h(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1h(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1h(s)ds+1Γ(α)∫0t(t-s)α-1f(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1f(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1f(s)ds.
Therefore,
(3.8)|h(t1)-h(t2)|≤a(t2α-1-t1α-1)+1Γ(α)∫t1t2φmax⁡{r*,k̃r*,l̃r*}(s)ds+1Γ(α)∫0t1((t2-s)α-1-(t1-s)α-1)φmax⁡{r*,k̃r*,l̃r*}(s)ds+∑i=1nbir*Γ(α-βi+1)(t2α-βi-t1α-βi+2(t2-t1)α-βi),
where
(3.9)a=∑i=1m∑j=1nγibjηiα-βjr*(1-∑i=1mγiηiα-1)Γ(α-βj+1)+∑i=1nbir*(1-∑i=1mγiηiα-1)Γ(α-βi+1)+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1φmax⁡{r*,k̃r*,l̃r*}(s)ds+1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1φmax⁡{r*,k̃r*,l̃r*}(s)ds.
The right-hand side of the above inequality tends to zeros independently ofh∈Br*, as t2-t1→0. This shows that N(Br*) is equicontinuous. Step 3.
N has a closed graph.
Letun→u*, hn∈N(un) and hn→h*. We will prove that h*∈N(u*). Now hn∈N(un) implies that there exists fn∈SF,un such that for each t∈[0,1],
(3.10)hn(t)=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1hn(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1hn(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1hn(s)ds+1Γ(α)∫0t(t-s)α-1fn(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1fn(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1fn(s)ds.
We need to show that there existsf*∈SF,u such that for each t∈[0,1],
(3.11)h*(t)=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1h*(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1h*(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1h*(s)ds+1Γ(α)∫0t(t-s)α-1f*(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1f*(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1f*(s)ds.
In (2.18), taking
(3.12)g(t)=1Γ(α)∫0t(t-s)α-1f(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1f(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1f(s)ds.
In view of Lemma 2.10, we know that for any f∈L([0,1],ℝ), the integral equation (2.18) has a unique solution, which we denote by hf. Because of this, we can define the operator
(3.13)Φ:L([0,1],R)⟶C([0,1],R),f⟼hf.
It is easy to verify that the operator Φ is linear. On the other hand, we can get
(3.14)‖hf‖≤1(1-λ)Γ(α)∫01(|t-s|)α-1|f(s)|ds+1(1-λ)(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1|f(s)|ds+∑i=1mγi(1-λ)(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1|f(s)|ds≤μ∫01|f(s)|ds,
where
(3.15)μ=1(1-λ)Γ(α)+∑i=1mγiηiα-1(1-λ)(1-∑i=1mγiηiα-1)Γ(α)+1(1-λ)(1-∑i=1mγiηiα-1)Γ(α),
that is, ∥Φ(f)∥≤μ∥f∥L1, which implies Φ is continuous. From Lemma 2.7, it follows that Φ∘SF,u is a closed graph operator. Moreover, we know hn∈Φ(SF,un). Since un→u*, it follows from Lemma 2.7 that (3.11) holds for some f*∈SF,u*.
Therefore,N is a compact multivalued map, u.s.c. with convex closed values. As a consequence of Lemma 2.8, we immediately conclude that N has a fixed-point u which is a solution of the problem (1.5) and (1.6). The proof is complete.As a direct corollary of Theorem3.1, we can immediately obtain the following result when the nonlinearity F has sublinear growth in the state variable.Corollary 3.2.
Suppose that (H1) and (H2) are satisfied. Then the problem (1.5) and (1.6) has at least one solution in C([0,1],ℝ), provided that (H4)
there exist functionsp(t),q(t),r(t),θ(t)∈L([0,1],ℝ), and σ1,σ2,σ3∈[0,1) such that(3.16)‖F(t,u,v,w)‖P≤p(t)|u|σ1+q(t)|v|σ2+r(t)|u|σ3+θ(t)
for each (t,u,v,w)∈[0,1]×ℝ×ℝ×ℝ.Proof.
We only need to verify that (H4) implies (H3) in Theorem3.1. Taking φ(t)=p(t)|u|σ1+q(t)|v|σ2+r(t)|u|σ3+θ(t), it is easy to prove
(3.17)τ=lim⁡r→∞inf⁡∫01max⁡{φr(s),φk̃r(s),φl̃r(s)}rds=0,μi=lim⁡r→∞inf⁡∫0ηi(ηi-s)α-1max⁡{φr(s),φk̃r(s),φl̃r(s)}rds=0,ω=lim⁡r→∞inf⁡∫01(1-s)α-1max⁡{φr(s),φk̃r(s),φl̃r(s)}rds=0,
because of σ1,σ2,σ3∈[0,1). Then (H3) is satisfied. The proof is finished.In the next part, we are concerned with the BVP (1.5) and (1.6) with nonconvex valued right-hand side. By using Covitz and Nadler fixed-point theorem, we obtain the following result.Theorem 3.3.
Assume that the following hypotheses hold:(A1)
F:[0,1]×ℝ×ℝ×ℝ→Pcl(ℝ); t→F(t,u,v,w) is measurable for each u,v,w∈ℝ;(A2)
There exist three functionsp,q,r∈L1([0,1],ℝ) such that for a.e. t∈[0,1] and u1,u2,v1,v2,w1,w2∈ℝ,(3.18)Hd(F(t,u1,v1,w1),F(t,u2,v2,w2))≤p(t)|u1-u2|+q(t)|v1-v2|+r(t)|w1-w2|,d(0,F(t,0,0,0))≤m(t).
Then the BVP (1.5) and (1.6) has at least one solution in C([0,1],ℝ) provided that
(3.19)ã0+b̃0+c̃0+∑i=1m+1ãi+k̃b̃i+l̃c̃i(1-∑i=1mγiηiα-1)<1-λ,
where
(3.20)ã0=1Γ(α)∫01p(s)ds,ãi=1Γ(α)∫0ηi(ηi-s)α-1p(s)ds,i=1,2,…,m+1,b̃0=1Γ(α)∫01q(s)ds,b̃i=1Γ(α)∫0ηi(ηi-s)α-1q(s)ds,i=1,2,…,m+1,c̃0=1Γ(α)∫01p(s)ds,c̃i=1Γ(α)∫0ηi(ηi-s)α-1r(s)ds,i=1,2,…,m+1,
ηm+1=1 and λ defined as in Lemma 2.10, k̃,l̃ defined by (3.1) in Theorem 3.1.Proof.
Transform the problem into a fixed-point problem. LetN:C([0,1],ℝ)→𝒫(C([0,1],ℝ) be defined as in Theorem 3.1. We shall show that N satisfies the assumptions of Lemma 2.9. The proof will be given in two steps.Step 1.
N(u)∈𝒫cl(C([0,1],ℝ)) for each u∈C([0,1],ℝ).
Indeed, let(un)n≥0∈N(u) such that un→ũ in C([0,1],ℝ). Then there exists gn∈SF,u such that for each t∈[0,1],
(3.21)un(t)=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1un(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1un(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1un(s)ds+1Γ(α)∫0t(t-s)α-1gn(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1gn(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1gn(s)ds.
For everyw∈F(t,u(t),(Tu)(t),(Su)(t)), from (A2), we have
(3.22)|w|≤d(0,F(t,0,0,0))+Hd(F(t,0,0,0),F(t,u(t),(Tu)(t),(Su)(t))).
Then,
(3.23)|gn(t)|≤m(t)+p(t)|u(t)|+q(t)|(Tu)(t)|+r(t)|(Su)(t)|a.e.t∈[0,1],
that is, gn(t)∈B(0,m(t)+p(t)|u(t)|+q(t)|(Tu)(t)|+r(t)|(Su)(t)|), where
(3.24)B(0,m(t)+p(t)|u(t)|+q(t)|(Tu)(t)|+r(t)|(Su)(t)|)={w∈R:|w|≤m(t)+p(t)|u(t)|+q(t)|(Tu)(t)|+r(t)|(Su)(t)|}:=Ψ(t).
It is clear thatΨ:[0,1]→𝒫cp,cv(ℝ) is a multivalued integrable bounded map. Since gn(·)∈Ψ(·),n≥1, we may pass to a subsequence if necessary to get that gn converges weakly to g in Lw1([0,1],ℝ). From Mauz's lemma [37], there exists g∈conv¯{gn(t):n≥1}, then there exists a subsequence {g¯n:n≥1} in conv¯{gn:n≥1}, such that g¯n converges strongly to g in L1([0,1],ℝ), which implies g∈L1([0,1],ℝ). Then for each t∈[0,1],
(3.25)un(t)⟶ũ(t)=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1ũ(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1ũ(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1ũ(s)ds+1Γ(α)∫0t(t-s)α-1g(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1g(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1g(s)ds.
So, ũ∈N(u)∈𝒫cl(C([0,1],ℝ)). Step 2.
There existsγ<1, such that
(3.26)Hd(N(u),N(u*))≤γ‖u-u*‖∞foreachu,u*∈C([0,1],R).
Letu,u*∈C([0,1],ℝ) and h∈N(u). Then there exists f(t)∈F(t,u(t),(Tu)(t),(Su)(t)) such that for each t∈[0,1](3.27)h(t)=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1h(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1h(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1h(s)ds+1Γ(α)∫0t(t-s)α-1f(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1f(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1f(s)ds.
It follows from (A2) that
(3.28)Hd(F(t,u(t),(Tu)(t),(Su)(t)),F(t,u*(t),(Tu*)(t),(Su*)(t)))≤p(t)|u(t)-u*(t)|+q(t)|(Tu)(t)-(Tu*)(t)|+r(t)|(Su)(t)-(Su*)(t)|,t∈[0,1].
Hence, there is g∈F(t,u*(t),(Tu*)(t),(Su*)(t)) such that
(3.29)|f(t)-g|≤p(t)|u(t)-u*(t)|+q(t)|(Tu)(t)-(Tu*)(t)|+r(t)|(Su)(t)-(Su*)(t)|,t∈[0,1].
ConsiderG:[0,1]→𝒫(ℝ), given by
(3.30)G(t)={g∈R:|f(t)-g|≤p(t)|u(t)-u*(t)|+q(t)|(Tu)(t)-(Tu*)(t)|+r(t)|(Su)(t)-(Su*)(t)|}.
Since the multivalued operatorG̃:t→G(t)∩F(t,u*(t),(Tu*)(t),(Su*)(t)) is measurable (see in [38, Proposition  III.4]), there exists a function g*, which is a measurable selection for G. So, g*(t)∈F(t,u*(t),(Tu*)(t),(Su*)(t)) and
(3.31)|f(t)-g*(t)|≤p(t)|u(t)-u*(t)|+q(t)|(Tu)(t)-(Tu*)(t)|+r(t)|(Su)(t)-(Su*)(t)|,t∈[0,1].
For eacht∈[0,1], define
(3.32)h*(t)=∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1h*(s)ds-∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1h*(s)ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1h*(s)ds+1Γ(α)∫0t(t-s)α-1g*(s)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1g*(s)ds-1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1g*(s)ds.
Then, fort∈[0,1](3.33)|h(t)-h*(t)|≤∑i=1m∑j=1nγibj(1-∑i=1mγiηiα-1)Γ(α-βj)∫0ηi(ηi-s)α-βj-1tα-1|h(s)-h*(s)|ds+∑i=1nbi(1-∑i=1mγiηiα-1)Γ(α-βi)∫01(1-s)α-βi-1tα-1|h(s)-h*(s)|ds+∑i=1nbiΓ(α-βi)∫0t(t-s)α-βi-1|h(s)-h*(s)|ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1|f(s)-g*(s)|ds+1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1|f(s)-g*(s)|ds+1Γ(α)∫0t(t-s)α-1|f(s)-g*(s)|ds≤λ‖h-h*‖+‖u-u*‖×(1Γ(α)∫0t(t-s)α-1(p(s)+q(s)k̃+r(s)l̃)ds+∑i=1mγi(1-∑i=1mγiηiα-1)Γ(α)∫0ηi(ηi-s)α-1tα-1(p(s)+q(s)k̃+r(s)l̃)ds+1(1-∑i=1mγiηiα-1)Γ(α)∫01(1-s)α-1tα-1(p(s)+q(s)k̃+r(s)l̃)ds).
So,(3.34)‖h-h*‖≤(ã0+b̃0+c̃01-λ+∑i=1m+1ãi+k̃b̃i+l̃c̃i(1-λ)(1-∑i=1mγiηiα-1))‖u-u*‖.
By an analogous relation, obtained by interchanging the roles ofu,u*, it follows that
(3.35)Hd(N(u),N(u*))≤C‖u-u*‖.
where C=(ã0+b̃0+c̃0)/(1-λ)+∑i=1m+1(ãi+k̃b̃i+l̃c̃i/(1-λ)(1-∑i=1mγiηiα-1))<1. Therefore, N is a contraction operator. With the help of Lemma 2.9, the problem (1.5) and (1.6) has at least one solution.Remark 3.4.
In (1.5) and (1.6), if bi=0, γj=0, i=1,…,n, j=1,…,m, k(t,s)=l(t,s)≡0. Then we obtain the following fractional boundary value problem of differential inclusions
(3.36)-Dαy(t)∈F(t,y(t)),a.e.t∈J=[0,1],0<α≤2,y(0)=y(1)=0
which has been studied in [26]. It is easy to see that our results (Theorems 3.1 and 3.3) partly generalize and improve the results obtained in it.



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*Source: 436045-2011-08-11.xml*