R E S E A R C H
Open Access
On a coupled system of fractional
compartmental models for a biological
system
Nana Jin and Shurong Sun
**Correspondence:
School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P.R. China
Abstract
In this paper, we investigate the existence of solutions for a coupled system of fractional compartmental models as differential inclusions with coupled nonlocal and integral boundary conditions, whose multivalued terms depend on lower-order fractional derivatives. By means of nonlinear alternative of Leray-Schauder type, continuous and measurable selection theorems together with Leray-Schauder degree theory, some sufficient conditions for the existence of solutions are presented, which extend some known results. Several examples are given to demonstrate the application of our main results.
MSC: 26A33; 34A12; 34A60; 34B10; 34A34
Keywords: fractional differential inclusions; nonlocal boundary value problems; existence of solutions; coupled system; compartmental model
1 Introduction
Fractional calculus has emerged as an interesting field of investigation in the last few decades. Fractional differential equations and inclusions arise in the mathematical model-ing of systems and processes occurrmodel-ing in many engineermodel-ing and scientific disciplines such as physics, chemistry, viscoelasticity, electrochemistry, electromagnetism, aerodynamics, economics, polymer rheology, control theory, signal and image processing, biophysics, blood flow phenomena, etc. [–]. In consequence, the subject of fractional differential equations and inclusions is gaining much importance and attention. For details and ex-amples, see [–] and the references therein.
On the other hand, coupled systems of fractional differential equations arise in various problems of applied nature, for example, HIV is a retrovirus that targets the CD+T lym-phocytes, which are the most abundant white blood cells of the immune system. Perelson [, ] developed a simple model for the primary infection with HIV. In this model, four categories of cells were defined: uninfected CD+T cells, latently infected CD+T cells, productively infected CD+T cells and virus population. AAM Arafal et al. introduced fractional-order model of infection of CD+T-cells. The coupled system is described by
the following set of fractional ordinary differential equations of orderα,α,α> :
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
Dα(T) =s–KVT–dT+bI,
Dα(I) =KVT– (b+δ)I,
Dα(I) =NδI–cV,
whereT,IandV denote the concentration of uninfected CD+T cells, infected CD+ T cells, and free HIV virus particles in the blood, respectively.δrepresents the death rate of infected T cells and includes the possibility of death by bursting of infected T cells, henceδd. The parameterbis the rate at which infected cells return to uninfected class, whilecis the death rate of virus andNis the average number of viral particles produced by an infected cell. Besides, [–] have established the existence and uniqueness for solutions of some systems of nonlinear fractional differential equations.
Compartmental analysis initially developed from studies of the uptake and distribution of radioactive tracers, but today it plays a fundamental role in many parts of medicine, bio-engineering and environmental science [–]. Compartmental models of pharmacoki-netics were recently generalized using fractional calculus to extend the governing systems for the form of fractional-order differential equations with specified initial conditions [, ].
In , Ivo Petráš and Richard L. Magin [] considered the two-compartmental phar-macokinetic model for oral drug administration as follows:
⎧ ⎨ ⎩
cDαq(t) = –k q(t), cDαq
(t) =kq(t) +kq(t),
(.)
with the initial condition
q() =d, q() =d, (.)
wherecDαi(i= , ) denotes the Caputo fractional derivative,α
,α> ,t≥,qi(t) (i= , ) denotes the amount of drug in a specific compartment,ki(i= , ) is the fractional rate of transfer to compartment. In their results, they assumedq() =q() = ifα,α∈(, ]. They pointed out a very effective numerical method for the solution of system (.)-(.), and the numerical solution is also created as a Matlab function.
Let q(t)∈F(t,q(t),q(t),cDγq(t)) and q(t)∈G(t,q
(t),cDδq(t),q(t)). We get the converted coupled system of nonlinear fractional differential inclusions:
⎧ ⎨ ⎩
cDαx(t)∈F(t,x(t),y(t),cDγy(t)), <α≤, <γ < ,
cDβy(t)∈G(t,x(t),cDδx(t),y(t)), <β≤, <δ< , (.)
supplemented with coupled nonlocal and integral boundary conditions of the form
⎧ ⎨ ⎩
x() =h(y), Ty(s)ds=μx(η), η∈(,T),
y() =φ(x), Tx(s)ds=μy(ξ), ξ∈(,T),
wheret∈J:= [,T],cDidenotes the Caputo fractional derivatives of orderi,i=α,β,γ,δ respectively,F,G:J×R→P(R),h,φ are given continuous functionals andμ,μ
are real constants.
To the best of our knowledge, there are few people to study the coupled system of frac-tional differential inclusions for compartmental models. For the part of theoretical results, our basic tools are the theory of fractional calculus, the methods and results for differential inclusions, and several fixed point theorems for multifunctions due to [, ].
Next, we compare our theoretical results with the other literature in details as follows:
(i) Our approach is adapted to the case when the right-hand sides have convex values as well as nonconvex for system (.)-(.), which was not considered in [, , , ].
(ii) The present work is an extension of a recent paper of the authors [], which was considered for a single-valued case. This adds to the uncertainty about the unknown function.
(iii) The fractional compartment form of (.)-(.) is a particular case of the coupled system (.)-(.) if we takeF={–kx(t)},G={kx(t) +ky(t)},
h(y) =φ(x) =μ=μ= .
(iv) Coupled system (.) is called a commensurate-order system ifα=β, otherwise it is an incommensurate-order system [].
(v) The coupled system for a single-valued map [] is a particular case of the corresponding multivalued map if we takeF={f},G={g}andf,gare continuous functions.
(vi) We adopt the ideas of selection theorems to reduce the condition that the right-hand side has convex values.
(vii) The mentioned methods are also useful for further investigations concerning the existence of solutions of a coupled system of fractional differential inclusions and other types.
The structure of this paper is as follows. In Section , we introduce some notations, definitions of fractional calculus and multivalued maps, together with some basic lemmas to prove our main results. In Section , we will consider the sufficient conditions for the existence results. Finally, in Section , several examples are given to illustrate our main results.
2 Preliminaries
This section is devoted to presenting some notation and preliminary lemmas that will be used in the proofs of the main results [, ].
Let (X, · ) be a normed space, and letYbe a subset ofX. We denote
(i) P(X) ={Y⊆X:Y=∅}; (ii) Pcl(X) ={Y∈P(X) :Yclosed};
(iii) Pb(X) ={Y∈P(X) :Ybounded};
(iv) Pcp(X) ={Y∈P(X) :Ycompact};
(v) Pcv(X) ={Y∈P(X) :Yconvex};
(vi) Pcp,cv(X) ={Y∈P(X) :Ycompact, convex}.
Consider the Pompeiu-Hausdorff metricHd:P(X)×P(X)→R∪ {∞}given by
Hd(A,B) =max
sup
a∈A
d(a,B),sup
b∈B
whered(A,b) =infa∈Ad(a;b),d(a,B) =infb∈Bd(a;b). Then (Pb,cl(X),Hd) is a metric space and (Pcl(X),Hd) is a generalized (complete) metric space (see [, ]).
A multivalued mapF:X→P(X)
(i) is convex (closed) valued ifF(x)is convex (closed) for allx∈X;
(ii) is bounded on bounded sets ifF(B) = x∈BF(x)is bounded inXfor allB∈Pb(X)
(i.e.,supx∈B{sup{|y|:y∈F(x)}}<∞);
(iii) is called upper semicontinuous(u.s.c)onXif, for eachx∈X, the setF(x)is a
nonempty closed subset ofXand if, for each open setNofXcontainingF(x), there exists an open neighborhoodNofxsuch thatF(N)⊆N;
(iv) is called lower semicontinuous if the set{x∈X:F(x)∩O=∅}is open for each open setOinX;
(v) is said to be completely continuous ifF(B)is relatively compact for every B∈Pb(X);
(vi) is said to be measurable if, for everyy∈R, the function
t→dy,F(t)=inf|y–z|:z∈G(t)
is measurable;
(vii) has a fixed point if there isx∈Xsuch thatx∈F(x). The fixed point set of the multivalued operatorFwill be denoted byFixF.
Definition .([]) The fractional integral of orderq> of a Lebesgue integrable func-tionf(·) : (,∞)→Ris defined by
Iqf(t) = (q)
t
f(s) (t–s)–qds,
provided that the right-hand side is pointwise defined on (,∞) and (·) is the (Euler) gamma function defined by (q) =∞tq–e–tdt.
Definition .([]) The Caputo fractional derivative of orderq> of a functionf(·) : [,∞)→Ris defined by
cDqf(t) = (n–q)
t
(t–s)n–q–f(n)(s)ds,
where n= [q] + . It is assumed implicitly that f(·) isntimes differentiable whose nth derivative is absolutely continuous.
Lemma .([]) Letα> .If we assume h∈ACn(, ),then the differential equation
cDα
h(t) =
has a unique solution
h(t) =c+ct+ct+· · ·+cn–tn–,
In view of Lemma.,it follows that
IαcDαh(t) =h(t) +c+ct+ct+· · ·+cn–tn–,
for some ci∈R,i= , , , . . . ,n– ,n is the smallest integer greater than or equal toα.
Lemma .([]) Letω,z∈L[,T]and x,y∈AC[,T].Then the unique solution of the
problem
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
cDαx(t) =ω(t), t∈(,T), <α≤,
cDβy(t) =z(t), t∈(,T), <β≤,
x() =h(y), Ty(s)ds=μx(η),
y() =φ(x), Tx(s)ds=μy(ξ), η,ξ∈(,T),
(.)
is
x(t) =
t
(t–s)α–
(α) ω(s)ds+ (σt+ )h(y) +tσφ(x)
+ t
μξ
μ
η
(η–s)α–
(α) ω(s)ds–
T
(T–s)β
(β+ )z(s)ds
+T
μ
ξ
(ξ–s)β– (β) z(s)ds–
T
(T–s)α
(α+ )ω(s)ds
and
y(t) =
t
(t–s)β–
(β) z(s)ds+ (σt+ )φ(x) +tσh(y)
+ t
μη
μ
ξ
(ξ–s)β–
(β) z(s)ds–
T
(T–s)α
(α+ )ω(s)ds
+T
μ
η
(η–s)α–
(α) ω(s)ds–
T
(T–s)β
(β+ )z(s)ds
,
where
=T – μ
μηξ
,
σ=
μμξ–T
,
σ=
Tμ(T– ξ)
,
σ=
μμη–T
,
σ=
Tμ(T– η)
.
and there exist functions f,g ∈ L(J,R) such that f(t)∈F(t,x(t),y(t),cDγy(t)), g(t) ∈
G(t,x(t),cDδx(t),y(t)) a.e. ont∈Jand
x(t) =
t
(t–s)α–
(α) f(s)ds+ (σt+ )h(y) +tσφ(x)
+ t
μξ
μ
η
(η–s)α– (α) f(s)ds–
T
(T–s)β
(β+ )g(s)ds
+T
μ
ξ
(ξ–s)β–
(β) g(s)ds–
T
(T–s)α
(α+ )f(s)ds
and
y(t) =
t
(t–s)β–
(β) g(s)ds+ (σt+ )φ(x) +tσh(y)
+ t
μη
μ
ξ
(ξ–s)β–
(β) g(s)ds–
T
(T–s)α
(α+ )f(s)ds
+T
μ
η
(η–s)α–
(α) f(s)ds–
T
(T–s)β
(β+ )g(s)ds
,
whereandσi(i= , , , ) are defined in Lemma ..
Definition . A multivalued mapF:X→Pcl(X) is called
() γ-Lipschitz if there existsγ> such that
Hd
F(x),F(y)≤γd(x,y) for eachx,y∈X;
() a contraction if it isγ-Lipschitz withγ < .
Definition . A multivalued mapF:J×R→P(R) is said to beL-Carathéodory if
() t→F(t,x,y,z)is measurable for eachx,y,z∈R;
() (x,y,z)→F(t,x,y,z)is upper semicontinuous for almostt∈J; () for eachl> , there existsϕl∈L(J,R+)such that
F(t,x,y,z)=sup|v|:v∈F(t,x,y,z)≤ϕl(t)
for allx ≤l,y ≤l,z ≤land for a.e.t∈J.
Lemma . ([]) Let X be a Banach space. Let F : J × X →Pcp,cv(X) be an L
-Carathéodory multivalued map and T be a linear continuous mapping from L(J,X)to
C(J,X).Then the operator
T◦SF:C(J,X)→Pcp,cv
C(J,X), y→(T◦SF)(y) =T(SF,y)
is a closed graph operator in C(J,X)×C(J,X).
Lemma .(Nonlinear alternative of Leray-Schauder type []) Let X be a Banach space,
C be a closed convex subset of X,andU be an open subset of C with∈U.Suppose that F:U¯→Pcp,cv(C)is an upper semicontinuous compact map.Then either()F has a fixed
point inU¯,or()there is x∈∂Uandλ∈(, )such that x∈λF(x).
Lemma .(Covitz and Nadler []) Let(X,d)be a complete metric space.If F :X→ Pcl(X)is a contraction,then F has a fixed point,i.e.,a point z∈X such that z∈F(z).
3 Main results
In this section, we will give some existence results for the coupled system (.)-(.). Let C(J,R) denote the space of all continuous functions defined on J. Let X ={x:
x ∈ C(J,R) andcDδx ∈ C(J,R)} be a Banach space endowed with the norm x
X =
x+cDδx=max
t∈J|x(t)|+maxt∈J|cDδx(t)|, where <δ < [], and let Y ={y:
y∈ C(J,R) andcDγy∈C(J,R)} be a Banach space equipped with the norm y
Y =
y+cDγy=max
t∈J|y(t)|+maxt∈J|cDγy(t)|, where <γ < . Obviously the product space (X×Y, · X×Y) is a Banach space with the norm(x,y)X×Y =xX+yY for (x,y)∈X×Y.
For each (x,y)∈X×Y, define the sets of selections ofF,Gby
SF,(x,y)=
f ∈L(J,R) :f(t)∈Ft,x(t),y(t),cDγy(t)for a.e.t∈J
and
SG,(x,y)=
g∈L(J,R) :g(t)∈Gt,x(t),cDδx(t),y(t)for a.e.t∈J.
In view of Lemma ., we define operatorsN:X×Y→P(X×Y) andN:X×Y→
P(X×Y) as
N(x,y) =h∈X×Y: there existf ∈SF,(x,y)andg∈SG,(x,y)
such thath(x,y)(t) =A(t,x,y),∀t∈J (.)
and
N(x,y) =h∈X×Y: there existf ∈SF,(x,y)andg∈SG,(x,y)
such thath(x,y)(t) =A(t,x,y),∀t∈J
, (.)
where
A(t,x,y) =
t
(t–s)α–
(α) f(s)ds+ (σt+ )h(y) +tσφ(x)
+ t
μξ
μ
η
(η–s)α–
(α) f(s)ds–
T
(T–s)β
(β+ )g(s)ds
+T
μ
ξ
(ξ–s)β–
(β) g(s)ds–
T
(T–s)α
(α+ )f(s)ds
and of the coupled system (.)-(.).
For computational convenience, we introduce the notations:
P=
3.1 The Carathéodory case
First we consider the case whenF,Gare convex valued. We give the following conditions:
(H) F,G:J×R→P
(H) There existm,m∈C(J,R+)andϕ,ϕ,ψ,ψ,ρ,ρ: [,∞)→(,∞)
continuous, nondecreasing such that
F(t,x,y,z)=sup|f|:f ∈F(t,x,y,z)
≤m(t)ϕ
|x|+ψ
|y|+ρ
|z|
and
G(t,x,y,z)=sup|g|:g∈G(t,x,y,z)
≤m(t)ϕ
|x|+ψ
|y|+ρ
|z|
forx,y,z∈Rand a.e.t∈J;
(H) h,φare continuous functionals withh() =φ() = , and there exist constants L> ,L> such that
h(x) –h(x)≤Lx–x,
φ(x) –φ(x)≤Lx–x, ∀x,x∈C(J,R).
Theorem . Assume that(H)-(H)are satisfied and there exists K> such that
K>
i=
imi
ϕi(K) +ψi(K) +ρi(K)
( –)–. (.)
If,in addition,< ,where
=P+P+
T–δ
( –δ)P+
T–γ
( –γ)P,
=P+P+
T–δ
( –δ)P+
T–γ
( –γ)P,
=Q+Q+
T–δ
( –δ)Q+
T–γ
( –γ)Q,
then the coupled system(.)-(.)has at least one solution on J.
Proof Consider the operatorsN:X×Y→P(X×Y) andN:X×Y→P(X×Y) defined by (.) and (.). From (H), we have that for each (x,y)∈X×Y, the setsSF,(x,y)andSG,(x,y) are nonempty []. For (x,y)∈X×Y, letf∈SF,(x,y),g∈SG,(x,y)and
h(x,y)(t)
=
t
(t–s)α–
(α) f(s)ds+ (σt+ )h(y) +tσφ(x)
+ t
μξ
μ
η
(η–s)α–
(α) f(s)ds–
T
(T–s)β
(β+ )g(s)ds
+T
μ
ξ
(ξ–s)β–
(β) g(s)ds–
T
(T–s)α
(α+ )f(s)ds
and
We will show thatN satisfies the requirements of the nonlinear alternative of Leray-Schauder type. The proof will be given in five steps.
and
Based on assumptions (H), we find the following estimates:
h(y)≤h(y) –h()+h()≤LyY≤Lr, ∀y∈R,
φ(x)≤φ(x) –φ()+φ()≤LxX≤Lr, ∀x∈R.
Using these estimates, we get
h(x,y)(t)
≤ Tα
(α+ )· m
In a similar manner, for anyt∈J, we obtain
In consequence, we get
Hence, we obtain
Step . Nmaps bounded sets into equicontinuous sets inX×Y.
Analogously, one can obtain
h(x,y)(t) –h(x,y)(t)
≤t
β
–t
β
β (β)· m
ϕ(r) +ψ(r) +ρ(r)+|σ|Lr+|σ|Lr
(t–t)
+(t–t)
β
β (β) · m
ϕ(r) +ψ(r) +ρ(r)
+t–t
||
|μμ|ηTβ (β+ ) · m
ϕ(r) +ψ(r) +ρ(r)
+|μ|ηT
α+ (α+ ) · m
ϕ(r) +ψ(r) +ρ(r)
+ |μ|T
α+ (α+ )· m
ϕ(r) +ψ(r) +ρ(r)
+ T
β+
(β+ )· m
ϕ(r) +ψ(r) +ρ(r)
,
h(x,y)(t) –h(x,y)(t)
≤ t
β–
–t
β–
(β– ) (β– )· m
ϕ(r) +ψ(r) +ρ(r)
and
cDγ
h(x,y)(t) –cDγh(x,y)(t)
≤ t
–γ
–t
–γ
( –γ) ( –γ)
P· m
ϕ(r) +ψ(r) +ρ(r)
+P· m
ϕ(r) +ψ(r) +ρ(r)
+Qr
.
Hence|h(x,y)(t) –h(x,y)(t)| → ast→t. Therefore, the operatorN(x,y) is equicon-tinuous.
From the foregoing arguments, we infer that the operatorN(x,y) is completely contin-uous by the Arzelá-Ascoli theorem.
Step . Nhas a closed graph.
Letting (xn,yn)→(x∗,y∗), (hn,hn)∈N(xn,yn) and (hn,hn)→(h∗,h∗), we need to show
(h∗,h∗)∈N(x∗,y∗). Now (hn,hn)∈N(xn,yn) implies that there existfn∈SF,(xn,yn)andgn∈
SG,(xn,yn)such that for allt∈J,
hn(xn,yn)(t) =
t
(t–s)α–
(α) fn(s)ds+ (σt+ )h(y) +tσφ(x)
+ t
μξ
μ
η
(η–s)α–
(α) fn(s)ds–
T
(T–s)β
(β+ )gn(s)ds
+T
μ
ξ
(ξ–s)β–
(β) gn(s)ds–
T
(T–s)α
(α+ )fn(s)ds
Now we set
U=(x,y)∈X×Y:(x,y)X×Y<K.
Clearly,U is an open subset ofX×Y and (, )∈U. As a consequence of Steps -, to-gether with the Arzelá-Ascoli theorem, we can conclude thatN:U→Pcp,cv(X)×Pcp,cv(Y) is upper semicontinuous and completely continuous. From the choice of U, there is no (x,y)∈∂U such that (x,y)∈λN(x,y) for someλ∈(, ). Therefore, by Theorem ., we deduce thatNhas a fixed point (x,y)∈U, which is a solution of the coupled system
(.)-(.). This completes the proof.
3.2 The lower semicontinuous case
Now we study the case whenF,Gare not necessarily convex valued. In this result, we need to give the following conditions:
(H) F,G:J×R→Pcp(R)are multivalued maps such that
() (t,x,y,z)→F(t,x,y,z)and(t,x,y,z)→G(t,x,y,z)are⊗B⊗B⊗B measurable;
() (x,y,z)→F(t,x,y,z)and(x,y,z)→G(t,x,y,z)are lower semicontinuous for a.e.t∈J.
Theorem . Assume that(H)-(H)and relation(.)hold.Then the coupled system
(.)-(.)has at least one solution on J.
Proof From (H), (H) and [], Lemma ., maps
F:X→P
L(J,R), x→F(x,y) =SF,(x,y),
F:Y→P
L(J,R), y→F(x,y) =SG,(x,y),
are lower semicontinuous and have nonempty closed and decomposable values. Then from the selection theorem due to Bressan and Colombo [], there exist continuous functionsf :X→L(J,R) andg:Y→L(J,R) such thatf ∈F(x,y) andg∈F(x,y) for allx∈X,y∈Y. That is to say, we havef(t,x(t),y(t),cDγy(t))∈F(t,x(t),y(t),cDγy(t)) and
g(t,x(t),cDδx(t),y(t))∈G(t,x(t),cDδx(t),y(t)) for a.e.t∈J. Now consider the problem
⎧ ⎨ ⎩
cDαx(t) =f(t,x(t),y(t),cDγy(t)), t∈J,
cDβy(t) =g(t,x(t),cDδx(t),y(t)), t∈J, (.)
with the boundary conditions (.). Note that if (x,y)∈X×Yis a solution of the coupled system (.), then (x,y) is a solution to the coupled system (.)-(.).
A solution of the boundary value problem (.), (.) is then reformulated as a fixed point problem for the operatorN:X×Y→X×Ydefined by
where
It can easily be shown thatNis continuous and completely continuous and satisfies all the conditions of the Leray-Schauder nonlinear alternative for single-valued maps []. The remaining part of the proof is similar to that of Theorem ., so we omit it. This
completes the proof.
3.3 The Lipschitz case
In this section, we need to give the following conditions:
(H) F,G:J×R→P
cp(R)are multivalued maps such that
where situation,Nsatisfies the requirements of Lemma ..
From (H)(), we deduce
Hd
Ft,x(t),y(t),cDγy(t),Ft,x(t),y(t),cDγy(t)
≤m(t)x(t) –x(t)+y(t) –y(t)+cDγy(t) –cDγy(t)
and
Hd
Gt,x(t),cDδx(t),y(t),Gt,x(t),cDδx(t),y(t)
≤m(t)x(t) –x(t)+cDδx(t) –cDδx(t)+y(t) –y(t).
Hence, for a.e. t∈J, there exist f ∈F(t,x(t),y(t),cDγy(t)) andg∈G(t,x(t),cDδx(t),y(t))
such that
f(t) –f≤m(t)x(t) –x(t)+y(t) –y(t)+cDγy(t) –cDγy(t) (.)
and
g(t) –g≤m(t)x(t) –x(t)+cDδx(t) –cDδx(t)+y(t) –y(t). (.)
Consider the multivalued mapsV,V:J→P(R) given by
V(t) =
f ∈R:
|f(t) –f| ≤m(t)(|x(t) –x(t)|+|y(t) –y(t)|+|cDγy(t) –cDγy(t)|)
and
V(t) =
g∈R:
|g(t) –g| ≤m(t)(|x(t) –x(t)|+|cDδx(t) –cDδx(t)|+|y(t) –y(t)|)
.
Define
κ(t) =m(t)x(t) –x(t)+y(t) –y(t)+cDγy(t) –cDγy(t),
κ(t) =m(t)x(t) –x(t)+cDδx(t) –cDδx(t)+y(t) –y(t).
Sincef(t),g(t),κ(t),κ(t) are measurable, [], Theorem III., implies thatVandV are measurable. It follows from (H) that the mapst→F(t,x(t),y(t),cDγy(t)) andt→
G(t,x(t),cDδx(t),y(t)) are measurable. Hence, by (.)-(.) and [], Proposition ..,
the multivalued maps t → V(t) ∩ F(t,x(t),y(t),cDγy(t)) and t → V(t)∩ G(t,x(t), cDδx(t),y(t)) are measurable and nonempty closed valued. Therefore, we can findf
(t)∈
F(t,x(t),y(t),cDγy(t)) andg(t)∈G(t,x(t),cDδx(t),y(t)) such that for a.e.t∈J,
f(t) –f(t)≤m(t)x(t) –x(t)+y(t) –y(t)+cDγy(t) –cDγy(t)
and
+|μμ|T
α+ξ
|| (α+ ) · m
x–xX+y–yY
+ |μ|T
β+ξ
|| (β+ )· m
x–xX+y–yY
+ |μ|T
β+
|| (β+ )· m
x–xX+y–yY
+ T
α+
|| (α+ )· m
x–xX+y–yY
+maxL|σ+ |,L|σ|
x–xX+y–yY
=·
x–xX+y–yY
,
h(x,y)(t) –h(x,y)(t)
≤ t
(t–s)α–
(α– )f(s) –f(s)ds+
|μμ|ξ
|| η
(η–s)α–
(α) f(s) –f(s)ds
+|μ|ξ
|| T
(T–s)β
(β+ )g(s) –g(s)ds+
|μ|T ||
ξ
(ξ–s)β–
(β) g(s) –g(s)ds
+ T
|| T
(T–s)α
(α+ )f(s) –f(s)ds+|σ| ·h(y) –h(y)+|σ| ·φ(x) –φ(x)
≤·
x–xX+y–yY
and
cDδh
(x,y)(t) –cDδh(x,y)(t)
≤
(t–s)–δ
( –δ)h(x,y)
(t) –h(x,y)(t)ds
≤ T–δ
( –δ)·
x–xX+y–yY
,
we obtain
h(x,y) –h(x,y)X≤
+
T–δ
( –δ)
·x–xX+y–yY
.
In a similar manner, we can have
h(x,y)(t) –h(x,y)(t)≤·
x–xX+y–yY
,
h(x,y)(t) –h(x,y)(t)≤·
x–xX+y–yY
and
cDγ
h(x,y)(t) –cDγh(x,y)(t)≤ T –γ
( –γ)·
x–xX+y–yY
.
We deduce that
h(x,y) –h(x,y)Y≤
+
T–γ
( –γ)
·x–xX+y–yY
Thus,
(h–h,h–h)X×Y
≤
+
T–δ
( –δ)++
T–γ
( –γ)
x–xX+y–yY
.
Denote
τ=+
T–δ
( –δ)++
T–γ
( –γ).
By using an analogous relation obtained by interchanging the roles of (x,y) and (x,y), we get
Hd
N(x,y),N(x,y)≤τx–xX+y–yY
.
Therefore, from (.), Lemma . implies thatNhas a fixed point, which is a solution of
the coupled system (.)-(.). This completes the proof.
Remark . Iff(t,x(t),y(t),cDγy(t)) andg(t,x(t),cDδx(t),y(t)) are continuous functions,
F(t,x(t),y(t),cDγy(t)) only contains the function f(t,x(t),y(t),cDγy(t)) and G(t,x(t),
cDδx(t),y(t)) only contains the functiong(t,x(t),cDδx(t),y(t)). Then the existence results
Theorem . and Theorem . are just the ones, respectively, in paper [].
Remark . The new existence results for a class of second-order coupled system dif-ferential inclusions with coupled nonlocal and integral boundary conditions follow as a special case by takingα=β= in the results of this paper.
4 Examples for fractional compartmental models
In this section, we will give some examples to illustrate our main results.
Example . Consider the following fractional differential inclusions:
⎧ ⎨ ⎩
cDx(t)∈F(t,x(t),y(t),cDγy(t)), t∈[, ],
cDy(t)∈G(t,x(t),cDδx(t),y(t)), (.)
with boundary conditions of the form
⎧ ⎨ ⎩
x() = y(t), Ty(s)ds=x(),
y() = x(t), Tx(s)ds=y(), (.)
whereα= , β=, γ = ,δ= ,T = ,μ=μ=,η=ξ =, L= ,L= , and
F,G: [, ]×R→Rare multivalued maps given by
Ft,x,y,cDγy=
f ∈R: ≤f≤ |x|
+|x|+siny+tan
–cDy+ e–t +t
,
Gt,x,cDδx,y=
g∈R: ≤g≤sinx+ cDx
+cDx
+tan–y+cost+
It is clear thatF,GareL-Carathéodory and have convex values satisfying
F(t,x,y,z)=sup|f|:f ∈F(t,x,y,z)≤, for each (t,x,y,z)∈J×R,
G(t,x,y,z)=sup|g|:f ∈G(t,x,y,z)≤, for each (t,x,y,z)∈J×R,
withm(t) =m(t) =ϕ(|x|) =ϕ(|x|) =ρ(|z|)≡,ψ(|y|) =ψ(|y|) =ρ(|z|)≡.
Using the given data, we find thatP≈.,P≈.,P≈.,P≈.,
P≈.,P≈.,P≈.,P≈.,Q≈.,Q≈.,Q≈ .,Q≈.,≈.,≈.,≈. < .
Furthermore, letKbe any number satisfying
K>.× + .×
– . > ..
Clearly, all the conditions of Theorem . are satisfied. So there exists at least one solu-tion of problem (.)-(.) on [, ].
Example . Consider the following coupled system of fractional compartmental mod-els:
⎧ ⎨ ⎩
cDx(t)∈F(t,x(t),y(t),cDγy(t)), t∈[, ],
cDy(t)∈G(t,x(t),cDδx(t),y(t)), (.)
with boundary conditions of the form
⎧ ⎨ ⎩
x() = , Ty(s)ds= ,
y() = , Tx(s)ds= , (.)
where α=β= ,γ =δ=, T= , μ=μ= , η=ξ =,L= ,L= andF,G: [, ]×R→Rare multivalued maps given by
F(t,x,y,z) =–kx(t)
, G(t,x,y,z) =kx(t) +ky(t)
, k,k,k∈R+.
It is clear thatF,Gsatisfy (H) and
F(t,x,y,z)≤kx, for each (t,x,y,z)∈J×R,
G(t,x,y,z)≤kx+kx ≤max{k,k}
x+y,
for each (t,x,y,z)∈J×R,
with m(t)≡ k, m(t)≡ max{k,k}, ϕ(|x|) =ϕ(|x|)≡ x, ψ(|y|)≡ y, ψ(|y|) =
ρ(|z|) =ρ(|z|)≡. Becausex(t),y(t) denote the amount of a drug in a specific com-partment in [], xandy are constants. Lettingϕ(|x|) =ϕ(|x|)≡c,ψ(|y|)≡c (c,care constants). Using the given data, we find thatP≈.,P≈,P≈.,
P≈,P≈.,P≈,P≈.,P≈,Q≈.,Q≈.,Q≈.,
Furthermore, letKbe any number satisfying
K>.kc+ .max{k,k} ·(c+c)
– . .
Clearly, all the conditions of Theorem . are satisfied. So there exists at least one solu-tion of problem (.)-(.) on [, ].
At the same time, we can see that the numerical solutions and simulations for problem (.)-(.) (single-valued) are obtained in []. That is, the conclusions we have gained are correct.
Example . Consider problem (.)-(.), whereF,G: [, ]×R→R+are multivalued maps given by
F(t,x,y,z) =
,|sinx+siny+sinz| ( +t)
,
G(t,x,y,z) =
,|cosx+cosy+cosz| ( +t)
.
Now
sup|f|:f ∈F(t,x,y,z)≤
( +t) ≤, for each (t,x,y,z)∈[, ]×R ,
sup|g|:g∈G(t,x,y,z)≤
( +t)≤, for each (t,x,y,z)∈[, ]×R ,
and
dH
F(t,x,y,z),F(t,x,y,z)≤ ( +t)
|x–x|+|y–y|+|z–z|
,
dH
G(t,x,y,z),G(t,x,y,z)
≤
( +t)
|x–x|+|y–y|+|z–z|
.
Herem(t) =(+t),m(t) =(+t) withm= andm= .
Using the given data, we find that≈.,≈.,≈.,≈. and
+
T–δ
( –δ)++
T–γ
( –γ)≈. < .
The compactness ofF,Gtogether with the above calculations leads to the existence of solution of problem (.)-(.) by Theorem ..
5 Conclusion
In this article, we present existence conditions of solutions, which are the prerequisites for solving the numerical solutions. Before solving the numerical solution, we can know whether there is a solution. This reduces a lot of unnecessary calculations to a certain extent. It is very significant for fractional compartmental model for a biological system.
Competing interests
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.
This research is supported by the Natural Science Foundation of China (11571202), by Shandong Provincial Natural Science Foundation (ZR2016AM17), and by Graduate Innovation Foundation of University of Jinan (YCXS15005).
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 23 November 2016 Accepted: 5 April 2017
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