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## differential equations with fractional order

### Khalid Hilal and Ahmed Kajouni

*

*Correspondence:

kajjouni@gmail.com Laboratoire de Mathématiques Appliquées & Calcul Scientiﬁque, Université Sultan Moulay Slimane, BP 523, Beni Mellal, 23000, Morocco

Abstract

This note is motivated by some papers treating the fractional hybrid diﬀerential equations involving Riemann-Liouville diﬀerential operators of order 0 <

### α

< 1. An existence theorem for this equation is proved under mixed Lipschitz and

Carathéodory conditions. Some fundamental fractional diﬀerential inequalities which are utilized to prove the existence of extremal solutions are also established.

Necessary tools are considered and the comparison principle is proved, which will be useful for further study of qualitative behavior of solutions.

Keywords: hybrid diﬀerential equation; Caputo fractional derivative; maximal and minimal solutions

1 Introduction

During the past decades, fractional diﬀerential equations have attracted many authors (see [–]). The diﬀerential equations involving fractional derivatives in time, compared with those of integer order in time, are more realistic to describe many phenomena in nature (for instance, to describe the memory and hereditary properties of various materials and processes), the study of such equations has become an object of extensive study during recent years.

The quadratic perturbations of nonlinear diﬀerential equations have attracted much attention. We call them fractional hybrid diﬀerential equations. There have been many works on the theory of hybrid diﬀerential equations, and we refer the readers to the arti-cles [–].

Dhage and Lakshmikantham [] discussed the following ﬁrst order hybrid diﬀerential equation:

⎧ ⎨ ⎩

d dt[

x(t)

f(t,x(t))] =g(t,x(t)) a.e.tJ= [,T], x(t) =x∈R,

wherefC(J×R,R\{}) andgC(J×R,R). They established the existence, uniqueness results and some fundamental diﬀerential inequalities for hybrid diﬀerential equations ini-tiating the study of theory of such systems and proved utilizing the theory of inequalities, its existence of extremal solutions and comparison results.

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Zhaoet al.[] have discussed the following fractional hybrid diﬀerential equations in-volving Riemann-Liouville diﬀerential operators:

⎧ ⎨ ⎩

Dq[ x(t)

f(t,x(t))] =g(t,x(t)) a.e.tJ= [,T], x() = ,

wherefC(J×R,R\{}) andgC(J×R,R). The authors of [] established the existence theorem for fractional hybrid diﬀerential equations and some fundamental diﬀerential inequalities, they also established the existence of extremal solutions.

Benchohraet al.[] discussed the following boundary value problems for diﬀerential equations with fractional order:

⎧ ⎨ ⎩

cDαy(t) =f(t,y(t)) for eachtJ= [,T],  <α< ,

ay() +by(T) =c,

wherecDαis the Caputo fractional derivative,f : [,T]×RRis a continuous function, a,b,care real constants witha+b= .

From the above works, we develop the theory of boundary fractional hybrid diﬀerential equations involving Caputo diﬀerential operators of order  <α< . An existence theorem for boundary fractional hybrid diﬀerential equations is proved under mixed Lipschitz and Carathéodory conditions. Some fundamental fractional diﬀerential inequalities which are utilized to prove the existence of extremal solutions are also established. Necessary tools are considered and the comparison principle is proved, which will be useful for further study of qualitative behavior of solutions.

2 Boundary value problems for hybrid differential equations with fractional order

In this section, we introduce notations, deﬁnitions, and preliminary facts which are used throughout this paper. ByX=C(J,R) we denote the Banach space of all continuous func-tions fromJ= [,T] intoRwith the norm

y=supy(t),tJ,

and letC(J×R,R) denote the class of functionsg:J×R→Rsuch that

(i) the maptg(t,x)is measurable for eachx∈R, and (ii) the mapxg(t,x)is continuous for eachtJ.

The classC(J×R,R) is called the Carathéodory class of functions onJ×Rwhich are Lebesgue integrable when bounded by a Lebesgue integrable function onJ.

ByL(J;R) denote the space of Lebesgue integrable real-valued functions onJequipped with the norm · Ldeﬁned by

xL=

T

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Deﬁnition .[] The fractional integral of the functionhL([a,b],R+) of orderα

R+is deﬁned by

ah(t) =

t

a

(t–s)α– (α) h(s)ds,

whereis the gamma function.

Deﬁnition .[] For a functionhgiven on the interval [a,b], the Caputo fractional-order derivative ofh, is deﬁned by

cDα

a+h

(t) =  (nα)

t

a

(t–s)n–α–h(n)(s)ds,

wheren= [α] +  and [α] denotes the integer part ofα.

In this paper we consider the boundary value problems for hybrid diﬀerential equations with fractional order (BVPHDEF for short) involving Caputo diﬀerential operators of or-der  <α< ,

⎧ ⎨ ⎩

( x(t)

f(t,x(t))) =g(t,x(t)) a.e.tJ= [,T], af(,x()x())+bf(Tx,(xT()T))=c,

()

wherefC(J×R,R\{}),gC(J×R,R) anda,b,care real constants witha+b= . By a solution of BVPHDEF () we mean a functionxC(J,R) such that

(i) the functiontf(xt,x)is continuous for eachx∈R, and (ii) xsatisﬁes the equations in().

The theory of strict and nonstrict diﬀerential inequalities related to the ODEs and hybrid diﬀerential equations is available in the literature (see [, ]). It is known that diﬀerential inequalities are useful for proving the existence of extremal solutions of the ODEs and hybrid diﬀerential equations deﬁned onJ.

3 Existence result

In this section, we prove the existence results for the boundary value problems for hybrid diﬀerential equations with fractional order () on the closed and bounded intervalJ= [,T] under mixed Lipschitz and Carathéodory conditions on the nonlinearities involved in it.

We deﬁne the multiplication inXby

(xy)(t) =x(t)y(t) forx,yX.

Clearly,X=C(J;R) is a Banach algebra with respect to the above norm and multiplication in it.

Theorem .[] Let S be a non-empty,closed convex and bounded subset of the Banach algebra X,and let A:XX and B:XX be two operators such that

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(c) x=AxByxSfor allyS,and

(d) Mψ(r) <r,whereM=B(S)=supB(x):xS;

then the operator equation AxBx=x has a solution in S.

We make the following assumptions:

(H) The functionxf(tx,x) is increasing inRalmost everywhere fortJ.

(H) There exists a constantL> such that f(t,x) –f(t,y)L|xy|

for alltJandx,y∈R.

(H) There exists a functionhL(J,R)such that g(t,x)h(t) a.e.tJ

for allx∈R.

Lemma . Assume that hypothesis(H)holds and a,b,c are real constants with a+b= . Then,for any hL(J;R),the function xC(J;R)is a solution of the BVPHDEF

⎧ ⎨ ⎩

( x(t)

f(t,x(t))) =h(t) a.e. tJ= [,T],

af(,x()x())+bf(Tx(,xT()T))=c ()

if and only if x satisﬁes the hybrid integral equation

x(t) =f t,x(t)(α)

t

(t–s)α–h(s)ds

–  a+b

b (α)

T

(T–s)α–h(s)dsc

. ()

Proof Assume thatxis a solution of problem (). By deﬁnition,f(xt,(xt()t))is continuous. Ap-plying the Caputo fractional operator of the orderα, we obtain the ﬁrst equation in (). Again, substitutingt=  andt=Tin () we have

x() f(,x())=

– a+b

b (α)

T

(T–s)α–h(s)dsc

,

x(T) f(T,x(T))=

(α)

T

(T–s)α–h(s)ds–  a+b

b (α)

T

(T–s)α–h(s)dsc

,

then

a x() f(,x())+b

x(T) f(T,x(T))=

–ab (a+b)(α)

T

(T–s)α–h(s)ds+ ac a+b

+ b (α)

T

(T–s)α–h(s)ds

b

 (a+b)(α)

T

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this implies that

Theorem . Assume that hypotheses(H)-(H)hold and a,b,c are real constants with a+b= .Further,if

then the hybrid fractional-order diﬀerential equation()has a solution deﬁned on J.

Proof We deﬁne a subsetSofXby

It is clear thatSsatisﬁes the hypothesis of Theorem .. By an application of Lemma ., equation () is equivalent to the nonlinear hybrid integral equation

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Deﬁne two operatorsA:XXandB:SXby

Then the hybrid integral equation () is transformed into the operator equation as

x(t) =Ax(t)Bx(t), tJ. ()

We shall show that the operatorsAandBsatisfy all the conditions of Theorem .. Claim. Letx,yX, then by hypothesis (H),

Ax(t) –Ay(t)=f t,x(t)f t,y(t)Lx(t) –y(t)xy

for alltJ. Taking supremum overt, we obtain

AxAyLxy

for allx,yX.

Claim. We show thatBis continuous inS.

Let (xn) be a sequence inSconverging to a pointxS. Then, by the Lebesgue dominated

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=  (α)

t

(t–s)α–g s,x(s)ds

–  a+b

b (α)

T

(T–s)α–g s,x(s)dsc

=Bx(t)

for alltJ. This shows thatBis a continuous operator onS. Claim.Bis a compact operator onS.

First, we show thatB(S) is a uniformly bounded set inX. LetxS. Then, by hypothesis (H), for alltJ,

Bx(t)≤  (α)

t

(t–s)α–g s,x(s)ds

+  a+b

b (α)

T

(T–s)α–g s,x(s)ds+|c|

Tα–

(α) t

h(s)ds+ bT α–

|a+b|(α) T

h(s)ds+ |c|

|a+b|

Tα–

(α)hL

 + |b|

|a+b|

+ |c|

|a+b|.

ThusBxT(α)α–hL( +| b a+b|) +

|c|

|a+b|for allxS. This shows thatBis uniformly bounded onS.

Next, we show thatB(S) is an equicontinuous set onX. We setp(t) =th(s)ds.

Lett,tJ, then for anyxS,

Bx(t) –Bx(t)=  (α)

t

(t–s)α–g s,x(s)ds–  (α)

t

(t–s)α–g s,x(s)ds

Tα–

(α) t

t

g s,x(s)ds

Tα–

(α)p(t) –p(t).

Sincepis continuous on compactJ, it is uniformly continuous. Hence

ε> ,∃η> : |tt|<ηBx(t) –Bx(t)<ε

for allt,t∈Jand for allxX.

This shows thatB(S) is an equicontinuous set inX.

Then, by the Arzelá-Ascoli theorem,Bis a continuous and compact operator onS. Claim. The hypothesis (c) of Theorem . is satisﬁed.

LetxXandySbe arbitrary such thatx=AxBy. Then

x(t)=Ax(t)By(t)

f t,x(t)(α)

t

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– 

Taking supremum overt,

xF(

ThenxSand the hypothesis (c) of Theorem . is satisﬁed. Finally, we have

Thus, all the conditions of Theorem . are satisﬁed and hence the operator equation AxBx=xhas a solution inS. As a result, BVPHDEF () has a solution deﬁned onJ. This

completes the proof.

4 Fractional hybrid differential inequalities

We discuss a fundamental result relative to strict inequalities for BVPHDEF (). We begin with the deﬁnition of the classCp([,T],R).

Deﬁnition . mCp([,T],R) means thatmC([,T],R) andtpm(t)C([,T],R).

Lemma .[] Let mCp([,T],R).Suppose that for any t∈(, +∞)we have m(t) = 

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Then it follows that

Dqm(t)≥.

Theorem . Assume that hypothesis(H)holds.Suppose that there exist functions y,zCp([,T],R)such that

y(t) f(t,y(t))

g t,y(t) a.e. tJ ()

and

z(t) f(t,z(t))

g t,z(t) a.e. tJ, ()

 <tT,with one of the inequalities being strict.Then

y<z,

where y=t–αy(t)|

t=and z=t–αz(t)|t=implies

y(t) <z(t)

for all tJ.

Proof Suppose that inequality () holds. Assume that the claim is false. Then, sincey<z andt–αy(t) andt–αz(t) are continuous functions, there existstsuch that  <tTwith y(t) =z(t) andy(t) <z(t), t<t.

Deﬁne

Y(t) = y(t)

f(t,y(t)) and Z(t) = z(t) f(t,z(t)).

Then we haveY(t) =Z(t), and by virtue of hypothesis (H), we getY(t) <Z(t) for all ≤t<t.

Settingm(t) =Y(t) –Z(t), tt, we ﬁnd thatm(t) < , t<tandm(t) =  with mCp([,T],R). Then, by Lemma ., we haveDqm(t)≥. By () and (), we obtain

g t,y(t)≥DqY(t)≥DqZ(t) >g t,z(t).

This is a contradiction withy(t) =z(t). Thus the conclusion of the theorem holds and the

proof is complete.

Theorem . Assume that hypothesis (H) holds and a, b, c are real constants with a+b= .Suppose that there exist functions y,zCp([,T],R)such that

y(t) f(t,y(t))

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and

z(t) f(t,z(t))

g t,z(t) a.e. tJ, ()

one of the inequalities being strict,and if a> ,b< and y(T) <z(T),then

a y() f(,y())+b

y(T) f(T,y(T))<a

z() f(,z())+b

z(T)

f(T,z(T)) ()

implies

y(t) <z(t) ()

for all tJ.

Proof We haveaf(,y()y())+bf(Ty(,yT(T)))<af(,z()z())+bf(Tz,(zT(T))). This impliesa(f(,y()y())f(,z()z())) <b(f(Tz(,zT()T))f(Ty(,yT()T))).

Sinceb<  andy(T) <z(T) by hypothesis (H), thenf(Tz(,zT(T)))f(Ty(,yT()T))> .

This shows that f(,y()y())f(,z()z())<  sincea> , and by hypothesis (H) we havey() < z().

Hence the application of Theorem . yields thaty(t) <z(t).

Theorem . Assume that the conditions of Theorem.hold with inequalities()and ().Suppose that there exists a real number M> such that

g(t,x) –g(t,x)≤ M  +

x f(t,x)

x f(t,x)

a.e. tJ ()

for all x,x∈Rwith x≥x.Then

a y() f(,y())+b

y(T) f(T,y(T))<a

z() f(,z())+b

z(T) f(T,z(T))

implies,provided M( +α),

y(t) <z(t)

for all tJ.

Proof We set f(t,(t()t))=f(tz,(zt()t))+ε( +tα) for smallε>  and letZε(t) = (t)

f(t,(t))andZ(t) =

z(t)

f(t,z(t))fortJ. So that we have

Zε(t) >Z(t)zε(t) >z(t).

Sinceg(t,x) –g(t,x)+Mtα(f(xt,x)f(xt,x)) and(f(zt,(zt()t)))≥g(t,z(t)) for alltJ, one has

DαZε(t) =DαZ(t) +εDαtα

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g t,zε(t)M

 +(Zε–Z) +ε( +α)

g t,zε(t)+ε( +α) –M

>g t,zε(t)

providedM( +α).

Also, we havezε() >z()y(). Hence, the application of Theorem . yields thaty(t) < zε(t) for alltJ.

By the arbitrariness ofε> , taking the limits asε→, we havey(t)z(t) for alltJ.

This completes the proof.

Remark . Letf(t,x) =  andg(t,x) =x. We can easily verify thatfandgsatisfy condition ().

5 Existence of maximal and minimal solutions

In this section, we shall prove the existence of maximal and minimal solutions for BVPHDEF () onJ= [,T]. We need the following deﬁnition in what follows.

Deﬁnition . A solutionrof BVPHDEF () is said to be maximal if for any other solution xto BVPHDEF () one hasx(t)r(t) for alltJ. Similarly, a solutionρof BVPHDEF () is said to be minimal ifρ(t)x(t) for alltJ, wherexis any solution of BVPHDEF () onJ.

We discuss the case of maximal solution only, as the case of minimal solution is similar and can be obtained with the same arguments with appropriate modiﬁcations. Given an arbitrarily small real number ε> , consider the following boundary value problem of BVPHDEF of order  <α< :

⎧ ⎨ ⎩

( x(t)

f(t,x(t))) =g(t,x(t)) +ε a.e.tJ= [,T],

af(,x()x())+bf(Tx(,xT()T))=c+ε, ()

wherefC(J×R,R\{}) andC(J×R,R).

An existence theorem for BVPHDEF () can be stated as follows.

Theorem . Assume that hypotheses(H)-(H)hold and a,b,c are real constants with a+b= .Suppose that inequality()holds.Then,for every small numberε> ,BVPHDEF ()has a solution deﬁned on J.

Proof By hypothesis, since

L

Tα–hL

(α)

 + |b|

|a+b|

+ |c|

|a+b|

< ,

there existsε>  such that

L

Tα–h

L+εTα

(α)

 + |b|

|a+b|

+|c|+ε

|a+b|

< 

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Our main existence theorem for maximal solution for BVPHDEF () is the following.

Theorem . Assume that hypotheses(H)-(H)hold with the conditions of Theorem. and a, b,c are real constants with a+b= .Furthermore, if condition()holds,then BVPHDEF()has a maximal solution deﬁned on J.

Proof Let{εn}∞ be a decreasing sequence of positive real numbers such thatlimn→∞εn=

, whereεis a positive real number satisfying the inequality

L

Tα–h

L+εTα

(α)

 + |b|

|a+b|

+|c|+ε

|a+b|

< .

The number ε exists in view of inequality (). By Theorem ., there exists a solution r(t,εn) deﬁned onJof the BVPHDEF

⎧ ⎨ ⎩

( x(t)

f(t,x(t))) =g(t,x(t)) +εn a.e.tJ, af(,x()x())+bf(Tx(,xT()T))=c+εn.

()

Then any solutionuof BVPHDEF () satisﬁes

u(t) f(t,u(t))

g t,u(t),

and any solution of auxiliary problem () satisﬁes

r(t,εn)

f(t,r(t,εn))

=g t,r(t,εn)

+εn>g t,r(t,εn)

,

whereaf(,u()u())+bf(Tu,(uT(T)))=cc+εn=af(,r(,εn)r(,εn))+bf(Tr(,rT(,εn)T,εn)). By Theorem ., we infer

that

u(t)r(t,εn) ()

for alltJandnIN. Since

c+ε =a

r(,ε) f(,r(,ε))+b

r(T,ε) f(T,r(T,ε))

a r(,ε) f(,r(,ε))+b

r(T,ε)

f(T,r(T,ε))=c+ε,

then by Theorem ., we infer thatr(t,ε)r(t,ε). Therefore,r(t,εn) is a decreasing

se-quence of positive real numbers, and the limit

r(t) = lim

n→∞r(t,εn) ()

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arbitrary. Then

Sincef is continuous on a compact setJ×[–N,N], it is uniformly continuous there. Hence,

f t,r(t,εn)

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uniformly for allnN. Therefore, from the above inequality, it follows that r(t,εn) –r(t,εn)→ astt

uniformly for allnN. Therefore,

r(t,εn)→r(t) asn→ ∞for alltJ.

Next, we show that the functionr(t) is a solution of BVPHDEF () deﬁned onJ. Now, since r(t,εn) is a solution of BVPHDEF (), we have

r(t,εn) =

f t,r(t,εn)

(α)

t

(t–s)α– g s,r(s,εn)

+εn

ds

–  a+b

b (α)

T

(T–s)α– g s,r(s,εn)

+εn

dscεn

for alltJ. Taking the limit asn→ ∞in the above equation yields

r(t) =f t,r(t)(α)

t

(t–s)α–g s,r(s)ds

–  a+b

b (α)

T

(T–s)α–g s,r(s)dsc

for alltJ. Thus, the functionris a solution of BVPHDEF () onJ. Finally, from inequality () it follows thatu(t)r(t) for alltJ. Hence, BVPHDEF () has a maximal solution

onJ. This completes the proof.

6 Comparison theorems

The main problem of the diﬀerential inequalities is to estimate a bound for the solution set for the diﬀerential inequality related to BVPHDEF (). In this section, we prove that the maximal and minimal solutions serve as bounds for the solutions of the related diﬀerential inequality to BVPHDEF () onJ= [,T].

Theorem . Assume that hypotheses(H)-(H)and condition()hold and a,b,c are real constants with a+b= .Suppose that there exists a real number M> such that

g(t,x) –g(t,x)M  +

xf(t,x)

xf(t,x)

a.e. tJ

for all x,x∈Rwith xx,where M( +α).Furthermore,if there exists a function uC(J,R)such that

⎧ ⎨ ⎩

( u(t)

f(t,u(t)))≤g(t,u(t)) a.e. tJ,

af(,u()u())+bf(Tu,(uT()T))c, ()

then

u(t)r(t) ()

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Proof Let ε>  be arbitrarily small. By Theorem .,r(t,ε) is a maximal solution of BVPHDEF () so that the limit

r(t) =lim

ε→r(t,ε) ()

is uniform onJand the functionris a maximal solution of BVPHDEF () onJ. Hence, we obtain

⎧ ⎨ ⎩

( r(t,ε)

f(t,r(t,ε))) =g(t,r(t,ε)) +ε a.e.tJ, af(,r(,ε)r(,ε))+bf(Tr(,Tr(T,ε),ε))=c+ε.

()

From the above inequality it follows that

⎧ ⎨ ⎩

( r(t,ε)

f(t,r(t,ε))) >g(t,r(t,ε)) a.e.tJ,

af(,r(,ε)r(,ε))+bf(Tr(,Tr(T,ε),ε))=c+ε. ()

Now we apply Theorem . to inequalities () and () and conclude thatu(t) <r(t,ε) for alltJ. This, in view of limit (), further implies that inequality () holds onJ. This

completes the proof.

Theorem . Assume that hypotheses(H)-(H)and condition()hold and a,b,c are real constants with a+b= .Suppose that there exists a real number M> such that

g(t,x) –g(t,x)≤ M  +

x f(t,x)

x f(t,x)

a.e. tJ

for all x,x∈Rwith x≥x,where M( +α).Furthermore,if there exists a function vC(J,R)such that

⎧ ⎨ ⎩

( v(t)

f(t,v(t)))≥g(t,v(t)) a.e. tJ, af(,v()v())+bf(Tv(,vT()T))>c,

then

ρ(t)v(t)

for all tJ,whereρis a minimal solution of BVPHDEF()on J.

Note that Theorem . is useful to prove the boundedness and uniqueness of the solu-tions for BVPHDEF () onJ. A result in this direction is as follows.

Theorem . Assume that hypotheses(H)-(H)and condition()hold and a,b,c are real constants with a+b= .Suppose that there exists a real number M> such that

g(t,x) –g(t,x)≤ M  +

x f(t,x)

x f(t,x)

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for all x,x∈Rwith xx,where M( +α).If an identically zero function is the only solution of the diﬀerential equation

m(t) = M

 +tαm(t) a.e. tJ, am() +bm(T) = , ()

then BVPHDEF()has a unique solution on J.

Proof By Theorem ., BVPHDEF () has a solution deﬁned onJ. Suppose that there are two solutions u andu of BVPHDEF () existing on Jwith u>u. Deﬁne a function m:J→Rby

m(t) = u(t) f(t,u(t))

u(t) f(t,u(t)).

In view of hypothesis (H), we conclude thatm(t) > . Then we have

m(t) =Dα

u(t) f(t,u(t))

u(t) f(t,u(t))

=g t,u(t)g t,u(t)

M

 +

u f(t,u(t))

u(t) f(t,u(t))

= M

 +tαm(t)

for almost everywheretJ, and sincem() = u() f(,u())–

u()

f(,u()) andm(T) = u(T) f(T,u(T))– u(T)

f(T,u(T))anda u()

f(,u())+b u(T)

f(T,u(T))=a u()

f(,u())+b u(T)

f(T,u(T)), we haveam() +bm(T) = .

Now, we apply Theorem . withf(t,x) =  andc=  to get thatm(t)≤ for alltJ, where an identically zero function is the only solution of the diﬀerential equation () m(t)≤ is a contradiction withm(t) > . Then we can getu=u. This completes the

proof.

7 Existence of extremal solutions in vector segment

Sometimes it is desirable to have knowledge of the existence of extremal positive solutions for BVPHDEF () onJ. In this section, we shall prove the existence of maximal and mini-mal positive solutions for BVPHDEF () between the given upper and lower solutions on J= [,T]. We use a hybrid ﬁxed point theorem of Dhage [] in ordered Banach spaces for establishing our results. We need the following preliminaries in what follows. A non-empty closed setKin a Banach algebraXis called a cone with vertex  if

(i) K+KK,

(ii) λKKforλ∈R,λ≥,

(iii) (–K)∩K= , where  is the zero element ofX,

(iv) a coneKis called positive ifKKK, where◦is a multiplication composition inX.

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that if the coneK is normal inX, then every order-bounded set inXis norm-bounded. The details of cones and their properties appear in Heikkila and Lakshmikantham [].

Lemma .[] Let K be a positive cone in a real Banach algebra X and let u,u,v,vK be such that u≤vand u≤v.Then uu≤vv.

For any a,bX,the order interval[a,b]is a set in X given by

[a,b] ={xX:axb}.

Deﬁnition . A mappingQ: [a,b]X is said to be nondecreasing or monotone in-creasing ifxyimpliesQxQyfor allx,y∈[a,b].

We use the following ﬁxed point theorems of Dhage [] for proving the existence of extremal solutions for problem () under certain monotonicity conditions.

Lemma .[] Let K be a cone in a Banach algebra X and let a,bX be such that ab. Suppose that A,B: [a,b]K are two nondecreasing operators such that

(a) Ais Lipschitzian with a Lipschitz constantα, (b) Bis complete,

(c) AxBx∈[a,b]for eachx∈[a,b].

Further, if the cone K is positive and normal, then the operator equation AxBx=x has the least and the greatest positive solution in [a,b], whenever αM< , whereM=

B([a,b])=sup{Bx:x∈[a,b]}.

We equip the spaceC(J,R) with the order relation≤with the help of coneKdeﬁned by

K=xC(J,R) :x(t)≥,∀tJ. ()

It is well known that the coneKis positive and normal inC(J,R). We need the following deﬁnitions in what follows.

Deﬁnition . A functionaC(J,R) is called a lower solution of BVPHDEF () de-ﬁned on Jif it satisﬁes (). Similarly, a functionaC(J,R) is called an upper solution of BVPHDEF () deﬁned onJif it satisﬁes (). A solution to BVPHDEF () is a lower as well as an upper solution for BVPHDEF () deﬁned onJandvice versa.

We consider the following set of assumptions:

(B) f :J×R→R+– ,g:J×R→R+.

(B) BVPHDEF () has a lower solutionaand an upper solutionbdeﬁned onJwithab.

(B) The functionxf(tx,x) is increasing in the interval[mintJa(t),maxtJb(t)]almost

everywhere fortJ.

(B) The functionsf(t,x)andg(t,x)are nondecreasing inxalmost everywhere fortJ.

(B) There exists a functionkL(J,R)such thatg(t,b(t))k(t).

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Theorem . Suppose that assumptions(H)and(B)-(B)hold and a,b,c are real

con-then BVPHDEF()has a minimal and a maximal positive solution deﬁned on J.

Proof Now, BVPHDEF () is equivalent to integral equation () deﬁned onJ. LetX= C(J,R). Deﬁne two operatorsAandBonXby () and (), respectively. Then the inte-gral equation () is transformed into an operator equationAx(t)Bx(t) =x(t) in the Banach algebraX. Notice that hypothesis (B) impliesA,B: [a,b]K. Since the coneK inXis normal, [a,b] is a norm-bounded set inX. Now it is shown, as in the proof of Theorem ., thatAis a Lipschitzian with the Lipschitz constantLandBis a completely continuous op-erator on [a,b]. Again, hypothesis (B) implies thatAandBare nondecreasing on [a,b]. To see this, letx,y∈[a,b] be such thatxy. Then, by hypothesis (B),

for alltJ. SoAandBare nondecreasing operators on [a,b]. Lemma . and hypothesis (B) together imply that

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for alltJandx∈[a,b]. As a resulta(t)Ax(t)Bx(t)b(t) for alltJandx∈[a,b]. Hence,AxBx∈[a,b] for allx∈[a,b]. Again,

M=B[a,b]=supBx:x∈[a,b]L

Tα–h

L

(α)

 + |b|

|a+b|

+ |c|

|a+b|

,

and soαML(Tα–hL(α) ( +

|b|

|a+b|) +

|c|

|a+b|) < . Now, we apply Lemma . to the

opera-tor equationAxBx=xto yield that BVPHDEF () has a minimal and a maximal positive solution in [a,b] deﬁned onJ. This completes the proof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors typed, read, and approved the ﬁnal manuscript.

Acknowledgements

The authors sincerely thank the referees for their constructive suggestions which improved the content of the paper.

Received: 23 February 2015 Accepted: 5 June 2015 References

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59(3), 1095-1100 (2010)

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3. Lakshmikantham, V, Vasundhara Devi, J: Theory of fractional diﬀerential equations in Banach space. Eur. J. Pure Appl. Math.1, 38-45 (2008)

4. Lakshmikantham, V, Vatsala, AS: Basic theory of fractional diﬀerential equations. Nonlinear Anal. TMA69, 2677-2682 (2008)

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6. Podlubny, I: Fractional Diﬀerential Equations. Mathematics in Sciences and Engineering, vol. 198. Academic Press, San Diego (1999)

7. Zhao, Y, Sun, S, Han, Z, Li, Q: Positive solutions to boundary value problems of nonlinear fractional diﬀerential equations. Abstr. Appl. Anal.2011, Article ID 390543 (2011)

8. Zhao, Y, Sun, S, Han, Z, Zhang, M: Positive solutions for boundary value problems of nonlinear fractional diﬀerential equations. Appl. Math. Comput.16, 6950-6958 (2011)

9. Dhage, BC: On a condensing mappings in Banach algebras. Math. Stud.63, 146-152 (1994)

10. Dhage, BC: A nonlinear alternative in Banach algebras with applications to functional diﬀerential equations. Nonlinear Funct. Anal. Appl.8, 563-575 (2004)

11. Dhage, BC: On a ﬁxed point theorem in Banach algebras with applications. Appl. Math. Lett.18, 273-280 (2005) 12. Dhage, BC, Lakshmikantham, V: Basic results on hybrid diﬀerential equations. Nonlinear Anal. Hybrid Syst.4, 414-424

(2010)

13. Dhage, BC, Lakshmikantham, V: Quadratic perturbations of periodic boundary value problems of second order ordinary diﬀerential equations. Diﬀer. Equ. Appl.2, 465-486 (2010)

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References

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