The existence of solutions for a nonlinear mixed problem of singular fractional differential equations

R E S E A R C H

Open Access

The existence of solutions for a nonlinear

mixed problem of singular fractional

differential equations

Dumitru Baleanu

_{, Hakimeh Mohammadi}

_{and Shahram Rezapour}

[email protected] 1_{Department of Chemical and}

Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia

2_{Department of Mathematics,}

Cankaya University, Ogretmenler Cad. 14, Balgat, Ankara, 06530, Turkey

Full list of author information is available at the end of the article

Abstract

By using fixed point results on cones, we study the existence of solutions for the singular nonlinear fractional boundary value problem

c_Dα_{u(t) =f(t,u(t),u(t),}c_Dβ_u(t)),

u(0) =au(1), u(0) =bcDβu(1), u(0) =u(0) =u(n–1)(0) = 0,

wheren≥3 is an integer,

α

β

(2 –

β),

derivative.

Keywords: boundary value problem; fixed point; fractional differential equation; Green function; regularization; singular

1 Introduction

Fractional differential equations (see, for example, [–] and references therein) started to play an important role in several branches of science and engineering. There are some works about existence of solutions for the nonlinear mixed problems of singular fractional boundary value problem (see, for example, [–] and []). Also, there are different meth-ods for solving distinct fractional differential equations (see, for example, [–] and []). By using fixed point results on cones, we focus on the existence of positive solutions for a nonlinear mixed problem of singular fractional boundary value problem. For the con-venience of the reader, we present some necessary definitions from fractional calculus theory (see, for example, []). The Caputo derivative of fractional orderαfor a function

f: [,∞)→Ris defined by

c_Dα

f(t) = 

(n–α) t



(t–s)n–α–f(n)(s)ds n–  <α<n,n= [α] + .

Letq≥. As you know,Lq_{[, ] denotes the space of functions, whoseqth powers of}

mod-ulus are integrable on [, ], equipped with the normxq= (



|x(t)|qdt)



q_{. We consider}

the sup norm

x=supx(t):t∈[, ]

on the spaceC[, ]. Also,AC[, ] is the set of absolutely continuous functions on [, ]. LetBbe a subset ofR_{. A functionf : [, ]×B→Ris called anL}q_{-Caratheodory function}

whenever the real-valued functionf(·,x,y,z) on [, ] is measurable for all (x,y,z)∈B, the function f(t,·,·,·) :B→Ris continuous for almost allt∈(, ], and for each compact setU⊂B, there exists a functionϕu∈Lq[, ] such that|f(t,x,y,z)| ≤ϕu(t) for almost all t∈[, ] and (x,y,z)∈U. Consider the nonlinear fractional boundary value problem

c_Dα

u(t) =ft,u(t),u(t),cDβu(t),

u() =au(), u() =bcDβu(), u() =u() =u(n–)() = ,

(∗)

wheren≥ is an integer,α∈(n– ,n),  <β< ,  <a< ,  <b<( –β) andq>_α_–. We say that the functionu: [, ]→Ris a positive solution for the problem wheneveru>  on [, ],c_Dα_{uis a function in L}q_{[, ], andusatisfies the boundary conditions almost}

everywhere on [, ]. In this paper, we suppose thatf is anLq_{-Caratheodory function on}

[, ]×B, whereB= (,∞)×(,∞)×(,∞), there exists a positive constantmsuch that

m≤f(t,x,y,z) for almost allt∈[, ] and (x,y,z)∈B,f satisfies the estimate

f(t,x,y,z)≤h(x) +r|y|+k|z|+γ(t)wx,|y|,|z|,

whereh,r,k∈C(,∞) are positive and non-increasing,γ ∈Lq_{[, ] andw∈C([,∞)×}

[,∞)×[,∞)) are positive, wis non-decreasing in all its variables, _hq(sα_)ds<∞,

 rq(s

α–_)ds<∞, kq(s

α–β_{)ds<∞, andlim}

x→∞w(x,x,x)_x = . Since we suppose that

prob-lem (∗) is singular, that is,f(t,x,y,z) may be singular at the value  of its space variablesx,

y,z, we use regularization and sequential techniques for the existence of positive solutions of the problem. In this way, for each natural numberndefine the functionfnby

fn(t,x,y,z) =f

t,χ_n+(x),χ_n+(y),χ_n+(z)

for allt∈[, ] and (x,y,z)∈R_{, where}

χ_n+(u) = ⎧ ⎨ ⎩

u, u≥ n, 

n, u<  n.

It is easy to see that eachfnis anLq-Caratheodory function on [, ]×R,m≤fn(t,x,y,z),

fn(t,x,y,z)≤h



n

+r



n

+k



n

+γ(t)w +x,  +|y|,  +|z|

and

fn(t,x,y,z)≤h(x) +r

|y|+k|z|+γ(t)w +x,  +|y|,  +|z|

Lemma .[] Letρ∈Lq_{[, ]and≤t}

<t≤.Then we have|

t (t–s)

α–_{ρ(s)ds| ≤}

(td d)

/p_ρ

qfor all t∈[, ]and

_t(t–s)α–ρ(s)ds–

t



(t–s)α–ρ(s)ds

≤

td

 + (t–t)d–td d

/p

ρq+

(t–t)d d

/p

ρq,

where d= (α– )p+ .

2 Main results

Now, we are ready to investigate the problem in regular and singular cases. First, we give the following result.

Lemma . Let y∈C[, ].Then the boundary value problem

c_Dα

u(t) =y(t) t∈(, ),

u() =au(), u() =bcDβu(), u() =u() =u(n–)() = 

is equivalent to the fractional integral equation u(t) =_G(t,s)y(s)ds,where

G(t,s) =(t–s)

α–

(α)

+a(α–β)(( –β) –b)( –s)

α–_{+b(α)( –β)(a+t–at)( –s)}α–β–

( –a)(α)(α–β)(( –β) –b)

whenever≤s≤t≤and

G(t,s) =a(α–β)(( –β) –b)( –s)

α–_{+b(α)( –β)(a+t–at)( –s)}α–β–

( –a)(α)(α–β)(( –β) –b)

whenever≤t≤s≤.

Proof Fromc_Dα_{u(t) =y(t) and the boundary conditions, we obtain}

u(t) =Iαy(t) +u() +u()t+u

_()

! t

_{+· · ·+}u(n–)()

(n– )! t

n–

= 

(α)

t



(t–s)α–_{y(s)ds+u() +u()t.}

By properties of the Caputo derivative, we get

c_Dβ_{u(t) =I}α–β_{y(t) +}c_Dβ_{u() +u()t}

= 

(α–β) t



Thus,u() =₍_α)( –s)α–_{y(s)ds+u() +u() and}

c_Dβ_{u() =}  (α–β)





( –s)α–β–_y(s)ds+ u() ( –β).

By using the boundary conditions u() =au() and u() =bcDβ_{u(), we get u() =} a(₍_α)( –s)α–_{y(s)ds+u() +u()) and}

u() =b



(α–β)





( –s)α–β–y(s)ds+ u

_()

( –β)

.

Hence,u() =_(α–b_β)((–_(–β)_β)–b)( –s)α–β–_y(s)dsand

u() = a ( –a)(α)





( –s)α–y(s)ds

+ ab( –β)

( –a)(α–β)(( –β) –b)





( –s)α–β–y(s)ds.

Thus,

u(t) = 

(α)

t



(t–s)α–y(s)ds+u() +u()t

=

t



(t–s)α– (α) +

a(α–β)(( –β) –b)( –s)α–

( –a)(α)(α–β)(( –β) –b)

+b(α)( –β)(a+t–at)( –s)

α–β–

( –a)(α)(α–β)(( –β) –b)

y(s)ds

+



t

a(α–β)(( –β) –b)( –s)α–

( –a)(α)(α–β)(( –β) –b)

+b(α)( –β)(a+t–at)( –s)

α–β–

( –a)(α)(α–β)(( –β) –b)

y(s)ds

=





G(t,s)y(s)ds.

This completes the proof.

Putk=(_(–a)α–β)(((–α)β(α)–b)+b–β)((–(αβ))–b)(–β)andk=

ab(–β)

(–a)(α–β)((–β)–b). It is easy to check that the

Green functionGin the last result belongs toC([, ]×[, ]),G(t,s) >  for all (t,s)∈ [, )×[, ),

G(t,s)≤k( –s)α–β–≤ and G(t,s)≥k( –s)α–β–

for all (t,s)∈[, ]×[, ]. Consider the Banach spaceX=C_{[, ] with the normx}

∗=

max{x,x}and the cone

For each natural numbern, define the operatorQnonPby

Now, we prove thatQnis a completely continuous operator (see []).

Lemma . The operator Qnis a completely continuous operator.

Proof Letx∈P. Then,c_Dβ_{x∈C[, ] and} c_Dβ_{x≥. Now, define ρ(t) =f}

n(t,u(t),u(t), c_Dβ_{u(t)) for almost allt∈[, ]. Thenρ∈L}q_{[, ] andρ(t)≥mfor almost allt∈[, ]. By}

using the properties of fractional integralIα_{, it is easy to see thatQ}

nx∈C[, ], (Qnx)(t)≥ Lq_{-Caratheodory function,{x}

m}is bounded inC[, ], and{cDβxm}is bounded inC[, ].

uniformly on [, ]. Hence,Qnis a continuous operator. Now, we have to show that for each

and

a completely continuous operator.

We need the following result (see [] and []).

Lemma .[] Let Y be a Banach space,P a cone in Y and and  bounded open

conditions. In this way, note that

and

there exists a positive constantLsuch that

for allt∈[, ] andx∈P. Since_t(t–s)–β_sα–_ds=(α)(–β)

Now, we give our last result.

Theorem . Problem (∗) has a solution u such that u(t)≥ mk

Note that R ∈ Lq_{[, ]. We show that {u}

using the Arzela-Ascoli theorem, without loss of generality, we can assume that{un}is

convergent inC[, ]. Letlimn→∞un=u. Then, it is easy to see thatcDβun(t) =(α–)

2.1 Examples for the problem

Example . Letρ,ρ∈Lq[, ],ρ(t)≥m>  for almost alltin [, ]. Suppose that

Example . Consider the nonlinear mixed problem of singular fractional boundary value problem

c_D_{u(t) =t+  +}  u(t)_–ρ

+ 

u(t)

+  (c_D_u(t))

+ u(t) _+u(t)₊c_D_u(t)  _{+ }

via boundary value conditionsu() =

u(),u() =  (

c_D_{)u() andu() =u() =· · ·=}

u(n–)() = , whereρ= ((_)u())_{. Let}

ft,u(t),u(t),cD_{=t+  +} 

u(t) –_ρ + 

u(t)

+  (c_Du(t))

+ u(t) _+u(t)₊c_D_u(t)  _{+ .}

Then the mapfis singular att= , andfsatisfies the desired conditions, whereh(x) = 

x–ρ

wheneverx _–ρ≥_{ andh(x) =  wheneverx} _{–ρ< ,r(x) =} 

x,k(x) =



x,w(x,y,z) =

x_+y_+z_{+ ,ρ(t) =t+  >  =m,ρ(t) =  andγ(t) =ρ(t) +}|_ρ(t)|_{. Then Theorem .} guarantees that this problem has a positive solution.

3 Conclusions

One of the most interesting branches is obtaining solutions of singular fractional differ-ential via boundary value problems. Having these things in mind, we study the existence of solutions for a singular nonlinear fractional boundary value problem. Two illustrative examples illustrate the applicability of the proposed method. It seems that the obtained results could be extended to more general functional spaces. Finally, note that all calcula-tions in proofs of the results depend on the definition of the fractional derivative.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have equal contributions. All authors read and approved the final manuscript.

Author details

1_{Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204,}

Jeddah, 21589, Saudi Arabia. 2_{Department of Mathematics, Cankaya University, Ogretmenler Cad. 14, Balgat, Ankara,} 06530, Turkey.3_{Institute of Space Sciences, Magurele, Bucharest, Romania.} 4_{Department of Mathematics, Azarbaijan} Shahid Madani University, Azarshahr, Tabriz, Iran.

Acknowledgements

Research of the second and third authors was supported by Azarbaijan Shahid Madani University. Also, the authors express their gratitude to the referees for their helpful suggestions, which improved the final version of this paper.

Received: 27 May 2013 Accepted: 28 August 2013 Published:11 Dec 2013

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