Fractional Navier boundary value problems

R E S E A R C H

Open Access

Fractional Navier boundary value

problems

Imed Bachar

_{, Habib Mâagli}

_{and Vicen¸tiu D. R˘adulescu}

*_{Correspondence:}

[email protected]

3_{Institute of Mathematics ‘Simion}

Stoilow’ of the Romanian Academy, P.O. Box 1-764, Bucharest, 014700, Romania

4_{Department of Mathematics,}

University of Craiova, Street A.I. Cuza No. 13, Craiova, 200585, Romania Full list of author information is available at the end of the article

Abstract

We study the following fractional Navier boundary value problem:

⎧ ⎪ ⎨ ⎪ ⎩

Dα_(Dβ_{u)(x) +u(x)f(x,u(x)) = 0, 0 <x< 1,}

limx→0+Dβ–1u(x) = 0, lim_x→0+Dα–1(Dβu)(x) =

ξ,

where

α,

β

ξ

ξ

ζ

Our purpose is to prove the existence, uniqueness, and global asymptotic behavior of a positive continuous solution, wheref: (0, 1)×[0,∞)→[0,∞) is continuous and dominated by a functionpsatisfying appropriate integrability condition.

MSC: 34A08; 34B15; 34B18; 34B27

Keywords: fractional Navier differential equations; positive solutions; Green’s function; perturbation arguments

1 Introduction

The existence, uniqueness, and global asymptotic behavior of positive continuous solu-tions related to fractional differential equasolu-tions have been studied by many researchers. Many fractional differential equations subject to various boundary conditions have been addressed; see, for instance, [–] and the references therein. It is known that fractional differential equations serve as a good tool to model many phenomena in various fields of science and engineering (see [–] and references therein for discussions of various applications).

In [], the authors proved the existence and uniqueness of a positive solution to the following fractional boundary value problem:

⎧ ⎨ ⎩

Dα_{u(x) =u(x)ϕ(x,u(x)),  <x< ,} lim_x→+Dα–u(x) = –ξ, u() =ζ,

(.)

where  <α≤,ξ,ζ ≥ are such thatξ+ζ > , andϕ(x,s)∈C+((, )×[,∞)) satisfies appropriate conditions. Inspired by the above-mentioned paper, we aim at studying similar problem in the case of fractional Navier boundary value problem. More precisely, we are

concerned with the problem

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Dα_(Dβ_{u)(x) +u(x)f(x,u(x)) = ,  <x< ,}

limx→+Dβ–u(x) = , lim_x→+Dα–(Dβu)(x) =ξ, u() = , Dβ_{u() = –ζ,}

(.)

whereα,β∈(, ], andξ,ζ ≥ are such thatξ+ζ > . The nonlinear termf(x,s) is re-quired to be a nonnegative continuous function in (, )×[,∞) dominated by a function pbelonging to the classJα,βdefined as follows.

Definition . Letα,β∈(, ]. A nonnegative measurable functionpon (, ) belongs to the classJα,βiff





tβ–_{( –t)}α_{p(t)dt<∞. (.)}

Next, we introduce the following notation.

(i) B+_{((, ))is the set of nonnegative measurable functions in(, ).}

(ii) LetXbe a metric space, we denote byC(X)(resp.C+(X)) the set of continuous (resp. nonnegative continuous) functions inX.

(iii) Forγ∈(, ],C–γ([, ]) ={w∈C((, ]) :x→x–γw(x)∈C([, ])}.

(iv) Forγ∈(, ],Gγ(x,s)is the Green function of the operatoru→–Dγu, with

boundary datalimx→+Dγ–u(x) =u() = . From [], Lemma , we have

Gγ(x,s) =



(γ)

xγ–( –s)γ––(x–s)+γ–, (.)

wherex+_{=max(x, ).}

Proposition .(see []) Let <γ ≤andϕ∈B+_{((, )).Then we have}

(i) For(x,s)∈(, ]×[, ],

(γ– )

(γ) H(x,s)≤Gγ(x,s)≤ 

(γ)H(x,s), (.)

whereH(x,s) :=xγ–_{( –s)}γ–_{( –max(x,s)).}

(ii) The functionx→Gγϕ(x) :=



Gγ(x,s)ϕ(s)dsbelongs toC–γ([, ])if and only if

 ( –s)

γ–_ϕ(s)ds<∞.

(iii) If the maps→( –s)γ–_{ϕ(s)∈C((, ))∩L}_{((, )),thenG}

γϕbelongs to

C–γ([, ]),and it is the unique solution of the problem

⎧ ⎨ ⎩

Dγ_{u(x) = –ϕ(x),  <x< ,} limx→+Dγ–u(x) =u() = .

Throughout this paper, forα,β∈(, ], letG(x,s) be the Green function of the operator u→Dα_(Dβ_{u) with Navier boundary conditionslim}

u() =Dβ_{u() = . Then we have}

G(x,s) = 



Gβ(x,t)Gα(t,s)dt. (.)

For a given functionpinB+_{((, )), we put}

κp:= sup x,s∈(,)





G(x,t)G(t,s)

G(x,s) p(t)dt, (.)

and we will prove thatκp<∞if and only ifp∈Jα,β.

From here on, letξ,ζbe two nonnegative constants such thatξ+ζ> , andθ(x) be the unique solution of the problem

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Dα_(Dβ_{u)(x) = ,  <x< ,}

limx→+Dβ–u(x) = , lim_x→+Dα–(Dβu)(x) =ξ, u() = , Dβ_{u() = –ζ.}

(.)

We can easily verify that, forx∈(, ],θ(x) =ξh(x) +ζh(x), where

h(x) = 



Gβ(x,t)Gα(t, )dt

= 

(α– )(α+β– )x

β–_{ –x}α₊ 

(α+β)x

β–_xα+_{–  (.)}

and

h(x) = 



Gβ(x,t)tα–dt=

(α– )

(α+β– )x

β–_{ –x}α

. (.)

Note that from (.), (.), and (.) it follows that there exists a constantc>  such that, for eachx∈(, ],

 cx

β–_{( –x)≤θ(x)≤cx}β–_{( –x). (.)}

To state our existence results, a combination of the following hypotheses are required.

(A) f is inC+((, )×[,∞)).

(A) There exists p ∈ Jα,β ∩C+((, )) with κp ≤ _ such that, for each x ∈ (, ),

the maps→s(p(x) –f(x,sθ(x)))is nondecreasing on[, ].

(A) For eachx∈(, ), the functions→sf(x,s)is nondecreasing on[,∞).

Our main results are the following.

Theorem . Under conditions(A)-(A),problem(.)admits a solution u∈C–β([, ])

such that

Moreover,this solution is unique if hypothesis(A)is also satisfied.

Corollary . Letα,β∈(, ],and h be a nonnegative function in C([,∞))such that the map s→(s) =sh(s)is nondecreasing on[,∞).Let q∈C+((, ))and assume that the functionq˜(x) :=q(x)max≤t≤θ(x)(t)belongs toJα,β.Then forλ∈[,_κ_˜

q),the problem

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Dα_(Dβ_{u)(x) +λq(x)u(x)h(u(x)) = ,  <x< ,} limx→+Dβ–u(x) = , lim_x→+Dα–(Dβu)(x) =ξ, u() = , Dβ_{u() = –ζ,}

(.)

admits a unique solution u∈C–β([, ])such that

( –λκq˜)θ(x)≤u(x)≤θ(x),  <x≤.

Our paper is organized as follows. In Section , we establish some properties ofG(x,s). In particular, we prove the existence of a constantc>  such that, for allx,t,s∈(, ),

 ct

β–_{( –t)}α_≤G(x,t)G(t,s)

G(x,s) ≤ct

β–_{( –t)}α

.

This implies thatκp<∞if and only ifp∈Jα,β. In Section , for a given functionp∈Jα,β

withκp≤_, we construct the Green functionH(x,s) of the operatoru→Dα(Dβu) +p(x)u

with boundary conditionslim_x→+Dβ–u(x) =lim_x→+Dα–(Dβu)(x) =u() =Dβu() = , and we derive some of its properties including the following:

( –κp)G(x,s)≤H(x,s)≤G(x,s) for all (x,s)∈(, ]×[, ]

and

Wϕ=Wpϕ+Wp(pWϕ) =Wpϕ+W(pWpϕ) forϕ∈B+

(, ),

whereWandWpare defined by

Wϕ(x) := 



G(x,s)ϕ(s)ds and Wpϕ(x) :=





H(x,s)ϕ(s)ds, x∈(, ].

Exploiting these results, we prove our main results by means of a perturbation argument.

2 Estimates on the Green function

We recall the definition of the Riemann-Liouville derivative.

Definition . (see [, , ]) The Riemann-Liouville derivative of fractional order

γ >  of a functiongis defined as

Dγg(x) := 

(n–γ)

d dx

n x



(x–s)n–γ–g(s)ds,

Next, we prove some properties ofG(x,s).

Proposition . Letα,β∈(, ].Then there exist two constants m> and M> such that,for all(x,s)∈(, ]×[, ],we have

mxβ–( –x)( –s)α–≤G(x,s)≤Mxβ–( –x)( –s)α–. (.)

Proof Using (.) and (.), we have

G(x,s) = 



Gβ(x,t)Gα(t,s)dt

≤ 

(β) 



xβ–( –x)( –t)β–Gα(t,s)dt

≤xβ–( –x)

(β)(α) 



( –t)β–_tα–_{( –s)}α–_dt

=Mxβ–( –x)( –s)α–.

On the other hand, using again (.), (.), and the inequality  –max(x,s)≥( –x)( –s), we get

G(x,s) = 



Gβ(x,t)Gα(t,s)dt

≥(β– )

(β) (α– )

(α) 



xβ–( –x)( –t)βtβ–( –s)α–dt

=mxβ–( –x)( –s)α–.

Using Proposition ., we deduce the following.

Corollary . Letα,β∈(, ].Then there exists a constant c> such that,for all x,t,s∈ (, ),we have

 ct

β–_{( –t)}α_≤G(x,t)G(t,s)

G(x,s) ≤ct

β–_{( –t)}α_{. (.)}

Proposition . Letα,β∈(, ],and p be a function in B+_{((, )).}

(i) There exists a constantc> such that

 c





tβ–_{( –t)}α_p(t)dt≤κ

p≤c





tβ–_{( –t)}α_{p(t)dt, (.)}

whereκpis given by(.).

In particular,

κp<∞ if and only if p∈Jα,β. (.)

(ii) Forx∈(, ],we have

W(θp)(x)≤κpθ(x), (.)

Proof Letpbe a function inB+_{((, )).}

(i) Inequalities in (.) follow immediately from (.) and (.).

(ii) Sinceθ(x) :=ξh(x) +ζh(x), it suffices to prove (.) forhandh. To this end,

observe that from (.) it follows that, for eachx,s∈(, ),

lim

r→ G(s,r) G(x,r)=

G(s, ) G(x, )=

h(s) h(x) .

So by Fatou’s lemma and (.) we deduce that





G(x,s)h(s) h(x)

p(s)ds≤lim inf

r→ 



G(x,s)G(s,r)

G(x,r)p(s)ds≤κp, that is,

W(hp)(x)≤κph(x) forx∈(, ].

Similarly, we prove thatW(hp)(x)≤κph(x) by observing that

lim

r→ G(s,r) G(x,r)=

h(s) h(x) .

This ends the proof.

Corollary . Letα,β∈(, ]andϕ∈B+((, )).Then x→Wϕ(x)∈C–β([, ])if and

only if _( –s)α–_ϕ(s)ds<∞.

Proof The assertion follows from (.) and the dominated convergence theorem.

Proposition . Let α,β ∈(, ] andϕ ∈B+_{((, )) be such that s→( –s)}α–_ϕ(s)∈ C((, ))∩L((, )).Then Wϕis the unique nonnegative solution in C–β([, ])of

⎧ ⎨ ⎩

Dα_(Dβ_{u)(x) =ϕ(x),  <x< ,}

limx→+Dβ–u(x) =lim_x→+Dα–(Dβu)(x) =u() =Dβu() = .

Proof Letϕ∈B+((, )). From (.) and the Fubini-Tonelli theorem we obtain

Wϕ(x) = 



Gβ(x,t)Gαϕ(t)dt, (.)

whereGαϕ(t) = _Gα(t,s)ϕ(s)ds.

Since the functions→( –s)α–_{ϕ(s)∈C((, ))∩L}_{((, )), we deduce by Proposition .} thatGαϕis the unique solution inC–α([, ]) of

⎧ ⎨ ⎩

Dα_{v(x) = –ϕ(x),  <x< ,} limx→+Dα–v(x) =v() = .

On the other hand, by using (.) we deduce that





( –t)β–Gαϕ(t)dt≤



(α) 



( –t)β–





tα–( –s)α–ϕ(s)ds

dt

≤ (β) (α– )(α+β– )





( –s)α–ϕ(s)ds<∞.

Hence, the functiont→( –t)β–_G

αϕ(t)∈C((, ))∩L((, )). Therefore, using (.)

and Proposition ., we deduce thatWϕis the unique solution inC–β([, ]) of

⎧ ⎨ ⎩

Dβ_{u(x) = –G}

αϕ(x),  <x< , limx→+Dβ–u(x) =u() = .

(.)

Combining (.) and (.), we obtain the required result.

3 Proofs of main results

Letα,β∈(, ]. For (x,s)∈(, ]×[, ], putH(x,s) =G(x,s) and

Hn(x,s) =





G(x,t)Hn–(t,s)p(t)dt, n≥. (.)

Now, letH: (, ]×[, ]→Rbe defined by

H(x,s) =

∞

n=

(–)nHn(x,s), (.)

provided that the series converges.

Lemma . Letα,β∈(, ]and m,M> be as in(.).Let p∈Jα,βwithκp< .Then on

(, ]×[, ],we have

(i) Hn(x,s)≤κpnG(x,s)for eachn∈N.

So,H(x,s)is well defined in(, ]×[, ]. (ii) For eachn∈N,

lnxβ–( –x)( –s)α–≤Hn(x,s)≤rnxβ–( –x)( –s)α–, (.)

where

ln=mn+





tβ–( –t)αp(t)dt n

and rn=Mn+





tβ–( –t)αp(t)dt n

.

(iii) Hn+(x,s) = Hn(x,t)G(t,s)p(t)dtfor eachn∈N.

(iv) _H(x,t)G(t,s)p(t)dt= _G(x,t)H(t,s)p(t)dt.

Proof By simple induction we prove (i), (ii), and (iii). (iv) By Lemma .(i) we have

Therefore, the series_{n≥ }Hn(x,t)G(t,s)p(t)dtconverges.

So by applying the dominated convergence theorem we deduce that





H(x,t)G(t,s)p(t)dt=

∞

n= 



(–)nHn(x,t)G(t,s)p(t)dt

=

∞

n= 



(–)nG(x,t)Hn(t,s)p(t)dt

= 



G(x,t)H(t,s)p(t)dt.

Proposition . Letα,β ∈(, ]and p∈Jα,β with κp < .Then the function (x,s)→

x–β_{H(x,s)∈C([, ]×[, ]).}

Proof Clearly, the function (x,s)→x–β_H

(x,s)∈C([, ]×[, ]). Assume that the function (x,s)→x–β_H

n–(x,s)∈C([, ]×[, ]).

Using Lemma .(i) and (.), we have, for all (x,s,t)∈[, ]×[, ]×(, ],

x–βG(x,t)Hn–(t,s)p(t)≤κpn–x–

β

G(x,t)G(t,s)p(t)

≤M( –x)( –t)α–_tβ–_{( –t)( –s)}α–_p(t) ≤Mtβ–( –t)αp(t).

So by (.) and the dominated convergence theorem we conclude that the function (x,s)→x–β_H

n(x,s)∈C([, ]×[, ]).

From Lemma .(i) and (.) we deduce that

x–βHn(x,s)≤κpnx–

β

G(x,s)≤Mκ_pn. (.)

Therefore, the series_n≥(–)n_x–β_H

n(x,s) is uniformly convergent on [, ]×[, ],

and so the function (x,s)→x–β_{H(x,s)∈C([, ]×[, ]).}

Lemma . Letα,β∈(, ]and p∈Jα,βwithκp≤_.Then for(x,s)∈(, ]×[, ],we

have

( –κp)G(x,s)≤H(x,s)≤G(x,s). (.)

Proof Letp∈Jα,βwithκp≤_. By Lemma .(i) we deduce that

_H(x,s)≤

∞

n=

(κp)nG(x,s) =

  –κp

G(x,s). (.)

Now, from the expression ofHwe have

H(x,s) =G(x,s) –

∞

n=

Since the series_{n≥ }G(x,t)Hn(t,s)p(t)dtconverges, we conclude by (.) and (.)

that

H(x,s) =G(x,s) –

∞

n= (–)n





G(x,t)Hn(t,s)p(t)dt

=G(x,s) – 

 G(x,t)

∞

n=

(–)nHn(t,s)

p(t)dt,

namely,

H(x,s) =G(x,s) –WpH(·,s)(x). (.)

On the other hand, since

WpH(·,s)(x)≤   –κp

WpG(·,s)(x)

=   –κp

H(x,s)≤

κp

 –κp

G(x,s), (.)

we deduce that

H(x,s)≥G(x,s) – κp  –κp

G(x,s) = – κp  –κp

G(x,s)≥.

Hence,H(x,s)≤G(x,s), and by (.) we have

H(x,s)≥G(x,s) –WpG(·,s)(x)≥( –κp)G(x,s).

Corollary . Letα,β∈(, ]and p∈Jα,βwithκp≤_.

Letϕ∈B+_{((, )).Then}

Wpϕ∈C–β

[, ] if and only if 



( –s)α–ϕ(s)ds<∞.

Proof The assertion follows from Proposition ., (.), and (.).

Lemma . Letα,β∈(, ]and p∈Jα,βwithκp≤_.

Let h∈B+_{((, )).Then we have,for x∈(, ],}

Wh(x) =Wph(x) +Wp(pWh)(x) =Wph(x) +W(pWph)(x). (.)

In particular,if W(ph) <∞,then

I–Wp(p·)

I+W(p·)h=I+W(p·)I–Wp(p·)

h=h. (.)

Proof Let (x,s)∈(, ]×[, ]. Then by (.) we have

Leth∈B+_{((, )). Using the Fubini theorem, we obtain}

Wh(x) = 



H(x,s) +WpH(·,s)(x)h(s)ds

=Wph(x) +W(pWph)(x).

Using Lemma .(iv) and again the Fubini theorem, we have



 



H(x,t)G(t,s)p(t)h(s)dt ds= 

 



G(x,t)H(t,s)p(t)h(s)dt ds,

that is,

Wp(pWh)(x) =W(pWph)(x).

So

Wh(x) =Wph(x) +W(pWph)(x) =Wph(x) +Wp(pWh)(x).

Proposition . Letα,β∈(, ]and p∈Jα,β∩C((, ))withκp≤_.Letϕ∈B+((, ))

be such that s→( –s)α–_{ϕ(s)∈C((, ))∩L}_{((, )).Then W}

pϕ∈C–β([, ]),and it is

the unique nonnegative solution of the problem ⎧

⎨ ⎩

Dα_(Dβ_{u)(x) +p(x)u(x) =ϕ(x),  <x< ,}

limx→+Dβ–u(x) =lim_x→+Dα–(Dβu)(x) =u() =Dβu() = ,

(.)

satisfying

( –κp)Wϕ≤u≤Wϕ. (.)

Proof By Corollary . the functionx→p(x)Wpϕ(x)∈C((, )).

Using (.) and (.), we have that there existsc≥ such that

Wpϕ(x)≤Wϕ(x)≤M





xβ–( –x)( –s)α–ϕ(s)ds=cxβ–( –x). (.)

Therefore,





( –s)α–p(s)Wpϕ(s)ds≤c





sβ–( –s)αp(s)ds<∞.

Hence, by Proposition . the functionu=Wpϕ=Wϕ–W(pWpϕ) satisfies the equation

⎧ ⎨ ⎩

Dα_(Dβ_{u)(x) =ϕ(x) –p(x)u(x),  <x< ,}

limx→+Dβ–u(x) =lim_x→+Dα–(Dβu)(x) =u() =Dβu() = .

Let us prove the uniqueness. Letv∈C–β([, ]) be another solution of problem (.)

satisfyingv≤Wϕ.

Putv˜:=v+W(pv). Since the functions→( –s)α–_{p(s)v(s)∈C((, ))∩L}_{((, )), then} by Proposition . it follows that

⎧ ⎨ ⎩

Dα_(Dβ_{v˜)(x) =ϕ(x),  <x< ,}

lim_x→+Dβ–v˜(x) =lim_x→+Dα–(Dβv˜)(x) =v˜() =Dβv˜() = .

From the uniqueness in Proposition . we conclude that

˜

v:=v+W(pv) =Wϕ.

So

I+W(p·)(v–u)+=I+W(p·)(v–u)–,

where (v–u)+_{=max(v–u, ) and (v–u)}–_{=max(u–v, ).}

From (.), (.), (.), and (.), there exists a constantc˜> , such that

Wp|v–u|≤˜cW(pθ)≤c˜κpθ<∞.

Therefore,u=vby Lemma ..

Proof of Theorem . Consider ξ ≥ andζ ≥ with ξ +ζ > . Let α,β ∈(, ] and p∈Jα,β∩C((, )) be such that (A) is satisfied.

Let

S:=u∈B+(, ): ( –κp)θ≤u≤θ

,

whereθ(x) :=ξh(x) +ζh(x), andhandhare defined respectively by (.) and (.). Define the operatorFonSby

Fu=θ–Wp(pθ) +Wp

p–f(·,u)u.

By (.) and (.) we have

Wp(pθ)≤W(pθ)≤κpθ≤θ. (.)

Using (A), we get

≤f(·,u)≤p for allu∈S. (.)

Next, we prove thatFS⊆S. Indeed, using (.) and (.), we have, foru∈S,

and

Fu≥θ–Wp(pθ)

≥( –κp)θ.

Observe that, by (A),Fbecomes nondecreasing onS.

Define the sequence{vn}byv= ( –κp)θ andvn+=Fvnforn∈N. SinceFS⊆S, we

havev=Fv≥v, and by the monotonicity ofFwe deduce that

( –κp)θ=v≤v≤ · · · ≤vn≤vn+≤θ.

Using (A)-(A) and the dominated convergence theorem, we deduce that the sequence {vn}converges to a functionu∈Ssatisfying

u=I–Wp(p·)

θ+Wp

p–f(·,u)u,

that is,

I–Wp(p·)

u=I–Wp(p·)

θ–Wp

uf(·,u),

and by (.) we haveW(pu)≤W(pθ)≤θ<∞. Therefore, by Lemma . we deduce that

u=θ–Wuf(·,u). (.)

We claim thatuis a solution.

Indeed, from (.) and (.), there exists a constantc>  such that

( –s)α–_{u(s)fs,u(s)≤( –s)}α–_{θ(s)p(s)≤cs}β–_{( –s)}α_{p(s). (.)}

So, by Proposition . the functionW(uf(·,u))∈C–β([, ]). This implies by (.) that

u∈C–β([, ]).

Now, since the functions→( –s)α–_{u(s)f(s,u(s))∈C((, ))∩L}_{((, )), we deduce by} Proposition . thatuis a solution.

It remains to prove the uniqueness. Letvbe another solution inC–β([, ]) to problem

(.) satisfying (.). Sincev≤θ, we deduce by (.) that

≤v(s)fs,v(s)≤θ(s)p(s)≤csβ–( –s)p(s).

This implies thats→( –s)α–_{v(s)f(s,v(s))∈C((, ))∩L}_{((, )). Let˜v:=v+W(vf(·,v)).} By Proposition ., we have

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Dα_(Dβ_{v˜)(x) = ,  <x< ,}

limx→+Dβ–v˜(x) = , lim_x→+Dα–(Dβv˜)(x) =_ξ, ˜

Hence,

v=θ–Wvf(·,v). (.)

Letω: (, )→Rbe defined by

ω(z) = ⎧ ⎨ ⎩

v(z)f(z,v(z))–u(z)f(z,u(z))

v(z)–u(z) ifv(z)=u(z),  ifv(z) =u(z).

By (A),ω∈B+((, )) and from (.) and (.) we deduce that

I+W(ω·)(v–u)+=I+W(ω·)(v–u)–, where (v–u)+_{=max(v–u, ) and (v–u)}–_{=max(u–v, ).}

From (A) we haveω≤p. So by using (.) and (.) we obtain Wω|v–u|≤W(pθ)≤κpθ<∞.

Hence,u=vby (.).

Proof of Corollary. The statement follows from Theorem . withf(x,t) =λq(x)h(t),

(t) =th(t) andp(x) :=λq(x)max≤t≤θ(x)(t). Example . Letσ≥,ν≥, andq∈C+_{((, )) be such that}





t(β–)(+σ+ν)_{( –t)}α+σ+ν_q(t)dt<∞.

Let(t) =tσ+_{ln( +t}ν_{) andq˜(t) :=q(t)max}

≤s≤θ(t)(s). Sinceq˜∈Jα,β, then forξ ≥,

ζ ≥ withξ+ζ >  andλ∈[,_κ

˜

q), the problem

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Dα_(Dβ_{u)(x) +λq(x)u}σ+_{(x)ln( +u}ν_{(x)) = ,  <x< ,} limx→+Dβ–u(x) = , lim_x→+Dα–(Dβu)(x) =ξ, u() = , Dβ_{u() = –ζ,}

has a unique solutionu∈C–β([, ]) such that

( –λαq_˜)θ(x)≤u(x)≤θ(x) forx∈(, ].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia.} 2_{College of Sciences and Arts, Department of Mathematics, King Abdulaziz University, Rabigh Campus, P.O. Box 344,}

Rabigh, 21911, Saudi Arabia. 3_{Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, P.O. Box 1-764,}

Bucharest, 014700, Romania.4_{Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, Craiova, 200585,}

Acknowledgements

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding this Research group NO (RG-1435-043). The authors would like to thank the anonymous referees for their careful reading of the paper.

Received: 16 February 2016 Accepted: 5 April 2016

References

1. Agarwal, RP, O’Regan, D, Stanˇek, S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl.371, 57-68 (2010)

2. Bachar, I, Mâagli, H: Positive solutions for superlinear fractional boundary value problems. Adv. Differ. Equ.2014, 240 (2014)

3. Bai, Z, Lü, H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl.311, 495-505 (2005)

4. Giacomoni, J, Kumar Mishra, P, Sreenadh, K: Fractional elliptic equations with critical exponential nonlinearity. Adv. Nonlinear Anal.5, 57-74 (2016)

5. Graef, JR, Kong, L, Kong, Q, Wang, M: Existence and uniqueness of solutions for a fractional boundary value problem with Dirichlet boundary condition. Electron. J. Qual. Theory Differ. Equ.2013, 55 (2013).

doi:10.14232/ejqtde.2013.1.55

6. Kaufmann, ER, Mboumi, E: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ.2008, 3 (2008)

7. Mâagli, H, Mhadhebi, N, Zeddini, N: Existence and estimates of positive solutions for some singular fractional boundary value problems. Abstr. Appl. Anal.2014, Article ID 120781 (2014)

8. Mâagli, H, Mhadhebi, N, Zeddini, N: Existence and exact asymptotic behavior of positive solutions for a fractional boundary value problem. Abstr. Appl. Anal.2013, Article ID 420514 (2013)

9. Metzler, R, Klafter, J: Boundary value problems for fractional diffusion equations. Physica A278, 107-125 (2000) 10. Pucci, P, Xiang, M, Zhang, B: Existence and multiplicity of entire solutions for fractionalp-Kirchhoff equations. Adv.

Nonlinear Anal.5, 27-55 (2016)

11. Xu, X, Jiang, D, Yuan, C: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Nonlinear Anal.71, 4676-4688 (2009)

12. Zhang, X, Liu, L, Wu, Y: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model.55, 1263-1274 (2012)

13. Diethelm, K, Freed, AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In: Keil, F, Mackens, W, Voss, H, Werther, J (eds.) Scientific Computing in Chemical Engineering. II. Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, pp. 217-224. Springer, Heidelberg (1999)

14. Gaul, L, Klein, P, Kempfle, S: Damping description involving fractional operators. Mech. Syst. Signal Process.5, 81-88 (1991)

15. Glockle, WG, Nonnenmacher, TF: A fractional calculus approach of self-similar protein dynamics. Biophys. J.68, 46-53 (1995)

16. Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

17. Kilbas, A, Srivastava, H, Trujillo, J: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

18. Kilbas, AA, Trujillo, JJ: Differential equations of fractional order: methods, results and problems. I. Appl. Anal.78, 153-192 (2001)

19. Kilbas, AA, Trujillo, JJ: Differential equations of fractional order: methods, results and problems. II. Appl. Anal.81, 435-493 (2002)

20. Mainardi, F: Fractional calculus: some basic problems in continuum and statical mechanics. In: Carpinteri, A, Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Calculus Mechanics, pp. 291-348. Springer, Vienna (1997)

21. Miller, K, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

22. Podlubny, I: Fractional Differential Equations. Academic Press, New York (1999)

23. Samko, S, Kilbas, A, Marichev, O: Fractional Integrals and Derivative. Theory and Applications. Gordon and Breach, Yverdon (1993)