**R E S E A R C H**

**Open Access**

## Positive solutions to fractional boundary

## value problems with nonlinear boundary

## conditions

### Wenquan Feng, Shurong Sun

*_{, Xinhui Li and Meirong Xu}

*_{Correspondence:}

sshrong@163.com

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P.R. China

**Abstract**

We consider the existence of at least one positive solution of the problem
–*Dα*_{0+}*u*(*t*) =*f*(*t*,*u*(*t*)), 0 <*t*< 1, under the circumstances that*u*(0) = 0,

*u*(1) =*H*1(

*ϕ*

(*u*)) +

*EH*2(*s*,*u*(*s*))*ds*, where 1 <

*α*

< 2,*D*

*α*

0+is the Riemann-Liouville
fractional derivative, and*u*(1) =*H*1(

*ϕ*

(*u*)) +

*EH*2(*s*,*u*(*s*))*ds*represents a nonlinear
nonlocal boundary condition. By imposing some relatively mild structural conditions
on*f*,*H*1,*H*2, and

*ϕ*

, one positive solution to the problem is ensured. Our results
generalize the existing results and an example is given as well.
**MSC:** 34A08; 34B18

**Keywords:** fractional diﬀerential equation; nonlinear boundary condition;
Krasnosel’skiˇi’s ﬁxed point theorem; positive solution

**1 Introduction**

In this paper we consider the existence of at least one positive solution of the fractional diﬀerential equation

–*Dα*_{+}*u*(*t*) =*ft*,*u*(*t*), <*t*< , ()

subject to the boundary conditions

*u*() = , *u*() =*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*, ()

here*E*(, ) is some measurable set, <*α*< ,*Dα*

+is the Riemann-Liouville fractional
derivative and*ϕ*is a linear functional having the form

*ϕ*(*u*) :=

*u*(*t*)*dθ*(*t*), ()

where the integral appearing in () is taken in the Lebesgue-Stieltjes sense,*θ*is a function
of bounded variation.

Let us review brieﬂy some recent results on such problems in order to see our problem ()-() in a more appropriate context.

So far, in view of their various applications in science and engineering, such as ﬂuid mechanics, control system, viscoelasticity, porous media, edge detection, optical systems, electromagnetism and so forth, see [–], fractional diﬀerential equations have attracted great attention of mathematicians.

There are a great number of works on the existence of solutions of various classes of ordinary diﬀerential equations and fractional diﬀerential equations; readers may refer to [–].

Some of them discussed two-point boundary value problems. For example, Bai and Lü [] studied the following two-point boundary value problem of fractional diﬀerential equations:

*Dα*_{+}*u*(*t*) +*ft*,*u*(*t*)= , <*t*< ,

*u*() =*u*() = ,

()

where <*α*≤,*Dα*

+is the standard Riemann-Liouville fractional derivative. By means of Guo-Krasnosel’skiˇi’s ﬁxed point theorem and the Leggett-Williams ﬁxed point theorem they obtained the existence of positive solutions.

Some authors discussed multi-point boundary value problems, for instance, by using ﬁxed point index theory, the Krein-Rutman theorem and some other methods, Jiang [] studied the eigenvalue interval of the multi-point boundary value problem

*Dα _{u}*

_{(}

_{t}_{) –}

_{Mu}_{(}

_{t}_{) =}

_{λ}_{f}_{t}_{,}

_{u}_{(}

_{t}_{)}

_{,}

_{t}_{∈}

_{[, ],}

*u*() =

*n*

*i*=
*βiu*(*ξi*),

()

where <*α*< ,*Dα*_{is the Caputo derivative,}_{M}_{≥}_{, <}_{ξ}

<*ξ*<· · ·<*ξn*≤.

There are also results on fractional boundary value problem with integral boundary con-ditions, let us refer to Vong []. He investigated positive solutions of the nonlocal bound-ary value problem for a class of singular fractional diﬀerential equations with an integral boundary condition,

*c _{D}α*

+*u*(*t*) +*f*

*t*,*u*(*t*)= ,

*u*() =· · ·=*u*(*n*–)() = , *u*() =

*u*(*s*)*dμ*(*s*),

()

where*n*≥,*α*∈(*n*– ,*n*) and*μ*is a function of bounded variation.

To proceed, Goodrich [] considered the existence of at least one positive solution of the ordinary diﬀerential equation

–*y*(*t*) =*ft*,*y*(*t*), <*t*< ,

*y*() =*H*

*ϕ*(*y*)+

*E*
*H*

*s*,*y*(*s*)*ds*, *y*() = ,

()

in which the boundary condition is more general.

frac-tional diﬀerential equation under the same boundary conditions. Secondly, the
bound-ary conditions are more ﬂexible and general than often. Let us take the condition*u*() =

*H*(*ϕ*(*u*)) +

*EH*(*s*,*u*(*s*))*ds*into consideration, where *H*,*H*,*ϕ*(*u*) are deﬁned in the
se-quel. If*H*(*ϕ*(*u*)) +

*EH*(*s*,*u*(*s*))*ds*= , the conditions are the standard Dirichlet boundary

conditions. Readers might refer to Bai and Lü []. As far as we are concerned,*ϕ*(*u*) varies
among many sorts of functionals. If*H*(*ϕ*(*u*)) =

*Fu*(*t*)*dt*(where*F*⊂*E*(, ) is deﬁned

in the sequel) or*H*(*ϕ*(*u*)) =

[,]*u*(*t*)*dθ*(*t*), our conditions reduce to integral boundary
conditions, while if*H*(*ϕ*(*u*)) =

*n*

*i*=|*ai*|*u*(*ξi*), we have multi-point boundary conditions.

Thirdly, compared to Goodrich [], we make an adjustment to the Green function and
deﬁne a function*r*(·) instead of a constant which aﬀects the deﬁned cone.

This paper is organized as follows. In Section , we review some preliminaries and lem-mas. In Section , a theorem and ﬁve corollaries about the existence of at least one positive solution of problem ()-() are obtained. Lastly, we give an example to illustrate the ob-tained theorem.

**2 Preliminaries and lemmas**

For the convenience of the readers, we give some background materials from fractional calculus theory to facilitate the analysis of the boundary value problem ()-().

**Deﬁnition .**([]) The Riemann-Liouville fractional integral of order*α*> of a function

*y*: (, +∞)→Ris given by

*Iα*

+*y*(*t*) =

(*α*)
*t*

(*t*–*s*)*α*–_{y}_{(}_{s}_{)}_{ds}_{,}

provided the right side is pointwise deﬁned on (, +∞).

**Deﬁnition .**([]) The Riemann-Liouville typed fractional derivative of order*α*(*α*> )
of a continuous function*f* : (, +∞)→Ris given by

*Dα*_{}+*f*(*t*) =

(*n*–*α*)

*d*
*dt*

*n* *t*

(*t*–*s*)*n*–*α*+*f*(*s*)*ds*,

where*n*= [*α*] + , [*α*] denotes the integer part of number*α*, provided that the right side is
pointwise deﬁned on (, +∞).

**Lemma .**([]) *Letα*> .*If we assume u*∈*C*(, )∩*L*(, ),*then the fractional diﬀerential*
*equation*

*Dα*_{}+*u*(*t*) =

*has u*(*t*) =*c**tα*–+*c**tα*–+· · ·+*cntα*–*n*,*ci*∈R,*i*= , , . . . ,*n*,*as a unique solution*,*where n*
*is the smallest integer greater than or equal toα*.

**Lemma .**([]) *Let u*∈*C*(, )∩*L*(, )*with a fractional derivative of orderα*(*α*> )

*that belongs to C*(, )∩*L*(, ).*Then*

*I*_{}*α*+*Dα*_{}+*u*(*t*) =*u*(*t*) +*c**tα*–+*c**tα*–+· · ·+*cntα*–*n*, *for some ci*∈R,*i*= , , . . . ,*n*,

**Remark .**([]) The Riemann-Liouville type fractional derivative and integral of order

*α*(*α*> ) have the following properties:

*Dα*

+*I*_{}*α*+*u*(*t*) =*u*(*t*), *I*_{}*α*+*I*_{}*β*+*u*(*t*) =*I*

*α*+*β*

+ *u*(*t*), *α*,*β*> ,*u*∈*L*(, ).

**Lemma .** *Let u*∈*C*[, ]*and* <*α*< .*Then the fractional diﬀerential equation *
*bound-ary value problem*

*Dα*_{}+*u*(*t*) +*y*(*t*) = , <*t*< ,

*u*() = , *u*() =*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*,

*has a unique solution*,

*u*(*t*) =*tα*–

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

+

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

*where*

*G*(*t*,*s*) =

(*α*)
⎧
⎨
⎩

*tα*–_{( –}_{s}_{)}*α*–_{– (}_{t}_{–}_{s}_{)}*α*–_{,} _{}_{≤}_{s}_{≤}_{t}_{≤}_{,}

*tα*–_{( –}_{s}_{)}*α*–_{,} _{}_{≤}_{t}_{≤}_{s}_{≤}_{.} ()

*Proof* We may apply Lemma . to reduce () to an equivalent integral equation,

*u*(*t*) = –*I*_{+}*αy*(*s*) +*c**tα*–+*c**tα*–, *c*,*c*∈R.

Consequently, the general solution of () is

*u*(*t*) = –
*t*

(*t*–*s*)*α*–

(*α*) *y*(*s*)*ds*+*c**t*

*α*–_{+}_{c}

*tα*–, *c*,*c*∈R.

By (), we have

*c*=*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*+

( –*s*)*α*–
(*α*) *y*(*s*)*ds*,

*c*= .

Therefore, the unique solution of problem () and () is

*u*(*t*) =*tα*–

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

–
*t*

(*t*–*s*)*α*–

(*α*) *y*(*s*)*ds*+*t*

*α*–

( –*s*)*α*–
(*α*) *y*(*s*)*ds*

=*tα*–

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

+

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

**Lemma .**([]) *Let*(*a*,*b*)⊂(, )*be an arbitrary but ﬁxed interval*.*Then the function*
*G*(*t*,*s*)*deﬁned by*()*satisﬁes the following conditions*:

() *G*(*t*,*s*) > ,*fort*,*s*∈(, );

() *there exists a positive functionγ*(·)∈*C*(, )*such that*

min

*a*≤*t*≤*bG*(*t*,*s*)≥*γ*(*s*)max≤*t*≤*G*(*t*,*s*) =*γ*(*s*)*G*(*s*,*s*), *for each*≤*s*≤. ()

**Lemma .**([]) *Let B be a Banach space*,*and let K*⊂*B be a cone*.*Assume*,*are*

*open and bounded subsets of B with*∈,¯⊂,*and let T*:*K*∩(¯\)→*K be a*

*completely continuous operator such that*

(i) *Tu* ≤ *u*,*u*∈*K*∩*∂*,*andTu* ≥ *u*,*u*∈*K*∩*∂*;*or*
(ii) *Tu* ≥ *u*,*u*∈*K*∩*∂*,*andTu* ≤ *u*,*u*∈*K*∩*∂*.

*Then T has a ﬁxed point in K*∩(¯\).

In order to get the main results, we ﬁrst need some structure on*H*,*H*,*ϕ*, and*f*
ap-pearing in problem ()-().

Let*B*be the Banach space on*C*([, ]) equipped with the usual supremum norm · .
Then deﬁne the cone*K*⊆*B*by

*K*=

*u*∈*B*|*u*(*t*)≥, min

*a*≤*t*≤*bu*(*t*)≥*γ*
∗_{}_{u}_{}_{,}_{ϕ}

(*u*),*ϕ*(*u*)≥

,

where*γ*∗=min{min*t*∈[*a*,*b*]*tα*–,min*s*∈[*a*,*b*]*γ*(*s*)}.
Deﬁne the operator*T*:*C*[, ]→*C*[, ] by

*Tu*(*t*) =*tα*–

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

+

*G*(*t*,*s*)*fs*,*u*(*s*)*ds*. ()

Here we come to the nine signiﬁcant assumptions.

(H) Let*H*: [, +∞)→[, +∞)and*H*: [, ]×[, +∞)→[, +∞)be real-valued
con-tinuous functions.

(H) The functional*ϕ*(*u*) :=

[,]*u*(*t*)*dθ*(*t*)can be written in the form

*ϕ*(*u*) =*ϕ*(*u*) +*ϕ*(*u*) :=

[,]

*u*(*t*)*dθ*(*t*) +

[,]

*u*(*t*)*dθ*(*t*), ()

where*θ*,*θ*,*θ*: [, ]→Rsatisfy*θ*,*θ*,*θ*∈*BV*([, ]), and*ϕ*,*ϕ*are continuous linear
functionals.

(H) There is a constant*C*∈[, )such that the functional*ϕ*in()satisﬁes the inequality

*ϕ*(*u*)≤*C**u* ()

for all*u*∈*C*([, ]). Furthermore, there is a constant*C*> such that the functional
*ϕ*in()satisﬁes|*ϕ*(*u*)| ≥*C**u*whenever*u*∈*K*.

(H) For each given*ε*> , there are*C*> and*Mε*> whenever*z*≥*Mε*and we have

(H) There exists a function*F*: [, +∞)→[, +∞)satisfying the growth condition

*F*(*u*) <*C**u* ()

for some*C*≥, having the property that for each given*ε*> , there is*Mε*> such

that

*H*(*x*,*u*) –*F*(*u*)<*εF*(*u*) ()

for all*x*∈[, ], whenever*u*≥*Mε*.

(H) Assume that the nonlinearity*f*(*t*,*u*)splits in the sense that*f*(*t*,*u*) =*a*(*t*)*g*(*u*), for
con-tinuous functions*a*: [, ]→[, +∞)and*g*:R→[, +∞)such that*a*is not
identi-cally zero on any subinterval of[, ].

(H) Supposelim*u*→+ *g*(*u*)

*u* = +∞.

(H) Supposelim*u*→+∞ *g*(*uu*) = .

(H) For each*i*= , both

[,]

*tα*–*dθi*(*t*)≥ ()

and

[,]

*G*(*t*,*s*)*dθi*(*t*)≥ ()

hold, where()holds for each*s*∈[, ].

**3 Main results**

In this section we state and prove the existence theorem of problem ()-().

**Lemma .** *Let T be the operator deﬁned in*().*Assume conditions*(H)*-*(H)*hold*.*Then*

*T*:*K*→*K*,*and the operator T is completely continuous*.

*Proof* Now we divide the proof into two steps; in the ﬁrst step we prove that*T*:*K*→*K*,
then in the next, the conclusion that the operator*T*is completely continuous is treated.

Step . Here we are going to show that*T* :*K*→*K*. In fact, since*H*,*H*,*a*, and*g* are
all nonnegative, it is easy to ﬁnd that whenever*u*∈*K*, it follows that (*Tu*)(*t*)≥, for each

*t*∈[, ], where we use the fact*ϕ*(*u*)≥, following*H*(*ϕ*(*u*))≥.
On the other hand, provided*u*∈*K*we get

min

*t*∈[*a*,*b*]*Tu*(*t*)≥*t*min∈[*a*,*b*]*t*

*α*–

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

+ min

*t*∈[*a*,*b*]

*G*(*t*,*s*)*fs*,*u*(*s*)*ds*

≥*γ*

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

+max

*t*∈[,]

*γ*(*s*)*G*(*t*,*s*)*fs*,*u*(*s*)*ds*

≥*γ*

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

+max

*t*∈[,]

*b*

*a*

*γ*(*s*)*G*(*t*,*s*)*fs*,*u*(*s*)*ds*

≥*γ*

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

+ min

*t*∈[*a*,*b*]*γ*(*t*)*t*max∈[,]

*b*

*a*

*G*(*t*,*s*)*fs*,*u*(*s*)*ds*

≥*γ*∗*Tu*, ()

where*γ*=min*t*∈[*a*,*b*]*tα*–.

Consequently, for each*i*= , we deduce that

*ϕi*(*Tu*) =

[,]

*tα*–

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

*dθi*(*t*)

+

[,]

*G*(*t*,*s*)*a*(*s*)*gu*(*s*)*ds* *dθi*(*t*)

=

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

[,]

*tα*–_{d}_{θ}

*i*(*t*)

+

[,]

*G*(*t*,*s*)*dθi*(*t*)

*a*(*s*)*gu*(*s*)*ds*

≥, ()

where the inequality follows from assumption (H). Thus,*ϕi*(*Tu*)≥, therefore as desired

we conclude*T*(*K*)⊂*K*.

Step . In this part we turn to the proof that*T*() in the sequel is bounded as well as
equicontinuous with the help of the Arzelà-Ascoli theorem.

With the continuity of*H*,*H*,*a*, and*g*, it is easy to ﬁnd*T*is continuous.

Let⊂*K*be bounded, namely, there exists a number*M*> , such that for each*u*∈

we have*u* ≤*M*. By the continuity of*H*,*H*, and*ϕ*, we ﬁnd*H*,*H*are bounded, so there
exist constants*P*> and*Q*> such that|*H*(*ϕ*())| ≤*P*and|*H*(*t*,*u*(*t*))| ≤*Q*.*E*(, ) is
a measurable set, take*L*=max*t*∈[,],*u*∈[,*M*]|*f*(*t*,*u*)|+ , then

(*Tu*)(*t*)=*tα*–

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

+

*G*(*t*,*s*)*fs*,*u*(*s*)*ds*

≤*H*

*ϕ*(*u*)+

*E*

_{H}_{}

*s*,*u*(*s*)*ds*+

*G*(*t*,*s*)*fs*,*u*(*s*)*ds*

≤*P*+*Qm*(*E*) +*L*

*G*(*t*,*s*)*ds*

≤*P*+*Qm*(*E*) +*L*

*G*(*s*,*s*)*ds*

=*P*+*Qm*(*E*) + *L*

(*α*+ ). ()

Hence*T*() is bounded.

For each*u*∈,*t*,*t*∈[, ],*t*<*t*, and assuming there is a*δ*> , such that*t*–*t*<*δ*,
for each*ε*> put

*δ*=min

*ε*,

*ε*
*N*
*α*–
,
*ε*

(*α*– )*N*

*α*–
,

*ε*

*Nα*
*α*

where*N*=*P*+*Qm*(*E*) +_{}_{(}_{α}L_{+)}, we ﬁnd|(*Tu*)(*t*) – (*Tu*)(*t*)|<*ε*, so that*T*() is
equicontin-uous.

Indeed,

(*Tu*)(*t*) – (*Tu*)(*t*)

=*t*_{}*α*–

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

+

*G*(*t*,*s*)*f*

*s*,*u*(*s*)*ds*

–*tα*_{}–

*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

–

*G*(*t*,*s*)*f*

*s*,*u*(*s*)*ds*

=*t*_{}*α*––*tα*_{}–*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*

+

*G*(*t*,*s*) –*G*(*t*,*s*)

*fs*,*u*(*s*)*ds*

≤*tα*_{}––*t*_{}*α*–*H*

*ϕ*(*u*)+

*E*

*H*

*s*,*u*(*s*)*ds*

+

*G*(*t*,*s*) –*G*(*t*,*s*)

*fs*,*u*(*s*)*ds*. ()

By the continuity of*H*,*H*, and*ϕ*, we ﬁnd*H*,*H* are bounded, so there exist constants

*P*> and*Q*> such that|*H*(*ϕ*())| ≤*P*and|*H*(*t*,*u*(*t*))| ≤*Q*.*E*(, ) is a measurable
set, take*L*=max*t*∈[,],*u*∈[,*M*]|*f*(*t*,*u*)|+ , then

(*Tu*)(*t*) – (*Tu*)(*t*)

≤*tα*_{}––*t*_{}*α*–*P*+*Qm*(*E*)+

* _{G}*(

*t*,

*s*) –

*G*(

*t*,

*s*)

*fs*,*u*(*s*)*ds*

≤*tα*_{}––*t*_{}*α*–*P*+*Qm*(*E*)

+ *L*

(*α*)
*t*

*t*_{}*α*–( –*s*)*α*–– (*t*–*s*)*α*––*t**α*–( –*s*)

*α*–_{+ (}_{t}

–*s*)*α*–*ds*

+
*t*

*t*

*t*_{}*α*–( –*s*)*α*–– (*t*–*s*)*α*––*tα*–( –*s*)

*α*–_{ds}

+

*t*

*tα*_{}–( –*s*)*α*––*t*_{}*α*–( –*s*)*α*–*ds*

≤*tα*_{}––*t*_{}*α*–*P*+*Qm*(*E*)+ *L*

(*α*)

_{}

*t*_{}*α*––*t*_{}*α*–

*t*

( –*s*)*α*–_{ds}

+
*t*

(*t*–*s*)*α*–– (*t*–*s*)*α*–

*ds*+*tα*_{}––*t*_{}*α*–

*t*

*t*

( –*s*)*α*–*ds*

+
*t*

*t*

(*t*–*s*)*α*–*ds*+*tα*––*t*

*α*–

*t*

( –*s*)*α*–*ds*

=*t*_{}*α*––*t*_{}*α*–*P*+*Qm*(*E*) + *L*

(*α*+ )

+ *L*

(*α*+ )

*tα*_{}–*t*_{}*α*. ()

Next we discuss the following three cases.
Case .*δ*≤*t*<*t*< .

*t*_{}*α*––*tα*_{}–≤*α*–

*δ*–*α*(*t*–*t*)≤(*α*– )*δ*

*α*–_{≤} *ε*
*N*,

*t*_{}*α*–*tα*_{} ≤ *α*

*δ*–*α*(*t*–*t*)≤*αδ*
*α*_{≤} *ε*

*N*.

Case . ≤*t*<*δ*<*t*< *δ*.

*t*_{}*α*––*t*_{}*α*–≤*t*_{}*α*–≤(*δ*)*α*–≤ *ε*

*N*, ()

*t*_{}*α*–*tα*_{} ≤*t*_{}*α*≤(*δ*)*α*≤ *ε*
*N*.

Case . ≤*t*<*t*<*δ*.

*tα*–

–*t**α*–≤*t**α*–≤*δα*–≤
*ε*

*N*, ()

*t*_{}*α*–*tα*_{} ≤*t*_{}*α*≤*δα*≤ *ε*

*N*.

Hence|(*Tu*)(*t*) – (*Tu*)(*t*)| ≤*ε*.

From the Arzelà-Ascoli theorem,*T* is completely continuous. The proof is complete.

With Lemma . in hand, we are now ready to present the ﬁrst existence theorem for problem () and ().

**Theorem .** *Assume that conditions*(H)*-*(H)*hold*.*Suppose that*

*C**C*+*C**m*(*E*) < ()

*and that E*(, ).*Then problem*()*-*()*has at least one positive solution*.

*Proof* Begin by selecting a number*η*such that

*η*

*b*

*a*

*γ*∗*G*

,*s* *a*(*s*)*ds*≥. ()

From (H), there exists a number*r*> such that*g*(*u*)≥*η**u*whenever <*u*≤*r*. Then
set

*r*:=

*u*∈*B*:*u*<*r*

, ()

for*u*∈*K*∩*∂r*we ﬁnd

(*Tu*)

≥

*G*

,*s* *a*(*s*)*g*

*u*(*s*)*ds*≥ *uη*

*b*

*a*

*γ*∗*G*

,*s* *a*(*s*)*ds*

≥ *u*, ()

from the deﬁnition of norm · , we get*Tu* ≥ *u*, and so,*T* is a cone expansion on

From condition (), we choose a positive number*ε*small enough, so that we may
as-sume

*C**C**ε*+*C**C*+*ε*+ (*C*+*C**ε*)*m*(*E*)≤. ()

Since*ϕ*(*u*)≥, it follows that*ϕ*(*u*)≥*ϕ*(*u*)≥*C**u*, if*u* ≥*M _{C}*

_{}

*ε*, then

*ϕ*(

*u*) >

*Mε*, and

then by condition (H), we have

*H*

*ϕ*(*u*)–*C**ϕ*(*u*)<*εC**ϕ*(*u*). ()

For*E*(, ), we may select <*a*<*b*< such that*E*⊆(*a*,*b*). By the deﬁnition of the
cone, we have

min

*t*∈*Eu*(*t*)≥*t*min∈[*a*,*b*]*u*(*t*)≥*r*

∗_{}_{u}_{}_{.} _{()}

If*u* ≥*Mε*

*r*∗, then

min

*t*∈*Eu*(*t*)≥*t*min∈[*a*,*b*]*u*(*t*)≥*r*

∗_{}_{u}_{ ≥}* _{r}*∗

*Mε*

*r*∗ =*Mε*. ()

Hence, by condition (H) we get

*H*

*x*,*u*(*s*)–*Fu*(*s*)<*εFu*(*s*). ()

Provided that

*u* ≥max

*Mε*
*C*

,*Mε*

*r*∗

, ()

both () and () hold.

Next we are going to discuss these two cases:*g*is bounded and unbounded on [, +∞),
respectively.

Now if*g*is bounded on [, +∞), then there exists*r*> suﬃciently large such that

*g*(*u*)≤*r*, for any*u*≥. ()

Indeed, we might assume without loss of generality that

*g*(*u*)≤_{} *r**ε*
*G*(*s*,*s*)*a*(*s*)*ds*

, ()

where*ε*is selected suﬃciently small such that both () and () hold. We deﬁne a number

*r*∗_{}:=max

*r*

*r*∗,*r*,
*Mε*
*C*

,*Mε*

*r*∗

.

Set

*r*∗_{}:=

Then for each*u*∈*K*∩*∂r*_{}∗we ﬁnd that

*Tu* ≤*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*+

*G*(*s*,*s*)*a*(*s*)*gu*(*s*)*ds*

≤*H*

*ϕ*(*u*)–*C**ϕ*(*u*)+*C**ϕ*(*u*)+

*E*

*H*

*s*,*u*(*s*)–*Fu*(*s*)*ds*

+

*E*

*Fu*(*s*)*ds*+

*r**εds*

≤*εC**C**u*+*C**C**u*+*m*(*E*)*C*( +*ε*)*u*+*εu*

=*C**C**ε*+*C**C*+*m*(*E*)*C*( +*ε*) +*ε*

*u*

≤ *u*, ()

whence*T* is a cone compression on*K*∩*∂r*∗_{}.

On the other hand, assume*g*is unbounded on [, +∞). From condition (H) there is a
number*r*> such that*g*(*u*)≤*η**u*whenever*u*>*r*, where*η*is picked with

*η*

*G*(*s*,*s*)*a*(*s*)*ds*≤*ε*. ()

Since*g*is unbounded on [, +∞), we can ﬁnd a number*r*∗_{}satisfying

*r*∗_{}>max

*r*

*r*∗ ,*r*,
*Mε*
*C*

,*Mε*

*r*∗

,

such that*g*(*u*)≤*g*(*r*_{}∗) for any*u*∈[,*r*_{}∗].
Take

*r*∗_{}:=

*u*∈*B*:*u*<*r*_{}∗. ()

Then for each*u*∈*K*∩*∂r*_{}∗we ﬁnd that

*Tu* ≤*H*

*ϕ*(*u*)+

*E*
*H*

*s*,*u*(*s*)*ds*+

*G*(*s*,*s*)*a*(*s*)*gu*(*s*)*ds*

≤*H*

*ϕ*(*u*)–*C**ϕ*(*u*)+*C**ϕ*(*u*)+

*E*

*H*

*s*,*u*(*s*)–*Fu*(*s*)*ds*

+

*E*

_{F}

*u*(*s*)*ds*+

*G*(*s*,*s*)*a*(*s*)*gr*_{}∗*ds*

≤*C**εϕ*(*u*) +*C**C**u*+

*E*

( +*ε*)*Fu*(*s*)*ds*+

*G*(*s*,*s*)*a*(*s*)*gr*_{}∗*ds*

≤*εC**C**u*+*C**C**u*+*m*(*E*)*C*( +*ε*)*u*+*εu*

=*C**C**ε*+*C**C*+*m*(*E*)*C*( +*ε*) +*ε*

*u*

≤ *u*, ()

Therefore, in either case, deﬁne*r*_{}∗=max{*r*∗_{},*r*_{}∗}, we ﬁnd*Tu* ≤ *u*whenever*u*∈*K*∩

*∂ _{r}*∗

. From Lemma ., we claim that problem ()-() has at least one positive solution;

the proof is complete.

Next we are going to give some corollaries since*ϕ*(*u*) admits a wide variety of
function-als. First we assume*H*(*ϕ*(*u*)) = , then, respectively, that*H*(*ϕ*(*u*)) =

*Fu*(*t*)*dt*,*H*(*ϕ*(*u*)) =

*n*

*i*=|*ai*|*u*(*ξi*) and*H*(*ϕ*(*u*)) =

[,]*u*(*t*)*dθ*(*t*), where*F*⊂*E*(, ) is not Lebesgue null. In
addition, we know*u*() = is also well deﬁned and if*u*() = the problem as well as the
boundary conditions are similar to [].

**Corollary .** *Assume that conditions*(H), (H)*-*(H)*hold*,*then the problem with *

*Dirich-let conditions*

–*Dα*_{+}*u*(*t*) =*ft*,*u*(*t*), <*t*< ,

*u*() = , *u*() = ,

()

*has at least one positive solution*.

**Corollary .** *Assume that conditions*(H)*-*(H)*hold*.*Suppose*,*in addition*,

*C**C*+*C**m*(*E*) < ,

*and F* ⊂*E*(, ) *is not Lebesgue null*,*and m*(*F*)≤*C*,*then the problem with integral*

*conditions*

–*Dα*_{+}*u*(*t*) =*ft*,*u*(*t*), <*t*< ,

*u*() =

*F*

*u*(*t*)*dt*+

*E*
*H*

*s*,*u*(*s*)*ds*, *u*() = ,

()

*has at least one positive solution*.

**Corollary .** *Assume that conditions*(H)*-*(H)*hold*.*Suppose*,*in addition*,

*C**C*+*C**m*(*E*) < ,

*and E*(, )*is not Lebesgue null*,*andn _{i}*

_{=}|

*ai*| ≤

*C*,

*then the problem with multi-point*

*conditions*

–*Dα*_{+}*u*(*t*) =*ft*,*u*(*t*), <*t*< ,

*u*() =

*n*

*i*=

|*ai*|*u*(*ξi*) +

*E*
*H*

*s*,*u*(*s*)*ds*, *u*() = ,

()

*has at least one positive solution*.

**Corollary .** *Assume that conditions*(H)*-*(H)*hold*.*Suppose*,*in addition*,

*and E*(, )*is not Lebesgue null*,*and the total variation ofθover*[, ]*satisﬁes V*[,](*θ*)≤

*C*,*then the problem with the Lebesgue-Stieltjes integral conditions*

–*Dα*_{+}*u*(*t*) =*ft*,*u*(*t*), <*t*< ,

*u*() =

[,]

*u*(*t*)*dθ*(*t*) +

*E*
*H*

*s*,*u*(*s*)*ds*, *u*() = ,

()

*has at least one positive solution*.

Specially, take*θ*(*t*) =*t*_{–}_{t}_{, for any}_{x}_{,}_{y}_{∈}_{[, ] we have}

*θ*(*x*) –*θ*(*y*)=(*x*–*y*)(*x*+*y*– )≤ |*x*–*y*|,

then

*V*[,](*θ*) =

*n*

*i*=

*θ*(*xi*) –*θ*(*xi*–)≤

*n*

*i*=

|*xi*–*xi*–|=
*n*

*i*=

(*xi*–*xi*–) = ,

thus we have the following corollary.

**Corollary .** *Assume that conditions*(H)*-*(H)*and*(H)*-*(H)*hold*.*Suppose*,*in *

*addi-tion*,

*C**C*< ,

*then the problem*

–*Dα*

+*u*(*t*) =*f*

*t*,*u*(*t*), <*t*< ,

*u*() =

[,]

*u*(*t*)*dt*–*t*, *u*() = ,

()

*has at least one positive solution*.

**4 Example**

In this part we give an example of Theorem ..
Deﬁne*ϕ*(*u*) in the ﬁrst place by

*ϕ*(*u*) =
*u*

–

*u*

+

[_{},_{}]

*u*(*t*)*dt*, ()

where

*ϕ*(*u*) =
*u*

–

*u*

, *ϕ*(*u*) =

[_{},_{}]

*u*(*t*)*dt*. ()

Then deﬁne*H*,*H*by

and

*H*(*x*,*u*) = *u*+ *x*+*ex*
√

*u*. ()

It is clear that

lim

*z*→∞ln(*z*+ ) +*z*

–*z*= , ()

and moreover

lim
*u*→∞

|(*u*+ *x*_{+}* _{e}x*√

_{u}_{) – }

_{u}_{|}

*u* = , ()

so conditions (H) and (H) hold, and we see*C*= ,*C*= , and*F*(*u*) = *u*.
Now we consider the boundary value problem

–*Dα*_{+}*u*(*t*) =*ft*,*u*(*t*), <*t*< ,

*u*() =ln*ϕ*(*u*) + +*ϕ*(*u*) +

*u*(*s*) + *s*+*esu*(*s*)*ds*, *u*() = ,

()

where*f*(*t*,*u*(*t*)) is a given function with conditions (H) and (H) satisﬁed. Here*E*= [_{},_{}]
is chosen such that*m*(*E*) =_{} and*E*⊂(, ).

What is more, for each*u*∈*K*

*ϕ*(*u*)≤
*u*+

*u*+

–

*u*=

*u* ()

and

*ϕ*(*u*)≥
*r*

∗_{}_{u}_{}_{.} _{()}

Then we ﬁnd that *C* =_{} ∈[, ] and*C*= _{}*r*∗> , so condition (H) is met as well.
Finally, after straightforward numerical calculations, condition (H) can also be achieved,
since

*C**C*+*C**m*(*E*) =
· + ·

< .

As a consequence, each of conditions (H)-(H) is satisﬁed. From Theorem ., problem () has at least one positive solution.

**Competing interests**

The authors declare that they have no competing interests.

**Authors’ contributions**

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

**Acknowledgements**

The authors sincerely thank the reviewers for their valuable suggestions and useful comments, which have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003), also supported by Graduate Innovation Foundation of University of Jinan (YCX13013).

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doi:10.1186/s13661-014-0225-0