R E S E A R C H
Open Access
Singular boundary value problems of
fractional differential equations with
changing sign nonlinearity and parameter
Yong Mu
1, Liying Sun
2and Zhenlai Han
2**Correspondence: [email protected]
2School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P.R. China
Full list of author information is available at the end of the article
Abstract
In this paper, we consider singular boundary value problems for the following nonlinear fractional differential equations with delay:
⎧ ⎨ ⎩
Dαx(t) +
λf
(t,x(t–τ
)) = 0, t∈(0, 1)\{τ},x(t) =
η
(t), t∈[–τ
, 0],x(1) =x(0) = 0,
where 2 <
α
≤3,Dαdenotes the Riemann-Liouville fractional derivative,λ
is a positive constant,f(t,x) may change sign and be singular att= 0,t= 1, andx= 0. By means of the Guo-Krasnoselskii fixed point theorem, the eigenvalue intervals of the nonlinear fractional functional differential equation boundary value problem are considered, and some positive solutions are obtained, respectively.MSC: 34A08; 34K37
Keywords: fractional functional differential equation; delay; boundary value problems; singular; positive solutions
1 Introduction
Fractional differential equations have been of increasing importance for the past decades due to their diverse applications in science and engineering, we can describe natural phe-nomena and mathematical models more accurately. Many researchers have shown their interest in fractional differential equations. The motivation for those works stems from both the intensive development of the theory of fractional calculus itself and the applica-tions such as in economics, engineering and other fields. Fractional differential equaapplica-tions have received much attention, the theory and its applications have been greatly developed; see [–].
There have been many papers dealing with boundary value problems of fractional dif-ferential equations [–] and initial value problems of fractional differential equations [–].
However, the results focused on the singular boundary value problems of fractional dif-ferential equations with delay are relatively scarce [–]. It is well known that in practical
problems, the behavior of systems not only depends on the status just at the present, but also on the status in the past.
Thus, in many cases, we must study fractional differential equations with delay in order to solve practical problems. Consequently, our aim in the paper is to consider the existence of solutions for singular boundary value problems of fractional differential equations with delay.
In , by means of the fixed point index theorem, Yuet al.[] investigate the exis-tence of multiple positive solutions for the third-order three-point singular semipositone boundary value problem
x(t) –λf(t,x) = , t∈(, ), x() =x(η) =x() = ,
where
<η< ,f(t,x) : (, )×(, +∞)→(–∞, +∞) is continuous and may be singular att= ,t= , andx= and also may be negative for some values oftandx;λis a positive parameter.
In , Zhaoet al.[] studied the existence on multiple positive solutions for the non-linear fractional differential equation boundary value problem
Dα
+u(t) +f(t,u(t)) = , t∈(, ), u() =u() =u() = ,
where <α≤,Dα
+is the Riemann-Liouville fractional derivative. By the lower and up-per solutions method and the fixed point theorem, they obtained some new existence criteria for singular and nonsingular fractional differential equation boundary value prob-lems.
In , Su [] studied the boundary value problem for a singular fractional differential equation with delay
⎧ ⎪ ⎨ ⎪ ⎩
Dαx(t) +f(t,x(t–τ)) = , t∈(, )\{τ},
x(t) =η(t), t∈[–τ, ], x() = ,
where <α ≤, Dα is the Riemann-Liouville fractional derivative,τ ∈(, ),f(t,x)∈
C((, )×R+,R) is continuous and may be singular att= , t= , and x= and may have negative values, whereR+= (, +∞). By the Guo-Krasnoselskii fixed point theorem, one obtained the existence results for positive solutions.
In , Vong [] considered the fractional differential equation with an integral boundary condition
cDα
+u(t) +f
t,u(t)= , u() =· · ·=un–() = , u() =
u(s)dμ(s),
wheren≥,n– <α<n,cDα
Motivated by the work mentioned above, in this paper, we study the existence of positive solutions of singular boundary value problems for nonlinear fractional functional differ-ential equation
⎧ ⎪ ⎨ ⎪ ⎩
Dαx(t) +λf(t,x(t–τ)) = , t∈(, )\{τ},
x(t) =η(t), t∈[–τ, ], x() =x() = ,
(.)
where <α≤,Dαdenote the Riemann-Liouville fractional derivative,λis a positive
parameter,τ∈(, ),η(t)∈C([–τ, ]), andη(t) > fort∈[τ, ),η() = ,fis a continuous functional defined on (, )×R+and which may be singular att= ,t= , andx= .
Whenτ= andλ= , problem (.) is reduced to the problem of fractional differential equations and has been studied by Zhaoet al.[]. To the best of our knowledge, no one has studied the existence of positive solutions for singular boundary value problem (.). Key tools in finding our main results are the Guo-Krasnoselskii fixed point theorem, and our main results of this paper are to extend and supplement some results in [, , ].
The paper is organized as follows. In Section , we shall introduce some definitions and lemmas to prove our main results. In Section , we investigate the existence of positive solution for boundary value problem (.) by the Guo-Krasnoselskii fixed point theorem.
2 Preliminaries
In the following section, we introduce the definitions and lemmas which are used through-out the paper. This material can be found in [, ].
The Riemann-Liouville fractional derivative of orderα(n– <α<n) of a functionf : (t, +∞)→Ris given by
Dα+f(t) = (n–α)
d dt
n t
t f(s)
(t–s)α–n+ds, t>t,
wherenis the smallest integer than or equal toαand(·) is the gamma function, provided that the right side is point wise defined on (t, +∞).
The Riemann-Liouville fractional integral of orderα(α> ) of a functionf : (t, +∞)→ Ris given by
Iαf(t) = (α)
t
t f(s)
(t–s)–αds, t>t,
where(·) is the gamma function, provided that the right side is point-wise defined on (t, +∞).
From the definition of the Riemann-Liouville derivative, we have the following state-ments.
Letα> . If we assumeu∈C(, )∩L(, ), then the fractional differential equation
Dα+u(t) =
hasu(t) =ctα–+ctα–+· · ·+cNtα–N,ci∈R,i= , , . . . ,N, as unique solutions, whereN
Assumeu∈C(, )∩L(, ) with a fractional derivative of orderα> . Then
Iα+Dα+u(t) =f(t) +ctα–+ctα–+· · ·+cntα–n
for someci∈R,i= , , . . . ,n, wherenis the smallest integer greater than or equal toα.
Next we introduce the Green function of boundary value problems for fractional differ-ential equations.
Lemma .[] Let <α≤and h: [, ]be continuous.Then the unique solution of the boundary value problem
Dα
+u(t) +λh(t) = , t∈(, ), u() =u() =u() = (.)
is
u(t) =
λG(t,s)h(s)ds, t∈[, ],
where
G(t,s) = (α)
tα–( –s)α–– (t–s)α–, ≤s≤t≤,
tα–( –s)α–, ≤t≤s≤. (.)
The following properties of the Green function play important roles in this paper.
Lemma .[] The function G(t,s)defined by(.)satisfies the following conditions: () G(t,s) > fort,s∈(, );
() q(t)G(,s)≤G(t,s)≤G(,s)fort,s∈(, ),whereq(t) =tα–.
Lemma . The function G∗(t,s) :=t–αG(t,s)satisfies the following conditions:
(α)t
s( –s)α–≤G∗(t,s)≤ (α)t
–α
s( –s)α– for t,s∈(, ).
Proof The proof can be obtained easily by Lemma ., so we omit it here.
The following lemma is fundamental in the proofs of our main results.
Lemma .[] Let E be a Banach space,and let K⊂E be a cone.Assume,are open and bounded subset of E 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∩(¯\).
3 Main results
(H) There exists a nonnegative functionρ∈C(, )∩L(, )such that
ϕ(t)h(x)≤f
t,v(t)x+ρ(t)≤ϕ(t)
g(x) +h(x)
for all(t,x)∈(, )×R+, whereϕ
,ϕ∈L(, )are nonnegative fort∈(, ),h,h∈ C(R+
,R+)are nondecreasing,g∈C(R+,R+)is nonincreasing,R+= [, +∞), and
v(t) =
, t∈(,τ], (t–τ)α–, t∈(τ, ).
(H)
<
τ
s( –s)α–ϕ(s)g
η(s–τ)ds< +∞,
and there exists a constantk> such that
τ
s( –s)α–ϕ(s)g
k (s–τ)
ds< +∞.
(H) There exists a subinterval[a,b]⊂(τ, )such that
b as( –s)
α–ϕ
(s)ds> .
Let X:={x(t) :x∈C([–τ, ],R),x(t) = fort∈[–τ, ],x() =x() = } be a Banach space with the maximum normx[–τ,]=max–τ≤t≤|x(t)|=max≤t≤|x(t)|forx∈X. Let Kbe a cone inXdefined by
K=x∈X;x(t)≥ fort∈[–τ, ].
Define
¯ η(t) =
η(t), t∈[–τ, ], , t∈(, ],
ω(t) =
, t∈[–τ, ],
λG(t,s)ρ(s)ds, t∈(, ],
and
x∗(t) =maxx(t) +η¯(t) –ω(t),
=
η(t), t∈[–τ, ],
max{x(t) –ω(t), }, t∈(, ]
for anyx∈K.
It is easy to know that the restrictionω|[,]ofωon [, ] is exactly the solution of
Dαx(t) +λρ(t) = , t∈(, ),α∈(, ],
Sincef : [, ]×C[–τ, ]→Ris a continuous function, settingf(t,x(t–τ)) :=h(t) in Lemma ., we see by Lemma . that a functionxis a solution of boundary value problem (.) if and only if it satisfies
x(t) =
λG(t,s)f(s,x(s–τ))ds, t∈(, ), η(t), t∈[–τ, ].
Consider the following operator:
(Tx)(t) =
λG(t,s)(f(s,x∗(s–τ)) +ρ(s))ds, t∈(, ],
, t∈[–τ, ]. (.)
Set
y(t) =
t–αx(t), t∈(, ),
, t∈[–τ, ]
and
y∗(t) =
max{tα–y(t) –ω(t), }, t∈(, ], η(t), t∈[–τ, ].
Then (.) is equivalent to
(Ty)(t) =
λG∗(t,s)(f(s,y∗(s–τ)) +ρ(s))ds, t∈(, ],
, t∈[–τ, ]. (.)
Obviously, ify˜is a fixed point of operatorTin (.), then
˜ x(t) =
tα–˜y(t), t∈(, ], , t∈[–τ, ],
is a fixed point of operatorTdefined by (.). Lemma . implies that
⎧ ⎪ ⎨ ⎪ ⎩
Dαx(t) +˜ λf((t,x˜∗(t–τ)) +ρ(t)) = , t∈(, )\{τ},
˜
x(t) = , t∈[–τ, ], ˜
x() =x˜() = .
Thus if
˜
x(t–τ) +η¯(t–τ) –ω(t–τ)≥ fort∈[, ], (.)
then
˜
x∗(t–τ) =x(t˜ –τ) +η¯(t–τ) –ω(t–τ).
Let
Then one can verify easily that the functionxsatisfied boundary value problem (.). As a result, in the following we will concentrate our study on finding the fixed points of operator T defined by (.).
Define the cone
K=
y∈K:y(t)≥tyfort∈[, ]
and
=
y∈K:y<r
,
=
y∈K:y<r
,
=
y∈K:y<R
,
=
y∈K:y<R
for anyr>r≥max{k, c},R>R≥max{k, c}, where
c:= λ (α)
s( –s)α–ρ(s)ds< +∞ (.)
andkis the constant in (H).
Lemma . Let(H)and(H)hold.Then the operator T:K∩(¯\)→Kis completely continuous.
Proof First we show that operatorT is well defined onK∩(¯\). For anyy∈K∩ (¯\), we know that
r≤ y ≤r
and
y(t)≥ty ≥tr fort∈[, ].
Then, fort∈[, ], we get
t–αω(t) =t–α
λG(t,s)ρ(s)ds
≤ t–α (α)
λs( –s)α–ρ(s)ds
≤ t (α)
λs( –s)α–ρ(s)ds
≤tc, (.)
wherecis defined as (.). Thus, fort∈[, ],
y(t) –t–αω(t)≥t(r–c)
≥r t
In view of (H), (H), and Lemma ., we show
(Ty)(t) =
τ
λG∗(t,s)fs,η(s–τ)+ρ(s)ds
+
τ
λG∗(t,s)fs, (s–τ)α–y(s–τ) –ω(s–τ)+ρ(s)ds
≤ λt–α (α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+λt –α
(α)
τ
s( –s)α–ϕ(s)
×
g
r (s–τ)
+h
y(s–τ) – (s–τ)–αω(s–τ)ds
≤ λt–α (α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+λt –α
(α)
τ
s( –s)α–ϕ(s)
g
r (s–τ)
+h
y(s–τ)ds
≤ λt–α (α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+λt –α
(α)
τ
s( –s)α–ϕ (s)
g
k (s–τ)
+h(r)
ds
< +∞.
Hence,T is uniformly bounded andTis well defined.
In fact, fory∈K∩(¯\),t∈[, ], in view of Lemma ., we have
Ty ≤ λt–α (α)
s( –s)α–fs,y∗(s–τ)+ρ(s)ds
≤ λt (α)
s( –s)α–fs,y∗(s–τ)+ρ(s)ds
and
(Ty)(t)≥ λt
(α)
s( –s)α–fs,y∗(s–τ)+ρ(s)ds
≥tTy.
Hence,T:K∩(¯\)→K.
Next we showT:K∩(¯\)→K is continuous and compact. For anyyn,y∈K∩ (¯\),n= , , . . . withyn–y[–τ,]→ asn→ ∞. Sincer≤ yn ≤randr≤ y ≤ r, fort∈(, ), we know
yn(t) –t–αω(t)≥
r t
and
y(t) –t–αω(t)≥r t
Then, fort∈[, ], we have
(Tyn)(t) – (Ty)(t)=
τλG∗(t,s)fs, (s–τ)α–yn(s–τ) –ω(s–τ)
+ρ(s)
–fs, (s–τ)α–y(s–τ) –ω(s–τ)–ρ(s)ds
≤λt–α (α)
τ
s( –s)α–fs, (s–τ)α–yn(s–τ) –ω(s–τ)
–fs, (s–τ)α–y(s–τ) –ω(s–τ)ds
≤λt–α (α)
τ
s( –s)α–ϕ (s)
g
k (s–τ)
+h(r)
ds
< +∞.
This implies thatTyn–Ty[–τ,]→ asn→ ∞. HenceTis continuous. Next we proveTis equicontinuous.
SinceG∗in uniformly continuous fort∈(, ), that is, for any> , there existsξ> , whent,t∈[, ] and|t–t|<ξ, we have
G∗(t,s) –G∗(t,s)=
τ
λϕ(s)
gη(s–τ)+h
η(s–τ)ds
+
τ
λϕ(s)
g
k (s–τ)
+h(r)
ds
–
.
Thus, for anyy∈K∩(¯\), we get
(Ty)(t) – (Ty)(t)≤ τ
λG∗(t,s) –G∗(t,s)ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+
τ
λG∗(t,s) –G∗(t,s)ϕ(s)
g
k (s–τ)
+h(r)
ds
< +
=.
ThusTis equicontinuous. Accordingly to the Ascoli-Arzelà theorem,Tis completely
con-tinuous. The proof is completed.
Now we prove the existence of positive solutions for boundary value problem (.) by using the Guo-Krasnoselskii fixed point theorem.
For convenience, we denote
ξ:= (α)
τ
s( –s)α–ϕ(s)
gη(s–τ)
+h
η(s–τ)ds+
τ
s( –s)α–ϕ(s)
g
k (s–τ)
+r
ds
and there exists a subinterval [β,γ]⊂(τ, ),
ζ:= min
t∈[β,γ](t–τ) , ζ
:= min
t∈[β,γ]t ,
ξ(t) := ζ (α)h
rζ
γ
β
s( –s)α–ϕ(s)ds> .
Theorem . Let(H), (H),andξ–r<ξ–r hold.Then the boundary value problem (.)has at least one positive solution if
lim
y→+∞ h(y)
y = (.)
for each
λ∈ξ–r,ξ–r
.
Proof Let> . Then in view of (.), there exists aM> such that
h(y)≤y fory>M. (.)
Choose
r≥max{M+ ,r+ },
then fory∈∂, like for (.), fort∈[, ], we obtain
y(t) –t–αω(t)≥t(r–c)
≥r t
. (.)
Then from (H), (.), (.), and Lemma ., we get
(Ty)(t)≤ λt –α
(α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+λt –α
(α)
τ
s( –s)α–ϕ (s)
×
g
r (s–τ)
+h
y(s–τ) – (s–τ)–αω(s–τ)ds
≤ λt–α (α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+λt –α
(α)
τ
s( –s)α–ϕ (s)
g
r (s–τ)
+h
y(s–τ)ds
≤ λ
(α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+ λ (α)
τ
s( –s)α–ϕ (s)
g
k (s–τ)
+h(r)
≤ λ (α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+ λ (α)
τ
s( –s)α–ϕ(s)
g
k (s–τ)
+r
ds
≤λξ<r.
Therefore, fory∈∂, we haveTy ≤ y.
On the other hand, fory∈∂, like for (.), fort∈[, ], we obtain
y(t) –t–αω(t)≥t(r–c)
≥r t
. (.)
Thus from (H), (.), and Lemma ., we get
Ty ≥ γ
β
λ min
t∈[β,γ]G
∗(t,s)fs, (s–τ)α–y(s–τ) –ω(s–τ)+ρ(s)ds
≥ γ
β
λ min
t∈[β,γ]G
∗(t,s)ϕ
(s)h
r (s–τ)
ds
≥ λζ (α)h
rζ
γ
β
s( –s)α–ϕ (s)ds
≥λξ>r.
Therefore, for y∈∂, we haveTy ≥ y. ThenT defined by (.) has a fixed point ˜
y∈K∩(¯\). In view of (.), we have
tα–y(t) –˜ ω(t) =tα–y(t) –˜ t–αω(t)
≥r t
α–
> .
It is easy to know (.) is satisfied. The proof is completed.
Denote
ξ:= (α)
τ
s( –s)α–ϕ (s)
gη(s–τ)+h
η(s–τ)ds
+
τ
s( –s)α–ϕ(s)
g
k (s–τ)
+h(R)
ds
> .
Theorem . Let(H)-(H)and
(α) M∗Aζ
b
a
s( –s)α–ϕ (s)ds
hold.Then boundary value problem(.)has at least one positive solution if
lim
y→+∞ h(y)
y = +∞ (.)
for each
λ∈
(α) M∗Aζ
b
a
s( –s)α–ϕ(s)ds
– ,ξ–R
, (.)
where A:=mint∈[a,b](t–τ),M∗is a positive constant.
Proof It follows from (.) that there exists aM∗> such that
h(y)≥M∗y fory>M∗. (.)
Choose
R≥max
R+ , M∗
A
.
Then fory∈∂, like for (.), fort∈[, ], we obtain
y(t) –t–αω(t)≥t(R–c)
≥R t
. (.)
Thus from (H), (.), (.), and Lemma ., we get
Ty ≥
b
a
λ min
t∈[a,b]G
∗(t,s)fs, (s–τ)α–y(s–τ) –ω(s–τ)+ρ(s)ds
≥ b
a
λ min
t∈[a,b]G
∗(t,s)ϕ
(s)h
R (s–τ)
ds
≥ b
a
λ min
t∈[a,b]G
∗(t,s)ϕ
(s)h
RA
ds
≥ M∗AλR
b
a
min
t∈[a,b]G
∗(t,s)ϕ
(s)ds
≥ λM∗ARζ (α)
b
a
s( –s)α–ϕ(s)ds
≥R.
Therefore, fory∈∂, we haveTy ≥ y.
On the other hand, fory∈∂, like for (.), fort∈[, ], we obtain
y(t) –t–αω(t)≥t(R–c)
≥R t
Thus from (H), (.), and (.), we have
(Ty)(t)≤ λt –α
(α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+λt –α
(α)
τ
s( –s)α–ϕ (s) × g R (s–τ)
+h
y(s–τ) – (s–τ)–αω(s–τ)ds
≤ λ
(α)
τ
s( –s)α–ϕ (s)
gη(s–τ)+h
η(s–τ)ds
+ λ (α)
τ
s( –s)α–ϕ (s)
g
k (s–τ)
+h(R)
ds
≤λξ<R.
Therefore, fory∈∂, we haveTy ≤ y. Arguments similar to those at the end of the proof of Theorem . show that boundary value problem (.) has a positive solution. The
proof is completed.
Theorem . Let(H)and(H)hold.Furthermore assume that
(H) there exists a subinterval[β,γ]⊂(τ, )and a positive constantrsuch that
r>max
k, c, λ (α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+
τ
s( –s)α–ϕ(s)
g
k (s–τ)
+h(r)ds
,
wherekis defined in(H),cis defined as(.)andλ∈(, +∞).
Then boundary value problem(.)has at least one positive solution y with <y<r.
Proof In view of (H), we choosen∈ {, , . . .}such that
r> λ (α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+
τ
s( –s)α–ϕ(s)
g
k (s–τ)
+h(r)
ds + n .
LetN={n,n+ , . . .}. Fixn∈Nand consider the family of integral equations
y(t) =
κλG∗(t,s)(fn(s,y∗(s–τ)) +ρ(s))ds+n, t∈(, ),
n, t∈[–τ, ],
(.)
whereκ∈(, ),
fn
t,y∗(t–τ)+ρ(t) =
We claim that any solutionyof (.) for anyκ∈(, ) must satisfyy =r. Otherwise, assume thatyis a solution of (.) for someκ∈(, ) such thaty=r. Theny∗(t–τ)≥n fort∈(, ). In view of Lemma ., we have
y ≤κλt –α
(α)
s( –s)α–fn
s,y∗(s–τ)+ρ(s)ds+
n. (.)
Thus, fort∈(, ), we have
y(t)≥ n+
κλt (α)
s( –s)α–fn
s,y∗(s–τ)+ρ(s)ds
≥ n+t
α–
y– n
≥ –tα– n+t
α–y
≥tα–y ≥tr.
Then like for (.), fort∈(, ), we have
y(t) –t–αω(t)≥t(r–c)
≥r t
.
Then from (H), fort∈(, ),κ∈(, ), we have
y(t) = n+κλ
G∗(t,s)fn
s,y∗(s–τ)+ρ(s)ds
= n+κλ
G∗(t,s)fs,y∗(s–τ)+ρ(s)ds
≤ n+
λt–α
(α)
s( –s)α–fs,y∗(s–τ)+ρ(s)ds
≤ n
+ λ (α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+ λ (α)
τ
s( –s)α–
×
ϕ(s)
g
r (s–τ)
+h
y(s–τ) – (s–τ)–αω(s–τ)ds.
Hence we obtain
r=y(t)
≤ n
+ λ (α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+ λ (α)
τ
s( –s)α–
×
ϕ(s)
g
r (s–τ)
+h
y(s–τ) – (s–τ)–αω(s–τ)ds.
Now the Leray-Schauder nonlinear alternative theorem guarantees that the equation
y(t) =
λG∗(t,s)fn
s,y∗(s–τ)+ρ(s)ds+ n
has a solutionyn, in¯r={y∈C[, ] :y ≤r}, fort∈(, ).
Next we claim that yn(t) has a uniform sharper lower bound. In view of (H) and yn(t) ≤r, we obtain
yn(t) =
n+λ
G∗(t,s)fn
s,y∗n(s–τ)+ρ(s)ds
≥ n+λ
γ
β
G∗(t,s)fs,yn∗(s–τ)+ρ(s)ds
≥ λt (α)
γ
β
s( –s)α–fs,yn∗(s–τ)+ρ(s)ds
≥ λt (α)
γ
β
s( –s)α–
ϕ(s)h
r (s–τ)
ds
≥ λt (α)h
rζ
γ
β
s( –s)α–ϕ(s)ds.
Choosingδ(t) =λ(tα)h(rζ)
γ β s(–s)
α–ϕ
(s)ds. Then we conclude that there exists a func-tionδ∈C(, ) that is unrelated tonsuch thatδ(t) > for a.e.t∈(, ) and for anyn∈N,
yn(t)≥δ(t).
Then we prove {yn}n∈N is an equicontinuous family on (, ). SinceG∗ in uniformly continuous fort∈(, ), that is, for any> , there existsζ> , whent,t∈[, ] and |t–t|<ζ, we have
G∗(t,s) –G∗(t,s)= λ
τ
ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+
τ
ϕ(s)
g
k (s–τ)
+h(r)
ds
–
.
Thus
(yn)(t) – (yn)(t)≤λ
τ
G∗(t
,s) –G∗(t,s)ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+λ
τ
G∗(t,s) –G∗(t,s)ϕ(s)
g
k (s–τ)
+h(r)
ds
< +
=.
convergence theorem, in view of
yn(t) =
λG∗(t,s)fn
s,y∗n(s–τ)+ρ(s)ds,
we have
y(t) =
λG∗(t,s)fs,y∗(s–τ)+ρ(s)ds.
Then boundary value problem (.) has one positive solution with <y<r. The proof
is completed.
Theorem . Let(H)and(H)hold.Assume that there exists a subinterval[β,γ]⊂(τ, ) satisfying
maxg(·) +maxh(·)≤
r
λ (α)
s( –s)α–ϕ(s)ds ,
min
≤y≤r h(y)≥
r
λζ
(α)
γ
β s( –s)α–ϕ(s)ds
,
whereλ∈(, +∞).Then boundary value problem(.)has at least one positive solution.
Proof In view of Theorem ., fory∈∂,t∈[, ], we obtain
y(t) –t–αω(t)≥r
t
. (.)
Then from (H), (.), and Lemma ., we get
(Ty)(t)≤ λt –α
(α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+λt –α
(α)
τ
s( –s)α–ϕ (s) × g r (s–τ)
+h
y(s–τ) – (s–τ)–αω(s–τ)ds
≤ λt–α (α)
τ
s( –s)α–ϕ(s)
gη(s–τ)+h
η(s–τ)ds
+λt –α
(α)
τ
s( –s)α–ϕ (s)
g
r (s–τ)
+h
y(s–τ)ds
≤ λ
(α)
τ
s( –s)α–ϕ (s)
gη(s–τ)+h
η(s–τ)ds
+ λ (α)
τ
s( –s)α–ϕ (s)
g
k (s–τ)
+h(r)
ds
≤ λ
(α)
r
λ (α)
s( –s)α–ϕ(s)ds
τ
s( –s)α–ϕ (s)ds
+ λ (α)
r
λ (α)
s( –s)α–ϕ(s)ds
τ
s( –s)α–ϕ(s)ds≤r.
On the other hand, fory∈∂, like for (.), fort∈[, ], we obtain
y(t) –t–αω(t)≥t(r–c)
≥r t
. (.)
Thus from (H), (.), and Lemma ., we get
Ty ≥ γ
β
λ min
t∈[β,γ]G
∗(t,s)fs, (s–τ)α–y(s–τ) –ω(s–τ)+ρ(s)ds
≥ γ
β
λ min
t∈[β,γ]G
∗(t,s)ϕ
(s)h
r (s–τ)
ds
≥ λζ (α)
r
λζ
(α)
γ
β s( –s)α–ϕ(s)ds γ
β
s( –s)α–ϕ (s)ds
≥r.
Therefore, fory∈∂, we haveTy ≥ y. Arguments similar to those at the end of the proof of Theorem . show that boundary value problem (.) has a positive solution. The
proof is completed.
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.
Author details
1Department of Basic Courses, Shandong Women’s University, Jinan, Shandong 250300, P.R. China.2School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P.R. China.
Acknowledgements
This research is supported by the Natural Science Foundation of China (11572205, 61374002, 61374074), and supported by Shandong Provincial Natural Science Foundation (ZR2013AL003).
Received: 23 October 2015 Accepted: 4 January 2016
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