R E S E A R C H
Open Access
Positive solutions for higher order
differential equations with integral boundary
conditions
Yude Ji
1*, Yanping Guo
2and Yukun Yao
3 *Correspondence:[email protected] 1College of Sciences, Hebei
University of Science and Technology, Shijiazhuang, Hebei 050018, P.R. China
Full list of author information is available at the end of the article
Abstract
In this paper, we consider the existence of at least three positive solutions for the 2nth order differential equations with integral boundary conditions
x(2n)(t) =f(t,x(t),x(t),. . .,x(2(n–1))(t)), 0≤t≤1,
x(2i)(0) =1
0 ki(s)x(2i)(s)ds, x(2i)(1) = 0, 0≤i≤n– 1,
where (–1)nf> 0 is continuous, andk
i(t)∈L1[0, 1] (i= 0, 1,. . .,n– 1) are nonnegative.
The associated Green’s function for the higher order differential equations with integral boundary conditions is first given, and growth conditions are imposed onf
which yield the existence of multiple positive solutions by using the Leggett-Williams fixed point theorem.
Keywords: boundary value problem; integral boundary conditions; positive solution; fixed point theorem
1 Introduction
The multi-point boundary value problems (BVPs) for ordinary differential equations arise in a variety of different areas of applied mathematics and physics. The study of nonlo-cal BVPs for second order ordinary differential equations has been widely investigated in [–]. Since then, nonlinear high order nonlocal BVPs have been studied by many authors. We refer the reader to [–] and references therein. Recently, Guoet al.[] used Leggett-Williams fixed point theorem to obtain the existence of at least three positive solutions for the nth orderm-point BVP
y(n)(t) =f(t,y(t),y(t), . . . ,y((n–))(t)), ≤t≤,
y(i)() = , y(i)() =m–
j= kijy(i)(ξj), ≤i≤n– ,
wherekij> (i= , , . . . ,n– ;j= , , . . . ,m– ), <ξ<ξ<· · ·<ξm–< , and (–)nf : [, ]×Rn→[, +∞) is continuous.
BVPs with integral boundary conditions for ordinary differential equations represent a very interesting and important class of problems and arise in the study of various phys-ical, biological and chemical processes, such as heat conduction, chemical engineering,
thermo-elasticity, underground water flow, population dynamics, and plasma physics. Such problems include two-, three-, multi-point and nonlocal BVPs as special cases. The existence and multiplicity of positive solutions for such problems have received a great deal of attention; see [–] and references therein. In particular, we would mention the result of [], Zhang and Ge investigated the existence and nonexistence of positive solu-tions of the following fourth-order BVP with integral boundary condisolu-tions
⎧ ⎪ ⎨ ⎪ ⎩
x()(t) =ω(t)f(t,x(t),x(t)), <t< ,
x() =g(s)x(s) ds, x() = ,
x() =h(s)x(s) ds, x() = ,
whereωmay be singular att= and (or)t= ,f ∈C([, ]×[, +∞)×(–∞, ], [, +∞)), andg,h∈L[, ] are nonnegative.
Motivated by [, ], in this paper, we consider the existence of at least three positive solutions for the nth order differential equations with integral boundary conditions
x(n)(t) =f(t,x(t),x(t), . . . ,x((n–))(t)), ≤t≤,
x(i)() =
ki(s)x(i)(s) ds, x(i)() = , ≤i≤n– ,
(.)
where (–)nf > is continuous, andki(t)∈L[, ] (i= , , . . . ,n– ) are nonnegative. For more precise conditions on f, let (–)j[a,b] = [a,b] ifj is even and (–)j[a,b] = [–b, –a] ifjis odd. Let
n–
j=
[aj,bj] = [a,b]× · · · ×[an–,bn–].
We shall require that
(–)nf : [, ]× n–
j=
(–)j[, +∞)→[, +∞).
We shall suppose the following conditions are satisfied:
(H) ki(t)∈L[, ]are nonnegative, andKi∈[, ), where
Ki=
( –s)ki(s) ds, ≤i≤n– ;
(H) (–)nf : [, ]× n–
j=(–)j[, +∞)→[, +∞)is continuous.
2 Preliminary results
Definition . LetEbe a Banach space overR. A nonempty convex closed setK⊂Eis said to be a cone provided that
(i) au∈Kfor allu∈Kand alla≥; (ii) u, –u∈Kimpliesu= .
Definition . The mapαis said to be a nonnegative continuous concave functional on
Kprovided thatα:K→[,∞) is continuous and
for allx,y∈K and ≤t≤. Similarly, we say the mapγ is a nonnegative continuous convex functional onKprovided thatγ:K→[,∞) is continuous and
γtx+ ( –t)y≤tγ(x) + ( –t)γ(y)
for allx,y∈Kand ≤t≤.
Definition . Let <a<b be given and letα be a nonnegative continuous concave functional onK. Define the convex setsPrandP(α,a,b) by
Pr=
x∈K| x <r and P(α,a,b) =x∈K|a≤α(x), x ≤b.
Theorem .(Leggett-Williams fixed point theorem []) Let A:Pc→Pcbe a completely
continuous operator and letαbe a nonnegative continuous concave functional on K such thatα(x)≤ x for all x∈Pc.Suppose there exist <a<b<d≤c such that
(C) {x∈P(α,b,d)|α(x) >b} =∅,andα(Ax) >bforx∈P(α,b,d),
(C) Ax <afor x ≤a,and
(C) α(Ax) >bforx∈P(α,b,c),with Ax >d.
Then A has at least three fixed points x,x,and xsuch that
x <a, b<α(x) and x >a withα(x) <b.
Remark . If we haved=c, then condition (C) of Theorem . implies condition (C) of Theorem ..
3 Preliminary lemmas
Lemma . Suppose(H)holds.Then gi(t,s)≤ (≤i≤n– ),where gi(t,s)is the Green’s
function for the problem
x(t) = , ≤t≤,
x() =ki(s)x(s) ds, x() = .
Proof It is easy to see thatgi(t,s)≤ (≤i≤n– ) by using Lemma . of [].
LetG(t,s) =gn–(t,s), then for ≤j≤n– , we recursively define
Gj(t,s) =
gn–j–(t,τ)Gj–(τ,s) dτ.
Lemma . Suppose(H)holds.If y∈C[, ],then the BVP
x(l)(t) =y(t), ≤t≤,
x(i)() =
kn–l+i–(s)x(i)(s) ds, x(i)() = , ≤i≤l– ,
(.)
Proof We will prove the result by using mathematical induction. Whenl= , which implies thati= , then the BVP (.) reduces to
x(t) =y(t), ≤t≤,
x() =kn–(s)x(s) ds, x() = .
(.)
By using Lemma ., it is easy to see that the BVP (.) has a unique solution
x(t) =
G(t,s)y(s) ds.
Therefore, the result holds forl= .
We assume that the result holds forl– . Now, we deal with the case forl. Letx(t) =u(t), then the BVP (.) is equivalent to the following BVPs:
x(t) =u(t), ≤t≤,
x() =kn–l–(s)x(s) ds, x() =
(.)
and
u((l–))(t) =y(t), ≤t≤,
u(i)() =
kn–l+i(s)u
(i)(s) ds, u(i)() = , ≤i≤l– . (.)
By applying Lemma ., the BVP (.) has a unique solution
x(t) =
gn–l–(t,r)u(r) dr. (.)
Replacinglbyl– andxbyuin (.), by applying the inductive hypothesis, the BVP (.) has also a unique solution
u(t) =
Gl–(t,s)y(s) ds. (.)
Substituting (.) into (.), we see that the BVP (.) has a unique solution
x(t) =
gn–l–(t,r)
Gl–(r,s)y(s) dsdr
=
gn–l–(t,r)Gl–(r,s) dr
y(s) ds
=
Gl(t,s)y(s) ds.
Therefore, the result holds forl. Lemma . is now completed.
For each ≤l≤n– , we defineAl:C[, ]→C[, ] by
Alu(t) =
With the use of Lemma ., for each ≤l≤n– , we have
(Alu)(l)(t) =u(t), ≤t≤, (Alu)(i)() =
kn–l+i–(s)(Alu)(i)(s) ds, (Alu)(i)() = , ≤i≤l– .
Therefore (.) has a solution if and only if the BVP
u(t) =f(t,An–u(t),An–u(t), . . . ,Au(t),u(t)), ≤t≤,
u() =kn–(s)u(s) ds, u() =
(.)
has a solution. Ifxis a solution of (.), thenu=x((n–))is a solution of (.). Conversely, ifuis a solution of (.), thenx=An–uis a solution of (.). In addition if (–)n–u(t)≥ (= ) on [, ], thenx=An–uis a positive solution of (.).
For ≤i≤n– , let
mi= min t∈[τ,–τ]
–τ
τ
gi(t,s)ds and Mi=max t∈[,]
gi(t,s)ds. (.)
Obviously, <mi<Mi. LetEdenote the Banach spaceC[, ] with the maximum norm
u =max
≤t≤ u(t),
and define the coneK⊂Eby
K=u∈E(–)n–u(t)≥, (–)n–u(t) is concave on [, ], and
min
t∈[τ,–τ](–)
n–u(t)≥τ u .
Finally, we define the nonnegative continuous concave functionalαonKby
α(u) = min
t∈[τ,–τ] u(t)
for eachu∈Kand it is easy to see thatα(u)≤ u .
4 Main results
Theorem . Suppose conditions(H), (H)hold.In addition assume there exist
nonneg-ative numbers a,b,and c such that <a<b≤min{τ,mn–
Mn–}c and f(t,un–,un–, . . . ,u,u)
satisfies the following growth conditions:
(H) (–)nf(t,un–,un–, . . . ,u,u) ≤ Mcn–, for (t,un–,un–, . . . ,u,u) ∈ [, ] ×
j=n–(–)n––j[, j+
i=Mn–ic]×(–)n–[,c];
(H) (–)nf(t,un–,un–, . . . ,u,u) < Man–, for (t,un–,un–, . . . ,u,u) ∈ [, ] ×
j=n–(–)n––j[, j+
i=Mn–ia]×(–)n–[,a];
(H) (–)nf(t,un–,un–, . . . ,u,u) ≥ mnb–, for (t,un–,un–, . . . ,u,u) ∈ [τ, – τ] ×
j=n–(–)n––j[ j+
i=mn–ib, j+
Then the BVP(.)has at least three positive solutions x,x,and x,which satisfy x(( n–))(t)<a, b<αx(( n–))(t) and
x(( n–))(t)>a withαx(( n–))(t)<b.
Proof Define the completely continuous operatorAby
Au(t) =
gn–(t,s)f
s,An–u(s), . . . ,Au(s),u(s)
ds.
We will first verify thatA:K→K. Letu∈K, then (–)n–Au(t)≥, ((–)n–Au)(t) = (–)n–f(t,A
n–u(t), . . . ,Au(t),u(t))≤, ≤t≤, and by Proposition . of [], we have
Au = max
t∈[,] Au(t)
= max
t∈[,]
gn–(t,s)f
s,An–u(s), . . . ,Au(s),u(s)
ds
= max
t∈[,]
h(t,s) + –t –Kn–
h(s,τ)kn–(τ) dτ
×fs,An–u(s), . . . ,Au(s),u(s)
ds
≤
h(s,s) + –Kn–
h(τ,τ)kn–(τ) dτ
×fs,An–u(s), . . . ,Au(s),u(s)ds.
On the other hand, by Proposition . of [], we obtain
min
t∈[τ,–τ](–) n–Au(t)
= min
t∈[τ,–τ](–) n–
gn–(t,s)f
s,An–u(s), . . . ,Au(s),u(s)
ds
= min
t∈[τ,–τ](–) n
h(t,s) + –t –Kn–
h(s,τ)kn–(τ) dτ
×fs,An–u(s), . . . ,Au(s),u(s)
ds
≥τ
h(s,s) + –Kn–
h(τ,τ)kn–(τ) dτ
×fs,An–u(s), . . . ,Au(s),u(s)ds
≥τ Au .
Consequently,A:K→K.
It is a standard argument to show that the operatorAis completely continuous. Equicon-tinuity and uniform boundedness follow readily from the properties ofGl, ≤l≤n– .
Ifu∈Pc, then u ≤c. For each ≤j≤n– , note that inductively (using (.)) we have
Aju =max t∈[,]
Gj(t,s)u(s) ds≤
j+
i=
From the condition (H) and (.), we obtain
Au =max
t∈[,] Au(t)
=max
t∈[,]
gn–(t,s)f
s,An–u(s),An–u(s), . . . ,Au(s),u(s)
ds
≤ c Mn–
max
t∈[,]
gn–(t,s)ds=c.
So,A:Pc→Pc.
In a completely analogous argument, the condition (H) implies the condition (C) of Theorem . is satisfied.
We now show that condition (C) is satisfied. Note that, for ≤t≤,
u(t) = (–)n– b
τ ∈P
α,b, b
τ
and α(u) = b
τ >b.
Thus,
u∈P
α,b, b
τ
α(u) >b
=∅.
Also, ifu∈P(α,b,τb), thenb≤(–)n–u(t)≤ τb fort∈[τ, –τ], implies for each ≤j≤
n– ,t∈[τ, –τ], inductively,
(–)n––jAju(t)≤ j+
i=
Mn–i
b τ,
(–)n––jAju(t) =
Gj(t,s)u(s) ds≥b
–τ
τ
Gj(t,s)ds≥
j+
i=
mn–ib.
With the use of condition (H) and (.), we get
α(Au) = min
t∈[τ,–τ]
gn–(t,s)f
s,An–u(s), . . . ,Au(s),u(s)
ds
> min
t∈[τ,–τ]
τ–τgn–(t,s)f
s,An–u(s), . . . ,Au(s),u(s)
ds
≥ b mn–
min
t∈[τ,–τ]
gn–(t,s)ds=b.
Therefore, condition (C) is satisfied.
Finally, we show that condition (C) is also satisfied. That is, we show that ifu∈P(α,b,c) and Au >d= b
τ, thenα(Au) >b. This follows sinceA:K→K. In particular, since
(–)n–(Au) is concave andmin
t∈[τ,–τ](–)n–(Au)(t)≥τ Au . That is,
α(Au) = min
t∈[τ,–τ]
Therefore, condition (C) is also satisfied. By Theorem ., there exist three solutions
u,u,u∈Kfor the BVP (.). Moreover, let
xi(t) =An–ui(t) =
Gn–(t,s)ui(s) ds, i= , , ,
thenx,x,xare three positive solutions for the BVP (.) and satisfy
x(( n–))(t)<a, b<αx(( n–))(t) and
x(( n–))(t)>a withαx(( n–))(t)<b.
Consider the following nth order differential equations with integral boundary condi-tions:
x(n)(t) =f(t,x(t),x(t), . . . ,x((n–))(t)), ≤t≤,
x(i)() = , x(i)() =
k∗i(s)x(i)(s) ds, ≤i≤n– ,
(.)
where (–)nf∈C([, ]×jn=–(–)j[, +∞)→[, +∞)) andki∗(t)∈L[, ] (i= , , . . . ,n– ) are nonnegative.
Now we deal with problem (.). The method is just similar to what we have done for the problem (.), so we omit the proof of main results in this section.
For convenience, we list the following assumptions:
(H∗) ki∗(t)∈L[, ]are nonnegative, andK∗
i ∈[, ), where
Ki∗=
ski∗(s) ds, ≤i≤n– .
By analogous methods, we have the following results.
Lemma . Suppose (H∗)holds. Then gi∗(t,s)≤ (≤i≤n– ),where gi∗(t,s)is the Green’s function for the problem
x(t) = , ≤t≤,
x() = , x() =ki∗(s)x(s) ds.
For≤i≤n– ,let
m∗i = min
t∈[τ,–τ] –τ
τ
g∗i(t,s)ds and Mi∗=max
t∈[,]
gi∗(t,s)ds.
Theorem . Suppose condition(H∗)holds.In addition assume there exist nonnegative numbers a,b,and c such that <a<b≤min{τ,m∗n–
M∗n–}c and f(t,un–,un–, . . . ,u,u)
sat-isfies the following growth conditions:
(H∗) (–)nf(t,u
n–,un–, . . . ,u,u) ≤ Mc∗
n–, for (t,un–,un–, . . . ,u,u) ∈ [, ] ×
j=n–(–)n––j[, j+
i=M∗n–ic]×(–)n–[,c];
(H∗) (–)nf(t,un–,un–, . . . ,u,u) < Ma∗
n–, for (t,un–,un–, . . . ,u,u) ∈ [, ] ×
j=n–(–)n––j[, j+
(H∗) (–)nf(t,u
n–,un–, . . . ,u,u) ≥ mb∗
n–, for (t,un–,un–, . . . ,u,u) ∈ [τ, – τ] ×
j=n–(–)n––j[ j+
i=m∗n–ib, j+
i=Mn∗–iτb]×(–)n–[b,τb].
Then the BVP(.)has at least three positive solutions x,x,and x,which satisfy
x((n–))
(t)<a, b<α
x(( n–))(t)
and
x(( n–))(t)>a withαx(( n–))(t)<b.
5 Example
Example . As an example of problem (.), consider the following sixth order BVP:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
x()(t) =f(t,x(t),x(t),x()(t)), ≤t≤,
x() =x(s) ds, x() = ,
x() =sx(s) ds, x() = ,
x()() =
sx()(s) ds, x()() = ,
(.)
where
f(t,u,u,u)
= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–, u·+sin(t+u+u), ≤u≤, –[(×,, –, )(u– ) +, ]·+sin(t+u+u), ≤u≤, –×,, u·+sin(t+u+u), ≤u≤, –[× ,, + (×, –×,, )(u– )]
·+sin(t+u+u)
, ≤u≤,
–×, ·+sin(t+u+u)
, u≥.
We notice thatn= ,ki(s) =si(i= , , ) andK=,K=,K=. If we takeτ=, by calculation we obtain
m= min t∈[,]
g(t,s)ds=
, M=tmax∈[,]
g(t,s)ds= ,
m= min t∈[,]
g(t,s)ds=
,, M=tmax∈[,]
g(t,s)ds= ,
m= min t∈[,]
g(t,s)ds= ,
,, M=tmax∈[,]
g(t,s)ds= ,.
In addition, if we takea= ,b= ,c= , then
<a= <b= ≤min
τ,m
M c =min ,,×, ,× × = ,
Therefore all the conditions of Theorem . are satisfied. Hence, the problem (.) has at least three positive solutionsx,x, andx, which satisfy
max
≤t≤ x()
(t)< , < min t∈[,]
x()
(t) and
max
≤t≤x ()
(t)> with min t∈[,]
x() (t)< .
Example . As another example of problem (.), consider the following sixth order BVP: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
x()(t) =f(t,x(t),x(t),x()(t)), ≤t≤,
x() = , x() =x(s) ds,
x() = , x() =sx(s) ds,
x()() = , x()() =
sx()(s) ds,
(.)
where
f(t,u,u,u)
= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–u·+sin(t+u+u), ≤u≤, –[(×,, –)(u– ) +]·+sin(t+u+u), ≤u≤, –×,,u·+sin(t+u+u), ≤u≤, –[× , , + ( × – × ,
,)(u– )]·
+sin(t+u+u)
, ≤u≤, –× ·+sin(t+u+u)
, u≥.
We notice thatn= ,k∗i(s) =si(i= , , ) andK∗
=,K∗=,K∗=. If we takeτ=, by calculation we obtain
m∗= min
t∈[,]
g∗(t,s)ds=
, M
∗
=max t∈[,]
g∗(t,s)ds= ,
m∗= min
t∈[,]
g∗(t,s)ds=
,, M
∗
=max t∈[,]
g∗(t,s)ds= ,
m∗= min
t∈[,]
g∗(t,s)ds= ,
,, M
∗
=max t∈[,]
g∗(t,s)ds= .
In addition, if we takea= ,b= ,c= , then
<a= <b= ≤min
τ,m
∗
M∗
c=min
,,× ,× × = ,
andf(t,u,u,u) satisfies the growth conditions (H∗)-(H∗).
Therefore all the conditions of Theorem . are satisfied. Hence, the problem (.) has at least three positive solutionsx,x, andx, which satisfy
max
≤t≤
x() (t)< , < min
t∈[,]
x() (t) and
max
≤t≤
x() (t)> with min
t∈[,]
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this article.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Author details
1College of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, P.R. China.2School of
Electrical Engineering, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, P.R. China.3Hebei Medical University, Shijiazhuang, Hebei 050200, P.R. China.
Acknowledgements
The authors express their sincere thanks to the referees for the careful and details reading of the manuscript and very helpful suggestions. The project was supported by the Natural Science Foundation of China (11371120), the Natural Science Foundation of Hebei Province (A2013208147, A2015208051).
Received: 13 July 2015 Accepted: 12 November 2015
References
1. Chasreechai, S, Tariboon, J: Positive solutions to generalized second-order three-point integral boundary-value problems. Electron. J. Differ. Equ.2011, 14 (2011)
2. Feng, M: Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions. Appl. Math. Lett.24, 1419-1427 (2011)
3. Galvis, J, Rojas, EM, Sinitsyn, AV: Existence of positive solutions of a nonlinear second-order boundary-value problem with integral boundary conditions. Electron. J. Differ. Equ.2015, 236 (2015)
4. Wu, H, Zhang, J: Positive solutions of higher-order four-point boundary value problem withp-Laplacian operator. J. Comput. Appl. Math.233, 2757-2766 (2010)
5. Jankowski, T: Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions. Nonlinear Anal., Theory Methods Appl.73, 1289-1299 (2010)
6. Zhao, J, Wang, P, Ge, W: Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces. Commun. Nonlinear Sci. Numer. Simul.16, 402-413 (2011)
7. Zhang, X, Ge, W: Symmetric positive solutions of boundary value problems with integral boundary conditions. Appl. Math. Comput.219, 3553-3564 (2012)
8. Yang, W: Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions. Comput. Math. Appl.63, 288-297 (2012)
9. Cabada, A, Wang, G: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl.389, 403-411 (2012)
10. Agarwal, RP, Wang, G, Ahmad, B, Zhang, L, Hobiny, A, Monaquel, S: On existence of solutions for nonlinear
q-difference equations with nonlocalq-integral boundary conditions. Math. Model. Anal.20, 604-618 (2015) 11. Pang, H, Xie, W, Cao, L: Successive iteration and positive solutions for a third-order boundary value problem involving
integral conditions. Bound. Value Probl.2015, 139 (2015)
12. Wang, Q, Guo, Y, Ji, Y: Positive solutions for fourth-order nonlinear differential equation with integral boundary conditions. Discrete Dyn. Nat. Soc.2013, Article ID 684962 (2013)
13. Jiang, W, Zhang, Y, Qiu, J: The existence of solutions forp-Laplacian boundary value problems at resonance on the half-line. Bound. Value Probl.2015, 179 (2015)
14. Guo, Y, Liu, X, Qiu, J: Three positive solutions for higher order m-point boundary value problems. J. Math. Anal. Appl.
289, 545-553 (2004)
15. Ahmad, B, Alsaedi, A, Alghamdi, BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Anal., Real World Appl.9, 1727-1740 (2008)
16. Ahmad, B, Ntouyas, SK, Alsulami, HH: Existence theory fornth order nonlocal integral boundary value problems and extension to fractional case. Abstr. Appl. Anal.2013, Article ID 183813 (2013)
17. Hao, X, Liu, L: Multiple monotone positive solutions for higher order differential equations with integral boundary conditions. Bound. Value Probl.2014, 74 (2014)
18. Li, Y, Zhang, X: Positive solutions for systems of nonlinear higher order differential equations with integral boundary conditions. Abstr. Appl. Anal.2014, Article ID 591381 (2014)
19. Zhang, X, Ge, W: Positive solutions for a class of boundary-value problems with integral boundary conditions. Comput. Math. Appl.58, 203-215 (2009)
