R E S E A R C H
Open Access
Existence of multiple positive solutions for
fourth-order boundary value problems in
Banach spaces
Yujun Cui
1*and Jingxian Sun
2*Correspondence:
1Department of Mathematics,
Shandong University of Science and Technology, Qingdao, 266590, P.R. China
Full list of author information is available at the end of the article
Abstract
This paper deals with the positive solutions of a fourth-order boundary value problem in Banach spaces. By using the fixed-point theorem of strict-set-contractions, some sufficient conditions for the existence of at least one or two positive solutions to a fourth-order boundary value problem in Banach spaces are obtained. An example illustrating the main results is given.
MSC: 34B15
Keywords: u0-positive operator; boundary value problem; positive solution; fixed-point theorem; measure of noncompactness
1 Introduction
In this paper, we consider the existence of multiple positive solutions for the fourth-order ordinary differential equation boundary value problem in a Banach spaceE
⎧ ⎨ ⎩
x()(t) =f(t,x(t), –x(t)), t∈(, ),
x() =x() =x() =x() =θ, (.)
wheref :I×E×E→Eis continuous,I= [, ],θ is the zero element ofE. This problem models deformations of an elastic beam in equilibrium state, whose two ends are simply supported. Owing to its importance in physics, the existence of this problem in a scalar space has been studied by many authors using Schauder’s fixed-point theorem and the Leray-Schauder degree theory (see [–] and references therein). On the other hand, the theory of ordinary differential equations (ODE) in abstract spaces has become an impor-tant branch of mathematics in last thirty years because of its application in partial differ-ential equations and ODEs in appropriately infinite dimensional spaces (see, for example, [–]). For an abstract space, it is here worth mentioning that Guo and Lakshmikantham [] discussed the multiple solutions of two-point boundary value problems of ordinary differential equations in a Banach space. Recently, Liu [] obtained the sufficient condi-tion for multiple positive solucondi-tions to fourth-order singular boundary value problems in an abstract space. In [], by using the fixed-point index theory in a cone for a strict-set-contraction operator, the authors have studied the existence of multiple positive solutions for the singular boundary value problems with an integral boundary condition.
However, the above works in a Banach space were carried out under the assumption that the second-order derivativexis not involved explicitly in the nonlinear termf. This is because the presence of second-order derivatives in the nonlinear functionf will make the study extremely difficult. As a result, the goal of this paper is to fill up the gap, that is, to investigate the existence of solutions for fourth-order boundary value problems of (.) in which the nonlinear functionf contains second-order derivatives,i.e.,f depends onx. The main features of this paper are as follows. First, we discuss the existence results in an abstract spaceE, notE=R. Secondly, we will consider the nonlinear term which is more extensive than the nonlinear term of [, ]. Finally, the technique for dealing with fourth-order BVP is completely different from [, ]. Hence, we improve and generalize the results of [, ] to some degree, and so, it is interesting and important to study the existence of positive solutions of BVP (.). The arguments are based upon theu-positive operator and the fixed-point theorem in a cone for a strict-set-contraction operator.
The paper is organized as follows. In Section , we present some preliminaries and lem-mas that will be used to prove our main results. In Section , various conditions on the existence of positive solutions to BVP (.) are discussed. In Section , we give an example to demonstrate our result.
2 Preliminaries
Let the real Banach spaceEwith norm · be partially ordered by a conePofE,i.e.,x≤y
if and only ify–x∈P.Pis said to be normal if there exists a positive constantNsuch that
θ≤x≤yimpliesx ≤Ny. We consider problem (.) inC[I,E]. Evidently,C[I,E] is a Banach space with normxC=maxt∈Ix(t)andQ={x∈C[I,E] :x(t)≥θ, fort∈I}is
a cone of the Banach spaceC[I,E]. In the following,x∈C[I,E]∩C[(, ),E] is called a solution of problem (.) if it satisfies (.).xis a positive solution of (.) if, in addition,x
is nonnegative and nontrivial,i.e.,x∈C[I,P] andx(t) ≡θ fort∈I.
For a bounded setV in a Banach space, we denote byα(V) the Kuratowski measure of noncompactness (see [–] for further understanding). In this paper, we denote byα(·)
the Kuratowski measure of noncompactness of a bounded set inEand inC[I,E].
Lemma .[] Let D⊂E and D be a bounded set,f be uniformly continuous and bounded from I×D into E,then
αf(I×V)=max
t∈I α
f(t,V), ∀V⊂D.
The key tool in our approach is the following fixed-point theorem of strict-set-contractions:
Theorem .[] Let P be a cone of a real Banach space E and Pr,R={u∈P|r≤ u ≤R}
with <r<R. Suppose that A:Pr,R→P is a strict set contraction such that one of the
following two conditions is satisfied:
(i) Au ≤uforu∈P,u=randAu ≥uforu∈P,u=R; (ii) Au ≥uforu∈P,u=randAu ≤uforu∈P,u=R;
then the operator A has at least one fixed point u∈Pr,Rsuch that r<u<R.
Definition . We say that a bounded linear operatorT:C[I,R]→C[I,R] isu-positive on a coneK={v∈C[I,R] :v(t)≥,∀t∈[, ]}if there existsu∈K\{θ}such that for everyu∈K\{θ}, there are positive constantsk(u),k(u) such that
k(u)u≤Tu≤k(u)u.
Lemma . Let T be u-positive on a cone K.If T is completely continuous,then r(L),
the spectral radius of T,is the unique positive eigenvalue of T with its eigenfunction in K.
Moreover,ifλTu=uholds withλ= (r(T))–,then for an arbitrary non-zero u∈K(u=
ku)the elements u andλTu are incomparable
λTu ≤u, λTu ≥u.
In the following, the closed balls in spacesE andC[I,E] are denoted, respectively, by
l={x∈E:x ≤l}(l> ) andBl={x∈C[I,E] :xC≤l}(l> ).
For convenience, let us list the following assumptions:
(H) f∈C[I×P×P,P],f is bounded and uniformly continuous intonI×(P∩r)for
anyr> , and there exist two nonnegative constantsl,lwithl+ l< such that
αf(t×D×D)
≤lα(D) +lα(D), t∈I,D,D⊂P∩r.
(H) There are three positive constantsa,b,csuch that
f(t,x,y)≤ax+by+c
N
for all(t,x,y)∈I×Pand a π+
b π < .
(H) There is aϕ∈P*(P*denotes the dual cone of P) withϕ(x) > for anyx>θ, two
nonnegative constantsa,band a real numberr> such that
ϕf(t,x,y)≥aϕ(x) +bϕ(y)
for all(t,x,y)∈I×(P∩r)and
a π+
b π > .
(H) There is aϕ∈P*withϕ(x) > for anyx>θand two nonnegative constantsa,band
a real numberR> such that
ϕf(t,x,y)≥aϕ(x) +bϕ(y), t∈I,x,y∈P,x,y ≥R
and a π +
b π > .
(H) There are three positive constantsa,b,rsuch that
f(t,x,y)≤ax+by
N
for all(t,x,y)∈I×(P∩r)and
a π+
(H) There existsη> such that
sup
t∈I,x,y∈P∩η
f(t,x,y)<η
N.
(H) There is aϕ∈P*withϕ= andϕ(x) > for anyx>θand a real numberη> such
that
ϕf(t,x,y)≥αη, t∈[τ, –τ],x,y∈P,τ( –τ)η
N ≤ x,y ≤η,
whereτ∈(,
),α= (τ( –τ)
–τ
τ s( –s)ds)
–.
Now, letG(t,s) be the Green’s function of the linear problemv= ,t∈(, ) together withv() =v() = , which can be explicitly given by
G(t,s) =
⎧ ⎨ ⎩
t( –s), ≤t≤s≤,
s( –t), ≤s≤t≤.
Obviously,G(t,s) have the following properties:
t( –t)s( –s)≤G(t,s)≤t( –t), ∀t,s∈[, ];
G(t,s)≥t( –t)G(τ,s), ∀τ,t,s∈[, ].
(.)
Set
(Su)(t) =
G(t,s)u(s)ds, t∈I,u∈C[I,E].
Obviously,S:C[I,E]→C[I,E] is continuous. Letu= –x. Sincex() =x() =θ, we have
x(t) = (Su)(t) =
G(t,s)u(s)ds. (.)
Using the above transformation and (.), BVP (.) becomes
–u(t) =ft, (Su)(t),u(t) (.)
with
u() =u() =θ. (.)
From (.) and (.), we have
u(t) =
Now, define an operatorAonQby
(Au)(t) =
G(t,s)f
s,
G(s,τ)u(τ)dτ,u(s)
ds. (.)
The following Lemma . can be easily obtained.
Lemma . Assume that(H)holds.Then A:Q→C[I,P]and
(i) A:Q→Qis continuous and bounded;
(ii) BVP(.)has a solution inC[I,P]∩C[(, ),P]if and only ifAhas a fixed point inQ.
Let
(Tψ)(t) =
G(t,s)ψ(s)ds, ∀ψ∈C[I,R].
It is easy to see thatT:C[I,R]→C[I,R] is au-positive operator withu(t) =sinπtand
r(T) =π.
Lemma . Suppose that(H)holds. Then for any l> ,A:Q∩Bl→Q is a strict set
contraction.
Proof For anyu∈Q∩Blandt∈[, ], by the expression ofS, we have
(Su)(t)≤
G(t,s)u(s)ds≤l,
and thusS(Q∩Bl)⊂Q∩Blis continuous and bounded. By the uniformly continuousf
and (H), and Lemma ., we have
αfI×S(D)×D=max
t∈I α
ft×S(D)×D≤lα
S(D)+lα(D).
Sincef is uniformly continuous and bounded onI×(Q∩l), we see from (.) thatAis
continuous and bounded onQ∩Bl. LetD⊂Q∩Bl, according to (.), it is easy to show
that the functions{Au:u∈D}are uniformly bounded and equicontinuous, and so in [] we have
αA(D)=max
t∈I α
AD(t),
where
AD(t)=Au(t) :u∈D,tis fixed⊂P∩l.
Using the obvious formula
and observing ≤G(t,s)≤, we find
αAD(t)≤αcoG(t,s)fs,Su(s),u(s):u∈D,s∈I
≤αcofs,Su(s),u(s)∪θ:u∈D,s∈I
=αfs,Su(s),u(s)∪θ:u∈D,s∈I
≤αfs,Su(s),u(s):u∈D,s∈I
≤αfI×S(B)×B≤lα
S(B)+lα(B)
≤lα(B) +lα(B), (.)
whereB={u(s)|s∈I,u∈D} ⊂Pl,S(B) ={
G(t,s)u(s)ds|t∈I,u∈D}. From the fact of [], we know
α(B)≤α(D). (.)
It follows from (.) and (.) that
αAD(t)≤(l+l)α(D),
and consequently,Ais a strict set contraction onQ∩Blbecause (l+l) < .
3 Main results
Theorem . Let a cone P be normal and condition(H)be satisfied.If(H)and(H)or
(H)and(H)are satisfied,then BVP(.)has at least one positive solution.
Proof Set
Q=
u∈Q:u(t)≥t( –t)u(s),∀t,s∈[, ].
It is clear thatQis a cone of the Banach spaceC[I,E] andQ⊂Q. For anyu∈Q, by (.), we can obtainA(Q)⊂Q, then
A(Q)⊂Q.
We first assume that (H) and (H) are satisfied. Let
W={u∈Q|Au≥u}.
In the following, we prove thatWis bounded.
For anyu∈W, we have θ≤u≤Au, that is,θ ≤u(t)≤Au(t),t∈I. And sou(t) ≤
NAu(t), setv(t) =u(t), by (H)
v(t)≤NAu(t) ≤N
G(t,s)f
s,
≤
G(t,s)
a
G(s,τ)u(τ)dτ+bu(s)+c
ds
≤
G(t,s)
a
G(s,τ)u(τ)dτ+bu(s)
ds+c. (.)
Forψ∈C[I,R], letLψ=aTψ+bTψ, thenL:C[I,R]→C[I,R] is a bounded linear operator. From (.), one deduces that
(I–L)v(t)≤c.
Sinceπis the first eigenvalue ofT, by (H), the first eigenvalue ofL,r(L) =πa+πb < .
Therefore, by [], the inverse operator (I–L)–exists and
(I–L)–=I+L+L+· · ·+Ln+· · ·.
It follows fromL(K)⊂K that (I–L)–(K)⊂K. So, we know thatv(t)≤(I–L)–c,
t∈[, ] andWis bounded.
TakingR>max{r,supW}, we have
Au ≥u, ∀u∈∂BR∩Q. (.)
Next, we are going to verify that for anyr∈(,r),
Au ≤u, ∀u∈∂Br∩Q. (.)
If this is false, then there existsu∈∂Br∩Qsuch thatAu≤u. This together with (H)
yields
ϕu(t)
≥ϕ(Au)(t)
=ϕ
G(t,s)fs, (Su)(s),u(s)
ds =
G(t,s)ϕfs, (Su)(s),u(s)
ds
≥
G(t,s)aϕ
(Su)(s)+bϕ
u(s)ds
=
G(t,s)
a
G(s,τ)ϕu(τ)dτ+bϕ
u(s)ds
=aT+bT
ϕu(t), t∈I.
Forψ∈C[I,R], letLψ=aTψ+bTψ, then the above inequality can be written in the form
ϕu(t)
≥L
ϕu(t)
. (.)
It is easy to see that
In fact,ϕ(u(t)) = implies u(t) =θ fort∈I, and consequently,uC = in
contra-diction touC=r. Now, notice thatL is a u-positive operator withu(t) =sinπt, then by Lemma ., we have ϕ(u(t)) =μsinπt for some μ> . This together with (a
π+
b
π)sinπt=L(sinπt) and (.) implies that
μsinπt=ϕ
u(t)≥L
ϕu(t)=L(μsinπt) =μ
a
π+
b
π
sinπt,
which is a contradiction to a π+
b
π > . So, (.) holds.
By Lemma .,Ais a strict set contraction onQr,R={u∈Q:r≤ uC≤R}.
Ob-serving (.) and (.) and using Theorem ., we see thatAhas a fixed point onQr,R.
Next, in the case that (H) and (H) are satisfied, by the method as in establishing (.), we can assert from (H) that for anyr∈(,r),
Au ≥u, ∀u∈∂Br∩Q. (.)
Let
(Lv)(t) =
G(t,s)
a
G(s,τ)v(τ)dτ+bv(s)
ds, v∈C[I,R].
It is clear thatL:C[I,R]→C[I,R] is a completely continuous linearu-operator with
u(t) =sinπtandL:K→Kin whichK={v∈K⊂C[I,R] :v(t)≥t( –t)v(s),∀t,s∈I}. In addition, the spectral radiusr(L) =πa +πb andsinπtis the positive eigenfunction of Lcorresponding to its first eigenvalueλ= (r(L))–.
Let
(Lδv)(t) = –δ
δ
G(t,s)
a
G(s,τ)v(τ)dτ+bv(s)
ds, v∈C[I,R],
whereδ∈(,). It is clear thatLδ:C[I,R]→C[I,R] is a completely continuous linear
u-operator withu(t) =t( –t) andLδ(K)⊂K. Thus, the spectral radiusr(Lδ)= and
Lδ has a positive eigenfunction corresponding to its first eigenvalueλδ= (r(Lδ))–.
Takeδn∈(, /) (n= , , . . .) satisfyingδ≥δ≥ · · · ≥δn≥ · · · andδn→ (n→ ∞).
Form>n,v∈K, we have
(Lδnv)(t)≤(Lδmv)(t)≤(Lv)(t), ∀t∈I.
By [], we haver(Lδn)≤r(Lδm)≤r(L). Letλδn= (r(Tδn))–, by Gelfand’s formula, we have λδn≥λδm≥λ. Letλδn→λasn→ ∞.
In the following, we prove thatλ=λ.
Letvδnbe the positive eigenfunction ofTδncorresponding toλδn,i.e.,
vδn(t) =λδn(Lδvδn)(t)
=λδn
–δn
δn
G(t,s)
a
G(s,τ)vδn(τ)dτ+bvδn(s)
satisfyingvδn= . Without loss of generality, by standard argument, we may suppose by
the Arzela-Ascoli theorem andλδn→λthatvδn→vasn→ ∞. Thus,v= and by
(.), we have
v(t) =λ
G(t,s)
a
G(s,τ)v(τ)dτ+bv(s)
ds,
that is,v(t) =λ(Lv)(t). This together with Lemma . guarantees thatλ=λ. By the above argument, it is easy to see that there exists aδ∈(,) such that
<r(Lδ) < a
π+
b
π.
Choose
R>max
R,r, NR
δ( –δ)
. (.)
Now, we assert that
Au ≤u, ∀u∈Q,uC=R. (.)
If this is not true, then there existsu∈Q with uC =R such thatAu ≤u, then
Au(t)≤u(t). Moreover, by the definition ofQ, we know
u(t)≥t( –t)u(s), ∀t,s∈I,
(Su)(t) =
G(t,s)u(s)ds≥t( –t)
s( –s)ds·u(τ), ∀t,τ∈I.
Thus,Nu(t) ≥t( –t)uC,N(Su)(t) ≥t(–t)uC, which implies by (.) we have
min
t∈[δ,–δ]
u(t)≥δ( –δ)
N uC≥R,
and
min
t∈[δ,–δ]
(Su)(t)≥δ( –δ)
N uC≥R.
So, by (H), we get
ϕu(t)≥ϕ(Au)(t) =ϕ
G(t,s)fs, (Su)(s),u(s)ds
≥ –δ
δ
G(t,s)ϕfs, (Su)(s),u(s)ds
≥ –δ
δ
G(t,s)aϕ
(Su)(s)+bϕ
≥ –δ
δ
G(t,s)
a
G(s,τ)ϕu(τ)dτ+bϕ
u(s)ds
=Lδϕu(t)
.
It is easy to see that
ϕu(t) = , t∈I.
In fact, ϕ(u(t)) = impliesu(t) =θ fort∈I, and consequently,uC= in
contra-diction touC=R. Now, notice thatLδ is au-positive operator withu(t) =t( –t).
Then by Lemma ., we haveϕ(u(t)) =μvδ(t) for someμ> , wherevδis the positive eigenfunction ofLδcorresponding toλδ. This together withr(Lδ)vδ(t) =Lδ(vδ(t)) implies
that
μvδ(t) =ϕ
u(t)≥Lδϕu(t)=L
μvδ(t)
=μr(Lδ)vδ(t),
which is a contradiction tor(Lδ) > . So, (.) holds.
By Lemma .,Ais a strict set contraction onQr,R={u∈Q:r≤ uC≤R}.
Ob-serving (.) and (.) and using Theorem ., we see thatAhas a fixed point onQr,R.
This together with Lemma . implies that BVP (.) has at least one positive solution.
Theorem . Let a cone P be normal.Suppose that conditions(H), (H), (H)and(H)
are satisfied.Then BVP(.)has at least two positive solutions.
Proof We can take the sameQ⊂C[I,E] as in Theorem .. As in the proof of Theo-rem ., we can also obtain thatA(Q)⊂Q. And we chooser,RwithR>η>r> such that
Au ≤u, ∀u∈∂Br∩Q, (.)
Au ≤u, ∀u∈∂BR∩Q. (.)
On the other hand, it is easy to see that
Au ≥u, ∀u∈∂Bη∩Q. (.)
In fact, if there exists u ∈ Q with uC =η such that Au ≥u, then observing maxt,s∈IG(t,s) = andSuC≤η, we get
θ ≤u(t)≤
G(t,s)fs, (Su)(s),u(s)
ds
≤
fs, (Su)(s),u(s)ds, ∀t∈I,
and so
u(t)≤N
fs, (Su)(s),u(s)ds≤
where, by virtue of (H),
M= sup
t∈I,x,y∈P∩η
f(t,x,y)<η
N. (.)
It follows from (.) and (.) that
η=uC≤
η NM<η,
a contradiction. Thus (.) is true.
By Lemma .,Ais a strict set contraction onQr,η={u∈Q|r≤ uC≤η}, and also on Qη,R={u∈Q|η≤ uC≤R}. Observing (.), (.), (.) and applying, respectively,
Theorem . toA,Qr,ηandQη,R, we assert that there existu∈Qr,ηandu∈Qη,Rsuch
thatAu=u andAu=uand, by Lemma . and (.),Su,Suare positive solutions
of BVP (.).
Theorem . Let a cone P be normal.Suppose that conditions(H), (H)and(H)and (H)are satisfied.Then BVP(.)has at least two positive solutions.
Proof We can take the sameQ⊂C[I,E] as in Theorem .. As in the proof of Theo-rem ., we can also obtain thatA(Q)⊂Q. And we chooser,RwithR>η>r> such that
Au ≥u, ∀u∈∂Br∩Q, (.)
Au ≥u, ∀u∈∂BR∩Q. (.)
On the other hand, it is easy to see that
Au ≤u, ∀u∈∂Bη∩Q. (.)
In fact, if there existsu∈QwithuC=ηsuch thatAu≥u, then
θ≤(Au)(t)≤u(t), t∈I.
Observingu(t)∈Qandt(–t)u(s)≤(Su)(t)≤u(s), we get
η=ϕuC≥ϕ
u(t)≥ϕ(Au)(t) =
G(t,s)ϕfs, (Su)(s),u(s)ds
≥ –τ
τ
G(t,s)ϕfs, (Su)(s),u(s)ds
>αητ( –τ) –τ
τ
s( –s)ds=η,
By Lemma .,Ais a strict set contraction onQr,η={u∈Q|r≤ uC≤η}and also on Qη,R={u∈Q|η≤ uC≤R}. Observing (.), (.), (.) and applying, respectively,
Theorem . toA,Qr,ηandQη,R, we assert that there existu∈Qr,ηandu∈Qη,R such
thatAu=uandAu=uand, by Lemma . and (.),Su,Suare positive solutions
of BVP (.).
4 One example
Now, we consider an example to illustrate our results.
Example . Consider the following boundary value problem of the finite system of scalar differential equations:
⎧ ⎨ ⎩
x()n (t) =fn(t,x(t), –x(t)), t∈(, ),
x() =x() =x() =x() = , (.)
where
fn(t,x,y) =g(yn) + ( +sinπt)
xn
( +xn+yn+)
, n= , , . . . , , (.)
f(t,x,y) =g(y) + ( +sinπt) x ( +x+y)
,
x= (x,x, . . . ,x), y= (y,y, . . . ,y),
g(u) =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u, ≤u≤., , .≤u≤., u– , .≤u≤, u, u≥.
(.)
Claim (.)has at least two positive solutions x*(t) = (x*
(t),x*(t), . . . ,x*(t))and x**(t) = (x**
(t),x**(t), . . . ,x**(t))such that
< max ≤i≤,t∈[,]
x*i(t)< . < max ≤i≤,t∈[,]
x**i(t).
Proof Let E=R ={x= (x,x, . . . ,x) :x
n ∈R,n= , , . . . , } with the norm x=
max≤n≤|xn|, and P={x= (x,x, . . . ,x) :xn≥,n= , , . . . , }. ThenP is a normal
cone inE, and the normal constant isN= . System (.) can be regarded as a boundary value problem of (.) inE, wherex= (x,x, . . . ,x),y= (y,y, . . . ,y),
f(t,x,y) =f(t,x,y),f(t,x,y), . . . ,f(t,x,y).
Evidently,f :I×P→Pis continuous. In this case, condition (H) is automatically sat-isfied. Sinceα(f(t,D,D)) is identical zero for anyt∈IandD,D⊂P∩l. Obviously,
P*=P, so we may chooseϕ= (, , . . . , ), then for anyx,y∈P, we have
ϕf(t,x,y)=
n=
Noticingx ≤ϕ(x)≤x, we have
ϕf(t,x,y)=
n=
fn(t,x,y)≥y ≥ϕ(y)
for all (t,x,y)∈I×Pwithx ≤. andy ≤., and
ϕf(t,x,y)=
n=
fn(t,x,y)≥y ≥ϕ(y),
for all (t,x,y)∈I×P withx ≥ and y ≥. So, the conditions (H) and (H) are satisfied witha=a= andb=b= .
Choosingη= . fort∈Iandx,y∈Pwithx,y ≤., we have
f(t,x,y)≤. < . = η.
So, condition (H) is satisfied. Thus, our conclusion follows from Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors typed, read and approved the final manuscript.
Author details
1Department of Mathematics, Shandong University of Science and Technology, Qingdao, 266590, P.R. China. 2Department of Mathematics, Xuzhou Normal University, Xuzhou, Jiangsu 221116, P.R. China.
Acknowledgements
The authors would like to thank the referees for carefully reading this article and making valuable comments and suggestions. This work is supported by the Foundation items: NSFC (10971179), NSF (BS2010SF023, BS2012SF022) of Shandong Province.
Received: 2 June 2012 Accepted: 24 September 2012 Published: 9 October 2012
References
1. Aftabizadeh, AR: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl.
116, 415-426 (1986)
2. Yang, Y: Fourth-order two-point boundary value problems. Proc. Am. Math. Soc.104, 175-180 (1988) 3. del Pino, MA, Manasevich, RF: Existence for a fourth-order boundary value problem under a two-parameter
nonresonance condition. Proc. Am. Math. Soc.112, 81-86 (1991)
4. Agarwal, RP: On fourth-order boundary value problems arising in beam analysis. Differ. Integral Equ.2, 91-110 (1989) 5. Cui, Y, Zou, Y: Existence and uniqueness theorems for fourth-order singular boundary value problems. Comput. Math.
Appl.58, 1449-1456 (2009)
6. Deimling, K: Ordinary Differential Equations in Banach Spaces. Springer, Berlin (1977)
7. Lakshmikantham, V, Leela, S: Nonlinear Differential Equations in Abstract Spaces. Pergamon, Oxford (1981) 8. Guo, D, Lakshmikantham, V, Liu, X: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht
(1996)
9. Guo, D, Lakshmikantham, V: Multiple solutions of two-point boundary value problem of ordinary differential equations in Banach space. J. Math. Anal. Appl.129, 211-222 (1988)
10. Liu, Y: Multiple positive solutions to fourth-order singular boundary value problems in abstract space. Electron. J. Differ. Equ.120, 1-13 (2004)
11. Zhang, X, Feng, M, Ge, W: Existence results for nonlinear boundary value problems with integral boundary conditions in Banach spaces. Nonlinear Anal.69, 3310-3321 (2008)
12. Krasnosel’skii, MA: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964) 13. Krasnosel’skii, MA, Zabreiko, PP: Geometrical Methods of Nonlinear Analysis. Springer, Berlin (1984)
14. Xia, D, Wu, Z, Yan, S, Shu, W: Theory of Real Variable Functions and Functional Analysis, 2nd edn. Higher Education Press, Beijing (1985)
doi:10.1186/1687-2770-2012-107
