R E S E A R C H
Open Access
Positive solutions for elastic beam equations
with nonlinear boundary conditions and a
parameter
Wenxia Wang
*, Yanping Zheng, Hui Yang and Junxia Wang
*Correspondence:
[email protected] Department of Mathematics, Taiyuan Normal University, Taiyuan, 030012, P.R. China
Abstract
This paper is concerned with the existence, nonexistence, and uniqueness of convex monotone positive solutions of elastic beam equations with a parameter
λ. The
boundary conditions mean that the beam is fixed at one end and attached to a bearing device or freed at the other end. By using fixed point theorem of cone expansion, we show that there existsλ
∗≥λ
∗> 0 such that the beam equation has at least two, one, and no positive solutions for 0 <λ
≤λ
∗,λ
∗<λ
≤λ
∗andλ
>λ
∗, respectively; furthermore, by using cone theory we establish some uniqueness criteria for positive solutions for the beam and show that such solutionxλdepends continuously on the parameterλ. In particular, we give an estimate for critical value of
parameterλ.
MSC: 34B18; 34B15
Keywords: elastic beam equation; positive solution; fixed point; cone
1 Introduction and preliminaries
In this paper, we consider the following nonlinear fourth-order two-point boundary value problem (BVP) for elastic beam equation:
x()(t) =λf(t,x(t)), <t< ,
x() =x() =x() =x() +q(x()) = , ()
whereλ≥ is a parameter. Throughout this paper, we assume thatf∈C([, ]×R+,R+),
q∈C(R+,R+), R+= [, +∞). x∈C[, ] is called a positive solution of BVP () ifx is a
solution of BVP () andx(t) > , <t< . A convex monotone positive solution means convex nondecreasing positive solution.
Because of characterization of the deformation of the equilibrium state, fourth-order boundary value problems for elastic beam equations are extensively applied to mechan-ics and engineering; see [–]. Some nonlinear elastic beam equations have been studied extensively. For a small sample of such work, we refer the reader to the work of Bai and Wang [], Bai [], Bonanno and Bellaa [], Li [], Liu and Li [], Liu [], Ma and Xu [], and Ma and Thompson [] on an elastic beam whose two ends are simply supported, the works of Yang [] and Zhang [] on an elastic beam of which one end is embedded and another end is fastened with a sliding clamp, and the work of Graefet al.[] on multipoint boundary value problems.
BVP () withq(x)≡ is called a cantilever beam equation, it describes the deflection of the elastic beam fixed at the left end and free at the right end. Existence and multiplicity of positive solutions of cantilever beam problems without parameter have been studied by some authors; see Yao [, ] and references therein. BVP () withq(x) ≡ describes the deflection of the elastic beam fixed at the left end and attached to a bearing device given by the function –qat the right end. When the elastic beam equation does not con-tain parameterλ, the existence of multiple positive solutions and unique positive solution was presented in [] by variational methods and in [] by a fixed point theorem, re-spectively; monotone positive solutions were obtained by using the monotone iteration method in []. However, there are few papers concerned with positive solutions for BVP () with parameter, especially with the solution’s dependence on parameterλin the exist-ing literature. The aim of this paper is to show that the existence and number of convex monotone positive solutions of BVP () are affected by the parameterλ.
The paper is organized as follows. In Section , we present that a nontrivial and non-negative solution of BVP () is convex monotone positive solution. In Section , we ob-tain some results on the existence, multiplicity and nonexistence of positive solutions for BVP (). These results show that the number of positive solutions for BVP () depends on the parameterλ. In Section , we establish some uniqueness criteria for positive solu-tions for BVP () and show that such a positive solutionxλdepends continuously on the parameterλ. In particular, we give an estimate for the critical value of the parameterλ.
In the rest of this section, we introduce some notations and known results. For the reader’s convenience, we suggest that one refer to [–], and [] for details.
LetEbe a real Banach space andθ denote the zero element ofE. A nonempty closed convex setP⊂E is called a cone ofE if it satisfies (i)x∈P,r> ⇒rx∈P; (ii)x∈P, –x∈P⇒x=θ.Eis partially ordered by the coneP,i.e.,x≤yiffy–x∈P. A conePis said to be normal if there exists a positive numberN, called the normal constant ofP, such thatθ≤x≤yimpliesx ≤Ny. Foru,v∈E,u≤v, denote [u,v] ={x∈E|u≤x≤v}.
For allx,y∈E, the notationx∼ymeans that there existμ> andμ> such that μx≤y≤μx. Clearly,∼is an equivalence relation. Givene> (i.e.,e∈Pande=θ), we
denote byPethe setPe={x∈E|x∼e}. It is easy to see thatPe⊂P.
LetD⊆E. An operatorT:D→Eis said to be increasing if forx,y∈D,x≤y⇒Tx≤Ty. An elementx∗∈Dis called a fixed point ofTifTx∗=x∗.
Lemma .(Fixed point theorem of cone expansion) [, ] Assume that and
are bounded open subsets of E withθ ∈⊂⊂.Let T :P∩(–)→P be a
completely continuous operator such thatTx ≤ xif x∈P∩∂ andTx ≥ x if
x∈P∩∂.Then T has a fixed point in P∩(–).
Lemma .[] Let P be a normal cone in E,T:Pe→Pebe increasing and for all x∈Pe and t∈(, ),there existsα(t)∈(, )such that T(tx)≥tα(t)Tx.Then T has a unique fixed
point x∗in Pe.Moreover,constructing successively the sequence wn=Twn–(n= , , . . .)for
any w∈Pe,we havelimn→+∞wn–x∗= .
2 Solutions
In what follows, setE=C[, ], the Banach space of all continuous functions on [, ] with the normx=max{|x(t)| |t∈[, ]}.P={x∈C[, ]|x(t)≥,t∈[, ]}. It is clear thatP
From [] and [], it is evident that BVP () has an integral formulation given by
x(t) =λ
G(t,s)fs,x(s)ds+qx() t
–
t
, ()
where
G(t,s) =
t(s–t), ≤t≤s≤,
s(t–s), ≤s≤t≤. ()
It is easy to see thatG(t,s)≥ and
t
s≤G(t,s)≤
t
s, t,s∈[, ]. ()
Define operatorsA,B,Cλ:P→C[, ] by
(Ax)(t) =
G(t,s)fs,x(s)ds, (Bx)(t) =qx() t
–
t
, Cλ=λA+B.
ThenA(P)⊂P,B(P)⊂P, andCλ(P)⊂P.
It is clear from () that solving BVP () is equivalent to finding fixed points of the oper-atorCλ. In particular,xis a fixed point ofBiffxis a solution of the following BVP:
x()(t) = , <t< ,
x() =x() =x() =x() +q(x()) = ,
andxis a fixed point ofλAiffxis a solution of the following cantilever beam problem:
x()(t) =λf(t,x(t)), <t< ,
x() =x() =x() =x() = . ()
Lemma . If x∈C[, ]satisfies
x()(t)≥, t∈(, ),
x() =x() =x() = , x()≤, ()
then
(i) x(t)is nondecreasing int∈[, ],moreover,≤x(t)≤x(),t∈[, ]; (ii) x(t)≥,t∈[, ],that is,x(t)is a convex function on[, ].
Proof From (), we havex(t)≤x()≤. Moreover,x(t)≥x() = . So,x(t)≥x() =
. Thus, we complete the proof of the lemma.
Now, let
K=
x∈P x(t) is nondecreasing, x(t)≥ t
x(),t∈[, ]
,
Lemma . Cλ(P)⊂K,A(P)⊂K,B(P)⊂K.
Proof x∈Pimpliesx(t)≥, sof(t,x(t))≥ andq(x())≥. Moreover, forx∈P,
(Cλx)()(t) =λf
t,x(t)≥, t∈(, ),
(Cλx)() = (Cλx)() = (Cλx)() = ,
(Cλx)() = –q
x()≤.
By Lemma ., (Cλx)(t) is convex and nondecreasing int∈[, ]. From () and () we have
(Cλx)(t) – t
(C
λx)()≥
λ
t
sfs,x(s)ds+qx() t
–
t
≥,
that is, t(Cλx)()≤(Cλx)(t) fort∈[, ]. Thus, we obtainCλ(P)⊂K. From the above proof, we can show thatA(P)⊂KandB(P)⊂K. This ends the proof.
Lemma .
(i) A:P→Kis a completely continuous operator;
(ii) ifq(x)is nondecreasing,thenB:P→Kis a completely continuous operator.
Proof Similarly to the proof of Theorem in [], applying the Arzela-Ascoli Theorem,
the proof can be completed.
From the proof of Lemma . we can show the following result.
Theorem . If x∈P\{θ}is a solution for BVP(),then x is a convex monotone positive solution for BVP().
So, in the following sections, we only need to study solutions for BVP () inP\{θ}.
3 Existence and nonexistence results
It is obvious from Lemma . that ifx∈P\{θ}is a solution for BVP () thenx∈K\{θ}. So in this section, we will apply Lemma . to study the existence, multiplicity and nonexistence of solutions for BVP () inK\{θ}. It is reasonable that the domain ofCλis restricted onK. The following conditions will be assumed:
(H) f(t,x)is nondecreasing inx∈[, +∞)for fixedt∈[, ]; (H) q(x)is nondecreasing inx∈[, +∞);
(H) F:=
sf(s, )ds> ; (H) q() < ;
(H) f∞:=limx→+∞mint∈[,]
f(t,x)
x = +∞; (H) q∞:=lim infx→+∞q(xx)> .
Set
=λ> |there existsxλ∈K\{θ}such thatCλxλ=xλ
()
Lemma . Suppose that(H)-(H)hold.Ifλ∈,then(,λ]⊂.
Proof λ∈means that there existsxλ∈K\{θ}such thatCλxλ=xλ. Therefore, for any
λ∈(,λ], we haveCλxλ≤Cλxλ=xλ. Setw=xλ,wn=Cλwn–,n= , , . . . . From (H)
and (H) we obtain w(t)≥w(t)≥ · · · ≥wn(t)≥ · · · ≥ Fλt. By Lemma . and (H),
{wn}converges to a fixed point ofCλinK\{θ}. Thus (,λ]⊂. This completes the proof.
Letλ∗=–Fq(),F=
sf(s, )ds,u(t) =Fλt,v(t) =tand
F∞=lim sup x→+∞
max
t∈[,]
f(t,x)
x , Q∞=lim supx→+∞
q(x)
x .
Theorem . Suppose that(H)-(H)hold.
(i) If(H)holds,thenCλhas minimal and maximal fixed points in[u,v]for λ∈(,λ∗].Moreover,there existsλ∗≥λ∗> such thatCλhas at least one and has no fixed points inK\{θ}for <λ<λ∗andλ>λ∗,respectively.
(ii) IfF∞< +∞,Q∞< ,then whenF∞> ,there existsλ∗≥(–F∞Q∞)> such thatCλ has at least one and no fixed points inK\{θ}for <λ<λ∗andλ>λ∗,respectively;
whenF∞= ,Cλhas at least one fixed point inK\{θ}forλ> .
Proof (i) From (H), (H), and (H) we haveλ∗> . For anyλ∈(,λ∗], we obtain
(Cλu)(t)≥ λ
t
sf(s, )ds=u(t), (Cλv)(t)≤
t
λ
∗F+q()
≤v(t).
Setun=Cλun–,vn=Cλvn–,n= , , . . . , then from (H) and (H) we have
u(t)≤u(t)≤ · · · ≤un(t)≤ · · · ≤vn(t)≤ · · · ≤v(t)≤v(t). ()
Lemma . implies that{un}and{vn}converge to fixed pointsuλandvλofCλ, respectively. From () it is evident thatuλ,vλ∈K\{θ}are the minimal fixed point and maximal fixed point ofCλin [u,v], respectively. From the definition ofλ∗we can complete the rest of
the proof.
(ii) For any << –Q∞, there existsN> such thatf(t,x)≤(F∞+)xandq(x)≤
(Q∞+)xforx>N,t∈[, ]. Letw(t) = Ntandλ=(–F∞Q∞+–), thenλ> and
(Cλw)(t)≤
w(t)
λ
(F∞+) +Q∞+
≤w(t).
Similarly to the proof of Lemma ., we can showλ∈. The conclusion (ii) follows from
Lemma . and the definition ofλ∗. This completes the proof of Theorem ..
Lemma . Suppose that(H)-(H) hold and that one of(H)and(H)holds. Ifis nonempty,then
Proof (i) Suppose to the contrary that there exists an increasing sequence{λn}+∞ ⊂
such thatlimn→+∞λn= +∞. Setxλn∈K\{θ}is a fixed point ofCλn, that is,Cλnxλn=xλn.
There are two cases to be considered.
Case .{xλn}+∞ is bounded, that is, there exists a constantM> such thatxλn ≤M
forn= , , . . . . Hence, from (H), (H), and () we have
M≥ xλn= (Cλnxλn)()≥
λn
sfs,xλn(s)
ds≥F
λn→+∞,
which is a contradiction.
Case .{xλn}+∞is unbounded, that is, there exists a subsequence of{xλn}+∞, still denoted
by{xλn}+∞, such thatlimn→+∞xλn= +∞.
When (H) holds, take L> λ
, there existsN > such thatf(t,x)≥Lxforx≥N, t∈[
, ]. Choosen such thatxλn> N. Thus, f(t,xλn)≥ Lxλn, t∈[, ].
Moreover, from (H) and the definition ofK, we have
xλn= (Cλnxλn)()≥
λ
sf
s, xλn
ds>
λLxλn>xλn,
which is a contradiction.
When (H) holds, choose> such that (q∞–) > . There existsN> such that
q(x)≥(q∞–)xforx≥N. Choosensuch thatxλn>N, so
qxλn()
=qxλn
≥(q∞–)xλn.
Moreover,
xλn= (Cλnxλn)()≥
q
xλn()
≥
(q∞–)xλn>xλn,
which is a contradiction.
Consequently, we find thatis bounded from above.
(ii) By the definition of λ∗, there exists a nondecreasing sequence {λn}+∞ such that
limn→+∞λn=λ∗. Letxλn∈K\{θ}be a fixed point ofCλn. Arguing similarly as above in
case , we can show that{xλn}+∞is a bounded subset inK, that is, there exists a constant
M> such thatxλn ≤M,n= , , . . . ; on the other hand, note that
xλn(t) –xλn(t) ≤λ∗
G(t,s) –G(t,s) f(s,M)ds+
q(M)|t–t|,
we see that{xλn}+∞is an equicontinuous subset inK. Consequently, by an application of
the Arzela-Ascoli Theorem we conclude that{xλn}+∞is a relatively compact set inK. So,
there exists a subsequence{xλni} ⊂ {xλn}converging tox∗∈K. Note that
xλni(t) =λni
G(t,s)fs,xλni(s)
ds+qxλni()
t
–
t
.
By taking the limit we havex∗(t) = (Cλ∗x∗)(t)≥λFt, that is,λ∗∈. The proof is
Theorem . Suppose that(H)-(H)hold and that one of(H)and(H)holds.Then,
there exists aλ∗≥λ∗> such that BVP()has at least two,one,and no positive solutions for <λ≤λ∗,λ∗<λ≤λ∗andλ>λ∗,respectively.
Proof Theorem . implies (,λ∗]⊂, soλ∗≥λ∗> . From Lemmas . and ., we have (,λ∗] =. Therefore, from the definition ofλ∗we only to prove thatCλhas at least two fixed points inK\{θ}forλ∈(,λ∗].
Now, givenλ∈(,λ∗]. Theorem . means thatCλhas at least one fixed pointxλ,∈
K\{θ}which satisfiesxλ, ≤.
LetK={x∈K| x< }. Note thatt( –t)≤ fort∈[, ], so forx∈Kwithx= ,
i.e.,x∈∂K, we have
Cλx= (Cλx)()≤
λ∗
s( –s)f(s, )ds+ q()
≤
<x. ()
When (H) holds, take L> λ, there exists N> such thatf(t,x)≥Lxforx≥N,
t∈[, ]. SetK={x∈K| x< N}. ThenK⊂K. Ifx∈∂K, we have
Cλx= (Cλx)()≥
λ
sf
s, x
ds>λL
x>x.
When (H) holds, from the proof of Lemma . we can setK={x∈K| x<N}. Then
K⊂K. Ifx∈∂K, we haveCλx=Cλx()≥q(x())≥(q∞–)x() >x.
Consequently, in virtue of Lemma . we find thatCλhas another fixed pointxλ,with
xλ,∈
K–K, as (H) holds,
K–K, as (H) holds.
Equation () implies thatCλhas no fixed points in∂K. In conclusion, forλ∈(,λ∗],Cλ has at least two fixed pointsxλ, andxλ,inK with <xλ,< <xλ,. The proof is
complete.
Remark . In the above results, we can replace (H) with the following condition: there exists∈(, ) such thatlimx→+∞mint∈[,]
f(t,x)
x = +∞.
In the following, we give some sufficient conditions that BVP () has no positive solu-tions.
Theorem . Suppose that there exists a nonnegative integrable function a(t)such that f(t,x)≥a(t)x,t∈[, ],x∈[, +∞)and a∗:=s( –s)a(s)ds> .Then BVP()has no
positive solutions forλ>
a∗.
Proof Assume to the contrary that xλ∈K\{θ} is a solution of BVP (), then xλ= (Cλxλ)()≥xλλ
s( –s)a(s)ds>xλ, which is a contradiction. The proof is
com-plete.
Theorem . Suppose that there exist an integrable function a(t)≥and a number
b∈[, ) such that f(t,x)≤a(t)x,q(x)≤bx, t∈[, ],x∈[, +∞)and a∗ :=
s( –
s)a(s)ds> .Then BVP()has no positive solutions for≤λ<–a∗b
.
Theorem . Suppose that q(x) > x,x∈[, +∞).Then BVP()has no positive solutions forλ≥.
Remark . Whenq(x)≡, BVP () becomes a cantilever beam problem (). In this case, we can delete the conditions onqin Theorems ., .-. and obtain the following cor-responding results for BVP ().
Suppose that (H) and (H) hold. Then BVP () has minimal and maximal solutions in [u,v] forλ∈(,F]. Further, if <F∞< +∞, then there existsλ∗≥max{F,F∞}such that
BVP () has at least one and has no positive solutions for <λ<λ∗andλ>λ∗, respectively; ifF∞= then BVP () has at least one positive solution forλ> .
Suppose that (H), (H), and (H) hold. Thenλ∗≥F
and BVP () has at least two, one
and has no positive solutions for <λ≤F ,
F <λ≤λ
∗andλ>λ∗, respectively.
Under the conditions in Theorem ., BVP () has no positive solutions forλ>a∗. Suppose thata(t) anda∗ satisfy the conditions in Theorem ., then BVP () has no
positive solutions for ≤λ<a∗
.
Remark . (i) We give an example to illustrate Theorem .. Letf(t,x) =t+
ln( +x),
and
q(x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
sinx, ≤x≤π
,
πx,
π
≤x≤π, π
≤x≤π,
+|sinx|, π≤x≤π, , x≥π.
By straightforward calculations we see thatF=,F=( +ln),q() =sin,λ∗=–Fq()=.
.,F∞= , andQ∞= . So the conditions in Theorem . are satisfied. Therefore,
by Theorem . we find that there existsλ∗ ≥F∞ = such that BVP () has minimal and maximal solutions in [u,v] for <λ≤., has at least one positive solution for
. <λ<λ∗and has no positive solutions forλ>λ∗, whereu(t) =λtandv(t) =t.
We give another example to illustrate Theorem .. Letf(t,x) = t( +x) +et√x, and
q(x) =
x
, ≤x≤,
, x≥.
A straightforward calculation can show thatF=,f∞= +∞,q() = ,F=
, andλ∗=
Therefore, the conditions (H)-(H) hold. Thus, by Theorem . we see that there exists
λ∗≥ such that BVP () has at least two, one, and no positive solutions for <λ≤ ,
<λ≤λ∗, andλ>λ∗, respectively.
(ii) In Theorems .-., we do not requiref andqto be monotone inx. For example, letf(t,x) = tx
+|cosx|and
q(x) =
Takea(t) =t,b= , then the conditions in Theorem . are satisfied anda∗ =. So by
Theorem . we find that BVP () has no positive solutions for ≤λ<.
4 Uniqueness and dependence on parameter
In this section, we will apply cone theory to further study the uniqueness of solution for BVP () inP\{θ}and the dependence of such a positive solution on the parameterλ. The following hypotheses are needed:
(H) q()= and for allx∈[, +∞)andr∈(, ), there existsα(r)∈(, )such that q(rx)≥rα(r)q(x);
(H) f(t, ) ≡andf(t,rx)≥rf(t,x)forr∈(, ),t∈[, ],x∈[, +∞);
(H) for allt∈[, ],x∈[, +∞)andr∈(, ), there existsβ(r)∈(, )such that f(t,rx)≥rβ(r)f(t,x).
Remark . The inequalities in (H), (H), and (H) are equivalent to the following in-equalities, respectively:
q(sx)≤sα(s)q(x), s> ,x∈[, +∞),
f(t,sx)≤sf(t,x), s> ,t∈[, ],x∈[, +∞),
f(t,sx)≤sβ(s)f(t,x), s> ,t∈[, ],x∈[, +∞).
Lete(t) =t and definePeas in Section . It is obvious thatPe⊂Pand ifx∈Pe then
x() = andx(t) > ,t∈(, ].
Remark . (H) and (H) implyq(x) > forx> . Moreover,q(x()) > forx∈Pe.
Remark . Letxλbe a solution for BVP () inP\{θ}. If (H) and (H) hold, thenxλ∈Pe. Indeed, from Theorem . we havexλ() =xλ. So Remark . impliesq(xλ()) > . Note that
q(xλ())
t
≤x
λ(t) = (Cλxλ)(t)≤
λ
sfs,xλ(s)
ds+qxλ()
t,
we concludexλ∈Pe.
So, in this section, we only need to consider the unique solution for BVP () inPe.
Lemma . Assume that(H)and(H)hold.Then B has a unique fixed point xin Pe,
moreover,constructing successively the sequence wn=Bwn– (n= , , . . .) for any initial
value w∈Pe,we havelimn→+∞wn–x= .
Proof For anyx∈Pe, we haveq(x())t≤Bx(t)≤q(x())t, which meansB(Pe)⊂Pe. For
allx∈Pe,r∈(, ), from (H) we haveB(rx)(t)≥rα(r)Bx(t). Consequently, the conclusion
follows from Lemma .. This completes the proof.
(ii) for anyλ≥andx∈Pe,there existsϕ(λ,x)∈(, )such thatBx≥ϕ(λ,x)Cλx;
(iii) for[u,v]⊂Peandr∈(, ),there existsη(r,u,v) > such that
Cλ(rx)≥r
+η(r,u,v)Cλx, ∀x∈[u,v].
Proof The conclusion (i) follows from (H), (H), (H), and (). The proof of (ii). For givenλ≥,x∈Pe, from (H) and () we have
(Cλx)(t)≤ t
λ
sfs,xds+qx()≤(λ
sf(s,x)ds+q(x())
q(x()) Bx(t).
Let
ϕ(λ,x) = q(x())
(λsf(s,x)ds+q(x())), ()
then <ϕ(λ,x) < and
Bx≥ϕ(λ,x)Cλx. ()
The proof of (iii). For anyx∈[u,v],u,v∈Pe, from () and () we have
Bx≥ q(x())
(λsf(s,x)ds+q(x())Cλx≥
q(u())
(λsf(s,v)ds+q(v()))Cλx.
Moreover, from (H) and (H) we have
Cλ(rx)≥λrAx+rα(r)Bx≥r
+η(r,u,v)Cλx, ∀r∈(, ),x∈[u,v],
whereη(r,u,v) = (rα(r)–r)q(u())
r(λsf(s,v)ds+q(v()))> . This completes the proof.
Lemma . Assume that(H), (H), (H),and(H)hold.Then Cλhas a unique fixed point
xλin Peiff there exists yλ∈Pesuch that Cλyλ≤yλ.Moreover,constructing successively the
sequence wn=Cλwn–(n= , , . . .)for any initial value w∈Pe,we have
lim
n→+∞wn–xλ= . ()
Proof ‘⇒’ Letxλbe a fixed point ofCλinPe,i.e.,Cλxλ=xλ. Takingyλ=xλ, we obtain
Cλyλ≤yλ.
‘⇐’ By virtue of Lemma .,Bhas a unique fixed pointxinPe. Moreover,
x≤Cλx, λ≥. ()
Now, we are going to prove
Letτ=inf{τ> |x≤τyλ}, thenτ≤. Otherwise,τ> , from Lemma . we have
x≤Cλx≤Cλ(τyλ)≤
τ
+η(τ
,x,τyλ) Cλyλ≤
τ
+η(τ
,x,τyλ) yλ.
By the definition ofτ, we get a contradictionτ≤+η(τ τ,x,τyλ)
. Thus, () holds.
Setxn=Cλxn–,yn=Cλyn–,y=yλ,n= , , . . . . From () and () we have
x≤x≤ · · · ≤xn≤ · · · ≤yn≤ · · · ≤y≤y=yλ. ()
Lemma . implies that{xn}and{yn}converge to fixed pointsx∗andy∗ofCλ, respectively. From (), we have
x≤x≤ · · · ≤xn≤ · · · ≤x∗≤y∗≤ · · · ≤yn≤ · · · ≤y≤yλ. ()
To prove thatCλhas only one fixed point in [x,yλ], let
μn=sup{τ> |xn≥τyn}, n= , , , . . . , ()
then
<μn≤, xn≥μnyn, n= , , . . . . ()
From ()-() we infer that < μ ≤ μ ≤ · · · ≤ μn ≤ · · · ≤ , which means that
limn→+∞μn=μ≤. We assert thatμ= . Otherwise, <μn≤μ< forn≥, then
by Lemma . we deduce that
xn+≥Cλ(μnyn)≥Cλ
μn μμyn
≥μn
μCλ(μyn)≥μn
+η(μ,x,yλ)
yn+.
By (), we haveμn+≥μn( +η(μ,x,yλ)), moreover,μ≥μ( +η(μ,x,yλ)), which is a contradiction. Soμ= . Thus, by () and () we have
y∗–x∗≤ yn–xn ≤( –μn)y → asn→+∞,
which means thatx∗=y∗:=xλis the unique fixed point ofCλin [x,yλ].
Now, we prove thatx∗is the unique fixed point ofCλinPe. By the above proof, we only
need to show thatCλdoes not have any fixed point inPe\[x,yλ]. Ifxis a fixed point of
CλinPe\[x,yλ]. Let
μ=sup
τ > τx∗≤x≤ τx
∗. ()
It is evident that <μ≤. If <μ< , thenx≤x∗≤μx∗≤
μyλ. By Lemma . we have
μ
+η
μ,x,
μyλ
x∗≤Cλ
μx∗≤x=Cλx≤Cλ
μx ∗
≤
μ( +η(μ,x,μyλ))
Thus, from () we haveμ≥μ( +η(μ,x,μyλ)), which is a contradiction. Soμ= . More-over,x=x∗, which implies the contradiction:x=x∗∈[x,yλ] andx∈Pe\[x,yλ].
Finally, the iterative scheme and () can be proved in a similar way to the proof of Theorem . of [], here it is omitted. The proof is complete.
Theorem . Assume that(H), (H), (H),and(H)hold.Then there exists aλ∗> such that BVP()has a unique solution xλin Peforλ∈[,λ∗)and does not have any solution
in Peforλ≥λ∗.Moreover,set wn=λAwn–+Bwn–(n= , , . . .)for any w∈Pe,then()
holds.
Proof By Lemma .,Bhas the unique fixed pointxinPe. Sox(t) =q(x())(t–t),
moreover,x=x() =q(x()) > . Letρ=
sf(s,x())
x() ds, we have
(Ax)(t)≤
t
sfs,x()
ds≤qx()
t
–
t
ρ=ρx(t). ()
Set={λ≥| there existsxλ∈Pesuch thatCλxλ=xλ}. Lemma . implies that
={λ≥| there existsyλ∈Pesuch thatCλyλ≤yλ}. ()
Similarly to the proof of Lemma ., we can show thatλ∈implies [,λ]⊂.
Now, takes> and letλ=ρ( –s
α(s)–
) andyλ=sx, thenλ> andyλ∈Pe. By
(H), (H), and (), we haveCλyλ≤λsρx+s
α(s)
x≤yλ, that is,λ∈. Moreover,
[,λ]⊂.
Letλ∗=sup, thenλ∗≥λ> . We assert thatλ∗∈/. Indeed, ifλ∗= +∞, from the
definition ofλ∗it is obvious thatλ∗∈/. Suppose thatλ∗< +∞andλ∗∈. Then by () and () there existsx≤yλ∗∈Pesuch thatCλ∗yλ∗≤yλ∗. Similarly to the proof of (), we have
(Ayλ∗)(t)≤
sf(s,yλ∗)
x()
ds
x(t)≤
sf(s,yλ∗)
x()
ds
(Byλ∗)(t).
Denoteρ=
sf(s,yλ∗)
x() ds, then
<ρ< +∞ and Ayλ∗≤ρByλ∗. ()
Setv=syλ∗for givens> , thenv∈Pe. Sinces
α(s )–
< , we can chooseδ> such that
sα( s)–
< –δρ. Therefore, from () we have
Cλ∗+δ(v)≤
λ∗+δsAyλ∗+s α(s
) Byλ∗ ≤s
λ∗Ayλ∗+δρByλ∗+Byλ∗–δρByλ∗
≤v.
This means thatλ∗+δ∈, which is a contradiction to the definition ofλ∗. So,= [,λ∗). Consequently, an application of Lemma . completes the proof.
Theorem . Assume that(H), (H), (H),and(H)hold.Then xλdepends upon the
parameterλas follows:
(i) xλis nondecreasing with respect toλforλ∈[,λ∗);
(ii) xλis continuous with respect toλforλ∈[,λ∗);
(iii) limλ→+xλ–x= andlimλ→λ∗–xλ= +∞.
Proof (i) Letλ,λ∈[,λ∗) withλ≤λ. SinceCλxλ ≤Cλxλ=xλ, from the proof of
Lemma ., we find that the unique fixedxλofCλbelongs to [x,xλ], which means that xλ≤xλ.
(ii) Letλ∈(,λ∗). In order to provelimλ→λ–xλ–xλ= , let sequence{λn}satisfy
<λ≤λ≤ · · · ≤λn≤ · · · ≤λ and lim
n→+∞λn=λ.
By virtue of the above conclusion (i) we have
xλ≤xλ≤ · · · ≤xλn≤ · · · ≤xλ, ()
which implies that{xλn}is a bounded subset inP. Further, similarly to the proof of the
conclusion (ii) in Lemma . we see that{xλn}converges tox∗∈P. From () we have x∗∈[xλ,xλ], which leads tox∗∈Pe. Note that
xλn=λnAxλn+Bxλn.
By taking the limit we havex∗=λAx∗+Bx∗=Cλx∗. SinceCλhas only one fixed point
inPe, thenx∗=xλ. This means thatxλ–xλ → asλ→λ – .
A similar argument can show that for anyλ∈[,λ∗),xλ–xλ → asλ→λ + . Thus,
the proof of (ii) is complete.
(iii) It is obvious from the above conclusion (ii) thatlimλ→+xλ–x= .
In order to finish the proof oflimλ→λ∗–xλ= +∞, we consider two cases. Case .λ∗= +∞.
Sincexλ=λAxλ+Bxλ≥λAx, thenxλ ≥λAx, which meanslimλ→λ∗–xλ= +∞. Case .λ∗< +∞.
By the above conclusion (i) we havelimλ→λ∗–xλ ≤+∞. Suppose to the contrary that
limλ→λ∗–xλ< +∞. Similarly to the case in the proof of Lemma ., we conclude that
Cλ∗has a fixed pointx∗∈P\{θ}. From Remark . we havex∗∈Pe. Soλ∗∈[,λ∗), which
is a contradiction. This ends the proof.
Now, we give an estimate for critical valueλ∗ in Theorem .. If (H) and (H) hold, then
f(t,x)
x ≤f(t, )≤tmax∈[,]f(t, ), x> ,t∈[, ].
Moreover,F∞=lim supx→+∞maxt∈[,]f(tx,x)∈[, +∞).
Theorem . Assume that(H), (H), (H),and(H)hold.Then
λ∗
≥
F∞, <F∞< +∞,
Proof For any> , there existsr∈(, ) such that
q()≤
r
and f
t,
r
≤
r(F∞+), r≤r,t∈[, ], ()
Note that r
α(r)
r > , we can choose a sufficiently large positive integer numberksuch that
(r
α(r)
r ) k≥
r, that is,
rα(r)
k
≤
rk–
. ()
Letw(t) = (r
)
ke(t), then, from (), (), and () we have
(Aw)(t)≤t
sf
s,
rk
ds≤ t
rk–
sf
s,
r
ds≤
(F∞+)w(t),
(Bw)(t)≤t
q rk ≤t
rα(r)
q
rk–
≤t
rα(r)
k q()≤t
r k = w(t).
Moreover, takingλ=F∞+, we have
(Cλw)(t) =λ(Aw)(t) + (Bw)(t)≤
λ(F∞+)w(t) +
w(t)≤w(t).
Consequently, from () we obtainλ= F∞+ ∈[,λ∗), that is,λ∗>F∞+, which implies
that () holds. This completes the proof.
Remark . Different from Theorems . and ., the estimate ofλ∗in Theorem . does not take into account effect ofq(x). This is valuable, because the conditions (H) and (H) cannot ensureQ∞< aslim supr→α(r) = . Certainly, ifQ∞< , thenλ∗≥(–F∞Q∞). In particular, ifQ∞= , thenλ∗≥F∞.
Corollary . Assume that(H), (H), (H),and(H)hold.Then
(i) BVP()has a unique positive solutionxλinPeforλ∈[, +∞).Moreover,for any w∈Pe,setwn=λAwn–+Bwn–(n= , , . . .),thenlimn→+∞wn–xλ= ;
(ii) xλis nondecreasing with respect toλforλ∈[, +∞);
(iii) xλis continuous with respect toλforλ∈[, +∞);
(iv) limλ→+xλ–x= andlimλ→+∞xλ= +∞.
Proof From (H), (H), (H), and (), we see thatCλ:Pe→Peis increasing for any given
λ≥. Further, for any givenλ≥ we have
Cλ(rx) =λA(rx) +B(rx)≥rδ(r)Cλx, x∈Pe,r∈(, ),
whereδ(r) =max{α(r),β(r)}. Thus, the conclusion (i) follows from Lemma ..
From (H), we havef(t,rx)≥rf(t,x) forr∈(, ),t∈[, ] andx∈[, +∞). Therefore,
Whenq(x)≡c> is a constant function,Q∞= andBx(t) =c(t–
t) :=x(t). It is
evident thatBsatisfies (H) and (H). So we can obtain the following two results.
Corollary . Assume that(H)and(H)hold.If F∞> ,then
(i) there existsλ∗≥F∞ > such that BVP()withq(x)≡chas a unique positive solutionxλinPeforλ∈[,λ∗)and does not have any solution inPeforλ≥λ∗. Moreover,for anyw∈Pe,setwn=x+λAwn–(n= , , . . .),then
limn→+∞wn–xλ= ;
(ii) xλis nondecreasing with respect toλforλ∈[,λ∗);
(iii) xλis continuous with respect toλforλ∈[,λ∗);
(iv) limλ→+xλ–x= andlimλ→λ∗–xλ= +∞.
Corollary . Assume that(H)and(H)hold.If F∞= ,then
(i) for anyλ∈[, +∞),BVP()withq(x)≡chas a unique positive solutionxλinPe, moreover,for anyw∈Pe,setwn=x+λAwn–(n= , , . . .),then
limn→+∞wn–xλ= ;
(ii) xλis nondecreasing inλforλ∈[, +∞);
(iii) xλis continuous with respect toλforλ∈[, +∞);
(iv) limλ→+xλ–x= andlimλ→+∞xλ= +∞.
Corollary . Assume that(H)and(H)hold.Then the conclusions(i), (ii), (iii),and
(iv)in Corollary.hold.
Finally, we give two concrete examples to illustrate those results in the section.
Example In BVP (), let
f(t,x) =
t
x, ≤x≤,
t
(x+
√
x), x> and q(x) =
x, ≤x≤,
, x> ,
it is obvious that the conditions (H) and (H) are satisfied. For anyr∈(, ),
as≤x≤, we havef(t,rx) = trx=rf(t,x); asx> and <rx≤, we havef(t,rx) =tr
x≥
rt
( x+
√
x)≥tr
(x+
√
x) =rf(t,x); asx> andrx> , we havef(t,rx) =t(rx+√rx)≥tr(x+√
r
√
x)≥rf(t,x),
that is,f(t,rx)≥rf(t,x) forx∈[, +∞) andt∈[, ]. Similarly, we can obtain q(rx)≥
rq(x) forx∈[, +∞). Therefore, the conditions (H) and (H) are satisfied. Note that
F∞=lim sup
x→+∞ tmax∈[,]
f(t,x)
x =
, Q∞=lim supx→+∞ q(x)
x = .
By Theorems ., . and Remarks ., . we see that there existsλ∗ ≥ such that BVP () has a unique positive solutionxλforλ∈[,λ∗) and does not have any positive so-lution forλ≥λ∗. Moreover, for anyw∈Pe, setwn(t) =λAwn–(t) +Bwn–(t) (n= , , . . .),
Figure 1 Solutions for BVP (1) withf(t,x) =t(1–1+xt)xandq(x) = 1
Example In BVP (), letf(t,x) =t(–+tx)x,q(x) = ,t∈[, ],x∈[, +∞), it is easy to see
that (H) holds. For anyr∈(, ), we have
f(rx) = t( –t)rx +rx ≥r·
t( –t)x
+x ≥rf(t,x), x∈[, +∞).
So, (H) holds. Note that
F∞=lim sup
x→+∞
max
t∈[,]
f(t,x)
x = ,
by Corollary . we find that BVP () has a unique positive solutionxλforλ≥. Moreover, for anyw∈Pe, setwn(t) = t–t+λAwn–(t) (n= , , . . .), thenlimn→∞wn–xλ= , and such a solutionxλ(t) satisfies the properties (ii), (iii), and (iv) in Corollary . with
x(t) =t–t.
In this example, by using Wolfram Mathematica ., we can plot the graphs of solutions
xλ(t) for BVP () withλ= , , , , , , as the Figure shows.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors participated in drafting, revising and commenting on the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments. This research was supported by the NNSF of China (11361047), the University Natural Science Research Develop Foundation of Shanxi Province of China (20111021, 2013156), Research Project Supported by Shanxi Scholarship Council of China (2013-102) and the Science Foundation of Qinghai Province of China (2012-Z-910).
Received: 7 October 2013 Accepted: 26 March 2014 Published:09 Apr 2014
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