R E S E A R C H
Open Access
On some properties of positive solutions
for a third-order three-point boundary value
problem with a parameter
Chengbo Zhai
1*, Li Zhao
1, Shunyong Li
1and HR Marasi
2*Correspondence:
1School of Mathematical Sciences,
Shanxi University, Taiyuan, Shanxi 030006, P.R. China
Full list of author information is available at the end of the article
Abstract
In this article, we discuss some properties of positive solutions for a third-order differential equation with three-point boundary conditions and a positive parameter. By using recent fixed point theory, we establish the existence and uniqueness of positive solutions for any given parameter, and we show that the positive solution is continuous, increasing with respect to the parameter. Moreover, we give some properties of limits for positive solutions. An example is provided to demonstrate the main result.
MSC: 34B10; 34B18
Keywords: third-order differential equation; three-point boundary value problem; fixed point theory; properties of positive solutions
1 Introduction
This article is concerned with some properties of positive solutions for the following third-order differential equation with three-point boundary conditions and a positive parame-ter:
u(t) +λft,u(t)= , t∈(, ), (.) u() =αu(), u() =βu(η), u() = , (.)
where <η< ,α,β> ,λis a positive parameter,f : [, ]×[, +∞)→[, +∞) is con-tinuous.
Third-order differential equations with boundary conditions are models for the deflec-tion of a curved beam having a constant or varying cross secdeflec-tion, three-layer beams, electromagnetic waves, gravity-driven flows (see []), and therefore have many important applications in some areas of applied mathematics and physics. Recently, more general problems about third-order differential equations with boundary conditions have been studied by many authors (see [–]). These papers are concerned with the existence or uniqueness of positive solutions for third-order differential equations, and their meth-ods are mainly the Krasnoselskii fixed point theorem, the Schauder fixed-order theorem, shooting method, fixed point theorems in cones and so on.
In a recent paper [], the authors studied the following third-order boundary value prob-lem:
u(t) +λq(t)ft,u(t)= , t∈(, ), (.) u() =αu(), u() =βu(η), u() = , (.)
where <η< ,α,β> ,λis a positive parameter,q: (, )→[,∞),f : [, ]×[,∞)→ [,∞) andq(t) may be singular att= and . The main result obtained in [] is the fol-lowing theorem.
Theorem . Assume that
(i) f: [, ]×[, +∞)→[, +∞)is continuous;
(ii) q: (, )→[, +∞)is continuous, <( –s)q(s)ds< +∞and there exists t∈(η, )such thatq(t) > ;
(iii) α,β> and +α
–η+η+α<β<–η+η. Moreover,if
Bf∞ <
Af,then,for any givenλ∈(Bf∞,Af),the problem(.), (.)has at least one positive solution,where
f= lim
x→+sup maxt∈[,] f(t,x)
x , f∞=xlim→∞inf mint∈[η,]
f(t,x) x .
The method used there is the Krasnoselskii fixed point theorem in a cone. From Theo-rem ., we can see that the positive solutions of the problem (.), (.) is related to the parameterλ. When the parameterλbelongs to a finite interval, the problem (.), (.) has a positive solution. But this article does not consider the uniqueness of positive so-lutions, also not consider the relation between positive solutions and the parameter. So, in this paper, we will use different methods to study the problem (.), (.). Under some new conditions, we study the existence and uniqueness of positive solutions, and discuss the relation between positive solutions and the parameter. For anyλ> , we cannot only get the existence and uniqueness of positive solutions, but also be able to get the relation between positive solutions and the parameter. In addition, the range of the parameterλ
that we discuss is bigger, that is,λ∈(, +∞).
2 Preliminaries
Lemma .([]) Assume thatβ= +α
–η+η+α,y: (, )→[, +∞)is continuous with <
( –s)y(s)ds< +∞,then the following boundary value problem:
u(t) +y(t) = , t∈(, ), (.) u() =αu(), u() =βu(η), u() = , (.)
has a unique solution,
u(t) =
G(t,s)y(s)ds+ (–t
+ t+ α)
β(–η+ η+ α) – ( + α)
G(,s)y(s)ds
– β(–t
+ t+ α)
β(–η+ η+ α) – ( + α)
where
G(t,s) =
⎧ ⎨ ⎩
t( –s), t≤s,
s(–t+ t–s), s≤t. (.)
Lemma .([]) The function G(t,s)given by(.)has the following properties: (i) ≤tG(,s)≤G(t,s)≤G(,s),(t,s)∈[, ]×[, ];
(ii) G(η,s)≤(–η+ η)G(,s),s∈[, ].
In this paper, we always assume
+ α
–η+ η+ α <β<
–η+ η. (.)
Lemma .([]) Assume(.)holds, y: (, )→[, +∞)is continuous with <( –
s)y(s)ds< +∞,then the solution u of the problem(.), (.)is nonnegative and satisfies
min
t∈[η,]u(t)≥γ u ,
where
γ =βη
(–η+ η+ α) –η( + α) + α[ –β(–η+ η)]
β(–η+ η+ α) .
For the convenience of calculation, we set
σ=β
–η+ η+ α– ( + α), σ= – βη+ βη.
Remark . From (.), we obtain
σ=β
–η+ η+ α– ( + α) > .
Moreover, byη∈(, ),
β> + α –η+ η+ α =
+ α
– ( –η)+ α >
+ α
+ α= .
In addition,
σ= – βη+ βη= –β
η– η= –βη– η–η
= –βη– η+βη= – ηβη–η+βη.
Also, by (.),β(–η+ η) < and thus +βη> βη. So
σ> βη– ηβ
To establish our main results, we need some results of operator equations involving monotone operators in ordered Banach spaces.
Suppose that (X, · ) is a real Banach space, andXis partially ordered by a coneK⊂X,θ
denotes the zero element ofX.Kis called normal if there is a constantN> , such that, for x,y∈X,θ≤x≤yimplies x ≤N y . Forx,y∈X, the notationx∼ymeans that there are
λ,μ> such thatλx≤y≤μx. Forh>θ (i.e. h≥θandh=θ), defineKh={x∈E|x∼h}. Obviously,Kh⊆K. An operatorA:X→Xis increasing ifx≤yimpliesAx≤Ay.
Theorem .(see Theorem . in []) K is a normal cone in a real Banach space X,h>θ.
A:K→K is an increasing operator.In addition, (i) there existsh∈Khsuch thatAh∈Kh;
(ii) forx∈Kandt∈(, ),there isϕ(t)∈(t, )such thatA(tx)≥ϕ(t)Ax. Then:
() operator equationAx=xhas a unique solutionx∗inKh;
() for any pointx∈Kh,making the sequencexn=Axn–,n= , , . . .,we getxn→x∗as
n→ ∞.
Theorem .(see Theorem . in []) Suppose that all the conditions of Theorem. hold. xλ(λ> )is the unique solution of operator equation Ax=λx.Then the following
conclusions hold:
(i) xλis strictly decreasing inλ,that is, <λ<λimpliesxλ>xλ; (ii) if there isγ ∈(, )such thatϕ(t)≥tγ fort∈(, ),thenx
λis continuous inλ,that
is,λ→λ(λ> )implies xλ–xλ →; (iii) limλ→+∞ xλ = ,limλ→+ xλ = +∞.
3 Main results
First, we consider a special function
h(t) =
G(t,s)ds+ (–t
+ t+ α)
β(–η+ η+ α) – ( + α)
G(,s)ds
– β(–t
+ t+ α)
β(–η+ η+ α) – ( + α)
G(η,s)ds, t∈[, ], (.)
whereG(t,s) is given in (.). Then we have
h(t) =
t
–t+ t–ss ds+
t
( –s)tds+(–t
+ t+ α)
σ
( –s)s ds
–β(–t
+ t+ α)
σ
η
–η+ η–ss ds+
η
( –s)ηds
=
( – βη+ βη) σ
α+ t–t+ t– t
=
σ σ
α+ t–t+ t– t
=
σ σ
α+σ
σ
t+
–σ
σ
t– t
Note thatσ,σ> , α+ t–t= α+t( –t) > and t– t=t( – t)≥, we have
h(t)≥,t∈[, ]. Further,
h(t) =
σ σ
–σ
σ
t+ t– t
≥, t∈[, ].
Consequently, we have the following conclusions.
Lemma . h(t)is continuous,nonnegative and increasing on t∈[, ].Moreover,
h(t)≥h() =ασ σ
> ;
h(t)≤h() =
σ σ
(α+ ) +
=ασ σ
+ σ σ
+
, t∈[, ].
And thus,h() >h().
The purpose of this section is to apply Theorems ., . to study the problem (.), (.), and for any parameterλ> , we will establish some new results on the existence and uniqueness of positive solutions. Further, we will show that the positive solution with re-spect toλhas some good properties such as the positive solution is continuous, increasing with respect to the parameter. Moreover, we give some properties of limits for positive so-lutions. It should be pointed out that the method used here is new to the literature and so are the existence and uniqueness results to the third-order differential equations.
We will discuss the problem (.), (.) in the Banach space C[, ] ={x: [, ]→ Ris continuous}, and the norm given as x =sup{|x(t)|:t∈[, ]}. The cone considered here isK={x∈C[, ]|x(t)≥,t∈[, ]}, a standard, normal cone inC[, ]. This space can be equipped with a partial order given byx,y∈C[, ],x≤y⇔x(t)≤y(t) fort∈[, ].
Theorem . Assume that:
(H) f : [, ]×[, +∞)→[, +∞)is continuous withf(t,ασσ) > ,t∈[, ]; (H) f(t,x)is increasing inxfor eacht∈[, ];
(H) for anyr∈(, ),there exists ϕ(r)∈(r, ) such thatf(t,rx)≥ϕ(r)f(t,x),∀t∈[, ],
x∈[, +∞).
Then the following conclusions hold:
() for any givenλ> ,the problem(.), (.)has a unique positive solutionu∗λinKh, whereh(t)is given as in(.).In addition,for any valueu∈Kh,using the sequence
un(t) =λ
G(t,s)fs,un–(s)
ds+λ(–t
+ t+ α)
σ
G(,s)f(s,un–(s))ds
–λβ(–t
+ t+ α)
σ
G(η,s)fs,un–(s)
ds,
we haveun(t)→u∗λ(t)asn→+∞,whereG(t,s)is given in(.);
() u∗λis strictly increasing inλ,that is, <λ<λimpliesu∗λ<u
∗
λ;
() if there existγ ∈(, )and a nonnegative functionψsuch thatϕ(t) =tγ[ +ψ(t)]for
(ii) for anyt∈[η, ], we have
three integrations of cases (i), (ii) are nonnegative. Therefore, from (H), (H) and the
above cases (i), (ii), we know thatT:K→Kis increasing. In the following, we check that T satisfies other conditions of Theorem ..
It follows from the above two cases (i), (ii) and (H) that
Evidently,u∗λis a unique positive solution of the problem (.), (.) for givenλ> . Further,
by Theorem .(i), it is easy to check thatu∗λis strictly increasing inλ, that is,u∗λ<u
∗
λfor <λ<λ; by Theorem .(iii), one haslimλ→+∞ u∗λ = +∞, limλ→+ u∗λ = . More-over, ifϕ(t) =tγ[ +ψ(t)], thenϕ(t)≥tγ fort∈(, ), Theorem .(ii) means thatu∗
λis
continuous inλ, that is,λ→λ(λ> ) implies u∗λ–u∗λ →.
In addition, letTλ=λT, thenTλ also satisfies all the conditions of Theorem . and
thus, for any initial value u∈Kh, constructing successively the sequenceun=Tλun–,
n= , , . . . , we haveun→u∗λasn→ ∞. That is,
un(t) =λ
G(t,s)fs,un–(s)
ds+λ–t
+ t+ α
σ
G(,s)fs,un–(s)
ds
–λβ(–t
+ t+ α)
σ
G(η,s)fs,un–(s)
ds, t∈[, ],n= , , . . . ,
andun(t)→u∗λasn→+∞.
Corollary . Assume that(H)-(H)hold.Then the following third-order boundary value
problem
u(t) +ft,u(t)= , t∈(, ),
u() =αu(), u() =βu(η), u() = ,
has a unique positive solution u∗in Kh,where h(t)is given as in(.).In addition,for any value u∈Kh,using the sequence
un(t) =
G(t,s)fs,un–(s)
ds+–t
+ t+ α
σ
G(,s)f(s,un–(s))ds
–β(–t
+ t+ α)
σ
G(η,s)fs,un–(s)
ds,
we have un(t)→u∗(t)as n→+∞,where G(t,s)is given in(.).
4 An example
We consider the following third-order boundary value problem with a parameter
u(t) +λ
u(t) +
+t= , t∈(, ), (.)
u() = u
(), u() =
u
, u() = . (.)
In this example, we know thatη=,α=,β=,σ=,σ= and
f(t,x) =
x +
+t.
After a simple computation, we get
=
+ α
–η+ η+ α <β<
–η+ η=
and (.) is satisfied. Moreover,
So all the conditions of Theorem . are satisfied. From Theorem ., we can claim that: (a) forλ> , the problem (.), (.) has a unique positive solutionu∗λinKh. Moreover,
Remark . From this example, we can see that there are many functions which satisfy the conditions of Theorem ..
Acknowledgements
The research was supported by the Youth Science Foundation of China (11201272) and Shanxi Province Science Foundation (2015011005), 131 Talents Project of Shanxi Province (2015). The third author was supported by Shanxi Scholarship Council of China (2016-009).
Competing interests
The authors declare that they have no competing interests.
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
1School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, P.R. China.2University of Tabriz, Tabriz, Iran.
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