R E S E A R C H
Open Access
Nontrivial solutions for a boundary value
problem with integral boundary conditions
Bingmei Liu
1*, Junling Li
1and Lishan Liu
2*Correspondence:
1College of Sciences, China
University of Mining and Technology, Xuzhou, Jiangsu 221116, China
Full list of author information is available at the end of the article
Abstract
This paper concerns the existence of nontrivial solutions for a boundary value problem with integral boundary conditions by topological degree theory. Here the nonlinear term is a sign-changing continuous function and may be unbounded from below.
1 Introduction
Consider the following Sturm-Liouville problem with integral boundary conditions
⎧ ⎪ ⎨ ⎪ ⎩
(Lu)(t) +h(t)f(t,u(t)) = , <t< , (cosγ)u() – (sinγ)u() =u(τ)dα(τ), (cosγ)u() + (sinγ)u() =
u(τ)dβ(τ),
(.)
where (Lu)(t) = (p˜(t)u(t))+q(t)u(t),p˜(t)∈C[, ],p˜(t) > ,q(t)∈C[, ],q(t) < ,αand β are right continuous on [, ), left continuous att= and nondecreasing on [, ] with α() =β() = ;γ,γ∈[,π/],
u(τ)dα(τ) and
u(τ)dβ(τ) denote the Riemann-Stieltjes integral of u with respect to α and β, respectively. Here the nonlinear term
f : [, ]×(–∞, +∞)→(–∞, +∞) is a continuous sign-changing function andf may be unbounded from below,h: (, )→[, +∞) with <h(s)ds< +∞is continuous and is allowed to be singular att= , .
Problems with integral boundary conditions arise naturally in thermal conduction prob-lems [], semiconductor probprob-lems [], hydrodynamic probprob-lems []. Integral BCs (BCs de-notes boundary conditions) cover multi-point BCs and nonlocal BCs as special cases and have attracted great attention, see [–] and the references therein. For more informa-tion about the general theory of integral equainforma-tions and their relainforma-tion with boundary value problems, we refer to the book of Corduneanu [], Agarwal and O’Regan []. Yang [], Boucherif [], Chamberlainet al.[], Feng [], Jianget al.[] focused on the existence of positive solutions for the cases in which the nonlinear term is nonnegative. Although many papers investigated two-point and multi-point boundary value problems with sign-changing nonlinear terms, for example, [–], results for boundary value problems with integral boundary conditions when the nonlinear term is sign-changing are rarely seen ex-cept for a few special cases [, , ].
Inspired by the above papers, the aim of this paper is to establish the existence of nontriv-ial solutions to BVP (.) under weaker conditions. Our findings presented in this paper have the following new features. Firstly, the nonlinear termf of BVP (.) is allowed to
be sign-changing and unbounded from below. Secondly, the boundary conditions in BVP (.) are the Riemann-Stieltjes integral, which includes multi-point boundary conditions in BVPs as special cases. Finally, the main technique used here is the topological degree theory, the first eigenvalue and its positive eigenfunction corresponding to a linear opera-tor. This paper employs different conditions and different methods to solve the same BVP (.) as []; meanwhile, this paper generalizes the result in [] to boundary value problems with integral boundary conditions. What we obtain here is different from [–].
2 Preliminaries and lemmas
LetE=C[, ] be a Banach space with the maximum normu=max≤t≤|u(t)|foru∈E. DefineP={u∈E|u(t)≥,t∈[, ]}andBr={u∈E| u<r}. ThenPis a total cone inE,
that is,E=P–P.P∗denotes the dual cone ofP, namely,P∗={g∈E∗|g(u)≥, for allu∈ P}. LetE∗denote the dual space ofE, then by Riesz representation theorem,E∗is given by
E∗=v|vis right continuous on [, ) and is bounded variation on [, ] withv() = .
We assume that the following condition holds throughout this paper.
(H) u(t)≡is the uniqueCsolution of the linear boundary value problem –(Lu)(t) = , <t< ,
(cosγ)u() – (sinγ)u() = , (cosγ)u() + (sinγ)u() = .
Letϕ,ψ∈C([, ],R+) solve the following inhomogeneous boundary value problems, respectively:
⎧ ⎪ ⎨ ⎪ ⎩
–(Lϕ)(t) = , <t< , (cosγ)ϕ() – (sinγ)ϕ() = , (cosγ)ϕ() + (sinγ)ϕ() =
and
⎧ ⎪ ⎨ ⎪ ⎩
–(Lψ)(t) = , <t< , (cosγ)ψ() – (sinγ)ψ() = , (cosγ)ψ() + (sinγ)ψ() = .
Letκ= –ϕ(τ)dα(τ),κ=ψ(τ)dα(τ),κ=ϕ(τ)dβ(τ),κ= –ψ(τ)dβ(τ).
(H) κ> ,κ> ,k=κκ–κκ> .
Lemma .([]) If(H)and(H)hold,then BVP(.)is equivalent to
u(t) =
G(t,s)h(s)fs,u(s)ds,
where G(t,s)∈C([, ]×[, ],R+)is the Green function for(.). Define an operatorA:E→Eas follows:
(Au)(t) =
G(t,s)h(s)fs,u(s)ds, u∈E. (.)
It is easy to show that A:E→Eis a completely continuous nonlinear operator, and if
For anyu∈E, define a linear operatorK:E→Eas follows:
(Ku)(t) =
G(t,s)h(s)u(s)ds, u∈E. (.)
It is easy to show thatK:E→Eis a completely continuous nonlinear operator andK(P)⊂
Pholds. By [], the spectral radiusr(K) ofKis positive. The Krein-Rutman theorem [] asserts that there areφ∈P\ {}andω∈P∗\ {}corresponding to the first eigenvalue λ= /r(K) ofKsuch that
λKφ=φ (.)
and
λK∗ω=ω, ω() = . (.)
HereK∗:E∗→E∗is the dual operator ofKgiven by:
K∗v(s) =
s
G(t,τ)h(τ)dv(t)dτ, v∈E∗.
The representation ofK∗, the continuity ofGand the integrability ofhimply thatω∈
C[, ]. Lete(t) :=ω(t). Thene∈P\ {}, and (.) can be rewritten equivalently as
r(K)e(s) =
G(t,s)h(s)e(t)dt,
e(t)dt= . (.)
Lemma .([]) If(H)holds,then there isδ> such that P={u∈P|
u(t)e(t)dt≥ δu}is a subcone of P and K(P)⊂P.
Lemma .([]) Let E be a real Banach space and⊂E be a bounded open set with
∈.Suppose that A:¯ →E is a completely continuous operator. ()If there is y∈E
with y= such that u=Au+μy for all u∈∂andμ≥,thendeg(I–A,, ) = . ()If Au=μu for all u∈∂andμ≥,thendeg(I–A,, ) = .Heredegstands for the Leray-Schauder topological degree in E.
Lemma . Assume that(H), (H)and the following assumptions are satisfied:
(C) There existφ∈P\ {},ω∈P∗\ {}andδ> such that(.), (.)hold andKmaps
PintoP.
(C) There exists a continuous operatorH:E→Psuch that
lim
u→+∞
Hu
u = .
(C) There exist a bounded continuous operatorF:E→Eandu∈Esuch thatFu+u+
Hu∈Pfor allu∈E.
Let A=KF,then there exists R> such that
deg(I–A,BR, ) = ,
where BR={u∈E| u<R}.
Proof Choose a constantL= (δλ)–( +ζ–) +K> . From (C), for <ε<L– , there existsR> such thatu>Rimplies
Hu<εu. (.)
Now we shall show
u=KFu+μφ for anyu∈∂BRandμ≥, (.)
provided thatRis sufficiently large.
In fact, if (.) is not true, then there existu∈∂BRandμ≥ satisfying
u=KFu+μφ. (.)
Sinceφ∈P\ {},e(t)∈P\ {},φ(t)e(t)dt> . Multiply (.) bye(t) on both sides and integrate on [, ]. Then, by (C), (.), we get
u(t)e(t)dt
=
(KFu)(t)e(t)dt+μ
φ(t)e(t)dt
≥λ( +ζ)
G(t,s)h(s)u(s)dse(t)dt–
(KHu)(t)e(t)dt–
v(t)e(t)dt
=λ( +ζ)
G(t,s)h(s)u(s)e(t)ds dt
–
G(t,s)h(s)(Hu)(s)e(t)ds dt–
v(t)e(t)dt
=λ( +ζ)
G(t,s)h(s)e(t)dt
u(s)ds
–
G(t,s)h(s)e(t)dt
(Hu)(s)ds–
v(t)e(t)dt
=λ( +ζ)r(K)
e(s)u(s)ds–r(K)
(Hu)(s)e(s)ds–
v(t)e(t)dt
= ( +ζ)
u(t)e(t)dt–r(K)
(Hu)(t)e(t)dt–
v(t)e(t)dt. (.)
Thus,
u(t)e(t)dt≤ζ–
r(K)
(Hu)(t)e(t)dt+
v(t)e(t)dt
By (.),(KHu)(t)e(t)dt=r(K)(Hu)(t)e(t)dtholds. Then (.), (.) and (.) im-ply
u(t) + (KHu)(t) + (Ku)(t)e(t)dt
≤ζ–
r(K)
(Hu)(t)e(t)dt+
v(t)e(t)dt
+r(K)
(Hu)(t)e(t)dt+
(Ku)(t)e(t)dt
≤ζ–( +ζ)r(K)
(Hu)(t)e(t)dt+ζ–
v(t)e(t)dt+
(Ku)(t)e(t)dt
≤ζ–( +ζ)r(K)εu+L, (.)
whereL=ζ–
v(t)e(t)dt+
(Ku)(t)e(t)dtis a constant.
(C) showsFu+u+Hu∈Pand (C) impliesμφ=μλKϕ∈P. Then (C), (.) and Lemma . tell us that
u+KHu+Ku=KFu+μφ+KHu+Ku=K(Fu+Hu+u) +μφ∈P.
The definition ofPyields
(u+KHu+Ku)(t)e(t)dt≥δu+KHu+Ku
≥δu–δKHu–δKu. (.)
It follows from (.), (.) and (.) that
u=δ–
(u+KHu+Ku)(t)e(t)dt+KHu+Ku
≤ε(δλ)– +ζ–u+Lδ–+εK · u+Ku
=εLu+L, (.)
whereL=Ku+Lδ–is a constant.
Since <εL< , then (.) deduces that (.) holds provided thatRis sufficiently large such thatR>max{L/( –εL),R}. By (.) and Lemma ., we have
deg(I–A,BR, ) = .
3 Main results
Theorem . Assume that(H), (H)hold and the following conditions are satisfied:
(A) There exist two nonnegative functionsb(t),c(t)∈C[, ]withc(t)≡and one
con-tinuous even functionB:R→R+ such that f(t,x)≥–b(t) –c(t)B(x)for allx∈R.
Moreover,Bis nondecreasing onR+and satisfieslimx→+∞B(x) x = .
(A) lim infx→+∞f(tx,x)>λuniformly ont∈[, ].
(A) lim supx→|
f(t,x)
x |<λuniformly ont∈[, ].
Hereλis the first eigenvalue of the operator K defined by(.).
Then BVP(.)has at least one nontrivial solution.
Proof We first show that all the conditions in Lemma . are satisfied. By Lemma ., condition (C) of Lemma . is satisfied. Obviously,B:E→Pis a continuous operator. By (A), for anyε> , there isL> such that whenx>L,B(x) <εxholds. Thus, foru∈E
withu>L,B(u) <εuholds. The fact thatBis nondecreasing onR+yields (Bu)(t)≤
B(u) for anyu∈P,t∈[, ]. SinceB:R→R+is an even function, for anyu∈Eand
t∈[, ], (Bu)(t)≤B(u) holds, which impliesBu ≤B(u) foru∈E. Therefore,
Bu ≤Bu<εu, ∀u∈Ewithu>L,
that is,limu→+∞Buu= . TakeHu=cBu, for anyu∈E, wherec=maxt∈[,]c(t) > . Obviously,limu→+∞Huu= holds. ThereforeHsatisfies condition (C) in Lemma .. Take u(t)≡b=maxt∈[,]b(t) > and (Fu)(t) =f(t,u(t)) for t∈[, ],u∈E, then it follows from (A) that
Fu+u+Hu∈P for allu∈E,
which shows that condition (C) in Lemma . holds.
By (A), there existε> and a sufficiently large numberl> such that
f(t,x)≥λ( +ε)x, ∀x≥l. (.)
Combining (.) with (A), there existsb≥ such that
f(t,x)≥λ( +ε)x–b–cB(x) for allx∈R, and so
Fu≥λ( +ε)u–b–Hu for allu∈E. (.)
SinceKis a positive linear operator, from (.) we have
(KFu)(t)≥λ( +ε)(Ku)(t) –Kb– (KHu)(t), ∀t∈[, ],u∈E. So condition (C) in Lemma . is satisfied.
According to Lemma ., we derive that there exists a sufficiently large numberR> such that
deg(I–A,BR, ) = . (.)
From (A) it follows that there exist <ε< and <r<Rsuch that
Thus
(Au)(t)≤( –ε)λK|u|(t), ∀t∈[, ],u∈Ewithu ≤r. (.)
Next we will prove that
u=μAu for allu∈∂Brandμ∈[, ]. (.)
If there existu∈∂Brandμ∈[, ] such thatu=μAu. Letz(t) =|u(t)|. Thenz∈Pand by (.),z≤( –ε)λKz. Thenth iteration of this inequality shows thatz≤( –ε)nλnKnz (n= , , . . .), soz ≤( –ε)nλn
Kn · z, that is, ≤( –ε)nλnKn. This yields –ε= ( –ε)λr(K) = ( –ε)λlimn→∞ n
√
Kn ≥, which is a contradictory inequality. Hence,
(.) holds.
It follows from (.) and Lemma . that
deg(I–A,Br, ) = . (.)
By (.), (.) and the additivity of Leray-Schauder degree, we obtain
deg(I–A,BR\Br, ) =deg(I–A,BR, ) –deg(I–A,Br, ) = –.
SoAhas at least one fixed point onBR\Br, namely, BVP (.) has at least one nontrivial
solution.
Corollary . Using(A∗)instead of(A),the conclusion of Theorem.remains true.
(A∗) There exist three constantsb> ,c> andα∈(, )such that
f(x)≥–b–c|x|α for anyx∈R.
Competing interests
The authors declare that no conflict of interest exists.
Authors’ contributions
All authors participated in drafting, revising and commenting on the manuscript. All authors read and approved the final manuscript.
Author details
1College of Sciences, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China.2School of
Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China.
Acknowledgements
The first two authors were supported financially by the National Natural Science Foundation of China (11201473, 11271364) and the Fundamental Research Funds for the Central Universities (2013QNA35, 2010LKSX09, 2010QNA42). The third author was supported financially by the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.
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10.1186/1687-2770-2014-15
Cite this article as:Liu et al.:Nontrivial solutions for a boundary value problem with integral boundary conditions.
