R E S E A R C H
Open Access
Bifurcation from interval at infinity for
discrete eigenvalue problems which are not
linearizable
Ruyun Ma
*and Yanqiong Lu
*Correspondence:
[email protected] Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P.R. China
Abstract
In this paper, we are concerned with the bifurcation from infinity for a class of discrete eigenvalue problems with nonlinear boundary conditions which are not linearizable and give a description of the behavior of the bifurcation components.
MSC: 34B10; 34B15
Keywords: difference equation; nonlinear boundary condition; bifurcation from interval; positive solutions
1 Introduction
For over a decade, there has been significant interest in positive solutions and multiple positive solutions for boundary value problems for finite difference equations; see, for ex-ample, [–]. Much of this interest has been spurred on by the applicability of the topo-logical method such as the upper and lower solutions technique [], a number of new fixed point theorems and multiple fixed point theorems [–] as applied to certain dis-crete boundary value problems. Quite recently, Rodríguez [] and Ma [] have given a topological proof and used a bifurcation theorem to study the structure of positive solu-tions of a difference equation. In this paper, we demonstrate a bifurcation technique that takes advantage of dealing with the discrete eigenvalue problem with nonlinear boundary conditions
–p(k– )y(k– )+q(k)y(k) =λa(k)fy(k), k∈I,
–y() +αgy()= , y(N) +βgy(N+ )= ,
(.)
which has different asymptotic linearizations at infinity. HereI={, , . . . ,N},is a for-ward difference operator withy(k) =y(k+ ) –y(k),α,β≥ are constants,λ> is a parameter, the functionsp:{, , . . . ,N} →(,∞),q,a:I→[,∞) witha(k) > onk∈I
and functionsf,gsatisfyf ∈C([,∞)) andg∈C([,∞)).
Since problem (.) has different linearizations at infinity, the standard global bifurcation results [] are not immediately applicable. However, Schmitt [] and Peitgen and Schmitt [] obtained a theorem on bifurcation from intervals at infinity. We can use this theorem to discuss the bifurcation from infinity for problem (.) and obtain further information on the location and behavior of the bifurcating sets of solutions.
We will make the following assumptions:
(H) p:{, , . . . ,N} →(,∞),q,a:I→[,∞)witha(k) > onk∈I,q≥andq≡. (H) f ∈C([,∞))and there exist constantsf
∞,f∞∈(,∞)and functions
h,h∈C([,∞))such that
f∞=lim inf
s→∞
f(s)
s , f
∞=lim sup s→∞
f(s)
s , (.)
f∞s+h(s)≤f(s)≤f∞s+h(s), s∈[,∞) (.)
with
hj(s) =o
|s| ass→ ∞,j= , .
(H) g∈C([,∞))and there exist constantsg
∞,g∞∈(,∞)and functions γ,γ∈C([,∞))such that
g∞=lim infs→∞ g(s)
s , g
∞=lim sup s→∞
g(s)
s ,
g∞s+γ(s)≤g(s)≤g∞s+γ(s), s∈[,∞) (.)
with
γj(s) =o
|s| ass→ ∞,j= , .
LetˆI={, , . . . ,N+ }andX={y|y:ˆI→R}be the space of all real-valued functions onˆI. Then it is a Banach space with the norm y =max{|y(k)||k∈ ˆI}.
Definition .([, p.], []) A solution setSof (.) is said to bifurcate from infinity in the interval [a,b] if
(i) the solutions of (.) area prioribounded inXforλ=aandλ=b. (ii) there exists{(μn,yn)} ⊂Ssuch that{μn} ⊂[a,b]and yn → ∞. Letλandμbe the first eigenvalues of
–p(k– )ϕ(k– )+q(k)ϕ(k) =λa(k)ϕ(k), k∈I,
–ϕ() +αg∞ϕ() = , ϕ(N) +βg∞ϕ(N+ ) = ,
(.)
and
–p(k– )ψ(k– )+q(k)ψ(k) =μa(k)ψ(k), k∈I,
–ψ() +αg∞ψ() = , ψ(N) +βg∞ψ(N+ ) = ,
(.)
respectively,ϕandψbe the corresponding eigenfunctions ofλandμ, respectively, and ϕ,ψ∈Xbe positive and normalized as ϕ = and ψ = .
Theorem . Assume that(H)-(H)hold.Then,for anyσ ∈(,fμ∞),the interval[fμ∞ – σ,fλ∞+σ]is a bifurcation interval from infinity of(.),and there exists no bifurcation in-terval from infinity of(.)in the set(,∞)\[μ
f∞–σ,
λ
f∞+σ].More precisely,there exists an
unbounded,closed and connected component in(,∞)×X,consisting of positive solutions of(.)and bifurcating from[μ
f∞–σ,
λ
f∞+σ]× {∞}.
Theorem . Assume that(H)-(H)hold.
(i) If
lim inf
u→∞ h(u) >
f∞
g∞lim supu→∞ γ(u), (.)
then the component obtained by Theorem.bifurcates into the regionλ< λ
f∞.
(ii) If
lim sup
u→∞ h(u) <
f∞
g∞lim infu→∞ γ(u), (.)
then the component obtained by Theorem.bifurcates into the regionλ> μ
f∞. Remark . Notice thatμ≤λ. Indeed, let (λ,ϕ) and (μ,ψ) satisfy (.) and (.), respectively. Multiplying (.) by ψ and (.) byϕ, summing fromk= tok=N and subtracting, we have that
(λ–μ) N
k=
a(k)ϕ(k)ψ(k)
= N
k=
p(k– )ψ(k– )
ϕ(k) –
p(k– )ϕ(k– )
ψ(k)
= N
k=
p(k)ψ(k+ )ϕ(k) –ψ(k)ϕ(k+ )
+p(k– )ψ(k– )ϕ(k) –ψ(k)ϕ(k– )
=p(N)ψ(N+ )ϕ(N) –ψ(N)ϕ(N+ )
+p()ψ()ϕ() –ψ()ϕ()
=αp()ϕ()ψ() +βp(N)ϕ(N+ )ψ(N+ )g∞–g∞≥.
This together withNk=a(k)ϕ(k)ψ(k) > impliesμ≤λ.
2 Bifurcation theorem and reduction to a compact operator equation
Our main tools in the proof of Theorems .-. are topological arguments [] and the global bifurcation theorem for mappings which are not necessary smooth [, ].
LetV be a real Banach space and F:R×V →V be completely continuous. Let us
consider the equation
u=F(λ,u). (.)
V forλ=a andλ=b,i.e.,there exists R> such that
u=F(a,u), u=F(b,u) for all u: u ≥R.
Furthermore,assume that
dI–F(a,·),BR(),
=dI–F(b,·),BR(),
for R> large.Then there exists a closed connected setC of solutions of(.)that is un-bounded in[a,b]×V,and either
(i) Cis unbounded inλdirection,or else
(ii) there exists an interval[c,d]such that(a,b)∩(c,d) =∅andCbifurcates from infinity in[c,d]×V.
To establish Theorem ., we begin with the reduction of (.) to a suitable equation for a compact operator and give some preliminary results.
Letuα¯(k),vβ¯(k) be the solutions of the initial value problems
–p(k– )uα¯(k– )
+q(k)uα¯(k) = fork∈I,
uα¯() = , uα¯() =α,¯
and
–p(k– )vβ¯(k– )
+q(k)vβ¯(k) = fork∈I,
vβ¯(N+ ) = , vβ¯(N) = –β,¯
respectively, hereα,¯ β¯∈[,∞). It is easy to compute and show that
(i) uα¯(k) = +α¯
k– s=
p() p(s) +
k– s=(
k–
j=s p(j))q(s)uα¯(s) > , anduis increasing onIˆ;
(ii) vβ¯(k) = +β¯
N s=k
p(N) p(s) +
N s=k+(
N–
j=k+p(j))q(s)vβ¯(s) > , andvis decreasing onˆI.
Lemma . Let h:I→R.Then the linear boundary value problem
–p(k– )y(k– )+q(k)y(k) =h(k), k∈I,
–y() +α¯y() = , y(N) +β¯y(N+ ) =
(.)
has a solution
y(k) = N
s=
Gα¯,β¯(k,s)h(s), k∈ ˆI, (.)
where
Gα¯,β¯(k,s) = ⎧ ⎨ ⎩
uα¯(s)vβ¯(k), ≤s≤k≤N+ ,
uα¯(k)vβ¯(s), ≤k≤s≤N.
Proof It is a consequence of Atici [, Section ], so we omit it.
Letω,ω∈(,∞). Then the linear boundary value problem
–p(k– )y(k– )+q(k)y(k) = , k∈I,
–y() +α¯y() =w, y(N) +β¯y(N+ ) =w
(.)
has a solution
y(k) = ωuα¯(k) ( +β)¯ uα¯(N+ ) –uα¯(N)
+ ωvβ¯(k) ( +α)¯ vβ¯() –vβ¯()
, k∈ ˆI.
From the properties ofuα¯(k),vβ¯(k), it follows that
y(k) > , k∈ ˆI.
Define the operatorT:X→Xas follows:
T[h](k) = N
s=
Gα¯,β¯(k,s)h(s), k∈ ˆI.
By a standard argument of compact operator, it is easy to show thatTis a compact operator and it is strong positive, meaning thatTh> onˆIfor anyh∈Xwith the condition that
h≥ andh≡ onI; see [, , ]. LetR[ω,ω] :R→Xbe defined as
R[ω,ω](k) =
ωuα¯(k)
( +β)¯uα¯(N+ ) –uα¯(N)
+ ωvβ¯(k) ( +α)¯ vβ¯() –vβ¯()
, k∈ ˆI. (.)
ThenR[ω,ω] is a linear bounded function.
From Lemma ., letT∞,T∞:X→Xdenote the resolvent of the linear boundary value problems
–p(k– )y(k– )+q(k)y(k) =h(k), k∈I,
–y() +αg∞y() = , y(N) +βg∞y(N+ ) = ,
(.)
and
–p(k– )y(k– )+q(k)y(k) =h(k), k∈I,
–y() +αg∞y() = , y(N) +βg∞y(N+ ) = ,
respectively. Taking into accountα¯=αg∞(α¯=αg∞),β¯=βg∞(β¯=βg∞), one can repeat the argument of the operatorT with some obvious changes. It follows thatT∞,T∞ are linear mappings ofXcompactly intoXand they are strong positive.
R[ω,ω] with some minor changes, we have thatR∞,R∞:R→Xis a linear, bounded mapping and
R∞[ω
,ω](k) =
ωuαg∞(k)
( +βg∞)uαg∞(N+ ) –uαg∞(N)
+ ωvβg∞(k)
( +αg∞)vβg∞() –vβg∞() ,
R∞[ω,ω](k) = ωuαg∞(k)
( +βg∞)uαg∞(N+ ) –uαg∞(N)
+ ωvβg∞(k)
( +αg∞)vβg∞() –vβg∞() .
Lemma . Let(H)-(H)hold.If[α,β]is a bifurcation interval from infinity of the set
of nonnegative solutions of(.),then we have(α,β)⊃[μ
f∞,
λ
f∞].Moreover,there exist
con-stants> small enough and M> large enough such that any nonnegative solution u of
(.)is positive onI wheneverˆ dist(λ, [μ
f∞,
λ
f∞]) <and u ≥M.
Proof Let (μj,yj) be a nonnegative solution of (.) withλ=μj∈[α,β] such that
yj → ∞, and μj→ ˆμ asj→ ∞. (.)
If
wj=
yj
yj
, (.)
then we have
wj=μjT∞
af(yj)
yj
+R∞
τ
g∞wj–
g(yj)
yj
inX, (.)
hereτ:{y(),y(), . . . ,y(N+ )} → {y(),y(N+ )}is a linear operator and
R∞τg∞w j–
g(yj)
yj
(k) = g ∞w
j(N+ ) –g(yj(N+ ))/ yj ( +βg∞)uαg∞(N+ ) –uαg∞(N)
uαg∞(k)
+ g
∞w
j() –g(yj())/ yj ( +αg∞)vβg∞() –vβg∞()
vβg∞(k).
From conditions (.) and (.), for any> , there exist constantsd,c> such that
f(s) –f∞s≤s+d, ∀u≥,
f(s) –f∞s≥–s–d, ∀u≥,
(.)
g(s) –g∞s≤s+c, ∀u≥,
g(s) –g∞s≥–s–c, ∀u≥.
(.)
Those imply that bothf(yj)/ yj andg(yj)/ yj are bounded. By the compactness ofT∞ andR∞, it follows from (.) that there exist a functionw∈Xand a subsequence of{wj}, still denoted by{wj}, such that
By (.), it follows from (.)-(.) that
Sinceis arbitrary, it follows that
lim sup
it follows from (.) that
Obviously,wsatisfies the following boundary value problem:
–p(k– )w(k– )+q(k)w(k) =μˆa(k)ν, k∈I,
–w() +αz= , w(N) +βz= .
(.)
This combined withϕ,ψsatisfying (.) and (.) can get that
(μˆf∞–λ) N
k=
a(k)ϕ(k)w(k)
≤
N
k=
a(k)ϕ(k)μνˆ (k) –λw(k)
= N
k=
–p(k– )w(k– )ϕ(k) +p(k– )ϕ(k– )w(k)
=p(N)ϕ(N+ )w(N) –w(N+ )ϕ(N)+p()ϕ()w() –ϕ()w()
=p(N)βϕ(N+ )z–g∞w(N+ )
+p()αϕ()z–g∞w()
≤p(N)βϕ(N+ )
g∞w(N+ ) –g∞w(N+ )
+p()αϕ()
g∞w() –g∞w()
= (.)
and
ˆ
μf∞–μ N
k=
a(k)ψ(k)w(k)
≥
N
k=
a(k)ψ(k)μν(ˆ k) –μw(k)
= N
k=
–p(k– )w(k– )ψ(k) +p(k– )ψ(k– )w(k)
=p(N)ψ(N+ )w(N) –w(N+ )ψ(N)
+p()ψ()w() –ψ()w()
=p(N)βψ(N+ )z–g∞w(N+ )+p()αψ()z–g∞w()
≥p(N)βψ(N+ )g∞w(N+ ) –g∞w(N+ )
+p()αψ()g∞w() –g∞w()
= . (.)
Thus
μ
f∞ ≤ ˆμ≤
λ
f∞.
Sincew> onIˆ, (.) implies thatwj> onIˆforjlarge enough, and so isyjfrom (.).
3 Existence of a bifurcation interval from infinity
This section is devoted to studying the existence of a bifurcation interval from infinity for (.). To do this, we associate with (.) a nonlinear mapping(λ,y) : (,∞)×X→Xas follows:
(λ,y) :=y–λT∞af|y|–R∞τg∞y–g|y|. (.)
We note that a nonnegativey∈Xattains (.) if and only if(λ,y) = . In this section, we shall apply Lemma . to show that for anyσ∈(, μ
f∞), the interval [fμ∞–σ,fλ∞ +σ] is a bifurcation interval from infinity for (.) and, consequently, [fμ∞ – σ,λ
f∞+σ] is a bifurcation interval from infinity for the nonnegative solutions of (.). In fact, if [fμ∞–σ,fλ∞+σ] is a bifurcation interval from infinity for (.), then, according to Definition ., we have
(i) the solutions of (.) area prioribounded inXforλ= μ
f∞ –σ andλ=
λ
f∞+σ.
(ii) there exists a sequence{(μn,yn)} ⊂Ssuch that{μn} ⊂[fμ∞ –σ,fλ∞ +σ]and
yn → ∞.
Let{μnj}be any convergent subsequence of{(μn,yn)}, and let
lim
j→∞μnj=μ
and lim
j→∞ ynj =∞.
We claim that
μ∈
μ
f∞,
λ
f∞
and ynj> onˆIifjis large enough. (.)
Indeed, as in the proof of Lemma ., we have the same conclusion that there exist some ν,z∈Xandμsuch that
w=μT∞
a|ν|
+R∞τg∞w–|z|
. (.)
Since
f∞w≤ |ν| ≤f∞w and g∞w≤ |z| ≤g∞w,
it follows from the strong positivity ofT∞and the positivity ofR∞that
μT∞[af∞w]≤w≤μT∞
af∞w
+R∞τg∞–g∞w
. (.)
This together with the strong positivity ofT∞implies that
w> onˆI. (.)
By using (.) and (.) with obvious changes, it follows that
μ
f∞ ≤μ ≤ λ
From (.), it follows thatwj> onˆIforjlarge enough and so isyjfrom (.). Therefore [fμ∞–σ,fλ∞ +σ] is actually an interval of bifurcation from infinity for (.).
In what follows, we shall apply Lemma . to show that [μ
f∞–σ,
λ
f∞+σ] is a bifurcation interval from infinity for (.), two lemmas on the nonexistence of solutions will be first shown. Letχ: (,∞)×X→Xbe defined as
χ(λ,u) :=u–λT∞
af|u|–χ(λ)R∞τg∞u–g|u|. (.)
Hereχ: [,fμ∞]→[, ] is a smooth cut-off function such that
χ(λ) =
⎧ ⎨ ⎩
nearλ= ,
nearλ= μ
f∞.
(.)
Lemma . Let(H)-(H)hold and⊂R+be a compact interval with∩[μ
f∞,
λ
f∞] =∅.
Then there exists a constant r> such that
χ(λ,y)= , λ∈,y∈X: y ≥r. (.)
Proof Assume on the contrary that there existλj≥,yj∈Xandλ∈such that
χ(λj,yj) = , λj→λ, yj → ∞asj→ ∞.
The same argument as in the proof of Lemma . gives a contradiction thatfμ∞ ≤λ≤fλ∞.
This is a contradiction. The proof of Lemma . is complete.
Lemma . Let(H)-(H)hold.Then for anyλ> λ
f∞ fixed,there exists a constant r>
such that
(λ,y)=tϕ, t∈[, ],y∈X: y ≥r. (.)
Proof Assume on the contrary that there existμ∈(fλ∞,∞),t,tj∈[, ], andyj∈Xcan be taken such that
(μ,yj) =tjϕ, tj→t, yj → ∞asj→ ∞.
Using the same argument as in the proof of Lemma ., we can obtain a subsequence of
{yj}, still denoted by{yj}, which may satisfy thatyj> onIˆfor allj> . It follows that
yj=μT∞
af(yj)
+R∞τg∞yj–g(yj)
+tjϕ,
tj→t∈[, ], yj → ∞asj→ ∞. (.)
Thus
–p(k– )yj(k– )
+q(k)yj(k) =μa(k)f(yj) +tjλϕ, k∈I,
–yj() +αg
yj()
= , yj(N) +βg
yj(N+ )
= .
Moreover, it follows fromϕsatisfies (.) and (.) that
Now use again for (.) the same procedure as in the proof of Lemma ., then we see that some subsequence of{yj/ yj }, still denoted by{yj/ yj }, tends to a positive function
The proof of Lemma . is complete.
Lemma . Letλ–
μ is an integer.Assume that
for any n large enough,
degλ–n,·,Brn,
= , (.)
degλ+n,·,Brn,
= . (.)
Proof First we show assertion (.). From Lemma ., there existsrn> such thatrn→
∞asn→ ∞satisfying
χ(λ,y)= , ∀λ∈
,λ–n,∀y∈X: y =rn.
Sinceχ() = andχ(λ–n) = fornlarge enough from (.), by the homotopy invariance and normalization of the topology degree, it follows that for anynlarge enough,
degλ–n,·,Brn,
=degχ
λ–n,·,Brn,
=degχ(,·),Brn,
=deg(IX,Brn, ) = .
Next, we show assertion (.). We may derive from Lemma . that
λ+n,y=tϕ, ∀t∈[, ],∀y∈X: y ≥rn.
So for anynlarge enough, by the homotopy invariance, it follows that
degλ+n,·,Brn,
=degλ+n,·–ϕ,Brn,
= .
Proof of Theorem. For any fixedn∈Nwith μ
f∞ –
n> , setαn=
μ
f∞ – n,βn=
λ
f∞ + n. It is easy to verify that for any fixednlarge enough, there existsrnsuch thatrn→ ∞as
n→ ∞satisfying that for anyr≥rn, it follows from Lemmas .-. that all conditions of Lemma . are satisfied. So there exists a closed connected componentCnof solutions (.) such thatCnis unbounded in [αn,βn]×Xand either
(i) Cnis unbounded inλdirection, or
(ii) there exists an interval[c,d]such that(αn,βn)∩(c,d) =∅andCnbifurcates from
infinity in[c,d]×X.
By Lemma ., the case (ii) cannot occur. Thus Cn is unbounded bifurcated from
[αn,βn]× {∞} in R× X. Furthermore, set σ = n for n large enough, we have from Lemma . that for any closed interval I ⊂ [fμ∞ – σ,fλ∞ +σ]\[fμ∞,fλ∞], if y ∈ {y ∈
X|(λ,y) is a solution of (.),λ∈I}, then y → ∞in X is impossible. SoC/σ must be
bifurcated from [μ
f∞ –σ,
λ
f∞+σ]× {∞}.
Next, we are devoted to the proof of Theorem ., which characterized the bifurcation components of (.).
Proof of Theorem . Under condition (.), assume to the contrary that there exists a positive solutionyjof (.) withλ=λj≥fλ∞, and
Ifwj= yj
yj , then the same argument as in the proof of Lemma . shows the existence of a
positive functionw∈Xsuch that a subsequence of{wj}, still denoted bywj, tends tow inX. It follows that for anyjlarge enough, we have
wj(k) >
On the other hand, we have
These two assertions combined, we obtain that for anyj≥j,
(λ–λjf∞) N
k=
a(k)yj(k)ϕ(k)
>h∗–
f∞ N
k=
q(k)ϕ(k)
+
λ(h∗–)g∞
f∞λ –
γ∗+αp()ϕ() +βp(N)ϕ(N+ )
> .
On the right-hand side, we see from (.) that
lim
→+
λ(h∗–)g∞
f∞λ
–γ∗+=h∗g ∞
f∞ –γ ∗ > .
This means that for anyjlarge enough,
(λ–λjf∞) N
k=
a(k)yj(k)ϕ(k) > ,
which contradicts the assumptionλj≥fλ∞. Case (.) has been proved. Case (.) can be
also verified by the same arguments, and the proof of Theorem . is complete.
Competing interests
The authors declare that they have no competing interests regarding the publication of this paper.
Authors’ contributions
RM completed the main study, carried out the results of this article and YL drafted the manuscript, checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11361054, No. 11201378), SRFDP (No. 20126203110004), Gansu provincial National Science Foundation of China (No. 1208RJZA258).
Received: 3 October 2013 Accepted: 4 April 2014 Published:06 May 2014 References
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