On Bifurcation from Infinity and Multipoint Boundary Value Problems ()
1. Introduction
In this paper, we generalize and improve a result of Coyle et al. [1] about the bifurcation from infinity after stating in the line of Nussbaum [2] , Schmitt [3] , etc., a type of nonlinear Krein-Rutman theorem for a class of positively 
-homogeneous, compact and continuous operators in Banach spaces leaving invariant cones.
Our method is motivated by the maximum principle of Degla [4] and a result on the principal eigenvalue of multi-point Boundary Value Problems (BVP’s) of Degla [5] which allow the use of cone theoretic arguments and of the well-known general result on bifurcation from infinity; see Coyle [1] , Mawhin [6] and Rabinowitz [7] .
Furthermore, in our abstract setting, the nonlinear Krein-Rutman Theorem resets an important result on the simplicity of positive eigenvalues [8] by avoiding some inconclusive argument [8] (page 3086, lines 29-37) also misused in [9] (page 550, lines 15-27). However the gap in their arguments under their assumptions, remains an open question.
2. Preliminary Definitions and Notations
We say that a nonempty subset 
of a Banach space 
is a cone if it is closed and 1)
2) 
and 3) 
In other words, the cones considered here are closed convex cones with vertex at 0.
A cone 
of a Banach space 
induces a partial ordering on 
by the relation
and it follows that
Therefore 
is called an ordered Banach space with 
as the positive cone of
. Note that we write 
when 
and
; i.e.,
A cone 
of a Banach space 
is said to be generating if
, and total if
.
Given a Banach space 
with dual
, if a cone 
of 
is generating, then the set defined by
is a cone of 
called the dual cone of
.
The positive cone 
of an ordered Banach space 
is said to be normal if there exists a positive constant 
such that
When
, such an ordered Banach space is said to be monotone.
Let 
be an ordered Banach space. Then
● A linear operator 
is said to be positive if
and strongly positive if
● 
● An arbitrary operator 
is said to be increasing if
strictly increasing if
and strongly increasing if
We shall say that 
is increasing on 
if
Observe that if an operator 
is increasing on 
and satisfies
, then it leaves invariant
.
Besides in our applications, we shall use the following terminology based on Degla [4] [5] , Elias [10] and Coppel [11] . Given fixed positive integers 
and 
such that
, and real numbers
, we shall denote by 
the Levin’s polynomial defined by 
and we shall deal with disconjugate 
order differential operators on 
of the form
where the coefficients 
are given continuous functions on
, that is, an 
-order differential linear operator 
such that every nontrivial solution of the differential equation 
has less than 
zeros counting their multiplicities.
Recall that an 
-order differential linear operator
;
with
, is disconjugate on 
if and only if 
has a Polya factorization; that is, there exist 
smooth positive functions
, 
, such that
where 
cf. [11] .
Furthermore 
will denote the Green function associated to the Boundary Value Problems (in short BVP’s)
and besides, given
, we shall adopt the notation 
and
As in [5] , we shall also consider the Banach space
equipped with the norm
and ordered by the cone
Now we are ready to state a variant of nonlinear Krein-Rutman theorems.
Proposition 1.1. Let 
be a real Banach space, 
a nontrivial cone in 
and assume that 
is a positively 1-homogeneous, compact and continuous operator.
a) If 
is increasing on 
and there exist a positive vector
, a positive real number 
and a positive integer
, such that
(i)
then 
has a positive eigenvalue 
with a positive eigenvector.
In case that 
is linear, its spectral radius 
is such a positive eigenvalue and satisfies
b) If 
has a nonempty interior 
and 
with the property
(ii)
then 
has a unique positive eigenvalue and a unique positive normalized eigenvector.
In case that 
is linear, this positive eigenvalue coincides with the spectral radius 
of
, is algebraically simple and has the following variational characterization:
Remark 1.1. For a linear operator
, the condition (ii) of b) is equivalent to
(iii)
Furthermore the conclusion of b) can be heuristically motivated by the application of the Krein-Rutman theorem to the quotient space
.
Remark 1.2. The above theorem is readily applicable to any positively 
-homogeneous, compact and continuous operators that are strongly positive on the cone of an ordered Banach space.
Remark 1.3. The proof of Theorem 2 of [8] does not fully hold but is valid for strongly increasing operators. The reason is that its conclusion (2.9) is not correct and should be read 
which does not contradict the inequality (2.10) therein; that is
.
The fact is that for instance in the Banach space 
ordered by the cone
we have
and so with 
it is clear that
Likewise the inequality “
” of the paragraph 4 of the proof of theorem 4.8 of [9] does not contradict the definition of “
” as can be seen with 
and 
for which 
by simply considering again the ordered Banach space
.
Therefore we are led to raise the following Open Question: Does there exist a strictly increasing and positively 1-homogeneous compact operator of which positive eigenvalue is not simple?
Remark 1.4. For a positive compact linear operator
, the condition (i) of Part a) of Proposition 1.1 is equivalent to
(iv)
The following example illustrates Proposition 1.1.
Example 1.5. Consider the system of boundary value problems:
with 
as a real parameter and
where the 
are assumed to be nonnegative continuous functions on 
such that on the one hand 
and 
have a common support
, and on the other hand 
and 
have a common support 
such that
; i.e.
, and where the unknown vector-valued function 
is clearly searched in
with zero Dirichlet boundary condition.
Then this system has a unique normalized solution with positive component functions on the interval 
corresponding to a unique positive value of the parameter
.
Justification. We shall make use of Proposition 1.1 for the sake of illustration that may motivate other interesting works. Indeed it is immediately seen that for nontrivial solutions, we have
, and the system of BVPs
is equivalent to the integral equation
with
Moreover by considering the special space of continuous vector-valued functions
endowed with the norm 
defined for any 
by
which contains all possible solutions of our eigenvalue problem, and by letting
we see that 
is a normal ordered Banach space. Furthermore the non-zero linear operator
; 
defined by
is compact and satisfies
with
by the strong classical maximum principle.
The conclusion follows. 
3. Bifurcation from Infinity of Conjugate Multipoint BVPs
This part can be considered as a more elaborated application of the main result of the previous section.
In the sequel we shall make use of the notations mentioned in Section 2. According to this,
equipped with the norm
and ordered by the cone
is an ordered Banach space.
Then the following theorem holds.
Theorem 2.1. Let 
satisfy
Moreover let 
be a continuous function such that
Then there exists a continuum 
of positive solutions of the BVPs
(El)
and 
such that 1) For each
, there is a corresponding subcontinuum 
contained in
which connects 
and
.
2) If 
with 
as
, then
(in fact in
) to the unique normalized nontrivial solution of
where
.
Remark 2.2. An analogue version of Theorem 2.1 can be stated with 
satisfying the following property:
Remark 2.3. It is worth observing that Theorem 2.1 is a generalized version of a result of [1] since this Theorem 2.1 concerns multipoint conjugate boundary conditions and deals with a function 
that may vanish on subintervals of 
For a proof of this Theorem 2.1, we need the lemma below which can also be deduced from Proposition 1.1.
Lemma A. [5]
If 
satisfies 
on a set of positive measure and 
for a.e.
, then the eigenvalue BVPs
(Ql)
has a positive eigenvalue 
which is simple with an eigenfunction 
such that
.
Now we recall a standard result on bifurcation theory which together with Lemma A will prove our Theorem 2.1 which is about a bifurcation from infinity for conjugate multipoint BVPs.
Lemma B. [1] [6] [7] [12]
Let 
be a real Banach space with norm
. Assume that
is such that for each
, 
is a compact linear operator, and for each
, 
is differentiable on
. Let 
be a cone with nonempty interior,
.
Moreover suppose that
is a completely continuous map satisfying
and consider the equation
If
, 
with
,
then there exist 
and a continuum 
such that for any
, there exists a corresponding subcontinuum 
contained in
which connects 
and
. Moreover if 
with 
as
; then
Proof of Theorem 2.1. First note that all possible solutions of the BVP’s (El) lie in 
since they are of the form
; where 
is continuous and 
is the Green function of the BVPs
with the property that 
is bounded on
.
Now (El) is equivalent, by the properties of the Green function
, to the following equation:
where 
and 
Moreover as seen in the proof of Lemma A [5] , the operator 
is a non-zero positive compact linear operator satisfying
while 
is completely continuous and satisfies
by the assumptions on
. Indeed:
1) We show that 
is completely continuous.
Step 1. 
maps bounded subsets into compact subsets. Let 
be a sequence of elements of 
of which norms are bounded, say by a constant real number
. Let
Then, on one hand,
and on the other hand, we have for all
:
Hence for all 
where 
is the modulus of continuity of
. Moreover as a continuous function, 
is uniformly continuous on the compact set
, and so
. Therefore the Ascoli theorem implies the existence of a subsequence 
of 
such that
for some
.
By applying again Ascoli theorem we see that there exists a subsequence of
, still denoted by
, such that
uniformly on 
for a suitable
. Indeed, to realize this claim, let 
and choose 
satisfying
where 
is a finite upper-bound of the ratio
.
Consider now on the compact 
the function 
extending continuously the quotient function
. The function 
is uniformly continuous on 
and so there exists 
for which
Therefore, denoting by 
the continuous extension of 
to
, i.e.,
we have for all 
satisfying
:
This shows that the sequence of functions 
is equicontinuous on 
and proves the claim since the functions 
are also uniformly bounded as
Now from the former convergence; i.e.
, we deduce that for all
,
Thus 
for 
and it follows that 
converges to 
in
.
Step 2. If 
converges to some 
in
, then the Lebesgue dominated convergence theorem implies that 
converges to 
for each
.
It follows from the combination of Steps 
that 
is completely continuous; i.e., 
maps bounded sets into compact sets and is continuous.
2) We show that 
as
.
To this end, let 
be arbitrary. Then by assumption there exists 
such that
where 
is an upper-bound of 
on
. By setting
, we have at once
Therefore for every
, we have on one hand
and on the other hand
Thus
Now by putting
we see clearly that
That is
The result follows by applying Lemma B. 
Acknowledgements
The author is grateful to Professor R. Agarwal for having given him the opportunity to attend the International Conference on the Theory, Methods and Applications of Nonlinear Equations from the 17th to the 21st December 2012.
The author would like also to thank the Abdus-Salam International Centre for Theoretical Physics (ICTP, Trieste, Italy) for its hospitality during his first visit as a Regular Associate.