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We generalize a result on bifurcation from infinity of high order ordinary differential equations with multi-point boundary conditions. Our abstract setting represents a variant of Nonlinear Krein-Ruthman theorems. Furthermore, an analysis of this abstract setting raises an open question motivated by some misunderstanding and inconclusive proofs about the simplicity of principal eigenvalues in some articles in the literature.

In this paper, we generalize and improve a result of Coyle et al. [

Our method is motivated by the maximum principle of Degla [

Furthermore, in our abstract setting, the nonlinear Krein-Rutman Theorem resets an important result on the simplicity of positive eigenvalues [

We say that a nonempty subset

In other words, the cones considered here are closed convex cones with vertex at 0.

A cone

and it follows that

Therefore

A cone

Given a Banach space

is a cone of

The positive cone

When

Let

● A linear operator

and strongly positive if

●

● An arbitrary operator

strictly increasing if

and strongly increasing if

We shall say that

Observe that if an operator

Besides in our applications, we shall use the following terminology based on Degla [

where the coefficients

Recall that an

with

where

cf. [

Furthermore

and besides, given

As in [

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

a) If

then

In case that

b) If

then

In case that

Remark 1.1. For a linear operator

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

Remark 1.3. The proof of Theorem 2 of [

The fact is that for instance in the Banach space

we have

and so with

Likewise the inequality “

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 following example illustrates Proposition 1.1.

Example 1.5. Consider the system of boundary value problems:

with

where the

Then this system has a unique normalized solution with positive component functions on the interval

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

is equivalent to the integral equation

with

Moreover by considering the special space of continuous vector-valued functions

endowed with the norm

which contains all possible solutions of our eigenvalue problem, and by letting

we see that

is compact and satisfies

with

by the strong classical maximum principle.

The conclusion follows.

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

Moreover let

Then there exists a continuum

and

which connects

2) If

(in fact in

where

Remark 2.2. An analogue version of Theorem 2.1 can be stated with

Remark 2.3. It is worth observing that Theorem 2.1 is a generalized version of a result 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. [

If

has a positive eigenvalue

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. [

Let

is such that for each

Moreover suppose that

is a completely continuous map satisfying

and consider the equation

If

then there exist

which connects

Proof of Theorem 2.1. First note that all possible solutions of the BVP’s (E_{l}) lie in

with the property that

Now (E_{l}) is equivalent, by the properties of the Green function

where

and

Moreover as seen in the proof of Lemma A [

while

by the assumptions on

1) We show that

Step 1.

Then, on one hand,

and on the other hand, we have for all

Hence for all

where

for some

By applying again Ascoli theorem we see that there exists a subsequence of

uniformly on

where

Consider now on the compact

Therefore, denoting by

we have for all

This shows that the sequence of functions

Now from the former convergence; i.e.

Thus

Step 2. If

It follows from the combination of Steps

2) We show that

To this end, let

where

Therefore for every

and on the other hand

Thus

Now by putting

we see clearly that

That is

The result follows by applying Lemma B.

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 17^{th} to the 21^{st} 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.