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.

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

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

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

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

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.

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

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

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 (E_l) 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 (E_l) 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.

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.