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.

Krein-Rutman Positivity Cone Homogeneity Compactness Eigenvalue Simplicity Bifurcation
1. Introduction

In this paper, we generalize and improve a result of Coyle et al.  about the bifurcation from infinity after stating in the line of Nussbaum  , Schmitt  , 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  and a result on the principal eigenvalue of multi-point Boundary Value Problems (BVP’s) of Degla  which allow the use of cone theoretic arguments and of the well-known general result on bifurcation from infinity; see Coyle  , Mawhin  and Rabinowitz  .

Furthermore, in our abstract setting, the nonlinear Krein-Rutman Theorem resets an important result on the simplicity of positive eigenvalues  by avoiding some inconclusive argument  (page 3086, lines 29-37) also misused in  (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   , Elias  and Coppel  . 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.  .

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  , 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  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  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.

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

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  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. 

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.    

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  , 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.

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