On Bifurcation from Infinity and Multipoint Boundary Value Problems

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.


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 1 -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.

Preliminary Definitions and Notations
We say that a nonempty subset  of a Banach space X is a cone if it is closed and 1) In other words, the cones considered here are closed convex cones with vertex at 0. A cone  of a Banach space X induces a partial ordering on X by the relation if and only if , x y y x − ∈  and it follows that 0 if and only if .
x ∈   Therefore ( ) , X  is called an ordered Banach space with  as the positive cone of X .Note that we write x y  when x y  and x y ≠ ; i.e.,

{ }
if and only if 0 .
A cone  of a Banach space X is said to be generating if X = −   , and total if X = −   .Given a Banach space X with dual X ′ , if a cone  of X is generating, then the set defined by The positive cone  of an ordered Banach space X is said to be normal if there exists a positive constant an ordered Banach space is said to be monotone.

Let ( )
, X  be an ordered Banach space.Then • A linear operator : T X X → is said to be positive if ( ) and strongly positive if Observe that if an operator T is increasing on  and satisfies ( ) 0 0 T = , 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 , n m and 1 , ,   [11].
Furthermore G will denote the Green function associated to the Boundary Value Problems (in short BVP's) , we shall adopt the notation ( ) As in [5], we shall also consider the Banach space T X X → is a positively 1-homogeneous, compact and continuous operator.a) If T is increasing on  and there exist a positive vector 0 u  , a positive real number ω and a positive integer m , such that then T has a positive eigenvalue with a positive eigenvector.
In case that T is linear, its spectral radius ( ) r T is such a positive eigenvalue and satisfies ( ) then T has a unique positive eigenvalue and a unique positive normalized eigenvector.
In case that T is linear, this positive eigenvalue coincides with the spectral radius ( ) r T of T , is alge- braically simple and has the following variational characterization: ( ) Furthermore the conclusion of b) can be heuristically motivated by the application of the Krein-Rutman theorem to the quotient space ker / T X .Remark 1.2.The above theorem is readily applicable to any positively 1 -homogeneous, compact and con- tinuous 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 0 λ λ ≤ which does not contradict the inequality (2.10) therein; that is 0 λ λ ≥ .The fact is that for instance in the Banach space 2   ordered by the cone and so with 1 0 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 T , 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 λ as a real parameter and where the , i j q are assumed to be nonnegative continuous functions on 1] [0, such that on the one hand 1,1 q and 2,1 q have a common support 1  , and on the other hand 1,2 q and 2,2 q have a common support 2  such that 1 2 ≠ ∅    ; i.e. 0 Q ≡ / , and where the unknown vector-valued function 0,1 ;   with zero Dirichlet boundary condition.
Then this system has a unique normalized solution with positive component functions on the interval ( ) 0,1 corresponding to a unique positive value of the parameter λ .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 0 λ ≠ , and the system of BVPs ( ) ( ) .
Moreover by considering the special space of continuous vector-valued functions which contains all possible solutions of our eigenvalue problem, and by letting X  is a normal ordered Banach space.Furthermore the non-zero linear operator : by the strong classical maximum principle.The conclusion follows.

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 x converges to q and converges uniformly x µ λ (in fact in P X ) to the unique normalized nontrivial solution of ( ) ( ) ( ) : q λ λ = .Remark 2.2.An analogue version of Theorem 2.1 can be stated with satisfying the following property: 0 on a set of positive measure and 0 for a.e. , .
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 q that may vanish 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

( ) ( ) ( )
, for almost every 0, 1 , 0 1 has a positive eigenvalue 1 λ which is simple with an eigenfunction 1 u 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 E be a real Banach space with norm ⋅ .Assume that [ ] , T λ ⋅ is a compact linear operator, and for each u E ∈ , ( ) and consider the equation , is equivalent, by the properties of the Green function G , to the following equation: ; .
is completely continuous and satisfies lim 0 by the assumptions on h .Indeed: 1) We show that : H X X → is completely continuous.
Step 1. H maps bounded subsets into compact subsets.Let ( ) where G ω is the modulus of continuity of G .Moreover as a continuous function, G is uniformly conti- By applying again Ascoli theorem we see that there exists a subsequence of ( )

,
we shall denote by P the Levin's polynomial defined by ( ) ( )

=
the paragraph 4 of the proof of theorem 4.8 of[9] does not contradict the definition of " by simply considering again the ordered Banach space ( )2 2

1 .
First note that all possible solutions of the BVP's (E λ ) lie in X since they are of the form f is continuous and G is the Green function of the BVPs ( ) nuous on the compact set [ ] [ ] Let X be a real Banach space,  a nontrivial cone in X and assume that : still denoted by ( )Hu converges to v in X .. It follows from the combination of Steps 1& 2 that H is completely continuous; i.e., H maps bounded sets into compact sets and is continuous.2) We show that