A Generalization of Mahadevan ’ s Version of the Krein-Rutman Theorem and Applications to p-Laplacian Boundary Value Problems

and Applied Analysis 3 2. Preliminaries Let X a Banach space, P ⊂ X be a cone in X. A cone P is called solid if it contains interior points, that is, o P / ∅. A cone is said to be reproducing if X P − P . Every cone P in E defines a partial ordering in E given by x ≤ y if and only if y − x ∈ P . If x ≤ y and x / y, we write x < y; if cone P is solid and y − x ∈ o P , we write x y. For the concepts and the properties about the cone we refer to 12, 13 . A mapping T : X → X is said to be increasing if x ≤ y implies Tx ≤ Ty and it is said to be strictly increasing if x < y implies Tx < Ty. The mapping is said to be compact if it takes bounded subsets of X into relatively compact subsets of X. We say that the mapping is positively 1-homogeneous if it satisfies the relation T tx tT x ∀x ∈ X, t ∈ R 0 0, ∞ . 2.1 We say that a real number λ is an eigenvalue of the operator if there exists a non-zero x ∈ X such that Tx λx. Definition 2.1. Let e ∈ P \ {θ}, a mapping T : P → P is called e−positive if for every non-zero x ∈ P a natural number n n x and two positive number c x , d x can be found such that c x e ≤ Tx ≤ d x e. 2.2 This is stronger than requiring that T is positive, that is, T P ⊂ P . It is always satisfied if P is a solid cone and T is strongly positive, that is, T P ⊂ o P , with any e ∈ P \ {θ}, but it can be satisfied more generally. For the application in the sequel, we state the following three lemmas which can be found in 14, Theorem 17.1 3, Lemma 1.2 15, Theorem 1.1 . The first one involves the global structure of the positive solution set for completely continuous map, the second one involve cones, and the last one involves the computation of fixed-point index. Lemma 2.2. Let F : R 0 × P → P be a compact, continuous map and such that F 0, x θ for all x ∈ P . Then, F λ, x x has a nontrivial connected unbounded component of solutions C ⊂ R 0 × P containing the point 0, θ . Lemma 2.3. Let x ∈ P . For an element y ∈ X, suppose a δ1 can be found such that y ≤ δ1x. Then a small δx y exists for which y ≤ δx y x. Lemma 2.4. Let Ω be a bounded open set in X, let P be a cone in X, and let A : P → P be a completely continuous map. Suppose that there is an increasing, positively 1-homogeneous mapping T and u∗ ∈ P \ {θ} such that Tu∗ ≥ u∗, and that Au ≥ Tu, Au/ u, ∀u ∈ ∂Ω ∩ P. 2.3 Then the fixed-point index i A,Ω ∩ P, P 0. 4 Abstract and Applied Analysis 3. Main Results Theorem 3.1. Let T : X → X be an increasing, positively 1-homogeneous, compact, continuous mapping. Suppose that for some non-zero element u v − w, where v,w ∈ P and −u / ∈ P , the following relation is satisfied: MTu ≥ u for some M > 0 , 3.1 where p is some positive integer. Then T has a non-zero eigenvector x0 in P : Tx0 λ0x0, 3.2 where the positive eigenvalue λ0 satisfies the inequality λ0 p √ M ≥ 1. 3.3 Proof. Let v ∈ P v / θ be as in the hypothesis of the theorem. For every positive integer n > 0, define Fn : R 0 × P → P by Fn λ, x : λTx 1 n λv. 3.4 Since T is compact and continuous, each of these operators Fn is clearly compact and continuous on R 0 × P . Also they map R 0 × P into P since T maps P into itself, which follows from the fact that T is increasing and Tθ θ. Let, by Lemma 2.2, C n ⊂ R 0 × P → P be a connected unbounded branch of solutions to the equation Fn λ, x x. 3.5 First we show that C n ⊂ 0, p √ M × P for all n > 0. Indeed, suppose that x is a fixed point of Fn λ, · for some λ ≥ 1. Then x Fn λ, x λTx 1/n λv and we obtain, from the properties of T and the inequalities v ≥ θ, v ≥ u, respectively, that x ≥ λTx, x ≥ 1 n λu. 3.6 Let τn sup{τ | x ≥ τu}. Obviously, τn ≥ 1/n λ > 0. Since T is increasing and 1homogeneous, by 3.4 , we have x λTx 1 n λv ≥ λ2T2x 1 n λv ≥ · · · ≥ λTx 1 n λv ≥ λTx ≥ λT τnu ≥ τn λ p M u. 3.7 Abstract and Applied Analysis 5 Consequently, by the definition of τn, λ M ≤ 1. 3.8and Applied Analysis 5 Consequently, by the definition of τn, λ M ≤ 1. 3.8 In other words, if λ > p √ M, then Fn λ, · has no fixed point. This implied thatC n ⊂ 0, p √ M ×P for every n > 0. Notice that the branch C n is connected and unbounded starting from 0, θ , there must necessarily exist xn with ‖xn‖ 1 and λn ∈ 0, p √ M such that λn, xn ∈ C n . That is, xn λnTxn 1 n λnv, λn ∈ [ 0, p √ M ] , ‖xn‖ 1. 3.9 Since the operator T is compact, a subsequence of indices ni i 1, 2, . . . can be chosen such that the sequence Txn strongly converges to some element y∗ ∈ P . By virtue of 3.9 , with this choice of the sequence ni, the convergence of the number λni to some λ∗ which satisfies the inequality 3.3 can be guaranteed simultaneously. Then xni will converge in norm to the element x0 λ∗y∗ with ‖x0‖ 1. Further, it follows from the fact ‖x0‖ 1 that λ∗ / 0. Let λ0 λ−1 ∗ . To obtain the equality 3.2 , it suffices to pass to the limits in the equality: xni λniTxni 1 ni λniv i 1, 2, . . . . 3.10 This completes the proof of the theorem. Example 3.2. Consider the positive 1-homogeneous map T :


Introduction
The Krein-Rutman theorem 1, 2 plays a very important role in nonlinear differential equations, as it provides the abstract basis for the proof of the existence of various principal eigenvalues, which in turn are crucial in bifurcation theory, in topological degree calculation, and in the stability analysis of solutions to elliptic equations.Owing to its importance, much attention has been given to the most general versions of the linear Krein-Rutman theorem by a number of authors, see 3-7 .For example, Krasnosel'skiȋ 3 introduced the concept of the e-positive linear operator and then used it to prove the following results concerning the eigenvalues of positive linear compact operator.
Theorem 1.1.Let X, a Banach space, P ⊂ X a cone in X.Let T : X → X be a linear, positive, and compact operator.Suppose that for some non-zero element u v − w, where v, w ∈ P and −u / ∈ P , the following relation is satisfied: MT p u ≥ u, for some M > 0, 1.1 where p is some positive integer.Then T has a non-zero eigenvector x 0 in P : where the positive eigenvalue λ 0 satisfies the inequality λ 0 p √ M ≥ 1.Furthermore, if P is a reproducing cone and T is e-positive for some e ∈ P \ {θ}, then 1 the positive eigenvalue λ 0 of T is simple; 2 the operator T has a unique positive eigenvector upto a multiplicative constant.
Recently, the nonlinear version of the Krein-Rutman theorem has been extended to positive eigenvalue problem for increasing, positively 1-homogeneous, compact, continuous operators by Mallet-Paret and Nussbaum 8, 9 , Mahadevan 10 , and Chang 11 .
The following nonlinear Krein-Rutman theorem has been established in 10 .
Theorem 1.2.Let X be a Banach space, P ⊂ X be a cone in X.Let T : X → X be an increasing, positively 1-homogeneous, compact, continuous operator for which there exists a non-zero u ∈ P and M > 0 such that Then T has a non-zero eigenvector x 0 in P .
Compared with Theorem 1.1, we note that the element u, appeared in Theorem 1.2, belongs to P .Consequently we put forward a problem: are the results in Theorem 1.2 valid if the condition u ∈ P is replaced with that in Theorem 1.1.The purpose of this study is to solve the above problem.By means of global structure of the positive solution set, we present a generalization of Mahadevan's version of the Krein-Rutman Theorem for a compact, positively 1-homogeneous operator on a Banach space having the properties of being increasing with respect to a convex cone P and such that there is a non-zero u ∈ P \ {θ} − P for which MT p u ≥ u for some positive constant M and some positive integer p.The method in this paper is somewhat different from that in 10 .
The paper is organized as follow.In Section 2, we give some basic definitions and state three lemmas which are needed later.In Section 3, we establish some results for the existence of the eigenvalues of positively compact, 1-homogeneous operator and deduce some results on the uniqueness of positive eigenvalue with positive eigenfunction.In Section 4, we present some new methods of computation of the fixed point index for cone mapping.The final section is concerned with applications to the existence of positive solutions for p-Laplacian boundary-value problems under some conditions concerning the positive eigenvalues corresponding to the relevant positively 1-homogeneous operators.

Preliminaries
Let X a Banach space, P ⊂ X be a cone in X.A cone P is called solid if it contains interior points, that is, o P / ∅.A cone is said to be reproducing if X P − P .Every cone P in E defines a partial ordering in E given by x ≤ y if and only if y − x ∈ P .If x ≤ y and x / y, we write x < y; if cone P is solid and y − x ∈ o P , we write x y.For the concepts and the properties about the cone we refer to 12, 13 .
A mapping T : X → X is said to be increasing if x ≤ y implies Tx ≤ Ty and it is said to be strictly increasing if x < y implies Tx < Ty.The mapping is said to be compact if it takes bounded subsets of X into relatively compact subsets of X.We say that the mapping is positively 1-homogeneous if it satisfies the relation We say that a real number λ is an eigenvalue of the operator if there exists a non-zero x ∈ X such that Tx λx.For the application in the sequel, we state the following three lemmas which can be found in 14, Theorem 17.1 3, Lemma 1.2 15, Theorem 1.1 .The first one involves the global structure of the positive solution set for completely continuous map, the second one involve cones, and the last one involves the computation of fixed-point index.
Lemma 2.2.Let F : R 0 × P → P be a compact, continuous map and such that F 0, x θ for all x ∈ P .Then, F λ, x x has a nontrivial connected unbounded component of solutions C ⊂ R 0 × P containing the point 0, θ .Lemma 2.3.Let x ∈ P .For an element y ∈ X, suppose a δ 1 can be found such that y ≤ δ 1 x.Then a small δ x y exists for which y ≤ δ x y x.Lemma 2.4.Let Ω be a bounded open set in X, let P be a cone in X, and let A : P → P be a completely continuous map.Suppose that there is an increasing, positively 1-homogeneous mapping T and u * ∈ P \ {θ} such that Tu * ≥ u * , and that Au ≥ Tu, Au / u, ∀u ∈ ∂Ω ∩ P.

2.3
Then the fixed-point index i A, Ω ∩ P, P 0.

Main Results
Theorem 3.1.Let T : X → X be an increasing, positively 1-homogeneous, compact, continuous mapping.Suppose that for some non-zero element u v − w, where v, w ∈ P and −u / ∈ P , the following relation is satisfied: where p is some positive integer.Then T has a non-zero eigenvector x 0 in P : where the positive eigenvalue λ 0 satisfies the inequality Proof.Let v ∈ P v / θ be as in the hypothesis of the theorem.For every positive integer n > 0, define F n : R 0 × P → P by Since T is compact and continuous, each of these operators F n is clearly compact and continuous on R 0 × P .Also they map R 0 × P into P since T maps P into itself, which follows from the fact that T is increasing and Tθ θ.Let, by Lemma 2.2, C n ⊂ R 0 × P → P be a connected unbounded branch of solutions to the equation x.

3.5
First we show that C n ⊂ 0, p √ M × P for all n > 0. Indeed, suppose that x is a fixed point of F n λ, • for some λ ≥ 1.Then x F n λ, x λT x 1/n λv and we obtain, from the properties of T and the inequalities v ≥ θ, v ≥ u, respectively, that Since T is increasing and 1homogeneous, by 3.4 , we have

3.7
Consequently, by the definition of τ n , In other words, if λ > p √ M, then F n λ, • has no fixed point.This implied that C n ⊂ 0, p √ M ×P for every n > 0.
Notice that the branch C n is connected and unbounded starting from 0, θ , there must necessarily exist x n with x n 1 and Since the operator T is compact, a subsequence of indices n i i 1, 2, . . .can be chosen such that the sequence Tx n strongly converges to some element y * ∈ P .By virtue of 3.9 , with this choice of the sequence n i , the convergence of the number λ n i to some λ * which satisfies the inequality 3.3 can be guaranteed simultaneously.Then x n i will converge in norm to the element x 0 λ * y * with x 0 1.Further, it follows from the fact x 0 1 that λ * / 0. Let λ 0 λ −1 * .To obtain the equality 3.2 , it suffices to pass to the limits in the equality: This completes the proof of the theorem.where G is a bounded closed set in a finite-dimensional space, the kernel K t, s is nonnegative, and p 2n 1 for some n ∈ N.
If there exists a system of points s 1 , s 2 , . . ., s p such that Then the map T defined by 3.11 has a nonnegative eigenfunction.In fact, it is easy to see that where  Theorem 3.4.Suppose that T is an increasing, positively 1-homogeneous, e-positive mapping.If there exist 3.17 then λ 2 ≥ λ 1 .Furthermore, if for x > y > θ, a positive number c x, y can be found such that Tx − Ty ≥ c x, y e, 3.18 Proof.It follows from the e-positiveness of T that there exist m, n such that

3.19
Then for t > 0, we have

3.20
From this and the fact that from which, by virtue of 3.17 , it follows that

3.23
This contradicts with the definition of δ.This shows that we must have δλ m 2 u 2 λ n 1 u 1 .This completes the proof of the theorem.

Computation for the Fixed-Point Index
We illustrate how e-positivity can be used to prove some fixed-point index results which can then be used to prove existence results for nonlinear equations.When Ω is a bounded open set in a Banach space X, we write Ω P : Ω ∩ P and ∂Ω P for its boundary relative to P .Theorem 4.1.Let Ω be a bounded open set in X containing θ, let P be a cone in X, and let A : P → P be a completely continuous map.Suppose that there is an increasing, positively 1-homogeneous, epositive mapping T such that Te ≤ e, and that Au ≤ Tu, Au / u ∀u ∈ ∂Ω P .

4.1
Then the fixed-point index i A, Ω P , P 1.
Proof.We show that Au / μu for all u ∈ ∂Ω P and all μ ≥ 1, from which the result follows by standard properties of fixed-point index see, e.g., 12-14 .Suppose that there exist u 0 ∈ ∂Ω P and μ 0 ≥ 1 such that Au 0 μ 0 u 0 , then μ 0 > 1.It follows from the e-positiveness of T that there exists a natural number n such that So, by induction, for all m ∈ N, we have which implies u 0 θ.This contradicts u 0 ∈ ∂Ω P .
Theorem 4.2.Let P be a normal cone in a real Banach space X and let A : P → P be a completely continuous map.Suppose that there is an increasing, positively 1-homogeneous, e-positive mapping T (with n 1 in Definition 2.1) which satisfies the following conditions: 1 there exists k ∈ 0, 1 such that Te ≤ ke, 4.4 2 there exists M > 0 such that Au ≤ Tu Me, u ∈ P.

4.5
Then there exists R 1 > 0 such that for any R > R 1 , the fixed-point index i A, B R ∩ P, P 1, where In the following, we prove that W is bounded.
For any u ∈ W \ {θ}, using the e-positiveness of T , we have

4.7
Let It is easy to see that 0 < μ 0 < ∞ and u ≤ μ 0 e.We now have u ≤ Tu Me ≤ kμ 0 M e, 4.9 which, by the definition of μ 0 , implies that μ 0 ≤ M/ 1 − k .So we know that u ≤ M/ 1 − k e and W is bounded by the normality of the cone P .Select R 1 > sup{ x | x ∈ W}.Then from the homotopy invariance property of fixedpoint index we have This completes the proof of the theorem.

Applications
In the following, we will apply the results in this paper to the existence of positive solution for two-point boundary-value problems for one-dimensional p-Laplacian: where φ p s |s| p−2 s, p ≥ 2, and φ p −1 φ q |s| q−2 s, 1/p 1/q 1.We make the following assumptions: Clearly, E, • is a Banach space and K is a cone of E. For any real constant r > 0, define B r {v ∈ E : x < r}.
Let the operators T and A be defined by: Under H 1 and H 2 , it is not difficult to verify that the non-zero fixed points of the operator A are the positive solutions of boundary-value problem 5.1 .In addition, we have from H 2 that T : K → E is a completely continuous, positively 1-homogeneous operator and T K ⊂ K. Lemma 5.1.Suppose that H 2 holds.Then for the operator T defined by 5.3 , there is a unique positive eigenvalue λ 1 of T with its eigenfunction in K.
Proof.First, we show that T is e-positive with e 1 − t, that is, for any v > θ from K, there exist α, β > 0 such that αe ≤ Tv ≤ βe.

Abstract and Applied Analysis
Clearly, we may take α Tv Tv 0 since T K ⊂ K.So 5.4 is proved.Now we need to show that for any u > v > θ, there always exists some c > 0 such that Tu − Tv ≥ ce. 5.6 In fact, we note that φ q is increasing, there exists an η ∈ 0, 1 such that Then for all t ∈ η, 1 , we have for all t ∈ 0, 1 .Therefore, the proof is complete and follows from Theorems 3.1 and 3.4.

5.10
Hence we obtained that T is ϕ * -positive operator.
Theorem 5.3.Suppose that the conditions H 1 and H 2 are satisfied, and where λ 1 is given in Lemma 5.1, then the boundary-value problem 5.1 has at least one positive solution.
Proof.It follows from 5.11 that there exists r > 0 such that

5.13
We may suppose that A has no fixed point on ∂B r ∩ K otherwise, the proof is finished .Therefore by 5.13 , Hence we have from Lemma 2.4 and Remark 5.2 that i A, B r ∩ K, K 0.

5.15
It follows from 5.12 that there exist 0 < σ < 1 and Thus, we have

5.17
Here we have used the following inequality: φ q a b ≤ φ q a φ q b , a,b > 0, 1 < q ≤ 2.

5.18
Thus by Theorem 4.2 and Remark 5.2, there exists

5.20
Thus, A has a fixed point in B R \ B r ∩ K. Consequently, 5.1 has a positive solution.
Theorem 5.4.Suppose that the conditions H 1 and H 2 are satisfied, and where λ 1 is given in Lemma 5.1, then the boundary-value problem 5.1 has at least one positive solution.
Proof.It follows from 5.21 that there exists r 2 > 0 such that

5.23
We may suppose that A has no fixed point on ∂B r 2 ∩ K otherwise, the proof is finished .Therefore by 5.23 ,

5.24
Hence we have from Theorem 4.1 and Remark 5.2 that i A, B r 2 ∩ K, K 1.

5.25
It follows from 5.22 that there exists ε > 0 such that f u ≥ λ 1−p 1 ε u p−1 when u is sufficiently large.We know from the continuity of f that there exists b ≥ 0 such that

5.31
Thus, A has a fixed point in B R 2 \ B r 2 ∩ K. Consequently, 5.1 has a positive solution.
Remark 5.5.p-Laplacian boundary-value problems have been studied by some authors 16, 17 and references therein .In preceding works mentioned, they study the existence of positive solutions by the shooting method, fixed-point theorem, or the fixed-point index under some different conditions.It is known that, when p 2, there are very good conditions imposed on f that ensure the existence of positive solution for two-point boundary-value problems 5.1 .In particular, some of those involving the first eigenvalues corresponding to the relevant linear operator are sharp conditions.So, Theorems 5.3 and 5.4 generalize a number of recent works about the existence of solutions for p-Laplacian boundary-value problems.

Abstract and Applied Analysis 13 For
v ∈ ∂B R 2 ∩ K, we have

1
Tv t , v ∈ ∂B R 2 ∩ K. 5.29 It follows from Lemma 2.4 that i A, B R 2 ∩ K, K 0, 5.30 and hence we obtained i A, B R 2 \ B r 2 ∩ K, K −1.
Definition 2.1.Let e ∈ P \ {θ}, a mapping T : P → P is called e−positive if for every non-zero x ∈ P a natural number n n x and two positive number c x , d x can be found such that • • • dt p−13.14ispositive at the point s 1 , s 1 of the topological product G × G.We denoted by G 1 ⊂ G a closed neighborhood of the point s 1 ∈ G 1 such that K p t, s > 0 when t, s ∈ G 1 .We denote by y t a continuous nonnegative function such that y s 1 > 0, y t Remark 3.3.Positive 1-homogeneous maps are usually only defined on a cone.In this case, Theorem 3.1 remains valid provided u ∈ P \ {θ}.Moreover, Theorem 2.1 and Corollary 2.1 of 5 already give a general result for a k-set contraction, positive 1-homogeneous maps.