# A Generalization of Mahadevan's Version of the Krein-Rutman Theorem and Applications top-Laplacian Boundary Value Problems

Authors: Yujun Cui; Jingxian Sun
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We will 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 coneP
and such that there is a nonzero u∈P∖{θ}−P
for which MTpu≥u for some
positive constant M and some positive integer p. Moreover, we give some new
results on the uniqueness of positive eigenvalue with positive eigenfunction and
computation of the fixed point index. As applications, the existence of positive
solutions for p-Laplacian boundary-value problems is considered under some
conditions concerning the positive eigenvalues corresponding to the relevant
positively 1-homogeneous operators.


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## Body

## 1. 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’skiĭ [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.
LetX, 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:
(1.1)MTpu≥u,forsomeM>0,
where p is some positive integer. Then T has a non-zero eigenvector x0 in P:
(1.2)Tx0=λ0x0,
where the positive eigenvalue λ0 satisfies the inequality λ0Mp≥1.
Furthermore, ifP is a reproducing cone and T is e-positive for some e∈P∖{θ}, then(1)
the positive eigenvalueλ0 of T is simple;(2)
the operatorT 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.
LetX 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
(1.3)MTu≥u,
Then T has a non-zero eigenvector x0 in P.Compared with Theorem1.1, 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 MTpu≥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 Section2, 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.


## 2. Preliminaries

LetX a Banach space, P⊂X be a cone in X. A cone P is called solid if it contains interior points, that is, Po≠∅. 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∈Po, we write x≪y. For the concepts and the properties about the cone we refer to [12, 13].A mappingT: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
(2.1)T(tx)=tT(x)∀x∈X,t∈R0+=[0,+∞).
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.
Lete∈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
(2.2)c(x)e≤Tnx≤d(x)e.This is stronger than requiring thatT 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)⊂Po, 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.
LetF:ℝ0+×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+⊂ℝ0+×P containing the point (0,θ).Lemma 2.3.
Letx∈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.
LetΩ 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
(2.3)Au≥Tu,Au≠u,∀u∈∂Ω∩P.
Then the fixed-point indexi(A,Ω∩P,P)=0.


## 3. Main Results

Theorem 3.1.
LetT: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:
(3.1)MTpu≥u(forsomeM>0),
where p is some positive integer. Then T has a non-zero eigenvector x0 in P:
(3.2)Tx0=λ0x0,
where the positive eigenvalue λ0 satisfies the inequality
(3.3)λ0Mp≥1.Proof.
Letv∈P(v≠θ) be as in the hypothesis of the theorem. For every positive integer n>0, define Fn:ℝ0+×P→P by
(3.4)Fn(λ,x):=λTx+1nλv.
Since T is compact and continuous, each of these operators Fn is clearly compact and continuous on ℝ0+×P. Also they map ℝ0+×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, Cn+⊂ℝ0+×P→P be a connected unbounded branch of solutions to the equation
(3.5)Fn(λ,x)=x.
First we show thatCn+⊂[0,Mp]×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
(3.6)x≥λTx,x≥1nλu.
Letτn=sup{τ|x≥τu}. Obviously, τn≥(1/n)λ>0. Since T is increasing and 1-homogeneous, by (3.4), we have
(3.7)x=λTx+1nλv≥λ2T2x+1nλv≥⋯≥λpTpx+1nλv≥λpTpx≥λpTp(τnu)≥τnλpMu.
Consequently, by the definition of τn,
(3.8)λpM≤1.
In other words, if λ>Mp, then Fn(λ,·) has no fixed point. This implied that Cn+⊂[0,Mp]×P for every n>0.
Notice that the branchCn+ is connected and unbounded starting from (0,θ), there must necessarily exist xn with ∥xn∥=1 and λn∈[0,Mp] such that (λn,xn)∈Cn+. That is,
(3.9)xn=λnTxn+1nλnv,λn∈[0,Mp],‖xn‖=1.
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:
(3.10)xni=λniTxni+1niλniv(i=1,2,…).
This completes the proof of the theorem.Example 3.2.
Consider the positive 1-homogeneous mapT:
(3.11)(Tx)(t)=(∫GK(t,s)xp(s)ds)1/p,
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 pointss1,s2,…,sp such that
(3.12)K(s1,s2)K(s2,s3)⋯K(sp,s1)>0.
Then the map T defined by (3.11) has a nonnegative eigenfunction. In fact, it is easy to see that
(3.13)(Tpx)(t)=(∫GK(p)(t,s)xp(s)ds)1/p,
where
(3.14)K(p)(t,s)=∫G⋯∫GK(t,t1)⋯K(tp-1,s)dt1⋯dtp-1
is positive at the point (s1,s1) of the topological product G×G. We denoted by G1⊂G a closed neighborhood of the point s1∈G1 such that K(p)(t,s)>0 when t,s∈G1. We denote by y(t) a continuous nonnegative function such that y(s1)>0,  y(t)=0 when t∉G1. Then
(3.15)(Tpy)(t)=(∫GK(p)(t,s)yp(s)ds)1/p>0
when t∈G1. From (3.15) it follows that there exists a number M>0 such that
(3.16)MTpy≥y.
This inequality is just the condition of Theorem 3.1.Remark 3.3.
Positive 1-homogeneous maps are usually only defined on a cone. In this case, Theorem3.1 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.Theorem 3.4.
Suppose thatT is an increasing, positively 1-homogeneous, e-positive mapping. If there exist
(3.17)u1∈P∖{θ},λ1>0suchthatλ1u1≤Tu1,u2∈P∖{θ},λ2>0suchthatλ2u2≥Tu2,
then λ2≥λ1. Furthermore, if for x>y>θ, a positive number c(x,y) can be found such that
(3.18)Tx-Ty≥c(x,y)e,
then λ1=λ2 implies that u1 is a scalar multiple of u2.Proof.
It follows from thee-positiveness of  T that there exist  m,n such that
(3.19)Tnu1≤d(u1)e,c(u2)e≤Tmu2,(c(u1),d(u2)>0).
Then for t>0, we have
(3.20)(tc(u2)-d(u1))e≤tTmu2-Tnu1≤tλ2mu2-λ1nu1.
From this and the fact that u1∈P∖{θ} we deduce that δ≜δλ2mu2(λ1nu1)>0, that is, δλ2mu2-λ1nu1∈P. Since T is increasing, T(δλ2mu2)≥T(λ1nu1), from which, by virtue of (3.17), it follows that
(3.21)λ2δλ2mu2≥T(δλ2mu2)≥T(λ1nu1)≥λ1λ1nu1.
By Lemma 2.3, we obtain that λ2≥λ1.
Now suppose thatλ1=λ2. According to the above proof, the number δ>0. Since δλ2mu2-λ1nu1∈P and (3.18), if δλ2mu2≠λ1nu1, then there exists c>0 such that
(3.22)T(δλ2mu2)-T(λ1nu1)≥ce.
Therefore, by (3.19)
(3.23)λ2δλ2mu2≥T(δλ2mu2)≥T(λ1nu1)+ce≥λ1λ1nu1+cd(u1)Tn(u1)≥λ1n+1u1+cd(u1)λ1nu1=(1+cλ1d(u1))λ2λ1nu1.
This contradicts with the definition of δ. This shows that we must have δλ2mu2=λ1nu1. This completes the proof of the theorem.


## 4. Computation for the Fixed-Point Index

We illustrate howe-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 ΩP:=Ω∩P and ∂ΩP for its boundary relative to P.Theorem 4.1.
LetΩ 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, e-positive mapping T such that Te≤e, and that
(4.1)Au≤Tu,Au≠u∀u∈∂ΩP.
Then the fixed-point index i(A,ΩP,P)=1.Proof.
We show thatAu≠μu for all u∈∂ΩP 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 u0∈∂ΩP and μ0≥1 such that Au0=μ0u0, then μ0>1. It follows from the e-positiveness of T that there exists a natural number n such that
(4.2)Tnu0≤d(u0)e,d(u0)>0.
So, by induction, for allm∈N, we have
(4.3)μ0mnu0≤Tmnu0≤d(u0)e
which implies u0=θ. This contradicts u0∈∂ΩP.Theorem 4.2.
LetP 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 existsk∈[0,1) such that
(4.4)Te≤ke,(2)
there existsM>0 such that
(4.5)Au≤Tu+Me,u∈P.Then there exists R1>0 such that for any R>R1, the fixed-point index i(A,BR∩P,P)=1, where BR={x∈X|∥x∥≤R}.Proof.
Let(4.6)W={u∈P∣u=μAu,0≤μ≤1}.
In the following, we prove that W is bounded.
For anyu∈W∖{θ}, using the e-positiveness of T, we have
(4.7)u=μAu≤Tu+Me≤(d(u)+M)e.
Let
(4.8)μ0=inf⁡{μ∣u≤μe}.
It is easy to see that 0<μ0<+∞ and u≤μ0e. We now have
(4.9)u≤Tu+Me≤(kμ0+M)e,
which, by the definition of μ0, implies that μ0≤M/(1-k). So we know that u≤(M/(1-k))e and W is bounded by the normality of the cone P.
SelectR1>sup{∥x∥|x∈W}. Then from the homotopy invariance property of fixed-point index we have
(4.10)i(A,BR∩P,P)=i(θ,BR∩P,P)=1,∀R>R1.
This completes the proof of the theorem.


## 5. 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-dimensionalp-Laplacian:
(5.1)(ϕp(v′))′+a(t)f(v(t))=0,t∈(0,1),v′(0)=0,v(1)=0,
where ϕp(s)=|s|p-2s,  p≥2, and (ϕp)-1=ϕq=|s|q-2s, (1/p)+(1/q)=1.We make the following assumptions:(H1)
f:[0,+∞)→[0,+∞) is continuous;(H2)
a:[0,1]→(0,+∞) is continuous.For eachv∈E:=C[0,1], we write ∥v∥=max{|v(t)|:t∈[0,1]}. Define
(5.2)K={v∈E:v(t)≥(1-t)‖v‖}.
Clearly, (E,∥·∥) is a Banach space and K is a cone of E. For any real constant r>0, define Br={v∈E:∥x∥<r}.Let the operatorsT and A be defined by:
(5.3)(Tv)(t)=∫t1ϕq(∫0s‍a(τ)vp-1(τ)dτ)ds,(Av)(t)=∫t1ϕq(∫0s‍a(τ)f(v(τ))dτ)ds,
respectively.Under(H1) and (H2), 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 (H2) that T:K→E is a completely continuous, positively 1-homogeneous operator and T(K)⊂K.Lemma 5.1.
Suppose that(H2) holds. Then for the operator T defined by (5.3), there is a unique positive eigenvalue λ1 of T with its eigenfunction in K.Proof.
First, we show thatT is e-positive with e=1-t, that is, for any v>θ from K, there exist α,β>0 such that
(5.4)αe≤Tv≤βe.
Let M1=maxt∈[0,1]a(t). Then
(5.5)(Tv)(t)≤∫t1ϕq(M1‖v‖p-1)ds=ϕq(M1‖v‖p-1)(1-t).
So, we may take β=ϕq(M1∥v∥p-1).
Clearly, we may takeα=∥Tv∥=(Tv)(0) since T(K)⊂K. So (5.4) is proved.
Now we need to show that for anyu>v>θ, there always exists some c>0 such that
(5.6)Tu-Tv≥ce.
In fact, we note that ϕq is increasing, there exists an η∈(0,1) such that
(5.7)m=min⁡s∈[η,1]⁡ϕq(∫0sa(τ)up-1(τ)dτ)-ϕq(∫0sa(τ)vp-1(τ)dτ)>0.
Then for all t∈[η,1], we have
(5.8)(Tu)(t)-(Tv)(t)≥∫η1ϕq(∫0sa(τ)up-1(τ)dτ)ds-∫η1ϕq(∫0sa(τ)vp-1(τ)dτ)ds≥m(1-t).
Since (Tu)(t)-(Tv)(t)≥(Tu)(η)-(Tv)(η)≥m(1-η) for all t∈[0,η], we have
(5.9)(Tu)(t)-(Tv)(t)≥m(1-η)(1-t),
for all t∈[0,1]. Therefore, the proof is complete and follows from Theorems 3.1 and 3.4.Remark 5.2.
Letφ* be the positive eigenfunction of T corresponding to λ1, thus λ1Tφ*=φ*. Then by Lemma 5.1, there exist α,β>0 such that
(5.10)αe≤Tφ*=λ1φ*≤βe.
Hence we obtained that T is φ*-positive operator.Theorem 5.3.
Suppose that the conditions(H1) and (H2) are satisfied, and
(5.11)lim⁡⁡inf⁡u→0+⁡f(u)up-1>λ11-p,(5.12)lim⁡⁡sup⁡u→+∞⁡f(u)up-1<λ11-p,
where λ1 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)f(u)≥λ11-pup-1,∀0≤u≤r.
We may suppose that A has no fixed point on ∂Br∩K (otherwise, the proof is finished). Therefore by (5.13),
(5.14)(Av)(t)≥∫t1ϕq(∫0sa(τ)λ11-pup-1(τ)dτ)ds=1λ1(Tv)(t),v∈∂Br∩K.
Hence we have from Lemma 2.4 and Remark 5.2 that
(5.15)i(A,Br∩K,K)=0.
It follows from (5.12) that there exist 0<σ<1 and M1>0 such that
(5.16)f(u)≤σp-1λ11-pup-1+M2,∀u≥0.
Thus, we have
(5.17)(Av)(t)≤∫t1ϕq(∫0sa(τ)[σp-1λ11-pup-1(τ)+M2]dτ)ds≤σλ1∫t1ϕq(∫0sa(τ)up-1(τ)dτ)ds+∫t1ϕq(M1M2)ds=σλ1(Tv)(t)+ϕq(M1M2)(1-t),v∈K.
Here we have used the following inequality:
(5.18)ϕq(a+b)≤ϕq(a)+ϕq(b),a,b>0,1<q≤2.
Thus by Theorem 4.2 and Remark 5.2, there exists R1>r such that
(5.19)i(A,BR∩K,K)=0,R>R1,
and hence we obtained
(5.20)i(A,(BR∖Br¯)∩K,K)=1.
Thus, A has a fixed point in (BR∖Br¯)∩K. Consequently, (5.1) has a positive solution.Theorem 5.4.
Suppose that the conditions(H1) and (H2) are satisfied, and
(5.21)lim⁡⁡sup⁡u→0+⁡f(u)up-1<λ11-p,(5.22)lim⁡⁡inf⁡u→+∞f(u)up-1>λ11-p,
where λ1 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 r2>0 such that
(5.23)f(u)≤λ11-pup-1,∀0≤u≤r2.
We may suppose that A has no fixed point on ∂Br2∩K (otherwise, the proof is finished). Therefore by (5.23),
(5.24)(Av)(t)≤∫t1ϕq(∫0s‍a(τ)λ11-pup-1(τ)dτ)ds=1λ1(Tv)(t),v∈∂Br2∩K.
Hence we have from Theorem 4.1 and Remark 5.2 that
(5.25)i(A,Br2∩K,K)=1.
It follows from (5.22) that there exists ε>0 such that f(u)≥(λ11-p+ε)up-1 when u is sufficiently large. We know from the continuity of f that there exists b≥0 such that
(5.26)f(u)≥(λ11-p+ε)up-1-b,∀u≥0.
Take
(5.27)R2>max⁡⁡{r2,(M1b2pm0ε)1/(p-1)},
where m0=mint∈[0,1]a(t), M1=maxt∈[0,1]a(t).
Forv∈∂BR2∩K, we have
(5.28)∫0sa(τ)f(v(τ))dτ≥∫0sa(τ)[(λ11-p+ε)vp-1(τ)-b]dτ≥λ11-p∫0sa(τ)vp-1(τ)dτ+m0ε∫0s(1-τ)p-1‖v‖p-1dτ-M1bs=λ11-p∫0sa(τ)vp-1(τ)dτ+m0εR2p-1∫0s(1-τ)p-1dτ-M1bs≥λ11-p∫0sa(τ)vp-1(τ)dτ+m0εR2p-1∫0min⁡⁡{s,1/2}(1-τ)p-1dτ-M1bs≥λ11-p∫0sa(τ)vp-1(τ)dτ+m0εR2p-1∫0min⁡⁡{s,1/2}(1-12)p-1dτ-M1bs≥λ11-p∫0sa(τ)vp-1(τ)dτ+(m0εR2p-112p-M1b)s≥λ11-p∫0sa(τ)vp-1(τ)dτ.
Thus, we have(5.29)(Av)(t)≥∫t1ϕq(∫0sa(τ)λ11-pup-1(τ)dτ)ds=1λ1(Tv)(t),v∈∂BR2∩K.
It follows from Lemma 2.4 that
(5.30)i(A,BR2∩K,K)=0,
and hence we obtained
(5.31)i(A,(BR2∖Br2¯)∩K,K)=-1.
Thus, A has a fixed point in (BR2∖Br2¯)∩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.



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