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

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 cone P and such that there is a nonzero u∈P\{θ}-P for which M[superscript]Tp[/superscript] u...5;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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Yujun Cui 1 and Jingxian Sun 2

Recommended by Lishan Liu

1, Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266590, China
2, Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China

Received 12 January 2012; Revised 24 March 2012; Accepted 13 July 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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'ski[ibreve] [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[arrow right]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: [figure omitted; refer to PDF] where p is some positive integer. Then T has a non-zero eigenvector _x0 in P : [figure omitted; refer to PDF] where the positive eigenvalue _λ0 satisfies the inequality _λ0 Mp...5;1 .

Furthermore, if P is a reproducing cone and T is e -positive for some e∈P\{θ} , then

(1) the positive eigenvalue _λ0 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[arrow right]X be an increasing, positively 1-homogeneous, compact, continuous operator for which there exists a non-zero u∈P and M>0 such that [figure omitted; refer to PDF] Then T has a non-zero eigenvector _x0 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 M^Tp u...5;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.

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, Po...0;∅ . 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...4;y if and only if y-x∈P . If x...4;y and x...0;y , we write x<y ; if cone P is solid and y-x∈ Po , we write x...a;y . For the concepts and the properties about the cone we refer to [12, 13].

A mapping T:X[arrow right]X is said to be increasing if x...4;y implies Tx...4;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 [figure omitted; refer to PDF] 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[arrow right]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 [figure omitted; refer to PDF]

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)⊂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.

Let F:^...0+ ×P[arrow right]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+ ⊂^...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...4;_δ1 x . Then a small _δx (y) exists for which y...4;_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[arrow right]P be a completely continuous map. Suppose that there is an increasing, positively 1-homogeneous mapping T and ^u* ∈P\{θ} such that T^u* ...5;^u* , and that [figure omitted; refer to PDF]

Then the fixed-point index i(A,Ω∩P,P)=0 .

3. Main Results

Theorem 3.1.

Let T:X[arrow right]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: [figure omitted; refer to PDF] where p is some positive integer. Then T has a non-zero eigenvector _x0 in P : [figure omitted; refer to PDF] where the positive eigenvalue _λ0 satisfies the inequality [figure omitted; refer to PDF]

Proof.

Let v∈P (v...0;θ) be as in the hypothesis of the theorem. For every positive integer n>0 , define _Fn :^...0+ ×P[arrow right]P by [figure omitted; refer to PDF] Since T is compact and continuous, each of these operators _Fn is clearly compact and continuous on ^...0+ ×P . Also they map ^...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, ^Cn+ ⊂^...0+ ×P[arrow right]P be a connected unbounded branch of solutions to the equation [figure omitted; refer to PDF]

First we show that ^Cn+ ⊂[0,Mp]×P for all n>0 . Indeed, suppose that x is a fixed point of _Fn (λ,·) for some λ...5;1 . Then x=_Fn (λ,x)=λTx+(1/n)λv and we obtain, from the properties of T and the inequalities v...5;θ , v...5;u , respectively, that [figure omitted; refer to PDF]

Let _τn =sup {τ | x...5;τu} . Obviously, _τn ...5;(1/n)λ>0 . Since T is increasing and 1-homogeneous, by (3.4), we have [figure omitted; refer to PDF] Consequently, by the definition of _τn , [figure omitted; refer to PDF] 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 branch ^Cn+ 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, [figure omitted; refer to PDF] Since the operator T is compact, a subsequence of indices _ni (i=1,2,...) can be chosen such that the sequence T_xn 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;0 . Let _λ0 =^λ*-1 . To obtain the equality (3.2), it suffices to pass to the limits in the equality: [figure omitted; refer to PDF] This completes the proof of the theorem.

Example 3.2.

Consider the positive 1-homogeneous map T : [figure omitted; refer to PDF] 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∈... .

If there exists a system of points _s1 ,_s2 ,...,_sp such that [figure omitted; refer to PDF] Then the map T defined by (3.11) has a nonnegative eigenfunction. In fact, it is easy to see that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] when t∈_G1 . From (3.15) it follows that there exists a number M>0 such that [figure omitted; refer to PDF] 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, 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.

Theorem 3.4.

Suppose that T is an increasing, positively 1-homogeneous, e -positive mapping. If there exist [figure omitted; refer to PDF] then _λ2 ...5;_λ1 . Furthermore, if for x>y>θ , a positive number c(x,y) can be found such that [figure omitted; refer to PDF] then _λ1 =_λ2 implies that _u1 is a scalar multiple of _u2 .

Proof.

It follows from the e -positiveness of T that there exist m,n such that [figure omitted; refer to PDF] Then for t>0 , we have [figure omitted; refer to PDF] From this and the fact that _u1 ∈P\{θ} we deduce that δ[triangle, =]_δ^λ2mu2 (^λ1n_u1 )>0 , that is, δ^λ2m_u2 -^λ1n_u1 ∈P . Since T is increasing, T(δ^λ2m_u2 )...5;T(^λ1n_u1 ) , from which, by virtue of (3.17), it follows that [figure omitted; refer to PDF] By Lemma 2.3, we obtain that _λ2 ...5;_λ1 .

Now suppose that _λ1 =_λ2 . According to the above proof, the number δ>0 . Since δ^λ2m_u2 -^λ1n_u1 ∈P and (3.18), if δ^λ2m_u2 ...0;^λ1n_u1 , then there exists c>0 such that [figure omitted; refer to PDF] Therefore, by (3.19) [figure omitted; refer to PDF] This contradicts with the definition of δ . This shows that we must have δ^λ2m_u2 =^λ1n_u1 . This completes the proof of the theorem.

4. 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 _ΩP :=Ω∩P and ∂_ΩP 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[arrow right]P be a completely continuous map. Suppose that there is an increasing, positively 1-homogeneous, e -positive mapping T such that Te...4;e , and that [figure omitted; refer to PDF] Then the fixed-point index i(A,_ΩP ,P)=1 .

Proof.

We show that Au...0;μu for all u∈∂_ΩP and all μ...5;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 ...5;1 such that A_u0 =_μ0u0 , then _μ0 >1 . It follows from the e -positiveness of T that there exists a natural number n such that [figure omitted; refer to PDF] So, by induction, for all m∈N , we have [figure omitted; refer to PDF] which implies _u0 =θ . This contradicts _u0 ∈∂_ΩP .

Theorem 4.2.

Let P be a normal cone in a real Banach space X and let A:P[arrow right]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 [figure omitted; refer to PDF]

(2) there exists M>0 such that [figure omitted; refer to PDF]

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||...4;R} .

Proof.

Let [figure omitted; refer to PDF] In the following, we prove that W is bounded.

For any u∈W\{θ} , using the e -positiveness of T , we have [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] It is easy to see that 0<_μ0 <+∞ and u...4;_μ0 e . We now have [figure omitted; refer to PDF] which, by the definition of _μ0 , implies that _μ0 ...4;M/(1-k) . So we know that u...4;(M/(1-k))e and W is bounded by the normality of the cone P .

Select _R1 >sup {||x|| | x∈W} . Then from the homotopy invariance property of fixed-point index we have [figure omitted; refer to PDF]

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-dimensional p -Laplacian: [figure omitted; refer to PDF] where _[varphi]p (s)=|s^|p-2 s , p...5;2 , and (_[varphi]p^)-1 =_[varphi]q =|s^|q-2 s , (1/p)+(1/q)=1 .

We make the following assumptions:

(_H1 ) : f:[0,+∞)[arrow right][0,+∞) is continuous;

(_H2 ) : a:[0,1][arrow right](0,+∞) is continuous.

For each v∈E:=C[0,1] , we write ||v||=max {|v(t)|:t∈[0,1]} . Define [figure omitted; refer to PDF] 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 operators T and A be defined by: [figure omitted; refer to PDF] 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[arrow right]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 that T is e -positive with e=1-t , that is, for any v>θ from K , there exist α,β>0 such that [figure omitted; refer to PDF] Let _M1 =_{max t∈[0,1]} a(t) . Then [figure omitted; refer to PDF] So, we may take β=_[varphi]q (_M1 ||v^||p-1 ) .

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 [figure omitted; refer to PDF] In fact, we note that _[varphi]q is increasing, there exists an η∈(0,1) such that [figure omitted; refer to PDF] Then for all t∈[η,1] , we have [figure omitted; refer to PDF] Since (Tu)(t)-(Tv)(t)...5;(Tu)(η)-(Tv)(η)...5;m(1-η) for all t∈[0,η] , we have [figure omitted; refer to PDF] for all t∈[0,1] . Therefore, the proof is complete and follows from Theorems 3.1 and 3.4.

Remark 5.2.

Let ^{[straight phi]*} be the positive eigenfunction of T corresponding to _λ1 , thus _λ1 T^{[straight phi]*} =^{[straight phi]*} . Then by Lemma 5.1, there exist α,β>0 such that [figure omitted; refer to PDF] Hence we obtained that T is ^{[straight phi]*} -positive operator.

Theorem 5.3.

Suppose that the conditions (_H1 ) and (_H2 ) are satisfied, and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] We may suppose that A has no fixed point on ∂_Br ∩K (otherwise, the proof is finished). Therefore by (5.13), [figure omitted; refer to PDF] Hence we have from Lemma 2.4 and Remark 5.2 that [figure omitted; refer to PDF]

It follows from (5.12) that there exist 0<σ<1 and _M1 >0 such that [figure omitted; refer to PDF] Thus, we have [figure omitted; refer to PDF] Here we have used the following inequality: [figure omitted; refer to PDF] Thus by Theorem 4.2 and Remark 5.2, there exists _R1 >r such that [figure omitted; refer to PDF] and hence we obtained [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] We may suppose that A has no fixed point on ∂_Br2 ∩K (otherwise, the proof is finished). Therefore by (5.23), [figure omitted; refer to PDF] Hence we have from Theorem 4.1 and Remark 5.2 that [figure omitted; refer to PDF]

It follows from (5.22) that there exists [straight epsilon]>0 such that f(u)...5;(^λ11-p +[straight epsilon])^up-1 when u is sufficiently large. We know from the continuity of f that there exists b...5;0 such that [figure omitted; refer to PDF] Take [figure omitted; refer to PDF] where _m0 =_{min t∈[0,1]} a(t) , _M1 =_{max t∈[0,1]} a(t) .

For v∈∂_BR2 ∩K , we have [figure omitted; refer to PDF]

Thus, we have [figure omitted; refer to PDF] It follows from Lemma 2.4 that [figure omitted; refer to PDF] and hence we obtained [figure omitted; refer to PDF] 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.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. The paper is supported by the National Science Foundation of China (10971179) and Research Award Fund for Outstanding Young Scientists of Shandong Province (BS2010SF023).

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