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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 305279, 14 pages
A Generalization of Mahadevan's Version of the Krein-Rutman Theorem and Applications to p-Laplacian Boundary Value Problems
1Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266590, China
2Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China
Received 12 January 2012; Revised 24 March 2012; Accepted 13 July 2012
Academic Editor: Lishan Liu
Copyright © 2012 Yujun Cui and Jingxian Sun. 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.
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 and such that there is a nonzero for which for some positive constant 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.
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  introduced the concept of the -positive linear operator and then used it to prove the following results concerning the eigenvalues of positive linear compact operator.
Theorem 1.1. Let , a Banach space, a cone in . Let be a linear, positive, and compact operator. Suppose that for some non-zero element , where and , the following relation is satisfied:
where is some positive integer. Then has a non-zero eigenvector in :
where the positive eigenvalue satisfies the inequality .
Furthermore, if is a reproducing cone and is -positive for some , then(1)the positive eigenvalue of is simple;(2)the operator 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 , and Chang .
The following nonlinear Krein-Rutman theorem has been established in .
Theorem 1.2. Let be a Banach space, be a cone in . Let be an increasing, positively 1-homogeneous, compact, continuous operator for which there exists a non-zero and such that Then T has a non-zero eigenvector in .
Compared with Theorem 1.1, we note that the element , appeared in Theorem 1.2, belongs to . Consequently we put forward a problem: are the results in Theorem 1.2 valid if the condition 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 and such that there is a non-zero for which for some positive constant and some positive integer . The method in this paper is somewhat different from that in .
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 -Laplacian boundary-value problems under some conditions concerning the positive eigenvalues corresponding to the relevant positively 1-homogeneous operators.
Let a Banach space, be a cone in . A cone is called solid if it contains interior points, that is, . A cone is said to be reproducing if . Every cone in defines a partial ordering in given by if and only if . If and , we write ; if cone is solid and , we write . For the concepts and the properties about the cone we refer to [12, 13].
A mapping is said to be increasing if implies and it is said to be strictly increasing if implies . The mapping is said to be compact if it takes bounded subsets of into relatively compact subsets of . We say that the mapping is positively 1-homogeneous if it satisfies the relation We say that a real number is an eigenvalue of the operator if there exists a non-zero such that .
Definition 2.1. Let , a mapping is called positive if for every non-zero a natural number and two positive number , can be found such that
This is stronger than requiring that is positive, that is, . It is always satisfied if is a solid cone and is strongly positive, that is, , with any , 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 be a compact, continuous map and such that for all . Then, has a nontrivial connected unbounded component of solutions containing the point .
Lemma 2.3. Let . For an element , suppose a can be found such that . Then a small exists for which .
Lemma 2.4. Let be a bounded open set in , let be a cone in , and let be a completely continuous map. Suppose that there is an increasing, positively 1-homogeneous mapping and such that , and that
Then the fixed-point index .
3. Main Results
Theorem 3.1. Let be an increasing, positively 1-homogeneous, compact, continuous mapping. Suppose that for some non-zero element , where and , the following relation is satisfied: where is some positive integer. Then has a non-zero eigenvector in : where the positive eigenvalue satisfies the inequality
Proof. Let be as in the hypothesis of the theorem. For every positive integer , define by
Since is compact and continuous, each of these operators is clearly compact and continuous on . Also they map into since maps into itself, which follows from the fact that is increasing and . Let, by Lemma 2.2, be a connected unbounded branch of solutions to the equation
First we show that for all . Indeed, suppose that is a fixed point of for some . Then and we obtain, from the properties of and the inequalities , , respectively, that
Let . Obviously, . Since is increasing and 1-homogeneous, by (3.4), we have Consequently, by the definition of , In other words, if , then has no fixed point. This implied that for every .
Notice that the branch is connected and unbounded starting from , there must necessarily exist with and such that . That is, Since the operator is compact, a subsequence of indices can be chosen such that the sequence strongly converges to some element . By virtue of (3.9), with this choice of the sequence , the convergence of the number to some which satisfies the inequality (3.3) can be guaranteed simultaneously. Then will converge in norm to the element with . Further, it follows from the fact that . Let . To obtain the equality (3.2), it suffices to pass to the limits in the equality: This completes the proof of the theorem.
Example 3.2. Consider the positive 1-homogeneous map :
where is a bounded closed set in a finite-dimensional space, the kernel is nonnegative, and for some .
If there exists a system of points such that Then the map defined by (3.11) has a nonnegative eigenfunction. In fact, it is easy to see that where is positive at the point of the topological product . We denoted by a closed neighborhood of the point such that when . We denote by a continuous nonnegative function such that , when . Then when . From (3.15) it follows that there exists a number such that 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 . Moreover, Theorem 2.1 and Corollary 2.1 of  already give a general result for a -set contraction, positive 1-homogeneous maps.
Theorem 3.4. Suppose that is an increasing, positively 1-homogeneous, -positive mapping. If there exist then . Furthermore, if for , a positive number can be found such that then implies that is a scalar multiple of .
Proof. It follows from the -positiveness of that there exist such that
Then for , we have
From this and the fact that we deduce that , that is, . Since is increasing, , from which, by virtue of (3.17), it follows that
By Lemma 2.3, we obtain that .
Now suppose that . According to the above proof, the number . Since and (3.18), if , then there exists such that Therefore, by (3.19) This contradicts with the definition of . This shows that we must have . This completes the proof of the theorem.
4. Computation for the Fixed-Point Index
We illustrate how -positivity can be used to prove some fixed-point index results which can then be used to prove existence results for nonlinear equations. When is a bounded open set in a Banach space , we write and for its boundary relative to .
Theorem 4.1. Let be a bounded open set in containing , let be a cone in , and let be a completely continuous map. Suppose that there is an increasing, positively 1-homogeneous, -positive mapping such that , and that Then the fixed-point index .
Proof. We show that for all and all , from which the result follows by standard properties of fixed-point index (see, e.g., [12–14]). Suppose that there exist and such that , then . It follows from the -positiveness of that there exists a natural number such that So, by induction, , we have which implies . This contradicts .
Theorem 4.2. Let be a normal cone in a real Banach space and let be a completely continuous map. Suppose that there is an increasing, positively 1-homogeneous, -positive mapping (with in Definition 2.1) which satisfies the following conditions:(1)there exists such that (2)there exists such that Then there exists such that for any , the fixed-point index , where .
In the following, we prove that is bounded.
For any , using the -positiveness of , we have Let It is easy to see that and . We now have which, by the definition of , implies that . So we know that and is bounded by the normality of the cone .
Select . Then from the homotopy invariance property of fixed-point index we have
This completes the proof of the theorem.
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 -Laplacian: where , , and , .
We make the following assumptions: is continuous; is continuous.
For each , we write . Define Clearly, is a Banach space and is a cone of . For any real constant , define .
Let the operators and be defined by: respectively.
Under and , it is not difficult to verify that the non-zero fixed points of the operator are the positive solutions of boundary-value problem (5.1). In addition, we have from that is a completely continuous, positively 1-homogeneous operator and .
Lemma 5.1. Suppose that holds. Then for the operator defined by (5.3), there is a unique positive eigenvalue of with its eigenfunction in .
Proof. First, we show that is -positive with , that is, for any from , there exist such that
Let . Then
So, we may take .
Clearly, we may take since . So (5.4) is proved.
Now we need to show that for any , there always exists some such that In fact, we note that is increasing, there exists an such that Then for all , we have Since for all , we have for all . Therefore, the proof is complete and follows from Theorems 3.1 and 3.4.
Remark 5.2. Let be the positive eigenfunction of corresponding to , thus . Then by Lemma 5.1, there exist such that Hence we obtained that is -positive operator.
Proof. It follows from (5.11) that there exists such that
We may suppose that has no fixed point on (otherwise, the proof is finished). Therefore by (5.13),
Hence we have from Lemma 2.4 and Remark 5.2 that
It follows from (5.12) that there exist and such that Thus, we have Here we have used the following inequality: Thus by Theorem 4.2 and Remark 5.2, there exists such that and hence we obtained Thus, has a fixed point in . Consequently, (5.1) has a positive solution.
Proof. It follows from (5.21) that there exists such that
We may suppose that has no fixed point on (otherwise, the proof is finished). Therefore by (5.23),
Hence we have from Theorem 4.1 and Remark 5.2 that
It follows from (5.22) that there exists such that when is sufficiently large. We know from the continuity of that there exists such that Take where , .
For , we have
Thus, we have It follows from Lemma 2.4 that and hence we obtained Thus, has a fixed point in . Consequently, (5.1) has a positive solution.
Remark 5.5. -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 , there are very good conditions imposed on 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 -Laplacian boundary-value problems.
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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