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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 305279, 14 pages
http://dx.doi.org/10.1155/2012/305279
Research Article

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: LishanΒ Liu

Copyright Β© 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.

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 π‘’βˆˆπ‘ƒβ§΅{πœƒ}βˆ’π‘ƒ for which 𝑀𝑇𝑝𝑒β‰₯𝑒 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.

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’skiΔ­ [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, π‘ƒβŠ‚π‘‹ a cone in 𝑋. Let π‘‡βˆΆπ‘‹β†’π‘‹ be a linear, positive, and compact operator. Suppose that for some non-zero element 𝑒=π‘£βˆ’π‘€, where 𝑣,π‘€βˆˆπ‘ƒ and βˆ’π‘’βˆ‰π‘ƒ, the following relation is satisfied: 𝑀𝑇𝑝𝑒β‰₯𝑒,π‘“π‘œπ‘Ÿπ‘ π‘œπ‘šπ‘’π‘€>0,(1.1) where 𝑝 is some positive integer. Then 𝑇 has a non-zero eigenvector π‘₯0 in 𝑃: 𝑇π‘₯0=πœ†0π‘₯0,(1.2) where the positive eigenvalue πœ†0 satisfies the inequality πœ†0π‘βˆšπ‘€β‰₯1.
Furthermore, if 𝑃 is a reproducing cone and 𝑇 is 𝑒-positive for some π‘’βˆˆπ‘ƒβ§΅{πœƒ}, then(1)the positive eigenvalue πœ†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, π‘ƒβŠ‚π‘‹ be a cone in 𝑋. Let π‘‡βˆΆπ‘‹β†’π‘‹ be an increasing, positively 1-homogeneous, compact, continuous operator for which there exists a non-zero π‘’βˆˆπ‘ƒ and 𝑀>0 such that 𝑀𝑇𝑒β‰₯𝑒,(1.3) Then T has a non-zero eigenvector π‘₯0 in 𝑃.

Compared with Theorem 1.1, we note that the element 𝑒, 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 π‘’βˆˆπ‘ƒ 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 π‘’βˆˆπ‘ƒβ§΅{πœƒ}βˆ’π‘ƒ for which 𝑀𝑇𝑝𝑒β‰₯𝑒 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, π‘ƒβŠ‚π‘‹ be a cone in 𝑋. A cone 𝑃 is called solid if it contains interior points, that is, π‘œπ‘ƒβ‰ βˆ…. A cone is said to be reproducing if 𝑋=π‘ƒβˆ’π‘ƒ. Every cone 𝑃 in 𝐸 defines a partial ordering in 𝐸 given by π‘₯≀𝑦 if and only if π‘¦βˆ’π‘₯βˆˆπ‘ƒ. If π‘₯≀𝑦 and π‘₯≠𝑦, we write π‘₯<𝑦; if cone 𝑃 is solid and π‘¦βˆ’π‘₯βˆˆπ‘œπ‘ƒ, we write π‘₯β‰ͺ𝑦. For the concepts and the properties about the cone we refer to [12, 13].

A mapping π‘‡βˆΆπ‘‹β†’π‘‹ is said to be increasing if π‘₯≀𝑦 implies 𝑇π‘₯≀𝑇𝑦 and it is said to be strictly increasing if π‘₯<𝑦 implies 𝑇π‘₯<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(𝑑π‘₯)=𝑑𝑇(π‘₯)βˆ€π‘₯βˆˆπ‘‹,π‘‘βˆˆβ„+0=[0,+∞).(2.1) We say that a real number πœ† is an eigenvalue of the operator if there exists a non-zero π‘₯βˆˆπ‘‹ such that 𝑇π‘₯=πœ†π‘₯.

Definition 2.1. Let π‘’βˆˆπ‘ƒβ§΅{πœƒ}, a mapping π‘‡βˆΆπ‘ƒβ†’π‘ƒ is called π‘’βˆ’positive if for every non-zero π‘₯βˆˆπ‘ƒ a natural number 𝑛=𝑛(π‘₯) and two positive number 𝑐(π‘₯), 𝑑(π‘₯) can be found such that 𝑐(π‘₯)𝑒≀𝑇𝑛π‘₯≀𝑑(π‘₯)𝑒.(2.2)

This is stronger than requiring that 𝑇 is positive, that is, 𝑇(𝑃)βŠ‚π‘ƒ. It is always satisfied if 𝑃 is a solid cone and 𝑇 is strongly positive, that is, 𝑇(𝑃)βŠ‚π‘œπ‘ƒ, with any π‘’βˆˆπ‘ƒβ§΅{πœƒ}, 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 πΉβˆΆβ„+0×𝑃→𝑃 be a compact, continuous map and such that 𝐹(0,π‘₯)=πœƒ for all π‘₯βˆˆπ‘ƒ. Then, 𝐹(πœ†,π‘₯)=π‘₯ has a nontrivial connected unbounded component of solutions 𝐢+βŠ‚β„+0×𝑃 containing the point (0,πœƒ).

Lemma 2.3. Let π‘₯βˆˆπ‘ƒ. For an element π‘¦βˆˆπ‘‹, suppose a 𝛿1 can be found such that 𝑦≀𝛿1π‘₯. Then a small 𝛿π‘₯(𝑦) exists for which 𝑦≀𝛿π‘₯(𝑦)π‘₯.

Lemma 2.4. Let Ξ© be a bounded open set in 𝑋, let 𝑃 be a cone in 𝑋, and let π΄βˆΆπ‘ƒβ†’π‘ƒ be a completely continuous map. Suppose that there is an increasing, positively 1-homogeneous mapping 𝑇 and π‘’βˆ—βˆˆπ‘ƒβ§΅{πœƒ} such that π‘‡π‘’βˆ—β‰₯π‘’βˆ—, and that 𝐴𝑒β‰₯𝑇𝑒,𝐴𝑒≠𝑒,βˆ€π‘’βˆˆπœ•Ξ©βˆ©π‘ƒ.(2.3)
Then the fixed-point index 𝑖(𝐴,Ξ©βˆ©π‘ƒ,𝑃)=0.

3. Main Results

Theorem 3.1. Let π‘‡βˆΆπ‘‹β†’π‘‹ be an increasing, positively 1-homogeneous, compact, continuous mapping. Suppose that for some non-zero element 𝑒=π‘£βˆ’π‘€, where 𝑣,π‘€βˆˆπ‘ƒ and βˆ’π‘’βˆ‰π‘ƒ, the following relation is satisfied: 𝑀𝑇𝑝𝑒β‰₯𝑒(π‘“π‘œπ‘Ÿπ‘ π‘œπ‘šπ‘’π‘€>0),(3.1) where 𝑝 is some positive integer. Then 𝑇 has a non-zero eigenvector π‘₯0 in 𝑃: 𝑇π‘₯0=πœ†0π‘₯0,(3.2) where the positive eigenvalue πœ†0 satisfies the inequality πœ†0π‘βˆšπ‘€β‰₯1.(3.3)

Proof. Let π‘£βˆˆπ‘ƒ(π‘£β‰ πœƒ) be as in the hypothesis of the theorem. For every positive integer 𝑛>0, define πΉπ‘›βˆΆβ„+0×𝑃→𝑃 by 𝐹𝑛1(πœ†,π‘₯)∢=πœ†π‘‡π‘₯+π‘›πœ†π‘£.(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 π‘‡πœƒ=πœƒ. Let, by Lemma 2.2, 𝐢+π‘›βŠ‚β„+0×𝑃→𝑃 be a connected unbounded branch of solutions to the equation 𝐹𝑛(πœ†,π‘₯)=π‘₯.(3.5)
First we show that 𝐢+π‘›βŠ‚[0,π‘βˆšπ‘€]×𝑃 for all 𝑛>0. Indeed, suppose that π‘₯ is a fixed point of 𝐹𝑛(πœ†,β‹…) for some πœ†β‰₯1. Then π‘₯=𝐹𝑛(πœ†,π‘₯)=πœ†π‘‡π‘₯+(1/𝑛)πœ†π‘£ and we obtain, from the properties of 𝑇 and the inequalities 𝑣β‰₯πœƒ, 𝑣β‰₯𝑒, respectively, that 1π‘₯β‰₯πœ†π‘‡π‘₯,π‘₯β‰₯π‘›πœ†π‘’.(3.6)
Let πœπ‘›=sup{𝜏|π‘₯β‰₯πœπ‘’}. Obviously, πœπ‘›β‰₯(1/𝑛)πœ†>0. Since 𝑇 is increasing and 1-homogeneous, by (3.4), we have 1π‘₯=πœ†π‘‡π‘₯+π‘›πœ†π‘£β‰₯πœ†2𝑇21π‘₯+π‘›πœ†π‘£β‰₯β‹―β‰₯πœ†π‘π‘‡π‘1π‘₯+π‘›πœ†π‘£β‰₯πœ†π‘π‘‡π‘π‘₯β‰₯πœ†π‘π‘‡π‘ξ€·πœπ‘›π‘’ξ€Έβ‰₯πœπ‘›πœ†π‘π‘€π‘’.(3.7) Consequently, by the definition of πœπ‘›, πœ†π‘π‘€β‰€1.(3.8) In other words, if πœ†>π‘βˆšπ‘€, then 𝐹𝑛(πœ†,β‹…) has no fixed point. This implied that 𝐢+π‘›βŠ‚[0,π‘βˆšπ‘€]×𝑃 for every 𝑛>0.
Notice that the branch 𝐢+𝑛 is connected and unbounded starting from (0,πœƒ), there must necessarily exist π‘₯𝑛 with β€–π‘₯𝑛‖=1 and πœ†π‘›βˆˆ[0,π‘βˆšπ‘€] such that (πœ†π‘›,π‘₯𝑛)∈𝐢+𝑛. That is, π‘₯𝑛=πœ†π‘›π‘‡π‘₯𝑛+1π‘›πœ†π‘›π‘£,πœ†π‘›βˆˆξ‚ƒ0,π‘βˆšπ‘€ξ‚„,β€–β€–π‘₯𝑛‖‖=1.(3.9) Since the operator 𝑇 is compact, a subsequence of indices 𝑛𝑖(𝑖=1,2,…) can be chosen such that the sequence 𝑇π‘₯𝑛 strongly converges to some element π‘¦βˆ—βˆˆπ‘ƒ. By virtue of (3.9), with this choice of the sequence 𝑛𝑖, the convergence of the number πœ†π‘›π‘– to some πœ†βˆ— which satisfies the inequality (3.3) can be guaranteed simultaneously. Then π‘₯𝑛𝑖 will converge in norm to the element π‘₯0=πœ†βˆ—π‘¦βˆ— with β€–π‘₯0β€–=1. Further, it follows from the fact β€–π‘₯0β€–=1 that πœ†βˆ—β‰ 0. Let πœ†0=πœ†βˆ—βˆ’1. To obtain the equality (3.2), it suffices to pass to the limits in the equality: π‘₯𝑛𝑖=πœ†π‘›π‘–π‘‡π‘₯𝑛𝑖+1π‘›π‘–πœ†π‘›π‘–π‘£(𝑖=1,2,…).(3.10) This completes the proof of the theorem.

Example 3.2. Consider the positive 1-homogeneous map 𝑇: ξ‚΅ξ€œ(𝑇π‘₯)(𝑑)=𝐺𝐾(𝑑,𝑠)π‘₯𝑝(𝑠)𝑑𝑠1/𝑝,(3.11) where 𝐺 is a bounded closed set in a finite-dimensional space, the kernel 𝐾(𝑑,𝑠) is nonnegative, and 𝑝=2𝑛+1 for some π‘›βˆˆβ„•.
If there exists a system of points 𝑠1,𝑠2,…,𝑠𝑝 such that 𝐾𝑠1,𝑠2𝐾𝑠2,𝑠3𝑠⋯𝐾𝑝,𝑠1ξ€Έ>0.(3.12) Then the map 𝑇 defined by (3.11) has a nonnegative eigenfunction. In fact, it is easy to see that (π‘‡π‘ξ‚΅ξ€œπ‘₯)(𝑑)=𝐺𝐾(𝑝)(𝑑,𝑠)π‘₯𝑝(𝑠)𝑑𝑠1/𝑝,(3.13) where 𝐾(𝑝)(ξ€œπ‘‘,𝑠)=πΊβ‹―ξ€œπΊπΎξ€·π‘‘,𝑑1ξ€Έξ€·π‘‘β‹―πΎπ‘βˆ’1ξ€Έ,𝑠𝑑𝑑1β‹―π‘‘π‘‘π‘βˆ’1(3.14) is positive at the point (𝑠1,𝑠1) of the topological product 𝐺×𝐺. We denoted by 𝐺1βŠ‚πΊ a closed neighborhood of the point 𝑠1∈𝐺1 such that 𝐾(𝑝)(𝑑,𝑠)>0 when 𝑑,π‘ βˆˆπΊ1. We denote by 𝑦(𝑑) a continuous nonnegative function such that 𝑦(𝑠1)>0, 𝑦(𝑑)=0 when π‘‘βˆ‰πΊ1. Then (π‘‡π‘ξ‚΅ξ€œπ‘¦)(𝑑)=𝐺𝐾(𝑝)(𝑑,𝑠)𝑦𝑝(𝑠)𝑑𝑠1/𝑝>0(3.15) when π‘‘βˆˆπΊ1. From (3.15) it follows that there exists a number 𝑀>0 such that 𝑀𝑇𝑝𝑦β‰₯𝑦.(3.16) 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 π‘’βˆˆπ‘ƒβ§΅{πœƒ}. Moreover, Theorem 2.1 and Corollary 2.1 of [5] already give a general result for a π‘˜-set contraction, positive 1-homogeneous maps.

Theorem 3.4. Suppose that 𝑇 is an increasing, positively 1-homogeneous, 𝑒-positive mapping. If there exist 𝑒1βˆˆπ‘ƒβ§΅{πœƒ},πœ†1>0π‘ π‘’π‘β„Žπ‘‘β„Žπ‘Žπ‘‘πœ†1𝑒1≀𝑇𝑒1,𝑒2βˆˆπ‘ƒβ§΅{πœƒ},πœ†2>0π‘ π‘’π‘β„Žπ‘‘β„Žπ‘Žπ‘‘πœ†2𝑒2β‰₯𝑇𝑒2,(3.17) then πœ†2β‰₯πœ†1. Furthermore, if for π‘₯>𝑦>πœƒ, a positive number 𝑐(π‘₯,𝑦) can be found such that 𝑇π‘₯βˆ’π‘‡π‘¦β‰₯𝑐(π‘₯,𝑦)𝑒,(3.18) then πœ†1=πœ†2 implies that 𝑒1 is a scalar multiple of 𝑒2.

Proof. It follows from the 𝑒-positiveness of 𝑇 that there exist π‘š,𝑛 such that 𝑇𝑛𝑒1𝑒≀𝑑1𝑒𝑒,𝑐2ξ€Έπ‘’β‰€π‘‡π‘šπ‘’2,𝑐𝑒1𝑒,𝑑2ξ€Έξ€Έ>0.(3.19) Then for 𝑑>0, we have 𝑒𝑑𝑐2ξ€Έξ€·π‘’βˆ’π‘‘1ξ€Έξ€Έπ‘’β‰€π‘‘π‘‡π‘šπ‘’2βˆ’π‘‡π‘›π‘’1β‰€π‘‘πœ†π‘š2𝑒2βˆ’πœ†π‘›1𝑒1.(3.20) From this and the fact that 𝑒1βˆˆπ‘ƒβ§΅{πœƒ} we deduce that π›Ώβ‰œπ›Ώπœ†π‘š2𝑒2(πœ†π‘›1𝑒1)>0, that is, π›Ώπœ†π‘š2𝑒2βˆ’πœ†π‘›1𝑒1βˆˆπ‘ƒ. Since 𝑇 is increasing, 𝑇(π›Ώπœ†π‘š2𝑒2)β‰₯𝑇(πœ†π‘›1𝑒1), from which, by virtue of (3.17), it follows that πœ†2π›Ώπœ†π‘š2𝑒2ξ€·β‰₯π‘‡π›Ώπœ†π‘š2𝑒2ξ€Έξ€·πœ†β‰₯𝑇𝑛1𝑒1ξ€Έβ‰₯πœ†1πœ†π‘›1𝑒1.(3.21) By Lemma 2.3, we obtain that πœ†2β‰₯πœ†1.
Now suppose that πœ†1=πœ†2. According to the above proof, the number 𝛿>0. Since π›Ώπœ†π‘š2𝑒2βˆ’πœ†π‘›1𝑒1βˆˆπ‘ƒ and (3.18), if π›Ώπœ†π‘š2𝑒2β‰ πœ†π‘›1𝑒1, then there exists 𝑐>0 such that π‘‡ξ€·π›Ώπœ†π‘š2𝑒2ξ€Έξ€·πœ†βˆ’π‘‡π‘›1𝑒1ξ€Έβ‰₯𝑐𝑒.(3.22) Therefore, by (3.19) πœ†2π›Ώπœ†π‘š2𝑒2ξ€·β‰₯π‘‡π›Ώπœ†π‘š2𝑒2ξ€Έξ€·πœ†β‰₯𝑇𝑛1𝑒1ξ€Έ+𝑐𝑒β‰₯πœ†1πœ†π‘›1𝑒1+𝑐𝑑𝑒1𝑇𝑛𝑒1ξ€Έβ‰₯πœ†1𝑛+1𝑒1+𝑐𝑑𝑒1ξ€Έπœ†π‘›1𝑒1=𝑐1+πœ†1𝑑𝑒1ξ€Έξƒͺπœ†2πœ†π‘›1𝑒1.(3.23) This contradicts with the definition of 𝛿. This shows that we must have π›Ώπœ†π‘š2𝑒2=πœ†π‘›1𝑒1. 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 Ξ© is a bounded open set in a Banach space 𝑋, we write Ξ©π‘ƒβˆΆ=Ξ©βˆ©π‘ƒ and πœ•Ξ©π‘ƒ for its boundary relative to 𝑃.

Theorem 4.1. Let Ξ© be a bounded open set in 𝑋 containing πœƒ, let 𝑃 be a cone in 𝑋, and let π΄βˆΆπ‘ƒβ†’π‘ƒ be a completely continuous map. Suppose that there is an increasing, positively 1-homogeneous, 𝑒-positive mapping 𝑇 such that 𝑇𝑒≀𝑒, and that 𝐴𝑒≀𝑇𝑒,π΄π‘’β‰ π‘’βˆ€π‘’βˆˆπœ•Ξ©π‘ƒ.(4.1) Then the fixed-point index 𝑖(𝐴,Ω𝑃,𝑃)=1.

Proof. We show that π΄π‘’β‰ πœ‡π‘’ for all π‘’βˆˆπœ•Ξ©π‘ƒ and all πœ‡β‰₯1, from which the result follows by standard properties of fixed-point index (see, e.g., [12–14]). Suppose that there exist 𝑒0βˆˆπœ•Ξ©π‘ƒ and πœ‡0β‰₯1 such that 𝐴𝑒0=πœ‡0𝑒0, then πœ‡0>1. It follows from the 𝑒-positiveness of 𝑇 that there exists a natural number 𝑛 such that 𝑇𝑛𝑒0𝑒≀𝑑0𝑒𝑒,𝑑0ξ€Έ>0.(4.2) So, by induction, forallπ‘šβˆˆπ‘, we have πœ‡0π‘šπ‘›π‘’0β‰€π‘‡π‘šπ‘›π‘’0𝑒≀𝑑0𝑒(4.3) which implies 𝑒0=πœƒ. This contradicts 𝑒0βˆˆπœ•Ξ©π‘ƒ.

Theorem 4.2. Let 𝑃 be a normal cone in a real Banach space 𝑋 and let π΄βˆΆπ‘ƒβ†’π‘ƒ 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 π‘˜βˆˆ[0,1) such that π‘‡π‘’β‰€π‘˜π‘’,(4.4)(2)there exists 𝑀>0 such that 𝐴𝑒≀𝑇𝑒+𝑀𝑒,π‘’βˆˆπ‘ƒ.(4.5)Then there exists 𝑅1>0 such that for any 𝑅>𝑅1, the fixed-point index 𝑖(𝐴,π΅π‘…βˆ©π‘ƒ,𝑃)=1, where 𝐡𝑅={π‘₯βˆˆπ‘‹|β€–π‘₯‖≀𝑅}.

Proof. Let π‘Š={π‘’βˆˆπ‘ƒβˆ£π‘’=πœ‡π΄π‘’,0β‰€πœ‡β‰€1}.(4.6) In the following, we prove that π‘Š is bounded.
For any π‘’βˆˆπ‘Šβ§΅{πœƒ}, using the 𝑒-positiveness of 𝑇, we have 𝑒=πœ‡π΄π‘’β‰€π‘‡π‘’+𝑀𝑒≀(𝑑(𝑒)+𝑀)𝑒.(4.7) Let πœ‡0=inf{πœ‡βˆ£π‘’β‰€πœ‡π‘’}.(4.8) It is easy to see that 0<πœ‡0<+∞ and π‘’β‰€πœ‡0𝑒. We now have 𝑒≀𝑇𝑒+π‘€π‘’β‰€π‘˜πœ‡0ξ€Έ+𝑀𝑒,(4.9) which, by the definition of πœ‡0, implies that πœ‡0≀𝑀/(1βˆ’π‘˜). So we know that 𝑒≀(𝑀/(1βˆ’π‘˜))𝑒 and π‘Š is bounded by the normality of the cone 𝑃.
Select 𝑅1>sup{β€–π‘₯β€–|π‘₯βˆˆπ‘Š}. Then from the homotopy invariance property of fixed-point index we have 𝑖𝐴,π΅π‘…ξ€Έξ€·βˆ©π‘ƒ,𝑃=π‘–πœƒ,π΅π‘…ξ€Έβˆ©π‘ƒ,𝑃=1,βˆ€π‘…>𝑅1.(4.10)
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: ξ€·πœ™π‘ξ€·π‘£ξ…žπ‘£ξ€Έξ€Έβ€²+π‘Ž(𝑑)𝑓(𝑣(𝑑))=0,π‘‘βˆˆ(0,1),ξ…ž(0)=0,𝑣(1)=0,(5.1) where πœ™π‘(𝑠)=|𝑠|π‘βˆ’2𝑠, 𝑝β‰₯2, and (πœ™π‘)βˆ’1=πœ™π‘ž=|𝑠|π‘žβˆ’2𝑠, (1/𝑝)+(1/π‘ž)=1.

We make the following assumptions:(𝐻1)π‘“βˆΆ[0,+∞)β†’[0,+∞) is continuous;(𝐻2)π‘ŽβˆΆ[0,1]β†’(0,+∞) is continuous.

For each π‘£βˆˆπΈβˆΆ=𝐢[0,1], we write ‖𝑣‖=max{|𝑣(𝑑)|βˆΆπ‘‘βˆˆ[0,1]}. Define 𝐾={π‘£βˆˆπΈβˆΆπ‘£(𝑑)β‰₯(1βˆ’π‘‘)‖𝑣‖}.(5.2) Clearly, (𝐸,β€–β‹…β€–) is a Banach space and 𝐾 is a cone of 𝐸. For any real constant π‘Ÿ>0, define π΅π‘Ÿ={π‘£βˆˆπΈβˆΆβ€–π‘₯β€–<π‘Ÿ}.

Let the operators 𝑇 and 𝐴 be defined by: ξ€œ(𝑇𝑣)(𝑑)=1π‘‘πœ™π‘žξ‚΅ξ€œπ‘ 0π‘Ž(𝜏)π‘£π‘βˆ’1ξ‚Άξ€œ(𝜏)π‘‘πœπ‘‘π‘ ,(𝐴𝑣)(𝑑)=1π‘‘πœ™π‘žξ‚΅ξ€œπ‘ 0ξ‚Άπ‘Ž(𝜏)𝑓(𝑣(𝜏))π‘‘πœπ‘‘π‘ ,(5.3) 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 π‘‡βˆΆπΎβ†’πΈ is a completely continuous, positively 1-homogeneous operator and 𝑇(𝐾)βŠ‚πΎ.

Lemma 5.1. Suppose that (𝐻2) holds. Then for the operator 𝑇 defined by (5.3), there is a unique positive eigenvalue πœ†1 of 𝑇 with its eigenfunction in 𝐾.

Proof. First, we show that 𝑇 is 𝑒-positive with 𝑒=1βˆ’π‘‘, that is, for any 𝑣>πœƒ from 𝐾, there exist 𝛼,𝛽>0 such that 𝛼𝑒≀𝑇𝑣≀𝛽𝑒.(5.4) Let 𝑀1=maxπ‘‘βˆˆ[0,1]π‘Ž(𝑑). Then ξ€œ(𝑇𝑣)(𝑑)≀1π‘‘πœ™π‘žξ€·π‘€1β€–π‘£β€–π‘βˆ’1𝑑𝑠=πœ™π‘žξ€·π‘€1β€–π‘£β€–π‘βˆ’1ξ€Έ(1βˆ’π‘‘).(5.5) So, we may take 𝛽=πœ™π‘ž(𝑀1β€–π‘£β€–π‘βˆ’1).
Clearly, we may take 𝛼=‖𝑇𝑣‖=(𝑇𝑣)(0) since 𝑇(𝐾)βŠ‚πΎ. So (5.4) is proved.
Now we need to show that for any 𝑒>𝑣>πœƒ, there always exists some 𝑐>0 such that π‘‡π‘’βˆ’π‘‡π‘£β‰₯𝑐𝑒.(5.6) In fact, we note that πœ™π‘ž is increasing, there exists an πœ‚βˆˆ(0,1) such that π‘š=minπ‘ βˆˆ[πœ‚,1]πœ™π‘žξ‚΅ξ€œπ‘ 0π‘Ž(𝜏)π‘’π‘βˆ’1ξ‚Ά(𝜏)π‘‘πœβˆ’πœ™π‘žξ‚΅ξ€œπ‘ 0π‘Ž(𝜏)π‘£π‘βˆ’1ξ‚Ά(𝜏)π‘‘πœ>0.(5.7) Then for all π‘‘βˆˆ[πœ‚,1], we have ξ€œ(𝑇𝑒)(𝑑)βˆ’(𝑇𝑣)(𝑑)β‰₯1πœ‚πœ™π‘žξ‚΅ξ€œπ‘ 0π‘Ž(𝜏)π‘’π‘βˆ’1ξ‚Άβˆ’ξ€œ(𝜏)π‘‘πœπ‘‘π‘ 1πœ‚πœ™π‘žξ‚΅ξ€œπ‘ 0π‘Ž(𝜏)π‘£π‘βˆ’1ξ‚Ά(𝜏)π‘‘πœπ‘‘π‘ β‰₯π‘š(1βˆ’π‘‘).(5.8) Since (𝑇𝑒)(𝑑)βˆ’(𝑇𝑣)(𝑑)β‰₯(𝑇𝑒)(πœ‚)βˆ’(𝑇𝑣)(πœ‚)β‰₯π‘š(1βˆ’πœ‚) for all π‘‘βˆˆ[0,πœ‚], we have (𝑇𝑒)(𝑑)βˆ’(𝑇𝑣)(𝑑)β‰₯π‘š(1βˆ’πœ‚)(1βˆ’π‘‘),(5.9) for all π‘‘βˆˆ[0,1]. Therefore, the proof is complete and follows from Theorems 3.1 and 3.4.

Remark 5.2. Let πœ‘βˆ— be the positive eigenfunction of 𝑇 corresponding to πœ†1, thus πœ†1π‘‡πœ‘βˆ—=πœ‘βˆ—. Then by Lemma 5.1, there exist 𝛼,𝛽>0 such that π›Όπ‘’β‰€π‘‡πœ‘βˆ—=πœ†1πœ‘βˆ—β‰€π›½π‘’.(5.10) Hence we obtained that 𝑇 is πœ‘βˆ—-positive operator.

Theorem 5.3. Suppose that the conditions (𝐻1) and (𝐻2) are satisfied, and liminf𝑒→0+𝑓(𝑒)π‘’π‘βˆ’1>πœ†11βˆ’π‘,(5.11)limsup𝑒→+βˆžπ‘“(𝑒)π‘’π‘βˆ’1<πœ†11βˆ’π‘,(5.12) where πœ†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 π‘Ÿ>0 such that 𝑓(𝑒)β‰₯πœ†11βˆ’π‘π‘’π‘βˆ’1,βˆ€0β‰€π‘’β‰€π‘Ÿ.(5.13) We may suppose that 𝐴 has no fixed point on πœ•π΅π‘Ÿβˆ©πΎ (otherwise, the proof is finished). Therefore by (5.13), ξ€œ(𝐴𝑣)(𝑑)β‰₯1π‘‘πœ™π‘žξ‚΅ξ€œπ‘ 0π‘Ž(𝜏)πœ†11βˆ’π‘π‘’π‘βˆ’1ξ‚Ά1(𝜏)π‘‘πœπ‘‘π‘ =πœ†1(𝑇𝑣)(𝑑),π‘£βˆˆπœ•π΅π‘Ÿβˆ©πΎ.(5.14) Hence we have from Lemma 2.4 and Remark 5.2 that 𝑖𝐴,π΅π‘Ÿξ€Έβˆ©πΎ,𝐾=0.(5.15)
It follows from (5.12) that there exist 0<𝜎<1 and 𝑀1>0 such that 𝑓(𝑒)β‰€πœŽπ‘βˆ’1πœ†11βˆ’π‘π‘’π‘βˆ’1+𝑀2,βˆ€π‘’β‰₯0.(5.16) Thus, we have ξ€œ(𝐴𝑣)(𝑑)≀1π‘‘πœ™π‘žξ‚΅ξ€œπ‘ 0ξ‚ƒπœŽπ‘Ž(𝜏)π‘βˆ’1πœ†11βˆ’π‘π‘’π‘βˆ’1(𝜏)+𝑀2ξ‚„ξ‚Άβ‰€πœŽπ‘‘πœπ‘‘π‘ πœ†1ξ€œ1π‘‘πœ™π‘žξ‚΅ξ€œπ‘ 0π‘Ž(𝜏)π‘’π‘βˆ’1ξ‚Άξ€œ(𝜏)π‘‘πœπ‘‘π‘ +1π‘‘πœ™π‘žξ€·π‘€1𝑀2ξ€Έ=πœŽπ‘‘π‘ πœ†1(𝑇𝑣)(𝑑)+πœ™π‘žξ€·π‘€1𝑀2ξ€Έ(1βˆ’π‘‘),π‘£βˆˆπΎ.(5.17) Here we have used the following inequality: πœ™π‘ž(π‘Ž+𝑏)β‰€πœ™π‘ž(π‘Ž)+πœ™π‘ž(𝑏),π‘Ž,𝑏>0,1<π‘žβ‰€2.(5.18) Thus by Theorem 4.2 and Remark 5.2, there exists 𝑅1>π‘Ÿ such that 𝑖𝐴,π΅π‘…ξ€Έβˆ©πΎ,𝐾=0,𝑅>𝑅1,(5.19) and hence we obtained 𝑖𝐡𝐴,π‘…β§΅π΅π‘Ÿξ‚ξ‚βˆ©πΎ,𝐾=1.(5.20) Thus, 𝐴 has a fixed point in (π΅π‘…β§΅π΅π‘Ÿ)∩𝐾. Consequently, (5.1) has a positive solution.

Theorem 5.4. Suppose that the conditions (𝐻1) and (𝐻2) are satisfied, and limsup𝑒→0+𝑓(𝑒)π‘’π‘βˆ’1<πœ†11βˆ’π‘,(5.21)liminf𝑒→+βˆžπ‘“(𝑒)π‘’π‘βˆ’1>πœ†11βˆ’π‘,(5.22) where πœ†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 π‘Ÿ2>0 such that 𝑓(𝑒)β‰€πœ†11βˆ’π‘π‘’π‘βˆ’1,βˆ€0β‰€π‘’β‰€π‘Ÿ2.(5.23) We may suppose that 𝐴 has no fixed point on πœ•π΅π‘Ÿ2∩𝐾 (otherwise, the proof is finished). Therefore by (5.23), ξ€œ(𝐴𝑣)(𝑑)≀1π‘‘πœ™π‘žξ‚΅ξ€œπ‘ 0π‘Ž(𝜏)πœ†11βˆ’π‘π‘’π‘βˆ’1ξ‚Ά1(𝜏)π‘‘πœπ‘‘π‘ =πœ†1(𝑇𝑣)(𝑑),π‘£βˆˆπœ•π΅π‘Ÿ2∩𝐾.(5.24) Hence we have from Theorem 4.1 and Remark 5.2 that 𝑖𝐴,π΅π‘Ÿ2ξ€Έβˆ©πΎ,𝐾=1.(5.25)
It follows from (5.22) that there exists πœ€>0 such that 𝑓(𝑒)β‰₯(πœ†11βˆ’π‘+πœ€)π‘’π‘βˆ’1 when 𝑒 is sufficiently large. We know from the continuity of 𝑓 that there exists 𝑏β‰₯0 such that ξ‚€πœ†π‘“(𝑒)β‰₯11βˆ’π‘ξ‚π‘’+πœ€π‘βˆ’1βˆ’π‘,βˆ€π‘’β‰₯0.(5.26) Take 𝑅2ξƒ―π‘Ÿ>max2,𝑀1𝑏2π‘π‘š0πœ€ξ‚Ά1/(π‘βˆ’1)ξƒ°,(5.27) where π‘š0=minπ‘‘βˆˆ[0,1]π‘Ž(𝑑), 𝑀1=maxπ‘‘βˆˆ[0,1]π‘Ž(𝑑).
For π‘£βˆˆπœ•π΅π‘…2∩𝐾, we have ξ€œπ‘ 0ξ€œπ‘Ž(𝜏)𝑓(𝑣(𝜏))π‘‘πœβ‰₯𝑠0πœ†π‘Ž(𝜏)11βˆ’π‘ξ‚π‘£+πœ€π‘βˆ’1ξ‚„(𝜏)βˆ’π‘π‘‘πœβ‰₯πœ†11βˆ’π‘ξ€œπ‘ 0π‘Ž(𝜏)π‘£π‘βˆ’1(𝜏)π‘‘πœ+π‘š0πœ€ξ€œπ‘ 0(1βˆ’πœ)π‘βˆ’1β€–π‘£β€–π‘βˆ’1π‘‘πœβˆ’π‘€1𝑏𝑠=πœ†11βˆ’π‘ξ€œπ‘ 0π‘Ž(𝜏)π‘£π‘βˆ’1(𝜏)π‘‘πœ+π‘š0πœ€π‘…2π‘βˆ’1ξ€œπ‘ 0(1βˆ’πœ)π‘βˆ’1π‘‘πœβˆ’π‘€1𝑏𝑠β‰₯πœ†11βˆ’π‘ξ€œπ‘ 0π‘Ž(𝜏)π‘£π‘βˆ’1(𝜏)π‘‘πœ+π‘š0πœ€π‘…2π‘βˆ’1ξ€œ0min{𝑠,1/2}(1βˆ’πœ)π‘βˆ’1π‘‘πœβˆ’π‘€1𝑏𝑠β‰₯πœ†11βˆ’π‘ξ€œπ‘ 0π‘Ž(𝜏)π‘£π‘βˆ’1(𝜏)π‘‘πœ+π‘š0πœ€π‘…2π‘βˆ’1ξ€œ0min{𝑠,1/2}ξ‚€11βˆ’2ξ‚π‘βˆ’1π‘‘πœβˆ’π‘€1𝑏𝑠β‰₯πœ†11βˆ’π‘ξ€œπ‘ 0π‘Ž(𝜏)π‘£π‘βˆ’1ξ‚€π‘š(𝜏)π‘‘πœ+0πœ€π‘…2π‘βˆ’112π‘βˆ’π‘€1𝑏𝑠β‰₯πœ†11βˆ’π‘ξ€œπ‘ 0π‘Ž(𝜏)π‘£π‘βˆ’1(𝜏)π‘‘πœ.(5.28)
Thus, we have ξ€œ(𝐴𝑣)(𝑑)β‰₯1π‘‘πœ™π‘žξ‚΅ξ€œπ‘ 0π‘Ž(𝜏)πœ†11βˆ’π‘π‘’π‘βˆ’1ξ‚Ά1(𝜏)π‘‘πœπ‘‘π‘ =πœ†1(𝑇𝑣)(𝑑),π‘£βˆˆπœ•π΅π‘…2∩𝐾.(5.29) It follows from Lemma 2.4 that 𝑖𝐴,𝐡𝑅2ξ€Έβˆ©πΎ,𝐾=0,(5.30) and hence we obtained 𝑖𝐡𝐴,𝑅2β§΅π΅π‘Ÿ2ξ‚ξ‚βˆ©πΎ,𝐾=βˆ’1.(5.31) Thus, 𝐴 has a fixed point in (𝐡𝑅2β§΅π΅π‘Ÿ2)∩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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