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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 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: 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.

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 𝑢𝑃{𝜃}𝑃 for which 𝑀𝑇𝑝𝑢𝑢 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 [37]. 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. Let 𝑋, a Banach space, 𝑃𝑋 a cone in 𝑋. Let 𝑇𝑋𝑋 be a linear, positive, and compact operator. Suppose that for some non-zero element 𝑢=𝑣𝑤, where 𝑣,𝑤𝑃 and 𝑢𝑃, the following relation is satisfied: 𝑀𝑇𝑝𝑢𝑢,𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑀>0,(1.1) where 𝑝 is some positive integer. Then 𝑇 has a non-zero eigenvector 𝑥0 in 𝑃: 𝑇𝑥0=𝜆0𝑥0,(1.2) where the positive eigenvalue 𝜆0 satisfies the inequality 𝜆0𝑝𝑀1.
Furthermore, if 𝑃 is a reproducing cone and 𝑇 is 𝑒-positive for some 𝑒𝑃{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, 𝑃𝑋 be a cone in 𝑋. Let 𝑇𝑋𝑋 be an increasing, positively 1-homogeneous, compact, continuous operator for which there exists a non-zero 𝑢𝑃 and 𝑀>0 such that 𝑀𝑇𝑢𝑢,(1.3) 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 𝑢𝑃 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 𝑢𝑃{𝜃}𝑃 for which 𝑀𝑇𝑝𝑢𝑢 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, 𝑃𝑋 be a cone in 𝑋. A cone 𝑃 is called solid if it contains interior points, that is, 𝑜𝑃. A cone is said to be reproducing if 𝑋=𝑃𝑃. Every cone 𝑃 in 𝐸 defines a partial ordering in 𝐸 given by 𝑥𝑦 if and only if 𝑦𝑥𝑃. If 𝑥𝑦 and 𝑥𝑦, we write 𝑥<𝑦; if cone 𝑃 is solid and 𝑦𝑥𝑜𝑃, we write 𝑥𝑦. For the concepts and the properties about the cone we refer to [12, 13].

A mapping 𝑇𝑋𝑋 is said to be increasing if 𝑥𝑦 implies 𝑇𝑥𝑇𝑦 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)𝑥𝑋,𝑡+0=[0,+).(2.1) We say that a real number 𝜆 is an eigenvalue of the operator if there exists a non-zero 𝑥𝑋 such that 𝑇𝑥=𝜆𝑥.

Definition 2.1. Let 𝑒𝑃{𝜃}, a mapping 𝑇𝑃𝑃 is called 𝑒positive if for every non-zero 𝑥𝑃 a natural number 𝑛=𝑛(𝑥) and two positive number 𝑐(𝑥), 𝑑(𝑥) can be found such that 𝑐(𝑥)𝑒𝑇𝑛𝑥𝑑(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)𝑜𝑃, with any 𝑒𝑃{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×𝑃𝑃 be a compact, continuous map and such that 𝐹(0,𝑥)=𝜃 for all 𝑥𝑃. 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 𝑥𝑃. For an element 𝑦𝑋, suppose a 𝛿1 can be found such that 𝑦𝛿1𝑥. Then a small 𝛿𝑥(𝑦) exists for which 𝑦𝛿𝑥(𝑦)𝑥.

Lemma 2.4. Let Ω be a bounded open set in 𝑋, let 𝑃 be a cone in 𝑋, and let 𝐴𝑃𝑃 be a completely continuous map. Suppose that there is an increasing, positively 1-homogeneous mapping 𝑇 and 𝑢𝑃{𝜃} such that 𝑇𝑢𝑢, and that 𝐴𝑢𝑇𝑢,𝐴𝑢𝑢,𝑢𝜕Ω𝑃.(2.3)
Then the fixed-point index 𝑖(𝐴,Ω𝑃,𝑃)=0.

3. Main Results

Theorem 3.1. Let 𝑇𝑋𝑋 be an increasing, positively 1-homogeneous, compact, continuous mapping. Suppose that for some non-zero element 𝑢=𝑣𝑤, where 𝑣,𝑤𝑃 and 𝑢𝑃, the following relation is satisfied: 𝑀𝑇𝑝𝑢𝑢(𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑀>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 𝜆0𝑝𝑀1.(3.3)

Proof. Let 𝑣𝑃(𝑣𝜃) be as in the hypothesis of the theorem. For every positive integer 𝑛>0, define 𝐹𝑛+0×𝑃𝑃 by 𝐹𝑛1(𝜆,𝑥)=𝜆𝑇𝑥+𝑛𝜆𝑣.(3.4) 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, 𝐶+𝑛+0×𝑃𝑃 be a connected unbounded branch of solutions to the equation 𝐹𝑛(𝜆,𝑥)=𝑥.(3.5)
First we show that 𝐶+𝑛[0,𝑝𝑀]×𝑃 for all 𝑛>0. Indeed, suppose that 𝑥 is a fixed point of 𝐹𝑛(𝜆,) for some 𝜆1. Then 𝑥=𝐹𝑛(𝜆,𝑥)=𝜆𝑇𝑥+(1/𝑛)𝜆𝑣 and we obtain, from the properties of 𝑇 and the inequalities 𝑣𝜃, 𝑣𝑢, respectively, that 1𝑥𝜆𝑇𝑥,𝑥𝑛𝜆𝑢.(3.6)
Let 𝜏𝑛=sup{𝜏|𝑥𝜏𝑢}. Obviously, 𝜏𝑛(1/𝑛)𝜆>0. Since 𝑇 is increasing and 1-homogeneous, by (3.4), we have 1𝑥=𝜆𝑇𝑥+𝑛𝜆𝑣𝜆2𝑇21𝑥+𝑛𝜆𝑣𝜆𝑝𝑇𝑝1𝑥+𝑛𝜆𝑣𝜆𝑝𝑇𝑝𝑥𝜆𝑝𝑇𝑝𝜏𝑛𝑢𝜏𝑛𝜆𝑝𝑀𝑢.(3.7) Consequently, by the definition of 𝜏𝑛, 𝜆𝑝𝑀1.(3.8) In other words, if 𝜆>𝑝𝑀, then 𝐹𝑛(𝜆,) has no fixed point. This implied that 𝐶+𝑛[0,𝑝𝑀]×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,𝑝𝑀] such that (𝜆𝑛,𝑥𝑛)𝐶+𝑛. That is, 𝑥𝑛=𝜆𝑛𝑇𝑥𝑛+1𝑛𝜆𝑛𝑣,𝜆𝑛0,𝑝𝑀,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 𝑦𝑃. 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=𝜆𝑦 with 𝑥0=1. Further, it follows from the fact 𝑥0=1 that 𝜆0. Let 𝜆0=𝜆1. To obtain the equality (3.2), it suffices to pass to the limits in the equality: 𝑥𝑛𝑖=𝜆𝑛𝑖𝑇𝑥𝑛𝑖+1𝑛𝑖𝜆𝑛𝑖𝑣(𝑖=1,2,).(3.10) This completes the proof of the theorem.

Example 3.2. Consider the positive 1-homogeneous map 𝑇: (𝑇𝑥)(𝑡)=𝐺𝐾(𝑡,𝑠)𝑥𝑝(𝑠)𝑑𝑠1/𝑝,(3.11) 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 𝐾𝑠1,𝑠2𝐾𝑠2,𝑠3𝑠𝐾𝑝,𝑠1>0.(3.12) Then the map 𝑇 defined by (3.11) has a nonnegative eigenfunction. In fact, it is easy to see that (𝑇𝑝𝑥)(𝑡)=𝐺𝐾(𝑝)(𝑡,𝑠)𝑥𝑝(𝑠)𝑑𝑠1/𝑝,(3.13) where 𝐾(𝑝)(𝑡,𝑠)=𝐺𝐺𝐾𝑡,𝑡1𝑡𝐾𝑝1,𝑠𝑑𝑡1𝑑𝑡𝑝1(3.14) is positive at the point (𝑠1,𝑠1) of the topological product 𝐺×𝐺. We denoted by 𝐺1𝐺 a closed neighborhood of the point 𝑠1𝐺1 such that 𝐾(𝑝)(𝑡,𝑠)>0 when 𝑡,𝑠𝐺1. We denote by 𝑦(𝑡) a continuous nonnegative function such that 𝑦(𝑠1)>0, 𝑦(𝑡)=0 when 𝑡𝐺1. Then (𝑇𝑝𝑦)(𝑡)=𝐺𝐾(𝑝)(𝑡,𝑠)𝑦𝑝(𝑠)𝑑𝑠1/𝑝>0(3.15) when 𝑡𝐺1. From (3.15) it follows that there exists a number 𝑀>0 such that 𝑀𝑇𝑝𝑦𝑦.(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 𝑢𝑃{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 𝑢1𝑃{𝜃},𝜆1>0𝑠𝑢𝑐𝑡𝑎𝑡𝜆1𝑢1𝑇𝑢1,𝑢2𝑃{𝜃},𝜆2>0𝑠𝑢𝑐𝑡𝑎𝑡𝜆2𝑢2𝑇𝑢2,(3.17) then 𝜆2𝜆1. Furthermore, if for 𝑥>𝑦>𝜃, a positive number 𝑐(𝑥,𝑦) can be found such that 𝑇𝑥𝑇𝑦𝑐(𝑥,𝑦)𝑒,(3.18) 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 𝑇𝑛𝑢1𝑢𝑑1𝑢𝑒,𝑐2𝑒𝑇𝑚𝑢2,𝑐𝑢1𝑢,𝑑2>0.(3.19) Then for 𝑡>0, we have 𝑢𝑡𝑐2𝑢𝑑1𝑒𝑡𝑇𝑚𝑢2𝑇𝑛𝑢1𝑡𝜆𝑚2𝑢2𝜆𝑛1𝑢1.(3.20) From this and the fact that 𝑢1𝑃{𝜃} we deduce that 𝛿𝛿𝜆𝑚2𝑢2(𝜆𝑛1𝑢1)>0, that is, 𝛿𝜆𝑚2𝑢2𝜆𝑛1𝑢1𝑃. Since 𝑇 is increasing, 𝑇(𝛿𝜆𝑚2𝑢2)𝑇(𝜆𝑛1𝑢1), from which, by virtue of (3.17), it follows that 𝜆2𝛿𝜆𝑚2𝑢2𝑇𝛿𝜆𝑚2𝑢2𝜆𝑇𝑛1𝑢1𝜆1𝜆𝑛1𝑢1.(3.21) By Lemma 2.3, we obtain that 𝜆2𝜆1.
Now suppose that 𝜆1=𝜆2. According to the above proof, the number 𝛿>0. Since 𝛿𝜆𝑚2𝑢2𝜆𝑛1𝑢1𝑃 and (3.18), if 𝛿𝜆𝑚2𝑢2𝜆𝑛1𝑢1, then there exists 𝑐>0 such that 𝑇𝛿𝜆𝑚2𝑢2𝜆𝑇𝑛1𝑢1𝑐𝑒.(3.22) Therefore, by (3.19) 𝜆2𝛿𝜆𝑚2𝑢2𝑇𝛿𝜆𝑚2𝑢2𝜆𝑇𝑛1𝑢1+𝑐𝑒𝜆1𝜆𝑛1𝑢1+𝑐𝑑𝑢1𝑇𝑛𝑢1𝜆1𝑛+1𝑢1+𝑐𝑑𝑢1𝜆𝑛1𝑢1=𝑐1+𝜆1𝑑𝑢1𝜆2𝜆𝑛1𝑢1.(3.23) This contradicts with the definition of 𝛿. This shows that we must have 𝛿𝜆𝑚2𝑢2=𝜆𝑛1𝑢1. 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=Ω𝑃 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 𝐴𝑃𝑃 be a completely continuous map. Suppose that there is an increasing, positively 1-homogeneous, 𝑒-positive mapping 𝑇 such that 𝑇𝑒𝑒, and that 𝐴𝑢𝑇𝑢,𝐴𝑢𝑢𝑢𝜕Ω𝑃.(4.1) Then the fixed-point index 𝑖(𝐴,Ω𝑃,𝑃)=1.

Proof. We show that 𝐴𝑢𝜇𝑢 for all 𝑢𝜕Ω𝑃 and all 𝜇1, from which the result follows by standard properties of fixed-point index (see, e.g., [1214]). Suppose that there exist 𝑢0𝜕Ω𝑃 and 𝜇01 such that 𝐴𝑢0=𝜇0𝑢0, then 𝜇0>1. It follows from the 𝑒-positiveness of 𝑇 that there exists a natural number 𝑛 such that 𝑇𝑛𝑢0𝑢𝑑0𝑢𝑒,𝑑0>0.(4.2) So, by induction, forall𝑚𝑁, we have 𝜇0𝑚𝑛𝑢0𝑇𝑚𝑛𝑢0𝑢𝑑0𝑒(4.3) 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 𝐴𝑃𝑃 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 𝑇𝑒𝑘𝑒,(4.4)(2)there exists 𝑀>0 such that 𝐴𝑢𝑇𝑢+𝑀𝑒,𝑢𝑃.(4.5)Then there exists 𝑅1>0 such that for any 𝑅>𝑅1, the fixed-point index 𝑖(𝐴,𝐵𝑅𝑃,𝑃)=1, where 𝐵𝑅={𝑥𝑋|𝑥𝑅}.

Proof. Let 𝑊={𝑢𝑃𝑢=𝜇𝐴𝑢,0𝜇1}.(4.6) In the following, we prove that 𝑊 is bounded.
For any 𝑢𝑊{𝜃}, using the 𝑒-positiveness of 𝑇, we have 𝑢=𝜇𝐴𝑢𝑇𝑢+𝑀𝑒(𝑑(𝑢)+𝑀)𝑒.(4.7) Let 𝜇0=inf{𝜇𝑢𝜇𝑒}.(4.8) It is easy to see that 0<𝜇0<+ and 𝑢𝜇0𝑒. We now have 𝑢𝑇𝑢+𝑀𝑒𝑘𝜇0+𝑀𝑒,(4.9) which, by the definition of 𝜇0, implies that 𝜇0𝑀/(1𝑘). So we know that 𝑢(𝑀/(1𝑘))𝑒 and 𝑊 is bounded by the normality of the cone 𝑃.
Select 𝑅1>sup{𝑥|𝑥𝑊}. Then from the homotopy invariance property of fixed-point index we have 𝑖𝐴,𝐵𝑅𝑃,𝑃=𝑖𝜃,𝐵𝑅𝑃,𝑃=1,𝑅>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: 𝜙𝑝𝑣𝑣+𝑎(𝑡)𝑓(𝑣(𝑡))=0,𝑡(0,1),(0)=0,𝑣(1)=0,(5.1) where 𝜙𝑝(𝑠)=|𝑠|𝑝2𝑠, 𝑝2, and (𝜙𝑝)1=𝜙𝑞=|𝑠|𝑞2𝑠, (1/𝑝)+(1/𝑞)=1.

We make the following assumptions:(𝐻1)𝑓[0,+)[0,+) is continuous;(𝐻2)𝑎[0,1](0,+) is continuous.

For each 𝑣𝐸=𝐶[0,1], we write 𝑣=max{|𝑣(𝑡)|𝑡[0,1]}. Define 𝐾={𝑣𝐸𝑣(𝑡)(1𝑡)𝑣}.(5.2) Clearly, (𝐸,) is a Banach space and 𝐾 is a cone of 𝐸. For any real constant 𝑟>0, define 𝐵𝑟={𝑣𝐸𝑥<𝑟}.

Let the operators 𝑇 and 𝐴 be defined by: (𝑇𝑣)(𝑡)=1𝑡𝜙𝑞𝑠0𝑎(𝜏)𝑣𝑝1(𝜏)𝑑𝜏𝑑𝑠,(𝐴𝑣)(𝑡)=1𝑡𝜙𝑞𝑠0𝑎(𝜏)𝑓(𝑣(𝜏))𝑑𝜏𝑑𝑠,(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 𝑇𝐾𝐸 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=1𝑡, that is, for any 𝑣>𝜃 from 𝐾, there exist 𝛼,𝛽>0 such that 𝛼𝑒𝑇𝑣𝛽𝑒.(5.4) Let 𝑀1=max𝑡[0,1]𝑎(𝑡). Then (𝑇𝑣)(𝑡)1𝑡𝜙𝑞𝑀1𝑣𝑝1𝑑𝑠=𝜙𝑞𝑀1𝑣𝑝1(1𝑡).(5.5) So, we may take 𝛽=𝜙𝑞(𝑀1𝑣𝑝1).
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 𝑇𝑢𝑇𝑣𝑐𝑒.(5.6) In fact, we note that 𝜙𝑞 is increasing, there exists an 𝜂(0,1) such that 𝑚=min𝑠[𝜂,1]𝜙𝑞𝑠0𝑎(𝜏)𝑢𝑝1(𝜏)𝑑𝜏𝜙𝑞𝑠0𝑎(𝜏)𝑣𝑝1(𝜏)𝑑𝜏>0.(5.7) Then for all 𝑡[𝜂,1], we have (𝑇𝑢)(𝑡)(𝑇𝑣)(𝑡)1𝜂𝜙𝑞𝑠0𝑎(𝜏)𝑢𝑝1(𝜏)𝑑𝜏𝑑𝑠1𝜂𝜙𝑞𝑠0𝑎(𝜏)𝑣𝑝1(𝜏)𝑑𝜏𝑑𝑠𝑚(1𝑡).(5.8) Since (𝑇𝑢)(𝑡)(𝑇𝑣)(𝑡)(𝑇𝑢)(𝜂)(𝑇𝑣)(𝜂)𝑚(1𝜂) for all 𝑡[0,𝜂], we have (𝑇𝑢)(𝑡)(𝑇𝑣)(𝑡)𝑚(1𝜂)(1𝑡),(5.9) for all 𝑡[0,1]. Therefore, the proof is complete and follows from Theorems 3.1 and 3.4.

Remark 5.2. Let 𝜑 be the positive eigenfunction of 𝑇 corresponding to 𝜆1, thus 𝜆1𝑇𝜑=𝜑. Then by Lemma 5.1, there exist 𝛼,𝛽>0 such that 𝛼𝑒𝑇𝜑=𝜆1𝜑𝛽𝑒.(5.10) Hence we obtained that 𝑇 is 𝜑-positive operator.

Theorem 5.3. Suppose that the conditions (𝐻1) and (𝐻2) are satisfied, and liminf𝑢0+𝑓(𝑢)𝑢𝑝1>𝜆11𝑝,(5.11)limsup𝑢+𝑓(𝑢)𝑢𝑝1<𝜆11𝑝,(5.12) 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 𝑓(𝑢)𝜆11𝑝𝑢𝑝1,0𝑢𝑟.(5.13) We may suppose that 𝐴 has no fixed point on 𝜕𝐵𝑟𝐾 (otherwise, the proof is finished). Therefore by (5.13), (𝐴𝑣)(𝑡)1𝑡𝜙𝑞𝑠0𝑎(𝜏)𝜆11𝑝𝑢𝑝11(𝜏)𝑑𝜏𝑑𝑠=𝜆1(𝑇𝑣)(𝑡),𝑣𝜕𝐵𝑟𝐾.(5.14) Hence we have from Lemma 2.4 and Remark 5.2 that 𝑖𝐴,𝐵𝑟𝐾,𝐾=0.(5.15)
It follows from (5.12) that there exist 0<𝜎<1 and 𝑀1>0 such that 𝑓(𝑢)𝜎𝑝1𝜆11𝑝𝑢𝑝1+𝑀2,𝑢0.(5.16) Thus, we have (𝐴𝑣)(𝑡)1𝑡𝜙𝑞𝑠0𝜎𝑎(𝜏)𝑝1𝜆11𝑝𝑢𝑝1(𝜏)+𝑀2𝜎𝑑𝜏𝑑𝑠𝜆11𝑡𝜙𝑞𝑠0𝑎(𝜏)𝑢𝑝1(𝜏)𝑑𝜏𝑑𝑠+1𝑡𝜙𝑞𝑀1𝑀2=𝜎𝑑𝑠𝜆1(𝑇𝑣)(𝑡)+𝜙𝑞𝑀1𝑀2(1𝑡),𝑣𝐾.(5.17) Here we have used the following inequality: 𝜙𝑞(𝑎+𝑏)𝜙𝑞(𝑎)+𝜙𝑞(𝑏),𝑎,𝑏>0,1<𝑞2.(5.18) Thus by Theorem 4.2 and Remark 5.2, there exists 𝑅1>𝑟 such that 𝑖𝐴,𝐵𝑅𝐾,𝐾=0,𝑅>𝑅1,(5.19) and hence we obtained 𝑖𝐵𝐴,𝑅𝐵𝑟𝐾,𝐾=1.(5.20) Thus, 𝐴 has a fixed point in (𝐵𝑅𝐵𝑟)𝐾. Consequently, (5.1) has a positive solution.

Theorem 5.4. Suppose that the conditions (𝐻1) and (𝐻2) are satisfied, and limsup𝑢0+𝑓(𝑢)𝑢𝑝1<𝜆11𝑝,(5.21)liminf𝑢+𝑓(𝑢)𝑢𝑝1>𝜆11𝑝,(5.22) 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 𝑓(𝑢)𝜆11𝑝𝑢𝑝1,0𝑢𝑟2.(5.23) We may suppose that 𝐴 has no fixed point on 𝜕𝐵𝑟2𝐾 (otherwise, the proof is finished). Therefore by (5.23), (𝐴𝑣)(𝑡)1𝑡𝜙𝑞𝑠0𝑎(𝜏)𝜆11𝑝𝑢𝑝11(𝜏)𝑑𝜏𝑑𝑠=𝜆1(𝑇𝑣)(𝑡),𝑣𝜕𝐵𝑟2𝐾.(5.24) Hence we have from Theorem 4.1 and Remark 5.2 that 𝑖𝐴,𝐵𝑟2𝐾,𝐾=1.(5.25)
It follows from (5.22) that there exists 𝜀>0 such that 𝑓(𝑢)(𝜆11𝑝+𝜀)𝑢𝑝1 when 𝑢 is sufficiently large. We know from the continuity of 𝑓 that there exists 𝑏0 such that 𝜆𝑓(𝑢)11𝑝𝑢+𝜀𝑝1𝑏,𝑢0.(5.26) Take 𝑅2𝑟>max2,𝑀1𝑏2𝑝𝑚0𝜀1/(𝑝1),(5.27) where 𝑚0=min𝑡[0,1]𝑎(𝑡), 𝑀1=max𝑡[0,1]𝑎(𝑡).
For 𝑣𝜕𝐵𝑅2𝐾, we have 𝑠0𝑎(𝜏)𝑓(𝑣(𝜏))𝑑𝜏𝑠0𝜆𝑎(𝜏)11𝑝𝑣+𝜀𝑝1(𝜏)𝑏𝑑𝜏𝜆11𝑝𝑠0𝑎(𝜏)𝑣𝑝1(𝜏)𝑑𝜏+𝑚0𝜀𝑠0(1𝜏)𝑝1𝑣𝑝1𝑑𝜏𝑀1𝑏𝑠=𝜆11𝑝𝑠0𝑎(𝜏)𝑣𝑝1(𝜏)𝑑𝜏+𝑚0𝜀𝑅2𝑝1𝑠0(1𝜏)𝑝1𝑑𝜏𝑀1𝑏𝑠𝜆11𝑝𝑠0𝑎(𝜏)𝑣𝑝1(𝜏)𝑑𝜏+𝑚0𝜀𝑅2𝑝10min{𝑠,1/2}(1𝜏)𝑝1𝑑𝜏𝑀1𝑏𝑠𝜆11𝑝𝑠0𝑎(𝜏)𝑣𝑝1(𝜏)𝑑𝜏+𝑚0𝜀𝑅2𝑝10min{𝑠,1/2}112𝑝1𝑑𝜏𝑀1𝑏𝑠𝜆11𝑝𝑠0𝑎(𝜏)𝑣𝑝1𝑚(𝜏)𝑑𝜏+0𝜀𝑅2𝑝112𝑝𝑀1𝑏𝑠𝜆11𝑝𝑠0𝑎(𝜏)𝑣𝑝1(𝜏)𝑑𝜏.(5.28)
Thus, we have (𝐴𝑣)(𝑡)1𝑡𝜙𝑞𝑠0𝑎(𝜏)𝜆11𝑝𝑢𝑝11(𝜏)𝑑𝜏𝑑𝑠=𝜆1(𝑇𝑣)(𝑡),𝑣𝜕𝐵𝑅2𝐾.(5.29) It follows from Lemma 2.4 that 𝑖𝐴,𝐵𝑅2𝐾,𝐾=0,(5.30) and hence we obtained 𝑖𝐵𝐴,𝑅2𝐵𝑟2𝐾,𝐾=1.(5.31) Thus, 𝐴 has a fixed point in (𝐵𝑅2𝐵𝑟2)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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