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 𝑀𝑇^𝑝𝑢≥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.

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. Let 𝑋, 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: 𝑀𝑇^𝑝𝑢≥𝑢,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 𝜆√₀^𝑝𝑀≥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→𝑋 be an increasing, positively 1-homogeneous, compact, continuous operator for which there exists a non-zero 𝑢∈𝑃 and 𝑀>0 such that 𝑀𝑇𝑢≥𝑢,(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 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 𝑀𝑇^𝑝𝑢≥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, 𝑃^𝑜≠∅. 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∈𝑃^𝑜, we write 𝑥≪𝑦. For the concepts and the properties about the cone we refer to [12, 13].

A mapping 𝑇∶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 𝑇(𝑡𝑥)=𝑡𝑇(𝑥)∀𝑥∈𝑋,𝑡∈ℝ⁺⁰=[0,+∞).(2.1) 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→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 𝑐(𝑥)𝑒≤𝑇^𝑛𝑥≤𝑑(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)⊂𝑃^𝑜, 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∶ℝ⁺⁰×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⁺⊂ℝ⁺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≤𝛿₁𝑥. Then a small 𝛿_𝑥(𝑦) exists for which 𝑦≤𝛿_𝑥(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→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 𝐴𝑢≥𝑇𝑢,𝐴𝑢≠𝑢,∀𝑢∈𝜕Ω∩𝑃.(2.3)
Then the fixed-point index 𝑖(𝐴,Ω∩𝑃,𝑃)=0.

3. Main Results

Theorem 3.1. Let 𝑇∶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: 𝑀𝑇^𝑝𝑢≥𝑢(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 𝜆√₀^𝑝𝑀≥1.(3.3)

Proof. Let 𝑣∈𝑃(𝑣≠𝜃) be as in the hypothesis of the theorem. For every positive integer 𝑛>0, define 𝐹_𝑛∶ℝ⁺⁰×𝑃→𝑃 by 𝐹_𝑛(𝜆,𝑥)∶=𝜆𝑇𝑥+1𝑛𝜆𝑣.(3.4) Since 𝑇 is compact and continuous, each of these operators 𝐹_𝑛 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 𝑇𝜃=𝜃. Let, by Lemma 2.2, 𝐶^+𝑛⊂ℝ⁺⁰×𝑃→𝑃 be a connected unbounded branch of solutions to the equation 𝐹_𝑛(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 𝐹_𝑛(𝜆,⋅) for some 𝜆≥1. Then 𝑥=𝐹_𝑛(𝜆,𝑥)=𝜆𝑇𝑥+(1/𝑛)𝜆𝑣 and we obtain, from the properties of 𝑇 and the inequalities 𝑣≥𝜃, 𝑣≥𝑢, respectively, that 𝑥≥𝜆𝑇𝑥,𝑥≥1𝑛𝜆𝑢.(3.6)
Let 𝜏_𝑛=sup{𝜏|𝑥≥𝜏𝑢}. Obviously, 𝜏_𝑛≥(1/𝑛)𝜆>0. Since 𝑇 is increasing and 1-homogeneous, by (3.4), we have 𝑥=𝜆𝑇𝑥+1𝑛𝜆𝑣≥𝜆²𝑇²𝑥+1≥⋯≥𝜆_𝑝𝑛𝜆𝑣𝑇^𝑝𝑥+1𝑛𝜆𝑣≥𝜆^𝑝≥𝜆_𝑝𝑇^𝑝𝑥𝑇^𝑝𝜏_𝑛𝑢≥𝜏_𝑛𝜆^𝑝𝑀𝑢.(3.7) Consequently, by the definition of 𝜏_𝑛, 𝜆^𝑝𝑀≤1.(3.8) In other words, if 𝜆>√^𝑝𝑀, then 𝐹_𝑛(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 𝑥_𝑛 with ‖𝑥_𝑛‖=1 and 𝜆_𝑛∈[0,√^𝑝𝑀] such that (𝜆_𝑛,𝑥_𝑛)∈𝐶^+𝑛. That is, 𝑥_𝑛=𝜆_𝑛𝑇𝑥_𝑛+1𝑛𝜆^𝑛𝑣,𝜆_𝑛∈0,√^𝑝𝑀,‖‖𝑥_𝑛‖‖=1.(3.9) Since the operator 𝑇 is compact, a subsequence of indices 𝑛_𝑖(𝑖=1,2,…) can be chosen such that the sequence 𝑇𝑥_𝑛 strongly converges to some element 𝑦^∗∈𝑃. By virtue of (3.9), with this choice of the sequence 𝑛_𝑖, the convergence of the number 𝜆_𝑛𝑖 to some 𝜆_∗ which satisfies the inequality (3.3) can be guaranteed simultaneously. Then 𝑥_𝑛𝑖 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: 𝑥_𝑛𝑖=𝜆_𝑛𝑖𝑇𝑥_𝑛𝑖+1𝑛_𝑖𝜆_𝑛𝑖𝑣(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)𝑥^𝑝(𝑠)𝑑𝑠^1/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∈ℕ.
If there exists a system of points 𝑠₁,𝑠₂,…,𝑠_𝑝 such that 𝐾𝑠₁,𝑠₂𝐾𝑠₂,𝑠₃⋯𝐾𝑠_𝑝,𝑠₁>0.(3.12) Then the map 𝑇 defined by (3.11) has a nonnegative eigenfunction. In fact, it is easy to see that (𝑇^𝑝𝑥)(𝑡)=_𝐺𝐾^(𝑝)(𝑡,𝑠)𝑥^𝑝(𝑠)𝑑𝑠^1/𝑝,(3.13) where 𝐾^(𝑝)(𝑡,𝑠)=_𝐺⋯_𝐺𝐾𝑡,𝑡₁⋯𝐾𝑡_𝑝−1,𝑠𝑑𝑡₁⋯𝑑𝑡_𝑝−1(3.14) 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 (𝑇^𝑝𝑦)(𝑡)=_𝐺𝐾^(𝑝)(𝑡,𝑠)𝑦^𝑝(𝑠)𝑑𝑠^1/𝑝>0(3.15) when 𝑡∈𝐺₁. From (3.15) it follows that there exists a number 𝑀>0 such that 𝑀𝑇^𝑝𝑦≥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 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 𝑢𝑢₁₂∈𝑃⧵{𝜃},𝜆₁>0𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝜆₁𝑢₁≤𝑇𝑢₁,∈𝑃⧵{𝜃},𝜆₂>0𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝜆₂𝑢₂≥𝑇𝑢₂,(3.17) then 𝜆₂≥𝜆₁. Furthermore, if for 𝑥>𝑦>𝜃, a positive number 𝑐(𝑥,𝑦) can be found such that 𝑇𝑥−𝑇𝑦≥𝑐(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 𝑇^𝑛𝑢₁≤𝑑𝑢₁𝑒,𝑐𝑢₂𝑒≤𝑇^𝑚𝑢₂,𝑐𝑢₁,𝑑𝑢₂>0.(3.19) Then for 𝑡>0, we have 𝑡𝑐𝑢₂−𝑑𝑢₁𝑒≤𝑡𝑇^𝑚𝑢₂−𝑇^𝑛𝑢₁≤𝑡𝜆^𝑚2𝑢₂−𝜆^𝑛1𝑢₁.(3.20) From this and the fact that 𝑢₁∈𝑃⧵{𝜃} we deduce that 𝛿≜𝛿_𝜆^𝑚2_𝑢2(𝜆^𝑛1𝑢₁)>0, that is, 𝛿𝜆^𝑚2𝑢₂−𝜆^𝑛1𝑢₁∈𝑃. Since 𝑇 is increasing, 𝑇(𝛿𝜆^𝑚2𝑢₂)≥𝑇(𝜆^𝑛1𝑢₁), from which, by virtue of (3.17), it follows that 𝜆₂𝛿𝜆^𝑚2𝑢₂≥𝑇𝛿𝜆^𝑚2𝑢₂≥𝑇𝜆^𝑛1𝑢₁≥𝜆₁𝜆^𝑛1𝑢₁.(3.21) By Lemma 2.3, we obtain that 𝜆₂≥𝜆₁.
Now suppose that 𝜆₁=𝜆₂. According to the above proof, the number 𝛿>0. Since 𝛿𝜆^𝑚2𝑢₂−𝜆^𝑛1𝑢₁∈𝑃 and (3.18), if 𝛿𝜆^𝑚2𝑢₂≠𝜆^𝑛1𝑢₁, then there exists 𝑐>0 such that 𝑇𝛿𝜆^𝑚2𝑢₂−𝑇𝜆^𝑛1𝑢₁≥𝑐𝑒.(3.22) Therefore, by (3.19) 𝜆₂𝛿𝜆^𝑚2𝑢₂≥𝑇𝛿𝜆^𝑚2𝑢₂≥𝑇𝜆^𝑛1𝑢₁+𝑐𝑒≥𝜆₁𝜆^𝑛1𝑢₁+𝑐𝑑𝑢₁𝑇^𝑛≥𝜆_1𝑛+1𝑢₁𝑢₁+𝑐𝑑𝑢₁𝜆^𝑛1𝑢₁=1+𝑐𝜆₁𝑑𝑢₁𝜆²𝜆^𝑛1𝑢₁.(3.23) This contradicts with the definition of 𝛿. This shows that we must have 𝛿𝜆^𝑚2𝑢₂=𝜆^𝑛1𝑢₁. 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 Ω_𝑃∶=Ω∩𝑃 and 𝜕Ω_𝑃 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→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 𝐴𝑢≤𝑇𝑢,𝐴𝑢≠𝑢∀𝑢∈𝜕Ω_𝑃.(4.1) Then the fixed-point index 𝑖(𝐴,Ω_𝑃,𝑃)=1.

Proof. We show that 𝐴𝑢≠𝜇𝑢 for all 𝑢∈𝜕Ω_𝑃 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∈𝜕Ω_𝑃 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 𝑇^𝑛𝑢₀≤𝑑𝑢₀𝑒,𝑑𝑢₀>0.(4.2) So, by induction, forall𝑚∈𝑁, we have 𝜇^0𝑚𝑛𝑢₀≤𝑇^𝑚𝑛𝑢₀≤𝑑𝑢₀𝑒(4.3) which implies 𝑢₀=𝜃. This contradicts 𝑢₀∈𝜕Ω_𝑃.

Theorem 4.2. Let 𝑃 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 exists 𝑘∈[0,1) such that 𝑇𝑒≤𝑘𝑒,(4.4)(2)there exists 𝑀>0 such that 𝐴𝑢≤𝑇𝑢+𝑀𝑒,𝑢∈𝑃.(4.5)Then there exists 𝑅₁>0 such that for any 𝑅>𝑅₁, the fixed-point index 𝑖(𝐴,𝐵_𝑅∩𝑃,𝑃)=1, where 𝐵_𝑅={𝑥∈𝑋|‖𝑥‖≤𝑅}.

Proof. Let 𝑊={𝑢∈𝑃∣𝑢=𝜇𝐴𝑢,0≤𝜇≤1}.(4.6) In the following, we prove that 𝑊 is bounded.
For any 𝑢∈𝑊⧵{𝜃}, using the 𝑒-positiveness of 𝑇, we have 𝑢=𝜇𝐴𝑢≤𝑇𝑢+𝑀𝑒≤(𝑑(𝑢)+𝑀)𝑒.(4.7) Let 𝜇₀=inf{𝜇∣𝑢≤𝜇𝑒}.(4.8) It is easy to see that 0<𝜇₀<+∞ and 𝑢≤𝜇₀𝑒. We now have 𝑢≤𝑇𝑢+𝑀𝑒≤𝑘𝜇₀+𝑀𝑒,(4.9) 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.
Select 𝑅₁>sup{‖𝑥‖|𝑥∈𝑊}. Then from the homotopy invariance property of fixed-point index we have 𝑖𝐴,𝐵_𝑅∩𝑃,𝑃=𝑖𝜃,𝐵_𝑅∩𝑃,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: 𝜙_𝑝𝑣^𝑣′+𝑎(𝑡)𝑓(𝑣(𝑡))=0,𝑡∈(0,1),_(0)=0,𝑣(1)=0,(5.1) where 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝≥2, and (𝜙_𝑝)⁻¹=𝜙_𝑞=|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 each 𝑣∈𝐸∶=𝐶[0,1], we write ‖𝑣‖=max{|𝑣(𝑡)|∶𝑡∈[0,1]}. Define 𝐾={𝑣∈𝐸∶𝑣(𝑡)≥(1−𝑡)‖𝑣‖}.(5.2) Clearly, (𝐸,‖⋅‖) is a Banach space and 𝐾 is a cone of 𝐸. For any real constant 𝑟>0, define 𝐵_𝑟={𝑣∈𝐸∶‖𝑥‖<𝑟}.

Let the operators 𝑇 and 𝐴 be defined by: (𝑇𝑣)(𝑡)=^1𝑡𝜙_𝑞^𝑠0(𝐴𝑣)(𝑡)=_1𝑡𝑎(𝜏)𝑣^𝑝−1(𝜏)𝑑𝜏𝑑𝑠,𝜙_𝑞^𝑠0𝑎(𝜏)𝑓(𝑣(𝜏))𝑑𝜏𝑑𝑠,(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 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 that 𝑇 is 𝑒-positive with 𝑒=1−𝑡, that is, for any 𝑣>𝜃 from 𝐾, there exist 𝛼,𝛽>0 such that 𝛼𝑒≤𝑇𝑣≤𝛽𝑒.(5.4) Let 𝑀₁=max_𝑡∈[0,1]𝑎(𝑡). Then (𝑇𝑣)(𝑡)≤^1𝑡𝜙_𝑞𝑀₁‖𝑣‖^𝑝−1𝑑𝑠=𝜙_𝑞𝑀₁‖𝑣‖^𝑝−1(1−𝑡).(5.5) So, we may take 𝛽=𝜙_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 𝑇𝑢−𝑇𝑣≥𝑐𝑒.(5.6) In fact, we note that 𝜙_𝑞 is increasing, there exists an 𝜂∈(0,1) such that 𝑚=min_{𝑠∈[𝜂,1]}𝜙_𝑞^𝑠0𝑎(𝜏)𝑢^𝑝−1(𝜏)𝑑𝜏−𝜙_𝑞^𝑠0𝑎(𝜏)𝑣^𝑝−1(𝜏)𝑑𝜏>0.(5.7) Then for all 𝑡∈[𝜂,1], we have (𝑇𝑢)(𝑡)−(𝑇𝑣)(𝑡)≥^1𝜂𝜙_𝑞^𝑠0−𝑎(𝜏)𝑢^𝑝−1(𝜏)𝑑𝜏𝑑𝑠_1𝜂𝜙_𝑞^𝑠0𝑎(𝜏)𝑣^𝑝−1(𝜏)𝑑𝜏𝑑𝑠≥𝑚(1−𝑡).(5.8) Since (𝑇𝑢)(𝑡)−(𝑇𝑣)(𝑡)≥(𝑇𝑢)(𝜂)−(𝑇𝑣)(𝜂)≥𝑚(1−𝜂) for all 𝑡∈[0,𝜂], we have (𝑇𝑢)(𝑡)−(𝑇𝑣)(𝑡)≥𝑚(1−𝜂)(1−𝑡),(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 𝛼𝑒≤𝑇𝜑^∗=𝜆₁𝜑^∗≤𝛽𝑒.(5.10) Hence we obtained that 𝑇 is 𝜑^∗-positive operator.

Theorem 5.3. Suppose that the conditions (𝐻₁) and (𝐻₂) are satisfied, and liminf_𝑢→0+𝑓(𝑢)𝑢^𝑝−1limsup_𝑢→+∞>𝜆^11−𝑝,(5.11)𝑓(𝑢)𝑢^𝑝−1<𝜆^11−𝑝,(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)≥𝜆^11−𝑝𝑢^𝑝−1,∀0≤𝑢≤𝑟.(5.13) We may suppose that 𝐴 has no fixed point on 𝜕𝐵_𝑟∩𝐾 (otherwise, the proof is finished). Therefore by (5.13), (𝐴𝑣)(𝑡)≥^1𝑡𝜙_𝑞^𝑠0𝑎(𝜏)𝜆^11−𝑝𝑢^𝑝−1(𝜏)𝑑𝜏𝑑𝑠=1𝜆₁(𝑇𝑣)(𝑡),𝑣∈𝜕𝐵_𝑟∩𝐾.(5.14) Hence we have from Lemma 2.4 and Remark 5.2 that 𝑖𝐴,𝐵_𝑟∩𝐾,𝐾=0.(5.15)
It follows from (5.12) that there exist 0<𝜎<1 and 𝑀₁>0 such that 𝑓(𝑢)≤𝜎^𝑝−1𝜆^11−𝑝𝑢^𝑝−1+𝑀₂,∀𝑢≥0.(5.16) Thus, we have (𝐴𝑣)(𝑡)≤^1𝑡𝜙_𝑞^𝑠0≤𝜎𝑎(𝜏)𝜎^𝑝−1𝜆^11−𝑝𝑢^𝑝−1(𝜏)+𝑀₂𝑑𝜏𝑑𝑠𝜆₁^1𝑡𝜙_𝑞^𝑠0=𝜎𝑎(𝜏)𝑢^𝑝−1(𝜏)𝑑𝜏𝑑𝑠+^1𝑡𝜙_𝑞𝑀₁𝑀₂𝑑𝑠𝜆₁(𝑇𝑣)(𝑡)+𝜙_𝑞𝑀₁𝑀₂(1−𝑡),𝑣∈𝐾.(5.17) Here we have used the following inequality: 𝜙_𝑞(𝑎+𝑏)≤𝜙_𝑞(𝑎)+𝜙_𝑞(𝑏),𝑎,𝑏>0,1<𝑞≤2.(5.18) Thus by Theorem 4.2 and Remark 5.2, there exists 𝑅₁>𝑟 such that 𝑖𝐴,𝐵_𝑅∩𝐾,𝐾=0,𝑅>𝑅₁,(5.19) and hence we obtained 𝑖𝐴,𝐵_𝑅⧵𝐵_𝑟∩𝐾,𝐾=1.(5.20) Thus, 𝐴 has a fixed point in (𝐵_𝑅⧵𝐵_𝑟)∩𝐾. Consequently, (5.1) has a positive solution.

Theorem 5.4. Suppose that the conditions (𝐻₁) and (𝐻₂) are satisfied, and limsup_𝑢→0+𝑓(𝑢)𝑢^𝑝−1liminf_𝑢→+∞<𝜆^11−𝑝,(5.21)𝑓(𝑢)𝑢^𝑝−1>𝜆^11−𝑝,(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)≤𝜆^11−𝑝𝑢^𝑝−1,∀0≤𝑢≤𝑟₂.(5.23) We may suppose that 𝐴 has no fixed point on 𝜕𝐵_𝑟2∩𝐾 (otherwise, the proof is finished). Therefore by (5.23), (𝐴𝑣)(𝑡)≤^1𝑡𝜙_𝑞^𝑠0𝑎(𝜏)𝜆^11−𝑝𝑢^𝑝−1(𝜏)𝑑𝜏𝑑𝑠=1𝜆₁(𝑇𝑣)(𝑡),𝑣∈𝜕𝐵_𝑟2∩𝐾.(5.24) Hence we have from Theorem 4.1 and Remark 5.2 that 𝑖𝐴,𝐵_𝑟2∩𝐾,𝐾=1.(5.25)
It follows from (5.22) that there exists 𝜀>0 such that 𝑓(𝑢)≥(𝜆^11−𝑝+𝜀)𝑢^𝑝−1 when 𝑢 is sufficiently large. We know from the continuity of 𝑓 that there exists 𝑏≥0 such that 𝑓(𝑢)≥𝜆^11−𝑝+𝜀𝑢^𝑝−1−𝑏,∀𝑢≥0.(5.26) Take 𝑅₂>max𝑟₂,𝑀¹𝑏2^𝑝𝑚₀𝜀^1/(𝑝−1),(5.27) where 𝑚₀=min_𝑡∈[0,1]𝑎(𝑡), 𝑀₁=max_𝑡∈[0,1]𝑎(𝑡).
For 𝑣∈𝜕𝐵_𝑅2∩𝐾, we have ^𝑠0𝑎(𝜏)𝑓(𝑣(𝜏))𝑑𝜏≥≥𝜆^𝑠0𝑎_11−𝑝(𝜏)𝜆^11−𝑝+𝜀𝑣^𝑝−1(𝜏)−𝑏𝑑𝜏^𝑠0=𝜆_11−𝑝𝑎(𝜏)𝑣^𝑝−1(𝜏)𝑑𝜏+𝑚₀𝜀^𝑠0(1−𝜏)^𝑝−1‖𝑣‖^𝑝−1𝑑𝜏−𝑀₁𝑏𝑠^𝑠0𝑎(𝜏)𝑣^𝑝−1(𝜏)𝑑𝜏+𝑚₀𝜀𝑅^2𝑝−1^𝑠0≥𝜆_11−𝑝(1−𝜏)^𝑝−1𝑑𝜏−𝑀₁𝑏𝑠^𝑠0𝑎(𝜏)𝑣^𝑝−1(𝜏)𝑑𝜏+𝑚₀𝜀𝑅^2𝑝−1^{0min{𝑠,1/2}}≥𝜆_11−𝑝(1−𝜏)^𝑝−1𝑑𝜏−𝑀₁𝑏𝑠^𝑠0𝑎(𝜏)𝑣^𝑝−1(𝜏)𝑑𝜏+𝑚₀𝜀𝑅^2𝑝−1^{0min{𝑠,1/2}}1−12^𝑝−1≥𝜆_11−𝑝𝑑𝜏−𝑀₁𝑏𝑠^𝑠0𝑎(𝜏)𝑣^𝑝−1(𝜏)𝑑𝜏+𝑚₀𝜀𝑅^2𝑝−112^𝑝≥𝜆_11−𝑝−𝑀₁𝑏𝑠^𝑠0𝑎(𝜏)𝑣^𝑝−1(𝜏)𝑑𝜏.(5.28)
Thus, we have (𝐴𝑣)(𝑡)≥^1𝑡𝜙_𝑞^𝑠0𝑎(𝜏)𝜆^11−𝑝𝑢^𝑝−1(𝜏)𝑑𝜏𝑑𝑠=1𝜆₁(𝑇𝑣)(𝑡),𝑣∈𝜕𝐵_𝑅2∩𝐾.(5.29) It follows from Lemma 2.4 that 𝑖𝐴,𝐵_𝑅2∩𝐾,𝐾=0,(5.30) and hence we obtained 𝑖𝐴,𝐵_𝑅2⧵𝐵_𝑟2∩𝐾,𝐾=−1.(5.31) Thus, 𝐴 has a fixed point in (𝐵_𝑅2⧵𝐵_𝑟2)∩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).