Global Structure of Positive Solutions of a Discrete Problem with Sign-Changing Weight

Abstract

Let 𝑇>5 be an integer, 𝕋={1,2,…,𝑇}. We are concerned with the global structure of positive solutions set of the discrete second-order boundary value problems Δ²𝑢(𝑡−1)+𝑟𝑚(𝑡)𝑓(𝑢(𝑡))=0,𝑡∈𝕋,𝑢(0)=𝑢(𝑇+1)=0, where 𝑟∈ℝ is a parameter, 𝑚∶𝕋→ℝ changes its sign and 𝑚(𝑡)≠0for𝑡∈𝕋.

1. Introduction

Let 𝑇>5 be an integer, 𝕋={1,2,…,𝑇}. In this paper, we are concerned with the global structure of positive solutions set of the discrete second-order boundary value problems Δ²𝑢(𝑡−1𝑢)(+0)𝑟𝑚=(𝑢𝑡)(𝑓𝑇+(𝑢1(𝑡)))==0,0,𝑡∈𝕋,((11..12)) where 𝑟∈ℝ is a parameter, 𝑚∶𝕋→ℝ changes its sign, and 𝑚(𝑡)≠0 for 𝑡∈𝕋.

The boundary value problems with sign-changing weight arise from a selection-migration model in population genetics, see Fleming [1]. 𝑚 changes sign corresponds to the fact that an allele 𝐴₁ holds an advantage over a rival allele 𝐴₂ at the same points and is at a disadvantage at others. The parameter 𝑟 corresponds to the reciprocal of the diffusion. So, the existence and multiplicity of positive solutions with sign-changing weight in continuous case has been studied by many authors, see, for example, [2–8] and the references therein.

For the discrete case, there are many literature dealing with difference equations similar to (1.1) subject to various boundary value conditions. We refer to [9–17] and the references therein. However, there are few papers to discuss the existence of positive solutions to (1.1) and (1.2) if 𝑚(𝑡) changes sign on 𝕋. Maybe the main reason is the spectrum of the following linear eigenvalue problems Δ²𝑢(𝑡−𝑢1()0+)=𝜆𝑚𝑢((𝑡𝑇)𝑢+(𝑡1))==00,𝑡∈𝕋,(1.3) is not clear when 𝑚 changes its sign on 𝕋.

In 2008, Shi and Yan [18] investigated the spectrum of the second-order boundary value problems of difference equations −Δ[𝑝(𝑡−1)Δ−𝑢𝑝𝑝((𝑡(0𝑇−))Δ1Δ𝑢)]𝑢(+0(𝑇)𝑞)(𝑡)=𝑢(𝐾𝑡)=𝑢𝑢𝜆((𝑚𝑇0)()𝑡)𝑢,(𝑡),𝑡∈𝕋,(1.4) where 𝐾=^𝑘11𝑘12_𝑘21𝑘22 is a symmetric and positive definite 2×2 matrix. 𝑝∶{0,1,…,𝑇}→ℝ, 𝑞∶𝕋→ℝ, and 𝑚∶𝕋→ℝ satisfy(A1)𝑚∶𝕋→ℝ changes its sign, and 𝑚(𝑡)≠0 for 𝑡∈𝕋.(A2)𝑝(𝑡−1)≥0,𝑞(𝑡)≥0,𝑝(𝑡−1)+𝑞(𝑡)>0,for𝑡∈𝕋, and 𝑝(0)+𝑘₁₁≠0, 𝑝(𝑇)>0.

They proved that (1.4) has 𝑇 eigenvalues 𝜆₁≤𝜆₂≤⋯≤𝜆_𝑇 which have 𝑇 corresponding linearly independent and orthogonal eigenfunctions.

However, it's easy to see that (1.3) is not included in (1.4) under the condition (A2). Also, Shi and Yan [18] provided no information about the sign of the eigenvalues and no information about the corresponding eigenfunctions.

In this paper, we will show that (1.3) has two principal eigenvalues 𝜆_𝑚,−<0<𝜆_𝑚,+, and the corresponding eigenfunctions we denote by 𝜓_𝑚,− and 𝜓_𝑚,+ don't change their sign on 𝕋∶={0,1,…,𝑇,𝑇+1}. Based on this result, using Rabinowitz's global bifurcation theorem [19], we will discuss the global structure of positive solutions set of (1.1) and (1.2).

The assumptions we are interested in this paper are as follows:(H1)𝑓∈𝐶(ℝ,ℝ) with 𝑠𝑓(𝑠)>0 for 𝑠≠0;(H2)𝑓₀=lim_|𝑠|→0(𝑓(𝑠)/𝑠)∈(0,∞);(H3)𝑓_∞=lim_{|𝑠|→+∞}(𝑓(𝑠)/𝑠)=0;(H3′)𝑓_∞=∞.

Our main result is the following.

Theorem 1.1. Assume that (H1), (H2) hold. (i)If (H3) holds, then there exist 0<𝜆^∗≤𝜆_𝑚,+/𝑓₀ and 𝜆_𝑚,−/𝑓₀≤𝜆_∗<0 such that (1.1), (1.2) has at least one positive solution for 𝑟∈(−∞,𝜆_∗)∪(𝜆^∗,∞).(ii)If (H3′) holds, then there exist 𝜌_∗≤𝜆_𝑚,−/𝑓₀<0 and 𝜌^∗≥𝜆_𝑚,+/𝑓₀>0 such that (1.1), (1.2) has at least one positive solution for 𝑟∈(𝜌_∗,0)∪(0,𝜌^∗).

The rest of the paper is arranged as follows. In Section 2, the existence of two principal eigenvalues of (1.3), and some properties of these two eigenvalues will be discussed. In Section 3, we will prove our main result.

2. Existence of Two Principal Eigenvalues to (1.3)

Recall that 𝕋={1,2,…,𝑇} and 𝕋={0,1,…,𝑇+1}. Let 𝑋={𝑢∶𝕋→ℝ∣𝑢(0)=𝑢(𝑇+1)=0}. Then 𝑋 is a Banach space under the norm ‖𝑢‖_𝑋=max_𝑡∈𝕋|𝑢(𝑡)|. Let 𝑌={𝑢∣𝑢∶𝕋→ℝ}. Then 𝑌 is a Banach space under the norm ‖𝑢‖_𝑌=max_𝑡∈𝕋|𝑢(𝑡)|.

It's well known that the operator 𝜒∶𝑋→𝑌, 𝜒(0,𝑢(1),𝑢(2),…,𝑢(𝑇),0)=(𝑢(1),𝑢(2),…,𝑢(𝑇))(2.1) is a homomorphism.

Define the operator 𝐿∶𝑋→𝑌 by 𝐿𝑢(𝑡)=−Δ²𝑢(𝑡−1),𝑡∈𝕋.(2.2)

In this section, we will discuss the existence of principal eigenvalues for (1.3) with 𝑚∶𝕋→ℝ that changes its sign. The main idea we will use aries from [20, 21].

Theorem 2.1. Equation (1.3) has two principal eigenvalues 𝜆_𝑚,+ and 𝜆_𝑚,− such that 𝜆_𝑚,−<0<𝜆_𝑚,+, and the corresponding eigenfunctions, we denoted by 𝜓_𝑚,+ and 𝜓_𝑚,− do not change sign on 𝕋.

Proof. Consider, for fixed 𝜆, the eigenvalue problems 𝐿𝑢−𝜆𝑢𝑚(0()𝑡)=𝑢(𝑢𝑡)(𝑇=+𝜇1𝑢)(𝑡=),0.𝑡∈𝕋,(2.3) By Kelley and Peterson [22, Theorem  7.6], for fixed 𝜆, (2.3) has 𝑇-simple eigenvalues 𝜇_𝑚,1(𝜆)<𝜇_𝑚,2(𝜆)<⋯<𝜇_𝑚,𝑇(𝜆),(2.4) and the corresponding eigenfunction 𝜓_𝑚,𝑘(𝜆,𝑡) has exactly 𝑘−1 simple generalized zeros.
Thus, 𝜆 is a principal eigenvalue of (1.3) if and only if 𝜇_𝑚,1(𝜆)=0.
On the other hand, let 𝑆_𝑚,𝜆=^𝑇_𝑡=0||Δ𝜙(𝑡)||²−𝜆^𝑇_𝑡=1𝑚(𝑡)𝜙(𝑡)²∶𝜙∈𝑋,^𝑇_𝑡=1𝜙(𝑡)²=1.(2.5) Clearly, 𝑆_𝑚,𝜆 is bounded below. Then 𝜇_𝑚,1(𝜆)=inf𝑆_𝑚,𝜆.
For fixed 𝜙∈𝑋, 𝜆→∑^𝑇𝑡=0|Δ𝜙(𝑡)|²−𝜆∑^𝑇𝑡=1𝑚(𝑡)𝜙²(𝑡) is an affine and so concave function. As the infimum of any collection of concave functions is concave, it follows that 𝜆→𝜇_𝑚,1(𝜆) is a concave function. Also, by considering test functions 𝜙₁,𝜙₂∈𝑋 such that ∑^𝑇𝑡=1𝑚(𝑡)𝜙²¹(𝑡)<0 and ∑^𝑇𝑡=1𝑚(𝑡)𝜙²²(𝑡)>0, it is easy to see that 𝜇_𝑚,1(𝜆)→−∞ as 𝜆→±∞. Thus, 𝜆→𝜇_𝑚,1(𝜆) is an increasing function until it attains its maximum and is a decreasing function thereafter.
Since 𝜇₁(0)>0, 𝜆→𝜇_𝑚,1(𝜆) must have exactly two zeros. Thus, (1.3) has exactly two principal eigenvalues, 𝜆_𝑚,+>0 and 𝜆_𝑚,−<0, and the corresponding eigenfunctions we denoted by 𝜓_𝑚,+ and 𝜓_𝑚,− don't change sign on 𝕋.

Remark 2.2. From the proof of Theorem 2.1, it is not difficult to see that the following results hold.(i)If 𝑚(𝑡)≥0 on 𝕋 and there exist at least one point 𝑡₀∈𝕋 such that 𝑚(𝑡₀)>0, then (1.3) has only one principal eigenvalue 𝜆_𝑚,+>0.(ii)If 𝑚(𝑡)≤0 on 𝕋 and there exist at least one point 𝑡₀∈𝕋 such that 𝑚(𝑡₀)<0, then (1.3) has only one principal eigenvalue 𝜆_𝑚,−<0.

Now, we give some properties for the above principal eigenvalue(s).

Theorem 2.3. Let 𝑚∶𝕋→ℝ change its sign. Assume that there exists 𝑚^∶𝕋→ℝ such that 𝑚(𝑡)≤𝑚^(𝑡) for 𝑡∈𝕋. Then the followings hold. (i)If 𝑚^ changes sign on 𝕋, then 𝜆_𝑚′,−≤𝜆_𝑚,−, 𝜆_𝑚′,+≤𝜆_𝑚,+;(ii)If 𝑚^≥0, then 0<𝜆_𝑚,+≤𝜆_𝑚,+.

Proof. For convenience, we only prove the case (i). It can be seen that for 𝜆>0, 𝑆_𝑚,𝜆≥𝑆_𝑚′,𝜆, which implies 𝜇_𝑚,1(𝜆)≥𝜇_𝑚′,1(𝜆) and consequently, 𝜆_𝑚,+≥𝜆_𝑚′,+.
On the other hand, for 𝜆<0, 𝑆_𝑚,𝜆≤𝑆_𝑚′,𝜆, which implies 𝜇_𝑚,1(𝜆)≤𝜇_𝑚′,1(𝜆) and consequently, 𝜆_𝑚,−≥𝜆_𝑚′,−.

Suppose that 𝕋₀={𝑎+1,𝑎+2,…,𝑏−1} is a strict subset of 𝕋, and 𝑚_𝕋0 denote the restriction of 𝑚 on 𝕋₀. Consider the linear eigenvalue problems Δ²𝑢(𝑡−1)𝑢+(𝑎𝜆)𝑚=(𝑡𝑢)(𝑢𝑏()𝑡)==0.0,𝑡∈𝕋₀,(2.6)

Then we get the following result.

Theorem 2.4. (i) If 𝑚(𝑡)>0 for 𝑡∈𝕋₀, then (2.6) has only one positive principal eigenvalue 𝜆_𝑚𝕋0,+ such that 0<𝜆_𝑚,+<𝜆_𝑚𝕋0,+.
(ii) If 𝑚(𝑡)<0 for 𝑡∈𝕋₀, then (2.6) has only one principal eigenvalue 𝜆_{𝑚𝕋0,−} such that 𝜆_{𝑚𝕋0,−}<𝜆_𝑚,−.
(iii) If 𝑚 changes its sign on 𝕋₀, then (2.6) has two principal eigenvalue 𝜆_{𝑚𝕋0,−}<0 and 𝜆_𝑚𝕋0,+>0 such that 𝜆_𝑚,+<𝜆_𝑚𝕋0,+,𝜆_{𝑚𝕋0,−}<𝜆_𝑚,−.(2.7)

Proof. Consider the following problems: −Δ²𝑢(𝑡−1)−𝑢𝜆(𝑎𝑚)(=𝑡)𝑢𝑢((𝑏𝑡))==𝜇0.𝑢(𝑡),𝑡∈𝕋₀,(2.8) Let 𝜇_{𝑚𝕋0,𝑘} denote the 𝑘th eigenvalue of (2.8), and 𝜓_{𝑚𝕋0,𝑘} the corresponding eigenfunction which has exactly 𝑘−1 generalized zeros in (𝑎,𝑏). Let 𝐸={𝑢∶{𝑎,…,𝑏}→ℝ∣𝑢(𝑎)=𝑢(𝑏)=0} be a Banach space under the norm ‖𝑢‖_𝐸=max{|𝑢(𝑡)|∶𝑡∈{𝑎,𝑎+1,…,𝑏−1,𝑏}}. Let 𝑆_{𝑚𝕋0,𝜆}=^𝑏−1_𝑡=𝑎||Δ𝜙(𝑡)||²−𝜆^𝑏−1_𝑡=_𝑎+1𝑚(𝑡)𝜙(𝑡)²∶𝜙∈𝐸,^𝑏−1_𝑡=_𝑎+1𝜙(𝑡)²=1.(2.9) We get 𝜇_𝑚𝕋0,1=inf𝑆_{𝑚𝕋0,𝜆}. Similar to the proof of Theorem 2.1, we get the following assertions.(i)If 𝑚(𝑡)>0 for 𝑡∈𝕋₀, then (2.6) has only one principal eigenvalue 𝜆_𝑚𝕋0,+>0.(ii)If 𝑚(𝑡)<0 for 𝑡∈𝕋₀, then (2.6) has only one negative principal eigenvalue 𝜆_{𝑚𝕋0,−}<0.(iii)If 𝑚 changes its sign on 𝕋₀, then (2.6) has two principal eigenvalue 𝜆_{𝑚𝕋0,−}<0 and 𝜆_𝑚𝕋0,+>0.
Now, we prove that the inequalities in (i), (ii), and (iii) hold. For convenience, suppose that ∑^{𝑏−1𝑡=𝑎+1}𝜓^2𝑚_𝕋0,1(𝜆,𝑡)=1.
Let 𝜓_𝑚𝕋0,1 denote the extension of 𝜓_𝑚𝕋0,1 by zero on 𝕋, that is, 𝜓_𝑚𝕋0,1(𝜆,𝑡)=𝜓^𝑚𝕋0,10,(𝜆,𝑡),𝑡𝑡∈∈𝕋𝕋₀⧵,𝕋₀.(2.10) Then, 𝜇_𝑚𝕋0,1(𝜆)=^𝑏−1_𝑡=𝑎|||Δ𝜓_𝑚𝕋0,1(𝜆,𝑡)|||²−𝜆^𝑏−1_𝑡=_𝑎+1=𝑚(𝑡)𝜓^2𝑚_𝕋0,1(𝜆,𝑡)_𝑇_𝑡=1|||Δ𝜓_𝑚𝕋0,1(𝜆,𝑡)|||²−𝜆^𝑇_𝑡=1>inf_𝑣∈𝑋𝑚(𝑡)𝜓^2𝑚_𝕋0,1(𝜆,𝑡)^𝑇_𝑡=1||Δ𝑣(𝜆,𝑡)||²−𝜆^𝑇_𝑡=1𝑚(𝑡)𝑣²(𝜆,𝑡),^𝑇_𝑡=1=𝜇_𝑚,1𝑣²(𝜆,𝑡)=1(𝜆),(2.11) which implies the desired results.

3. The Proof of the Main Result

First, we deal with the case 𝑟>0.

Recall that 𝐿∶𝑋→𝑌, 𝐿𝑢(𝑡)=−Δ²𝑢(𝑡−1).(3.1) Let 𝜁,𝜉∈𝐶(ℝ,ℝ) be such that 𝑓(𝑢)=𝑓₀𝑢+𝜁(𝑢),𝑓(𝑢)=𝑓_∞𝑢+𝜉(𝑢).(3.2) Clearly, lim_|𝑢|→0𝜁(𝑢)𝑢=0,lim^|𝑢|→∞𝜉(𝑢)𝑢=0.(3.3) Let 𝜉̃(𝑢)=max_{0≤𝑠≤𝑢}||𝜉(𝑠)||.(3.4) Then, 𝜉̃ is nondecreasing and lim_|𝑢|→∞𝜉̃(𝑢)𝑢=0.(3.5) Let us consider 𝐿𝑢−𝜆𝑚(𝑡)𝑟𝑓₀𝑢−𝜆𝑚(𝑡)𝑟𝜁(𝑢)=0,(3.6) as a bifurcation problem from the trivial solution 𝑢≡0.

Equation (3.6) can be converted to the equivalent equation 𝑢(𝑡)=𝜆𝐿⁻¹𝑚(⋅)𝑟𝑓₀𝑢(⋅)+𝑚(⋅)𝑟𝜁(𝑢(⋅))(𝑡).(3.7) Further, we note that ‖𝐿⁻¹[𝑚(⋅)𝜁(𝑢(⋅))]‖=𝑜(‖𝑢‖) for 𝑢 near 0 in 𝑋, since ‖‖𝐿⁻¹[𝑚(⋅)𝜁(𝑢(⋅))]‖‖=max_𝑡∈𝕋|||||_𝑇_𝑠=1≤𝐶⋅max_𝑠∈𝕋𝐺(𝑡,𝑠)𝑚(𝑠)𝜁(𝑢(𝑠))|||||||𝑚(𝑠)||‖𝜁(𝑢(⋅))‖,(3.8) where 𝐺(𝑡,𝑠)=1𝑇+1((𝑇𝑇++11−−𝑠𝑡))𝑠𝑡,,00≤≤𝑡𝑠≤≤𝑠𝑡≤≤𝑇𝑇++11,.(3.9)

The results of Rabinowitz [19] for (3.6) can be stated as follows: from (𝜆_𝑚,+/𝑟𝑓₀,0), there emanates an unbounded continua 𝒞₊ of positive solutions in ℝ×𝑋.

It is clear that any solution of (3.6) of the form (1,𝑢) yields a solution 𝑢 of (1.1) and (1.2). So, we focus on the shape of 𝒞₊ under the conditions (H1)–(H3) or (H1)–(H3′), and we will show that 𝒞₊ crosses the hyperplane {1}×𝑋 in ℝ×𝑋.

Lemma 3.1. Suppose that (H1)–(H3) hold. Let 𝐽=[𝑎,𝑏] be a given compact interval in (0,∞). Then for all 𝑟∈𝐽, there exists 𝑀_𝐽>0 such that all possible positive solution 𝑢 of (1.1) and (1.2) satisfy ‖𝑢‖_𝑋≤𝑀_𝐽.

Proof. Suppose on the contrary that there exists a sequence {𝑦_𝑛} of positive solutions for (3.6) with {𝜇_𝑛}⊂𝐽 and ‖𝑦_𝑛‖_𝑋→∞. Let 𝛼∈(0,1/𝑏𝑄), where 𝑄=∑^𝑇𝑠=1𝐺(𝑠,𝑠)|𝑚(𝑠)|. Then, by (H3), there exists 𝑢_𝛼>0 such that 𝑢>𝑢_𝛼 implies 𝑓(𝑢)<𝛼𝑢.
Let 𝐾_𝛼=max_{𝑢∈[0,𝑢𝛼]}𝑓(𝑢) and let 𝐴_𝑛={𝑡∈𝕋∣𝑦_𝑛(𝑡)≤𝑢_𝛼} and 𝐵_𝑛={𝑡∈𝕋∣𝑦_𝑛(t)>𝑢_𝛼}. Then we have 𝑦_𝑛(𝑡)=𝜇_𝑛𝑇_𝑠=1=𝜇_𝑛𝐺(𝑡,𝑠)𝑚(𝑠)𝑓𝑦_𝑛(𝑠)_𝐴𝑛𝐺(𝑡,𝑠)𝑚(𝑠)𝑓𝑦_𝑛(𝑠)+𝜇_𝑛_𝐵𝑛≤𝜇_𝑛𝐺(𝑡,𝑠)𝑚(𝑠)𝑓𝑦_𝑛(𝑠)𝐾_𝛼𝑄+𝜇_𝑛_𝐵𝑛𝐺(𝑡,𝑠)𝑚(𝑠)𝑓𝑦_𝑛(𝑠)(3.10) for 𝑡∈𝕋. Thus, 1𝜇_𝑛≤𝐾^𝛼𝑄‖‖𝑦_𝑛‖‖_𝑋+_𝐵𝑛𝐺(𝑡,𝑠)𝑚(𝑠)𝑓𝑦^𝑛(𝑠)‖‖𝑦_𝑛‖‖_𝑋.(3.11) On 𝐵_𝑛,𝑦_𝑛(𝑠)>𝑢_𝛼 implies 𝑓(𝑦_𝑛(𝑠))/‖𝑦_𝑛‖_𝑋<𝑓(𝑦_𝑛(𝑠))/𝑦_𝑛(𝑠)<𝛼. Thus, 1𝜇_𝑛≤𝐾^𝛼𝑄‖‖𝑦_𝑛‖‖_𝑋+𝛼_𝐵𝑛𝐺(𝑠,𝑠)||𝑚(𝑠)||≤𝐾^𝛼𝑄‖‖𝑦_𝑛‖‖_𝑋+𝛼𝑄.(3.12) Since 0<𝑎≤𝜇_𝑛≤𝑏 for all 𝑛, we have 1/𝜇_𝑛≥1/𝑏 for all 𝑛, and, thus, 1𝑏≤𝐾^𝛼𝑄‖‖𝑦_𝑛‖‖_𝑋+𝛼𝑄.(3.13) By the fact ‖𝑦_𝑛‖_𝑋→∞ as 𝑛→∞, we get 1𝑏≤𝛼𝑄<1𝑏.(3.14) This contradiction completes the proof.

Lemma 3.2. Suppose that (H1)–(H3) hold. Then Proj_ℝ𝒞₊⊃[𝜆_𝑚,+/𝑟𝑓₀,+∞).

Proof. Assume on the contrary that sup{𝜆∣(𝜆,𝑦)∈𝒞₊}<∞, then, there exists a sequence {(𝜇_𝑛,𝑦_𝑛)}∈𝒞₊ such that lim_𝑛→∞‖‖𝑦_𝑛‖‖_𝑋=∞,𝜇_𝑛<𝐶₀(3.15) for some positive constant 𝐶₀ independent of 𝑛, since 𝒞₊ is unbounded. On the other hand, 𝜇_𝑛>0 for all 𝑛∈ℕ, since (0,0) is the only solution of (3.6) for 𝜆=0 and 𝒞₊∩({0}×𝑋)=∅. Meanwhile, {(𝜇_𝑛,𝑦_𝑛)} satisfy Δ²𝑦_𝑛(𝑡−1)+𝜇_𝑛𝑚(𝑡)𝑓𝑦^𝑛(𝑡)𝑦_𝑛(𝑡)𝑦^𝑛𝑦(𝑡)=0,𝑡∈𝕋,_𝑛(0)=𝑦_𝑛(𝑇+1)=0.(3.16) By (H3), there exist a positive constant 𝐿_𝑓>0 such that 𝑓(𝑢)≤𝐿_𝑓𝑢. Define a function 𝜒∶𝕋→[0,∞) by 𝜒(𝑡)=𝐿^𝑓0,,𝑚𝑚((𝑡𝑡))><00,.(3.17) Then, 𝑚(𝑡)(𝑓(𝑦_𝑛(𝑡))/𝑦_𝑛(𝑡))≤𝜒(𝑡)𝑚(𝑡). Let 𝛾₊ be the principal eigenvalue of linear eigenvalue problem Δ²𝑣(𝑡−1)𝑣+(0𝛾)𝜒=(𝑡𝑣)(𝑚𝑇(+𝑡)1𝑣)(𝑡=)=0.0,𝑡∈𝕋,(3.18) Then, by Remark 2.2, 𝛾₊>0. Subsequently, by Theorem 2.3, we know that 𝛾₊≤𝜇_𝑛.(3.19) This combine with Lemma 3.1, lim_𝑛→∞𝜇_𝑛=∞, which contradicts (3.15). Thus, Proj_ℝ𝒞₊⊃𝜆^𝑚,+𝑟𝑓₀,∞.(3.20)

Now, by Lemma 3.2, 𝒞₊ crosses the hyperplane {1}×𝑋 in ℝ×𝑋, and, then, Theorem 1.1(i) holds. To obtain Theorem 1.1(ii), we need to prove the following Lemma.

Lemma 3.3. Suppose that (H1), (H2), and (H3′) hold. Then Proj_ℝ𝒞₊⊃(0,𝜆_𝑚,+/𝑟𝑓₀).

Proof. Let {(𝜇_𝑛,𝑦_𝑛)}⊂𝒞₊ be such that |𝜇_𝑛|+‖𝑦_𝑛‖_𝑋→∞ as 𝑛→∞. Then, Δ²𝑦_𝑛(𝑡−1𝑦)+𝜇_𝑛𝑚(𝑡)𝑓𝑦_𝑛(𝑡)=0,𝑡∈𝕋,_𝑛(0)=𝑦_𝑛(𝑇+1)=0.(3.21) If {‖𝑦_𝑛‖} is bounded, say, ‖𝑦_𝑛‖≤𝑀₁, for some 𝑀₁ independent of 𝑛, then we may assume that lim_𝑛→∞𝜇_𝑛=∞.(3.22) Note that 𝑓𝑦_𝑛(𝑡)𝑦_𝑛(𝑡)≥inf𝑓𝑠(𝑠)∣0<𝑠≤𝑀¹>0.(3.23) Then, there exist two constants 𝑀₃>0,𝑀₂>0 such that 0<𝑀₂<𝑓𝑦^𝑛(𝑡)𝑦_𝑛(𝑡)<𝑀³.(3.24) Define two functions 𝜒₁,𝜒₂∶𝕋→(0,∞) by 𝜒₁=𝑀²𝑀,𝑚(𝑡)>0,₃,𝑚(𝑡)<0.𝜒²=𝑀³𝑀,𝑚(𝑡)>0,₂,𝑚(𝑡)<0.(3.25) Let 𝜂_∗,𝜂^∗ be the positive principal eigenvalue of the linear eigenvalue problems ΔΔ²𝑣(𝑡−1)𝑣+(0𝜂)𝜒=₂(𝑣𝑡)(𝑇𝑚+(𝑡)1𝑣)(=𝑡)0=,0,𝑡∈𝕋,((33..2267))₂𝑣(𝑡−1)𝑣+(0𝜂)𝜒=₁(𝑣𝑡)(𝑚𝑇(+𝑡)1𝑣)(=𝑡)0=,0,𝑡∈𝕋,((33..2289)) respectively.
Combining (3.22) and (3.24) with the relation Δ²𝑦_𝑛(𝑡−1)+𝑟𝜇_𝑛𝑚(𝑡)𝑓𝑦^𝑛(𝑡)𝑦_𝑛(𝑡)𝑦^𝑛𝑦(𝑡)=0,𝑡∈𝕋,_𝑛(0)=𝑦_𝑛(𝑇+1)=0,(3.30) using Theorem 2.3, we get 𝜂_∗𝑟≤𝜇^𝑛≤𝜂^∗𝑟.(3.31) This contradicts (3.22). So, {‖𝑦_𝑛‖_𝑋} is bounded uniformly for all 𝑛∈ℕ.
Now, taking {(𝜇_𝑛,𝑦_𝑛)}⊂𝒞₊ be such that ‖‖𝑦_𝑛‖‖_𝑋⟶+∞,as𝑛⟶+∞.(3.32) We show that lim_𝑛→∞𝜇_𝑛=0.
Suppose on the contrary that, choosing a subsequence and relabeling if necessary, 𝜇_𝑛≥𝑏₀ for some constant 𝑏₀>0. By (3.32), there exists 𝑡₀∈𝕋 such that 𝑦_𝑛(𝑡₀)=‖𝑦_𝑛‖_𝑋 and 𝑦_𝑛(𝑡₀)→+∞ as 𝑛→+∞. Thus, 𝜇_𝑛𝑓𝑦_𝑛𝑡₀𝑦_𝑛𝑡₀⟶+∞,as𝑛⟶+∞.(3.33)
Now, the proof can be divided into two cases.
Case 1 (𝑚(𝑡₀)>0). Consider the following linear eigenvalue problems Δ²𝑣(𝑣𝑡−𝑡₀1)+𝛼𝑚(𝑡)𝑣(𝑡)=0,𝑡=𝑡₀,−1=𝑣𝑡₀+1=0.(3.34) By Theorem 2.4, (3.34) has a positive principal eigenvalue 𝛼₊, and 𝜇_𝑛𝑓𝑦_𝑛𝑡₀𝑦_𝑛𝑡₀≤𝛼⁺,(3.35) which contradicts (3.33).Case 2 (𝑚(𝑡₀)<0). Since (𝜇_𝑛,𝑦_𝑛) is a solution of (3.7), we get 0≥Δ²𝑦_𝑛𝑡₀−1=−𝜇_𝑛𝑟𝑚𝑡₀𝑓𝑦_𝑛𝑡₀>0.(3.36) This is a contradiction.
Thus, lim_𝑛→∞𝜇_𝑛=0.
At last, we deal with the case 𝑟<0.
Let us consider 𝐿𝑢−𝜆𝑟𝑚(𝑡)𝑓₀𝑢−𝜆𝑟𝑚(𝑡)𝜁(𝑢)=0(3.37) as a bifurcation problem from the trivial solution 𝑢≡0. The results of Rabinowitz [19] for (3.37) can be stated as follows: from (𝜆_𝑚,−/−𝑟𝑓₀,0), there emanates an unbounded continua 𝒞₋ of positive solutions in ℝ×𝑋.
It is clear that any solution of (3.37) of the form (−1,𝑢) yields a solution 𝑢 of (1.1) and (1.2). Now, our proofs focus on the shape of 𝒞₋. It will be proved that when (H1)–(H3) hold, then Proj_ℝ𝒞₋⊃(−∞,𝜆_𝑚,−/−𝑟𝑓₀), and when (H1)–(H3′) hold, Proj_ℝ𝒞₋⊃(𝜆_𝑚,−/−𝑟𝑓₀,0), that is, 𝒞₋ crosses the hyperplane {−1}×𝑋 in ℝ×𝑋. Since the proof is similar to the case 𝑟>0, we omit it.

Remark 3.4. As an application of Theorem 1.1, let us consider nonlinear discrete problem with indefinite weight Δ²𝑢(𝑡−𝑢1)(0+)𝑟=𝑚0(,𝑡)𝑓(𝑢𝑢((3𝑡)))==00,,𝑡∈𝕋,(3.38) where 𝕋={1,2}, 𝑚∶𝕋→ℝ with 𝑚(1)=1 and 𝑚(2)=−1, and 𝑓(𝑠)=⎧⎪⎨⎪⎩𝑠,2𝑠𝑠∈[−1,1],1+𝑠⁴,𝑠∈(−∞,−1)∪(1,∞).(3.39) Then, 𝑓₀=1,𝑓_∞=0.(3.40)
To find the principal eigenvalues of linear eigenvalue problem Δ²𝑢(𝑡−𝑢(10))+=𝜆0𝑚,(𝑡)𝑢𝑢((𝑡3))==00,,𝑡∈𝕋,((33..4412)) we rewrite (3.41) to the recursive sequence 𝑢(𝑡+1)=2−𝜆𝑚(𝑡)𝑢(𝑡)−𝑢(𝑡−1).(3.43) This together with the initial value condition 𝑢(0)=0,𝑢(1)=1(3.44) imply that 𝑢𝑢((32))==22−−𝜆𝜆𝑚𝑚((21))𝑢𝑢((21))−−𝑢𝑢((10))==32−−𝜆𝜆₂,.(3.45) The last equation together with the boundary value condition 𝑢(3)=0 imply that 𝜆𝑚_,−=−√3,𝜆𝑚_,+=√3.(3.46) Thus by Theorem 1.1(i), (3.38), has a positive solution if 𝑟∈(−∞,−√3)∪(√3,∞).

Acknowledgments

The authors are grateful to the anonymous referee for their valuable suggestions. R. Ma is supported by NSFC (11061030).