International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

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Volume 2011 |Article ID 989401 | https://doi.org/10.5402/2011/989401

Wanjun Li, "Eigenvalue of Coupled Systems for Hammerstein Integral Equation with Two Parameters", International Scholarly Research Notices, vol. 2011, Article ID 989401, 11 pages, 2011. https://doi.org/10.5402/2011/989401

Eigenvalue of Coupled Systems for Hammerstein Integral Equation with Two Parameters

Academic Editor: Z. Bai
Received29 Dec 2010
Accepted24 Feb 2011
Published11 Oct 2011

Abstract

By using the fixed-point index theory, we discuss the existence, multiplicity, and nonexistence of positive solutions for the coupled systems of Hammerstein integral equation with parameters.

1. Introduction

In recent years, the study of solutions for Hammerstein integral equations has been an interesting topic, since the solution of some boundary value problems for differential equations are usually equivalent to solutions of Hammerstein integral equations [1–6]. And many results concerning the existence of solutions for Hammerstein integral equations have been obtained by many authors [1, 7–9]. For example, in [1] the Hammerstein integral equation:ξ€œπœ“(π‘₯)=πΊπ‘˜(π‘₯,𝑦)𝑓(𝑦,πœ“(𝑦))𝑑𝑦,(1.1) was considered, where πΊβŠ‚β„π‘› is a bounded domain. When the nonlinear term is of the form 𝑓(𝑒) or βˆ‘π‘“(π‘₯,𝑒)=𝑛𝑖=1π‘Žπ‘–(π‘₯)𝑒𝛼𝑖𝑖, 𝛼𝑖>0;𝑖=1,2,…,𝑛, some existence results of nonnegative solutions in 𝐢(𝐺) for (1.1) were obtained; when the nonlinear term is a general 𝑓(π‘₯,𝑒), some multiple results for (1.1) in space 𝐿𝑝(𝐺)(𝑝β‰₯1) were derived. In [8], by means of the decomposition of the operator and the critical point theory, the existence of infinitely many solutions for (1.1) was considered. In [7] the integral equationξ€œπœ“(π‘₯)=πœ†[]π‘₯,π‘₯+π‘‡π‘˜(π‘₯,𝑦)𝑓(𝑦,πœ“(π‘¦βˆ’πœ(𝑦)))𝑑𝑦(1.2) was studied, where πœ† is a parameter. Using the Leggett-Williams fixed-point theorem, when πœ† belongs to some intervals, the existence of triple positive solutions for (1.2) was proved.

More recently, in [9] the integral equationξ€œπ‘’(π‘₯)=πœ†10π‘˜(π‘₯,𝑦)𝑓(𝑦,𝑒(𝑦))𝑑𝑦,(1.3) was studied and obtained that there exists a πœ†βˆ—>0 such that (1.3) has at least two, one and no positive solutions for πœ†βˆˆ(0,πœ†βˆ—), πœ†=πœ†βˆ—, and πœ†>πœ†βˆ—, respectively.

Motivated by the papers mentioned above, we consider the coupled systems of Hammerstein integral equation (1.4) in this paper,ξ€œπ‘’(π‘₯)=πœ†10π‘˜1(π‘₯,𝑦)𝑓1π‘£ξ€œ(𝑦,𝑒(𝑦),𝑣(𝑦))𝑑𝑦,(π‘₯)=πœ‡10π‘˜2(π‘₯,𝑦)𝑓2(𝑦,𝑒(𝑦),𝑣(𝑦))𝑑𝑦,(1.4) where πœ†, πœ‡βˆˆβ„2+⧡{(0,0)}, ℝ+=[0,+∞). Under some new assumptions, we show that there exists a continuous curve Ξ“ separating ℝ2+⧡{(0,0)} into two disjoint subsets π’ͺ1 and π’ͺ2 such that problem (1.4) has at least two, one and no positive solutions for π’ͺ1, Ξ“ and π’ͺ2, respectively. Proofs of our results are mainly based on the fixed-point index theory, for this type of results see [10–12].

The rest of the paper is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. In Section 3, we discuss the existence, multiplicity, and nonexistence of positive solution of the systems (1.4).

The vector (π‘’βˆ—,π‘£βˆ—) is said to be a positive solution of problem (1.4) if and only if (π‘’βˆ—,π‘£βˆ—) satisfies problem (1.4) and π‘’βˆ—(π‘₯)β‰₯0,π‘£βˆ—(π‘₯)>0, or π‘’βˆ—(π‘₯)>0,π‘£βˆ—(π‘₯)β‰₯0 for any π‘₯∈[0,1].

2. Preliminaries and Lemmas

In the rest of the paper, we always suppose the following assumptions hold:(𝐻1)π‘“π‘–βˆˆπΆ[[0,1]×ℝ+×ℝ+,ℝ+],𝑖=1,2; (𝐻2)π‘˜π‘–βˆˆπΆ[[0,1]Γ—[0,1],ℝ+],𝑖=1,2; (𝐻3) there exist 𝜎, 𝛼, π›½βˆˆ(0,1), 𝛼<𝛽 such that π‘˜π‘–(π‘₯,𝑦)β‰₯πœŽπ‘˜π‘–(𝑦,𝑦) for π‘¦βˆˆ(0,1) and π‘₯∈[𝛼,𝛽],𝑖=1,2;(𝐻4)𝑓𝑖 are nondecreasing on ℝ2+ for π‘₯∈[0,1], that is, 𝑓𝑖(π‘₯,𝑒1,𝑣1)≀𝑓𝑖(π‘₯,𝑒2,𝑣2), 𝑖=1,2 whenever (𝑒1,𝑣1)≀(𝑒2,𝑣2), where the inequality on ℝ2+ can be understood componentwise and 𝑓1(π‘₯,0,0)>0 or 𝑓2(π‘₯,0,0)>0 for all π‘₯∈[0,1];(𝐻5) there exist constants 𝑐𝑓𝑖>0,𝑖=1,2, such that 𝑓𝑖(π‘₯,𝑒,𝑣)β‰₯𝑐𝑓𝑖(𝑒+𝑣) for all π‘₯∈[0,1];(𝐻6)limβ€–(𝑒,𝑣)β€–β†’βˆž(𝑓𝑖(π‘₯,𝑒,𝑣)/(𝑒+𝑣))=∞,𝑖=1,2 uniformly for π‘₯∈[0,1].

We will consider the Banach space 𝐸=𝐢[0,1]×𝐢[0,1] equipped with the standard norm β€–β€–(𝑒,𝑣)=‖𝑒‖+‖𝑣‖=max0≀π‘₯≀1||||𝑒(π‘₯)+max0≀π‘₯≀1||||[][].𝑣(π‘₯),(𝑒,𝑣)∈𝐢0,1×𝐢0,1(2.1)

Define𝑃=(𝑒,𝑣)βˆˆπΈβˆΆπ‘’(π‘₯)β‰₯0,𝑒(π‘₯)β‰₯0,minπ‘₯∈[𝛼,𝛽]β€–β€–ξ‚Ό(𝑒(π‘₯)+𝑣(π‘₯))β‰₯𝜎(𝑒,𝑣).(2.2) It is easy to see that 𝑃 is a cone in 𝐸.

We define the operators π‘‡πœ†,π‘‡πœ‡βˆΆπ‘ƒβ†’πΆ[0,1] and π‘‡βˆΆπ‘ƒβ†’πΆ[0,1]×𝐢[0,1] byπ‘‡πœ†ξ€œ(𝑒,𝑣)(π‘₯)=πœ†10π‘˜1(π‘₯,𝑦)𝑓1𝑇(π‘₯,𝑒(𝑦),𝑣(𝑦))𝑑𝑦,πœ‡ξ€œ(𝑒,𝑣)(π‘₯)=πœ‡10π‘˜2(π‘₯,𝑦)𝑓2[],𝑇(𝑦,𝑒(𝑦),𝑣(𝑦))𝑑𝑦,π‘₯∈0,1𝑇(𝑒,𝑣)=πœ†(𝑒,𝑣),π‘‡πœ‡ξ€Έ(𝑒,𝑣),βˆ€(𝑒,𝑣)βˆˆπ‘ƒ.(2.3)

Obvious, the existence of a positive solution of problem (1.4) is equivalent to the existence of a nontrivial fixed point of 𝑇 in 𝑃. It is easy to prove that the following lemma is true.

Lemma 2.1. Assume that (𝐻1)–(𝐻3) hold. Then 𝑇(𝑃)βŠ‚π‘ƒ and π‘‡βˆΆπ‘ƒβ†’π‘ƒ is completely continuous.

Finally we list two lemmas, which are crucial to prove our main results.

Lemma 2.2 (see [13]). Let 𝐸 be a Banach space, 𝐾 a cone and Ξ© an open bounded subset of 𝐸. Let πœƒβˆˆΞ© and π‘‡βˆΆπΎβˆ©Ξ©β†’πΎ be a completely continuous mapping. Suppose that 𝑇π‘₯β‰ πœ›π‘₯ for all π‘₯βˆˆπΎβˆ©πœ•Ξ© and all πœ›β‰₯1. Then 𝑖(𝑇,𝐾∩Ω,𝐾)=1.

Lemma 2.3 (see [13]). Let 𝐸 be a Banach space and 𝐾 a cone in 𝐸. Assume that π‘‡βˆΆπΎπ‘Ÿβ†’πΎ(πΎπ‘Ÿ={π‘₯∈𝐾,β€–π‘₯β€–<π‘Ÿ,π‘Ÿ>0}) is a compact map such that 𝑇π‘₯β‰ π‘₯ for π‘₯βˆˆπœ•πΎπ‘Ÿ. If β€–π‘₯‖≀‖𝑇π‘₯β€– for all π‘₯βˆˆπœ•πΎπ‘Ÿ, then 𝑖(𝑇,πΎπ‘Ÿ,𝐾)=0.

3. Main Results

In this section, we consider the existence of positive solutions for (1.4) in 𝐸.

Lemma 3.1. Assume that (𝐻1)–(𝐻6) hold, and let Ξ£ be a compact subset of ℝ2+⧡{(0,0)}. Then there exists a constant 𝐢Σ>0 such that for all (πœ†,πœ‡)∈Σ and all possible positive solutions (𝑒(π‘₯),𝑣(π‘₯)) of (1.4) at (πœ†,πœ‡), one has β€–(𝑒,𝑣)‖≀𝐢Σ.

Proof. Suppose by contradiction that there is a sequence (𝑒𝑛,𝑣𝑛) of positive solutions of (1.4) at (πœ†π‘›,πœ‡π‘›) such that (πœ†π‘›,πœ‡π‘›)∈Σ for all 𝑛 and β€–(𝑒𝑛,𝑣𝑛)β€–β†’βˆž. Then (𝑒𝑛,𝑣𝑛)βˆˆπ‘ƒ and thus minπ‘₯∈[𝛼,𝛽]𝑒𝑛(π‘₯)+𝑣𝑛‖‖𝑒(π‘₯)β‰₯πœŽπ‘›,𝑣𝑛‖‖.(3.1) Since Ξ£ is compact, the sequence {(πœ†π‘›,πœ‡π‘›)}βˆžπ‘›=1 has a convergent subsequence which we denote without loss of generality still by {(πœ†π‘›,πœ‡π‘›)}βˆžπ‘›=1 such that limπ‘›β†’βˆžπœ†π‘›=πœ†βˆ—, limπ‘›β†’βˆžπœ‡π‘›=πœ‡βˆ—, and at least one πœ†βˆ—>0 or πœ‡βˆ—>0, hence for 𝑛 sufficiently large, we have πœ†π‘›β‰₯πœ†βˆ—/2>0.
Then by (𝐻6), there exists 𝑅𝑓1>0 such that 𝑓1(𝑑,𝑒,𝑣)β‰₯𝐾𝑓1(𝑒+𝑣),βˆ€π‘’+𝑣β‰₯𝑅𝑓1,(3.2) where 𝐾𝑓1 satisfies πœ†βˆ—πœŽ2𝐾𝑓12ξ€œπ›½π›Όπ‘˜1(𝑦,𝑦)𝑑𝑦>1.(3.3)
Thus for π‘₯∈[𝛼,𝛽] and βˆ€π‘’π‘›+𝑣𝑛β‰₯𝑅𝑓1, by using (3.1) and (3.2), we get ‖‖𝑒𝑛‖‖β‰₯𝑒𝑛(π‘₯)=πœ†π‘›ξ€œ10π‘˜1(π‘₯,𝑦)𝑓1𝑦,𝑒𝑛(𝑦),𝑣𝑛β‰₯πœ†(𝑦)π‘‘π‘¦βˆ—2ξ€œπ›½π›Όπ‘˜1(π‘₯,𝑦)𝑓1𝑦,𝑒𝑛(𝑦),𝑣𝑛β‰₯πœ†(𝑦)π‘‘π‘¦βˆ—πΎπ‘“12ξ€œπ›½π›Όπ‘˜1𝑒(π‘₯,𝑦)𝑛(𝑦)+𝑣𝑛β‰₯πœ†(𝑦)π‘‘π‘¦βˆ—πœŽ2𝐾𝑓12ξ€œπ›½π›Όπ‘˜1‖‖𝑒(𝑦,𝑦)𝑛‖‖+‖‖𝑣𝑛‖‖>‖‖𝑒𝑑𝑦𝑛‖‖+‖‖𝑣𝑛‖‖β‰₯‖‖𝑒𝑛‖‖(3.4) for all 𝑛 sufficiently large. This is a contraction.

Lemma 3.2. Assume that (𝐻1)–(𝐻6) hold, and let (1.4) have a positive solution at (πœ†,πœ‡). Then the problem also has a positive solution at (πœ†,πœ‡) for all (πœ†,πœ‡)≀(πœ†,πœ‡).

Proof. Let (𝑒,𝑣) be a positive solution of (1.4) at (πœ†,πœ‡), and let (πœ†,πœ‡)βˆˆβ„2+⧡{(0,0)} with (πœ†,πœ‡)≀(πœ†,πœ‡). First, we assume πœ†βˆˆ(0,πœ†), and πœ‡βˆˆ(0,πœ‡), then π‘’πœ†(π‘₯)=πœ†ξ€œ10π‘˜1(π‘₯,𝑦)𝑓1𝑦,π‘’πœ†(𝑦),π‘£πœ‡ξ€Έξ€œ(𝑦)𝑑𝑦β‰₯πœ†10π‘˜1(π‘₯,𝑦)𝑓1𝑦,π‘’πœ†(𝑦),π‘£πœ‡ξ€Έπ‘£(𝑦)𝑑𝑦,πœ‡(π‘₯)=πœ‡ξ€œ10π‘˜2(π‘₯,𝑦)𝑓2𝑦,π‘’πœ†(𝑦),π‘£πœ‡ξ€Έξ€œ(𝑦)𝑑𝑦β‰₯πœ‡10π‘˜2(π‘₯,𝑦)𝑓2𝑦,π‘’πœ†(𝑦),π‘£πœ‡ξ€Έ(𝑦)𝑑𝑦.(3.5) Set π‘‡πœ†ξ€œ(𝑒,𝑣)(π‘₯)=πœ†10π‘˜1(π‘₯,𝑦)𝑓1𝑇(𝑦,𝑒(𝑦),𝑣(𝑦))𝑑𝑦,πœ‡ξ€œ(𝑒,𝑣)(π‘₯)=πœ‡10π‘˜2(π‘₯,𝑦)𝑓2(𝑦,𝑒(𝑦),𝑣(𝑦))𝑑𝑦,(3.6) and let 𝑒0(π‘₯)=π‘’πœ†(π‘₯), 𝑣0(π‘₯)=π‘£πœ‡(π‘₯), 𝑒𝑛(π‘₯)=π‘‡πœ†(π‘’π‘›βˆ’1(π‘₯),π‘£π‘›βˆ’1(π‘₯)), 𝑣𝑛(π‘₯)=π‘‡πœ‡(π‘’π‘›βˆ’1(π‘₯),π‘£π‘›βˆ’1(π‘₯)), 𝑇(𝑒𝑛(π‘₯),𝑣𝑛(π‘₯))=(π‘‡πœ†(𝑒𝑛(π‘₯),𝑣𝑛(π‘₯)), π‘‡πœ‡(𝑒𝑛(π‘₯),𝑣𝑛(π‘₯))), 𝑛=1,2,…. Then 𝑒0β‰₯𝑒1β‰₯β‹―β‰₯𝑒𝑛β‰₯β‹―β‰₯πœŽπœ†π‘π‘“1𝐾1𝑣>πœƒ,0β‰₯𝑣1β‰₯β‹―β‰₯𝑣𝑛β‰₯β‹―β‰₯πœŽπœ‡π‘π‘“2𝐾2>πœƒ,(3.7) where 𝐾𝑖=βˆ«π›½π›Όπ‘˜π‘–(𝑦,𝑦)𝑑𝑦, 𝑛=1,2. By the compactness of the operator π‘‡πœ†(𝑒,𝑣)(π‘₯), π‘‡πœ‡(𝑒,𝑣)(π‘₯), Lemma 2.1, and the Lebesgue dominated convergence theorem, the sequence {𝑒𝑛}βˆžπ‘›=0={π‘‡πœ†(π‘’π‘›βˆ’1,π‘£π‘›βˆ’1)}βˆžπ‘›=0, and {𝑣𝑛}βˆžπ‘›=0={π‘‡πœ‡(π‘’π‘›βˆ’1,π‘£π‘›βˆ’1)}βˆžπ‘›=0 converges to π‘’βˆ— and π‘£βˆ—, respectively. It is clear that (π‘’βˆ—,π‘£βˆ—)βˆˆπ‘ƒβ§΅{(0,0)} and is a solution (1.4) at (πœ†,πœ‡). Proof of the case πœ‡=0, and πœ†βˆˆ(0,πœ†) or πœ†=0, and πœ‡βˆˆ(0,πœ†) can be done similarly. The proof is complete.

Lemma 3.3. Assume that (𝐻1)–(𝐻4) hold. Then there exists (πœ†βˆ—,πœ‡βˆ—)>(0,0) such that (1.4) has a positive solution for all (πœ†,πœ‡)≀(πœ†βˆ—,πœ‡βˆ—).

Proof. Let 𝑒0∫(π‘₯)=10π‘˜1(π‘₯,𝑦)𝑑𝑦, 𝑣0∫(π‘₯)=10π‘˜2(π‘₯,𝑦)𝑑𝑦, then (𝑒0(π‘₯),𝑣0(π‘₯))βˆˆπ‘ƒβ§΅{(0,0)}. Take 𝑀𝑓𝑖=maxπ‘₯∈[0,1]𝑓𝑖(π‘₯,𝑒0(π‘₯),𝑣0(π‘₯)), 𝑖=1,2. Then 𝑀𝑓𝑖>0 and at (πœ†βˆ—,πœ‡βˆ—)=(1/𝑀𝑓1,1/𝑀𝑓2), we get π‘‡πœ†βˆ—(𝑒,𝑣)(π‘₯)=πœ†βˆ—ξ€œ10π‘˜1(π‘₯,𝑦)𝑓1𝑇(𝑦,𝑒(𝑦),𝑣(𝑦))𝑑𝑦,πœ‡βˆ—(𝑒,𝑣)(π‘₯)=πœ‡βˆ—ξ€œ10π‘˜2(π‘₯,𝑦)𝑓2(𝑦,𝑒(𝑦),𝑣(𝑦))𝑑𝑦.(3.8) This shows that (πœ†βˆ—,πœ‡βˆ—)>(0,0) and 𝑒0(π‘₯)β‰₯πœ†βˆ—ξ€œ10π‘˜1(π‘₯,𝑦)𝑓1𝑦,𝑒0(𝑦),𝑣0𝑣(𝑦)𝑑𝑦,0(π‘₯)β‰₯πœ‡βˆ—ξ€œ10π‘˜2(π‘₯,𝑦)𝑓2𝑦,𝑒0(𝑦),𝑣0ξ€Έ(𝑦)𝑑𝑦(3.9) for π‘₯∈[0,1]. Set 𝑒𝑛(π‘₯)=π‘‡πœ†βˆ—ξ€·π‘’π‘›βˆ’1(π‘₯),π‘£π‘›βˆ’1ξ€Έ(π‘₯),𝑣𝑛(π‘₯)=π‘‡πœ‡βˆ—ξ€·π‘’π‘›βˆ’1(π‘₯),π‘£π‘›βˆ’1ξ€Έ,𝑇𝑒(π‘₯)𝑛(π‘₯),𝑣𝑛(ξ€Έ=𝑇π‘₯)πœ†βˆ—ξ€·π‘’π‘›(π‘₯),𝑣𝑛(ξ€Έπ‘₯),π‘‡πœ‡βˆ—ξ€·π‘’π‘›(π‘₯),𝑣𝑛(π‘₯)ξ€Έξ€Έ,𝑛=1,2,…(3.10) for π‘₯∈[0,1]. Then 𝑒0β‰₯𝑒1β‰₯β‹―β‰₯𝑒𝑛β‰₯β‹―β‰₯πœŽπœ†βˆ—ξ€œ10π‘˜1𝑣(𝑦,𝑦)𝑑𝑦>πœƒ,0β‰₯𝑣1β‰₯β‹―β‰₯𝑣𝑛β‰₯β‹―β‰₯πœŽπœ‡βˆ—ξ€œ10π‘˜2(𝑦,𝑦)𝑑𝑦>πœƒ.(3.11) We conclude the proof similarly to Lemma 3.2.

Define 𝒫={(πœ†,πœ‡)βˆˆβ„2+⧡{(0,0)}∢ (1.4) has a positive solution at (πœ†,πœ‡)}; then by Lemma 3.3, π’«β‰ πœ™, and it is easy to see that (𝒫,≀) is a partially ordered set.

Lemma 3.4. Assume that (𝐻1)–(𝐻6) hold. Then (𝒫,≀) is bounded above.

Proof. Suppose to the contrary that there exists a fixed-point sequence {(𝑒𝑛,𝑣𝑛)}(𝑛=1,2,…) of 𝑇(𝑒,𝑣) at (πœ†π‘›,πœ‡π‘›) such that limπ‘›β†’βˆžβ€–(πœ†π‘›,πœ‡π‘›)β€–=∞. Considering a subsequence if necessary, we assume limπ‘›β†’βˆžπœ†π‘›=∞. The proof for the case limπ‘›β†’βˆžπœ‡π‘›=∞ can be shown by an analogous way. Then there are two cases to be considered:(i)there exists a constant 𝐻>0 such that β€–(𝑒𝑛,𝑣𝑛)‖≀𝐻,𝑛=0,1,2,…;(ii)there exists a subsequence {(𝑒𝑛𝑖,𝑣𝑛𝑖)}βˆžπ‘–=1 such that limπ‘–β†’βˆžβ€–(𝑒𝑛𝑖,𝑣𝑛𝑖)β€–=∞,which is impossible by Lemma 3.1. So we only consider (i). In view of (𝐻4)–(𝐻6) we can choose 𝑙0>0 such that 𝑓1(𝑦,0,0)>𝑙0𝐻 or 𝑓2(𝑦,0,0)>𝑙0𝐻 and further 𝑓1(𝑦,𝑒𝑛,𝑣𝑛)>𝑙0𝐻 or 𝑓2(𝑦,𝑒𝑛,𝑣𝑛)>𝑙0𝐻 for π‘¦βˆˆ[0,1]. Thus for π‘₯∈[𝛼,𝛽], we know 𝑒𝑛(π‘₯)+𝑣𝑛(π‘₯)=πœ†π‘›ξ€œ10π‘˜1(π‘₯,𝑦)𝑓1𝑦,𝑒𝑛(𝑦),𝑣𝑛(𝑦)𝑑𝑦+πœ‡π‘›ξ€œ10π‘˜2(π‘₯,𝑦)𝑓2𝑦,𝑒𝑛(𝑦),𝑣𝑛(𝑦)𝑑𝑦β‰₯πœ†π‘›πœŽξ€œπ›½π›Όπ‘˜1(𝑦,𝑦)𝑓1𝑦,𝑒𝑛(𝑦),𝑣𝑛(𝑦)𝑑𝑦+πœ‡π‘›πœŽξ€œπ›½π›Όπ‘˜2(𝑦,𝑦)𝑓2𝑦,𝑒𝑛(𝑦),𝑣𝑛(𝑦)𝑑𝑦.(3.12) Now we will distinguish two cases.Case 1. If 𝑓1(𝑦,𝑒𝑛,𝑣𝑛)>𝑙0𝐻 and 𝑓2(𝑦,𝑒𝑛,𝑣𝑛)>𝑙0𝐻, we have 𝑒𝑛(π‘₯)+𝑣𝑛(π‘₯)β‰₯πœ†π‘›πœŽπ‘™0𝐻𝐾1+πœ‡π‘›πœŽπ‘™0𝐻𝐾2β‰₯ξ€·πœ†π‘›+πœ‡π‘›ξ€ΈπœŽπ‘™0𝐻𝐾0,(3.13) where 𝐾0=min(𝐾1,𝐾2), which implies that 𝐻β‰₯(πœ†π‘›+πœ‡π‘›)πœŽπ‘™0𝐻𝐾1 or (πœ†π‘›+πœ‡π‘›)≀1/πœŽπ‘™0𝐾1, which is a contradiction.Case 2. If that one of 𝑓1(𝑦,𝑒𝑛,𝑣𝑛)>𝑙0𝐻 and 𝑓2(𝑦,𝑒𝑛,𝑣𝑛)>𝑙0𝐻 is satisfied, without loss of generality, we assume 𝑓1(𝑦,𝑒𝑛,𝑣𝑛)>𝑙0𝐻. The proof for the case 𝑓2(𝑦,𝑒𝑛,𝑣𝑛)>𝑙0𝐻 can be shown by an analogous way. By (𝐻5) we have 𝑒𝑛(π‘₯)+𝑣𝑛(π‘₯)β‰₯πœ†π‘›πœŽπ‘™0𝐻𝐾1+πœ‡π‘›πœŽ2‖‖𝑒𝑛,𝑣𝑛‖‖𝑐𝑓2𝐾2β‰₯πœ†π‘›πœŽπ‘™0‖‖𝑒𝑛,𝑣𝑛‖‖𝐾1+πœ‡π‘›πœŽ2‖‖𝑒𝑛,𝑣𝑛‖‖𝑐𝑓2𝐾2β‰₯ξ€·πœ†π‘›+πœ‡π‘›ξ€ΈπΎ3‖‖𝑒𝑛,𝑣𝑛‖‖,(3.14) where 𝐾3=min(πœŽπ‘™0𝐾1,𝜎2𝑐𝑓2𝐾2); which implies that β€–(𝑒𝑛,𝑣𝑛)β€–β‰₯(πœ†π‘›+πœ‡π‘›)𝐾3β€–(𝑒𝑛,𝑣𝑛)β€– or (πœ†π‘›+πœ‡π‘›)≀1/𝐾3, which is a contradiction. The proof is complete.

Lemma 3.5. Assume that (𝐻1)–(𝐻6) hold. Then every chain in 𝒫 has a unique supremum in 𝒫.

Proof. Let π’ž be a chain in 𝒫. Since 𝒫 is a partially ordered set, it is enough to show that π’ž has an upper bound in 𝒫. Without loss of generality, we may choose a distinct sequence {(πœ†π‘›,πœ‡π‘›)}βˆˆπ’ž such that (πœ†π‘›,πœ‡π‘›)≀(πœ†π‘›+1,πœ‡π‘›+1),𝑛=1,2,…. By Lemma 3.4, two sequences {πœ†π‘›} and {πœ‡π‘›} converge to, say, πœ†π’ž and πœ‡π’ž, respectively. If (πœ†π’ž,πœ‡π’ž)βˆˆπ’«, then the proof is done. Since the sequence {(πœ†π‘›,πœ‡π‘›)} is bounded above, we may assume that the sequence belongs to a compact rectangle in ℝ2+⧡{(0,0)} and Lemma 3.1 implies that the corresponding solutions {(𝑒𝑛,𝑣𝑛)} are uniformly bounded in 𝐸. By the compactness of the integral operators π‘‡πœ† and π‘‡πœ‡, the sequence {(𝑒𝑛,𝑣𝑛)} has a subsequence converging to, say, {(𝑒𝑛,𝑣𝑛)}∈𝐸. We can easily show, by the Lebesgue convergence theorem, that (π‘’π’ž,π‘£π’ž) is a solution of (1.4) at (πœ†π’ž,πœ‡π’ž). Thus (πœ†π’ž,πœ‡π’ž)βˆˆπ’« and this completes the proof.

Lemma 3.6. Assume that (𝐻1)–(𝐻6) hold. Then there exists π‘ βˆ—βˆˆ[πœ‡βˆ—,πœ‡π‘’] such that (1.4) has a positive solution at (0,𝑠) for all 0<π‘ β‰€π‘ βˆ—, and no solution at (0,𝑠) for all 𝑠>π‘ βˆ—. Similarly, there exists π‘Ÿβˆ—βˆˆ[πœ†βˆ—,πœ†π‘’] such that (1.4) has a positive solution at (π‘Ÿ,0) for all 0<π‘ β‰€π‘ βˆ—, and no solution at (π‘Ÿ,0) for all π‘Ÿ>π‘Ÿβˆ—, where (πœ†π‘’,πœ‡π‘’) is upper bound of (𝒫,≀).

Proof. We know by Lemma 3.3 that (1.4) has a positive solution at (0,𝑠) for all 0<π‘ β‰€πœ‡βˆ—. Thus {(0,𝑠)βˆΆπ‘ >0}βˆ©π’« is a nonempty chain in 𝒫 and by Lemma 3.5, it has a unique supremum of the form (0,π‘ βˆ—) in 𝒫. The proof of the second part is similar.

Lemma 3.7. Assume that (𝐻1)–(𝐻6) hold. Then there exists a continuous curve Ξ“ separating ℝ2+⧡{(0,0)} into two disjoint subsets π’ͺ1 and π’ͺ2 such that problem (1.4) has at least one positive solution for (πœ†,πœ‡)∈π’ͺ1βˆͺΞ“ and no solution for (πœ†,πœ‡)∈π’ͺ2.

Proof. We first construct the curve Ξ“ on 𝐑2+⧡{(0,0)}. Define 𝐿π‘₯=ξ€½(π‘Ÿ,𝑠)βˆˆπ‘2+⧡{(0,0)}βˆΆπ‘ =π‘Ÿβˆ’π‘₯,π‘₯βˆˆπ‘….(3.15) At π‘₯=βˆ’π‘ βˆ—, we know by Lemma 3.6, (0,π‘ βˆ—)βˆˆπΏβˆ’π‘ βˆ—βˆ©π’«. Thus πΏβˆ’π‘ βˆ—βˆ©π’« is a nonempty chain in 𝒫 and Lemma 3.5 implies that the chain has a unique supremum. We show sup{πΏβˆ’π‘ βˆ—βˆ©π’«}=(0,π‘ βˆ—). Indeed, otherwise, we may choose (Μƒπ‘Ÿ,̃𝑠)(β‰ (0,π‘ βˆ—))βˆˆπΏβˆ’π‘ βˆ— such that (0,π‘ βˆ—)<(Μƒπ‘Ÿ,̃𝑠) and (1.4) has a solution at (Μƒπ‘Ÿ,̃𝑠). Thus by Lemma 3.2, (1.4) has a solution at (0,̃𝑠) and this contradicts Lemma 3.6. Similarly, we get sup{πΏπ‘Ÿβˆ—βˆ©π’«}=(π‘Ÿβˆ—,0).
For βˆ’π‘ βˆ—<π‘₯<π‘Ÿβˆ—, we know by Lemmas 3.3, 3.5, and 3.6 that 𝐿π‘₯βˆ©π’« is a nonempty chain in 𝒫 and thus the chain also has a unique supremum.
We notice that 𝐿π‘₯βˆ©π’«=βˆ…, for π‘₯<βˆ’π‘ βˆ— or π‘₯>π‘Ÿβˆ—. Now for π‘₯∈[βˆ’π‘ βˆ—,π‘Ÿβˆ—], let us define Γ𝐿(π‘₯)=supπ‘₯ξ€Έβˆ©π’«.(3.16) Then Ξ“βˆΆ[βˆ’π‘ βˆ—,π‘Ÿβˆ—]→𝐑2+⧡{(0,0)} is well defined and Ξ“(βˆ’π‘ βˆ—)=(0,π‘ βˆ—) and Ξ“(π‘Ÿβˆ—)=(π‘Ÿβˆ—,0). Similar to [10, Theorem  3.1], it is easy to prove that Ξ“ is continuous on [βˆ’π‘ βˆ—,π‘Ÿβˆ—].
Consequently, the curve Ξ“ separates 𝐑2+⧡{(0,0)} into two disjoint subsets π’ͺ1 and π’ͺ2, where π’ͺ1 is bounded and π’ͺ2 is unbounded. It is obvious that (1.4) has a positive solution at Ξ“(π‘₯) for all π‘₯∈[βˆ’π‘ βˆ—,π‘Ÿβˆ—]. If (πœ†,πœ‡)∈π’ͺ1 and so if (πœ†,πœ‡)∈𝐿π‘₯0, then π‘₯0∈[βˆ’π‘ βˆ—,π‘Ÿβˆ—] and (πœ†,πœ‡)<Ξ“(π‘₯0). Thus by Lemma 3.2, (πœ†,πœ‡)βˆˆπ’«. On the other hand, if (πœ†,πœ‡)∈π’ͺ2 and if (πœ†,πœ‡)∈𝐿π‘₯0, then either Ξ“(π‘₯0) is not defined when π‘₯0∈[βˆ’π‘ βˆ—,π‘Ÿβˆ—] or (πœ†,πœ‡)>Ξ“(π‘₯0) when π‘₯0βˆ‰[βˆ’π‘ βˆ—,π‘Ÿβˆ—]. We get (πœ†,πœ‡)βˆ‰π’« for both cases and the proof is done.

Now, we show the existence of the second positive solution for (πœ†,πœ‡)∈π’ͺ1. Let (πœ†,πœ‡)∈π’ͺ1, then we may choose π‘₯0∈[βˆ’π‘ βˆ—,π‘Ÿβˆ—] such that (πœ†,πœ‡)∈𝐿π‘₯0. We know by Lemma 3.7 that (1.4) has a positive solution at Ξ“(π‘₯0), so let (π‘’βˆ—,π‘£βˆ—) be the solution at Ξ“(π‘₯0) and let us denote Ξ“(π‘₯0)=(πœ†βˆ—,πœ‡βˆ—). Then obviously (πœ†,πœ‡)<(πœ†βˆ—,πœ‡βˆ—) and we have the following theorem.

Lemma 3.8. Let (πœ†,πœ‡)∈π’ͺ1. Then there exists πœ€0>0 and for all π‘₯∈[0,1], 0<πœ€β‰€πœ€0 such that π‘’βˆ—πœ€ξ€œ(π‘₯)>πœ†10π‘˜1(π‘₯,𝑦)𝑓1𝑦,π‘’βˆ—πœ€(𝑦),π‘£βˆ—πœ€ξ€Έπ‘£(𝑦)𝑑𝑦,βˆ—πœ€ξ€œ(π‘₯)>πœ‡10π‘˜2(π‘₯,𝑦)𝑓2𝑦,π‘’βˆ—πœ€(𝑦),π‘£βˆ—πœ€ξ€Έ(𝑦)𝑑𝑦,(3.17) where π‘’βˆ—πœ€(π‘₯)=π‘’βˆ—(π‘₯)+πœ€ and π‘£βˆ—πœ€(π‘₯)=π‘£βˆ—(π‘₯)+πœ€.

Proof. From (𝐻4), there exists constant 𝑀>0 such that min[]π‘₯∈0,1𝑓𝑖π‘₯,π‘’βˆ—(π‘₯),π‘£βˆ—ξ€Έ(π‘₯)β‰₯𝑀>0,𝑖=1,2.(3.18) Then by the uniform continuity of 𝑓𝑖 on a compact set, there exists πœ€0>0 small enough such that for all π‘₯∈[0,1], 0<πœ€β‰€πœ€0, ||𝑓1ξ€·π‘₯,π‘’βˆ—πœ€(π‘₯),π‘£βˆ—πœ€ξ€Έ(π‘₯)βˆ’π‘“1ξ€·π‘₯,π‘’βˆ—(π‘₯),π‘£βˆ—ξ€Έ||ξ‚΅πœ†(π‘₯)<π‘€βˆ—πœ†ξ‚Ά,||π‘“βˆ’12ξ€·π‘₯,π‘’βˆ—πœ€(π‘₯),π‘£βˆ—πœ€ξ€Έ(π‘₯)βˆ’π‘“2ξ€·π‘₯,π‘’βˆ—(π‘₯),π‘£βˆ—ξ€Έ||ξ‚΅πœ‡(π‘₯)<π‘€βˆ—πœ‡ξ‚Ά,βˆ’1πœ†π‘“1ξ€·π‘₯,π‘’βˆ—πœ€(π‘₯),π‘£βˆ—πœ€ξ€Έ(π‘₯)βˆ’πœ†βˆ—π‘“1ξ€·π‘₯,π‘’βˆ—(π‘₯),π‘£βˆ—ξ€Έ<ξ€·πœ†(π‘₯)βˆ—ξ€·βˆ’πœ†ξ€Έξ€·π‘€βˆ’π‘“π‘₯,π‘’βˆ—(π‘₯),π‘£βˆ—(π‘₯)≀0,πœ‡π‘“2ξ€·π‘₯,π‘’βˆ—πœ€(π‘₯),π‘£βˆ—πœ€ξ€Έ(π‘₯)βˆ’πœ‡βˆ—π‘“2ξ€·π‘₯,π‘’βˆ—(π‘₯),π‘£βˆ—ξ€Έ<ξ€·πœ‡(π‘₯)βˆ—βˆ’πœ‡ξ€Έξ€·π‘€βˆ’π‘“2ξ€·π‘₯,π‘’βˆ—(π‘₯),π‘£βˆ—(π‘₯)≀0.(3.19) Furthermore, we have for all π‘₯∈[0,1]π‘’βˆ—πœ€(π‘₯)>πœ†βˆ—ξ€œ10π‘˜1(π‘₯,𝑦)𝑓1𝑦,π‘’βˆ—(𝑦),π‘£βˆ—ξ€Έξ€œ(𝑦)𝑑𝑦>πœ†10π‘˜1(π‘₯,𝑦)𝑓1𝑦,π‘’βˆ—πœ€(𝑦),π‘£βˆ—πœ€ξ€Έ(𝑦)𝑑𝑦.(3.20) The inequalities for π‘£βˆ—πœ€ can be shown similarly and the proof is done.

We now state and prove our main result in this section.

Theorem 3.9. Assume (𝐻1)–(𝐻6). Then there exists a continuous curve Ξ“ separating ℝ2+⧡{(0,0)} into two disjoint subsets π’ͺ1 and π’ͺ2 such that problem (1.4) has at least two positive solution on π’ͺ1, at least one positive solution on Ξ“ and no solution on π’ͺ2.

Proof. By Lemma 3.7, it is enough to show the existence of the second positive solution of (1.4) for (πœ†,πœ‡)∈π’ͺ1. Let (πœ†,πœ‡)∈π’ͺ1. Let (πœ†,πœ‡)∈𝐿π‘₯0 and (π‘’βˆ—,π‘£βˆ—) be a positive solution of (1.4) at Ξ“(π‘₯0)=∢(πœ†βˆ—,πœ‡βˆ—) and let π‘’βˆ—πœ€(π‘₯)=π‘’βˆ—+πœ€,π‘£βˆ—πœ€(π‘₯)=π‘£βˆ—+πœ€, where πœ€ is given in Lemma 3.8. Let ξ€½Ξ©=(𝑒,𝑣)βˆˆπΈβˆΆβˆ’πœ€<𝑒(π‘₯)<π‘’βˆ—πœ€(π‘₯),βˆ’πœ€<𝑣(π‘₯)<π‘£βˆ—πœ€[]ξ€Ύ(π‘₯),π‘₯∈0,1,(3.21) then Ξ© is bounded open in 𝐸, 0∈Ω and π‘‡βˆΆπ‘ƒβˆ©Ξ©β†’π‘ƒ be a completely continuous mapping, since it is completely continuous. Let (𝑒,𝑣)βˆˆπ‘ƒβˆ©πœ•Ξ©. Then there exists π‘₯π‘œβˆˆ[0,1] such that either 𝑒(π‘₯π‘œ)=π‘’βˆ—πœ€(π‘₯π‘œ) or 𝑣(π‘₯π‘œ)=π‘£βˆ—πœ€(π‘₯π‘œ). Suppose that 𝑒(π‘₯π‘œ)=π‘’βˆ—πœ€(π‘₯π‘œ). Then by (𝐻4) and Lemma 3.8, π‘‡πœ†ξ€·π‘₯(𝑒,𝑣)π‘œξ€Έξ€œ=πœ†10π‘˜1ξ€·π‘₯π‘œξ€Έπ‘“,𝑦1ξ€œ(𝑦,𝑒(𝑦),𝑣(𝑦))π‘‘π‘¦β‰€πœ†10π‘˜1ξ€·π‘₯π‘œξ€Έπ‘“,𝑦1𝑦,π‘’βˆ—πœ€(𝑦),π‘£βˆ—πœ€ξ€Έ(𝑦)𝑑𝑦<π‘’βˆ—πœ€ξ€·π‘₯π‘œξ€Έξ€·π‘₯=π‘’π‘œξ€Έξ€·π‘₯β‰€πœ›π‘’π‘œξ€Έ,(3.22) for all πœ›β‰₯1. Similarly, for the case 𝑣(π‘₯π‘œ)=π‘£βˆ—πœ€(π‘₯π‘œ), we can get π‘‡πœ‡(𝑒,𝑣)(π‘₯π‘œ)<πœ›π‘£(π‘₯π‘œ) for all πœ›β‰₯1. Thus 𝑇(𝑒,𝑣)=(π‘‡πœ†(𝑒,𝑣),π‘‡πœ‡(𝑒,𝑣))β‰ πœ›(𝑒,𝑣), for all (𝑒,𝑣)βˆˆπ‘ƒβˆ©Ξ© and all πœ›β‰₯1 and by Lemma 2.2, 𝑖(𝑇,π‘ƒβˆ©Ξ©,𝑃)=1.(3.23) Modifying (3.2) for πœ†βˆ—=πœ†, that is, there exists 𝑅𝑓1>0 such that 𝑓1(π‘₯,𝑒,𝑣)β‰₯𝐾𝑓1(𝑒+𝑣),βˆ€π‘’+𝑣β‰₯𝑅𝑓1,(3.24) where 𝐾𝑓1 satisfies πœ†πœŽ2𝐾𝑓12ξ€œπ›½π›Όπ‘˜1(𝑦,𝑦)𝑑𝑦>1.(3.25) Let 𝑅0=max{𝐢Σ,(1/𝜎)𝑅𝑓1,β€–(π‘’βˆ—πœ€,π‘£βˆ—πœ€)β€–}, where 𝐢Σ is given in Lemma 3.1 with Ξ£ a compact rectangle in ℝ2+⧡{(0,0)} containing (πœ†,πœ‡). Let 𝑃𝑅0={(𝑒,𝑣)βˆˆπ‘ƒ,β€–(𝑒,𝑣)β€–<𝑅0}. Then (𝑒,𝑣)≠𝑇(𝑒,𝑣) for (𝑒,𝑣)βˆˆπœ•π‘ƒπ‘…0, by Lemma 3.1. Furthermore, if (𝑒,𝑣)βˆˆπ‘ƒπ‘…0, then 𝑒(𝑑)+𝑣(𝑑)β‰₯minπ‘₯∈[𝛼,𝛽](𝑒(π‘₯)+𝑣(π‘₯))β‰₯πœŽβ€–(𝑒,𝑣)β€–β‰₯𝑅𝑓1.(3.26) Thus by (3.24), 𝑓1(π‘₯,𝑒(π‘₯),𝑣(π‘₯))β‰₯𝐾𝑓1(𝑒(π‘₯)+𝑣(π‘₯)), for all π‘₯∈[𝛼,𝛽] and π‘‡πœ†ξ€œ(𝑒,𝑣)(π‘₯)=πœ†10π‘˜1(π‘₯,𝑦)𝑓1ξ€œ(𝑦,𝑒(𝑦),𝑣(𝑦))𝑑𝑦β‰₯πœ†πœŽπ›½π›Όπ‘˜1(𝑦,𝑦)𝑓1(𝑦,𝑒(𝑦),𝑣(𝑦))𝑑𝑦β‰₯πœ†πœŽπΎπ‘“1ξ€œπ›½π›Όπ‘˜1(β‰₯𝑦,𝑦)(𝑒(𝑦)+𝑣(𝑦))π‘‘π‘¦πœ†πœŽ2𝐾𝑓12ξ€œπ›½π›Όπ‘˜1(𝑦,𝑦)β€–(𝑒,𝑣)‖𝑑𝑠>β€–(𝑒,𝑣)β€–.(3.27) Therefore ‖𝑇(𝑒,𝑣)β€–β‰₯β€–π‘‡πœ†(𝑒,𝑣)β€–>β€–(𝑒,𝑣)β€– and by Lemma 2.3, 𝑖𝑇,𝑃𝑅0ξ€Έ,𝑃=0.(3.28) Consequently by the additivity of the fixed-point index, ξ€·0=𝑖𝑇,𝑃𝑅0ξ€Έξ‚€,𝑃=𝑖(𝑇,π‘ƒβˆ©Ξ©,𝑃)+𝑖𝑇,𝑃𝑅0β§΅ξ‚π‘ƒβˆ©Ξ©,𝑃.(3.29) Since 𝑖(𝑇,π‘ƒβˆ©Ξ©,𝑃)=1,𝑖(𝑇,𝑃𝑅0β§΅π‘ƒβˆ©Ξ©,𝑃)=βˆ’1 and thus 𝑇 has a fixed point on π‘ƒβˆ©Ξ© and another on 𝑃𝑅0β§΅π‘ƒβˆ©Ξ©, and this completes the proof.

Acknowledgments

The author is very grateful to the referee for her/his important comments and suggestions. This work is sponsored by the Tutorial Scientific Research Program Foundation of Education Department of Gansu Province, China (0810-03).

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Copyright Β© 2011 Wanjun Li. 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.


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