On Six Solutions for m-Point Differential Equations System with Two Coupled Parallel Sub-Super Solutions

Abstract

Under the assumption of two coupled parallel subsuper solutions, the existence of at least six solutions for a kind of second-order m-point differential equations system is obtained using the fixed point index theory. As an application, an example to demonstrate our result is given.

1. Introduction

In this paper, we consider the following second-order 𝑚-point boundary value problems of nonlinear equations system −𝜑^′′(𝑡)=𝑓₁𝜑(𝑡)+𝑓₂𝜓(𝑡),𝑡∈[0,1],−𝜓^′′(𝑡)=𝑔₁𝜓(𝑡)+𝑔₂𝜑(𝑡)[],𝜑,𝑡∈0,1_′(0)=0,𝜑(1)=^𝑚−2_𝑖=1𝛼_𝑖𝜑𝜉_𝑖,𝜓^′(0)=0,𝜓(1)=^𝑚−2_𝑖=1𝛼_𝑖𝜓𝜉_𝑖, (1.1) where 𝑓_𝑖,𝑔_𝑖∶ℝ¹→ℝ¹(𝑖=1,2) are continuous and 𝛼_𝑖,𝜉_𝑖 satisfying(𝐻₀)∑^𝑚−2_𝑖=1𝛼_𝑖∈(0,1) with 𝛼_𝑖∈(0,+∞) for 𝑖=1,2,…,𝑚−2 and 0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<1.

Multipoint boundary value problems arise in many applied sciences for example, the vibrations of a guy wire composed of 𝑁 parts with a uniform cross-section throughout, but different densities in different parts can be set up as a multipoint boundary value problems (see [1]). Many problems in the theory of elastic stability can be modelled by multipoint boundary value problems (see [2]). The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev [3]. Subsequently, Gupta [4] studied certain three-point boundary value problems for nonlinear second-order ordinary differential equations. Since then, the solvability of more general nonlinear multipoint boundary value problems has been discussed by several authors using various methods. We refer the readers to [5–12] and the references therein.

In the recent years, many authors have studied existence and multiplicity results for solutions of multipoint boundary value problems via the well-ordered upper and lower solutions method, see [8, 13, 14] and the references therein. However, only in very recent years, some authors considered the multiplicity of solutions under conditions of non-well-ordered upper and lower solutions. For some abstract results concerning conditions of non-well-ordered upper and lower solutions, the readers are referred to recent works [15–18].

In [19], Xu et al. considered the following second-order three-point boundary value problem 𝑦^′′(𝑡)[],𝑦(0)(1)𝜂+𝑓𝑡,𝑦=0,𝑡∈0,1=0,𝑦−𝛼𝑦=0, (1.2) where 0<𝜂<1,0<𝛼<1,𝑓∈𝐶([0,1]×ℝ¹,ℝ¹). He obtained the following result. First, let us give the following condition (𝐻₀)^′ to be used later.

(𝐻₀)^′ There exists 𝑀>0 such that 𝑓𝑡,𝑥₂−𝑓𝑡,𝑥₁≥−𝑀𝑥₂−𝑥₁,𝑡∈[0,1],𝑥₂≥𝑥₁. (1.3) Let the function 𝑒 be 𝑒=𝑒(𝑡)=𝑡 for 𝑡∈[0,1].

Theorem 1.1. Suppose that (𝐻₀)^′ holds, 𝑢₁ and 𝑢₂ are two strict lower solutions of (1.2), 𝑣₁ and 𝑣₂ are two strict upper solutions of (1.2), and 𝑢₁<𝑣₁,𝑢₂<𝑣₂,𝑢₂≰𝑣₁,𝑢₁≰𝑣₂. Moreover, assume −𝜉₀𝑒≤𝑢₂−𝑢₁≤𝜉₀𝑒,−𝜉₀𝑒≤𝑣₂−𝑣₁≤𝜉₀𝑒, (1.4) for some 𝜉₀>0. Then, the three-point boundary value problem (1.2) has at least six solutions.

We would also like to mention the result of Yang [20], in [20]. Yang studied the following integral boundary value problem −𝑎𝑢^′′+𝑏𝑢=𝑔(𝑡)𝑓(𝑡,𝑢),cos𝛾₀𝑢(0)−sin𝛾₀𝑢^′(0)=¹₀𝑢(𝜏)𝑑𝛼(𝜏),cos𝛾₁𝑢(1)+sin𝛾₁𝑢^′(1)=¹₀𝑢(𝜏)𝑑𝛽(𝜏), (1.5) where 𝛾₀∈[0,𝜋/2] and 𝛾₁∈[0,𝜋/2], ∫¹₀𝑢(𝜏)𝑑𝛼(𝜏) and ∫¹₀𝑢(𝜏)𝑑𝛽(𝜏) denote the Riemann-Stieltjes integrals of 𝑢 with respect to 𝛼 and 𝛽, respectively. Some sufficient conditions for the existence of either none, or one, or more positive solutions of the problem (1.5) were established. The main tool used in the proofs of existence results is a fixed point theorem in a cone, due to Krasnoselskii and Zabreiko.

At the same time, we note that Webb and Lan [21] have considered the first eigenvalue of the following linear problem 𝑢^′′(𝑡)(𝑡)𝑢(0)(1)=+𝜆𝑢=0,0<𝑡<1,=0,𝑢^𝑚−2_𝑖=1𝛼_𝑖𝑢𝜂_𝑖, (1.6) they also investigated the existence and multiplicity of positive solutions of several related nonlinear multipoint boundary value problems. Furthermore, Ma and O'Regan [22] studied the spectrum structure of the problem (1.6), and the authors obtained the concrete computational method and the corresponding properties of real eigenvalue of (1.6) by constructing an auxiliary function. Their work is very fundamental to further study for multipoint boundary value problems. By extending and improving the work in [22], Rynne [23] showed that the associated Sturm-Liouville problem consisting of (1.6) has a strictly increasing sequence of simple eigenvalues {𝜆_𝑛}^∞_𝑛=0 with eigenfunctions 𝜙_𝑛(𝑡)=sin(𝜆_𝑛𝑡).

Very recently, Kong et al. [24] were concerned with the general boundary value problem with a variable 𝑤𝑢^′′(𝑡)(𝑡)𝑓(𝑢)(),(𝑎)+𝑤=0,𝑡∈𝑎,𝑏cos𝛼𝑢−sin𝛼𝑢^′(𝑎)=0,𝛼∈[0,𝜋),𝑢(𝑏)=^𝑚−2_𝑖=1𝑘_𝑖𝑢𝜂_𝑖 (1.7) By relating (1.7) to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, the existence and nonexistence of nodal solutions of (1.7) were obtained. We also point out that Webb [25] made the excellent remark on some existence results of symmetric positive solutions obtained in some recent papers and the author also corrected the values of the principle eigenvalue previously given in some examples.

In this paper, by means of two coupled parallel subsuper solutions, we obtain some sufficient conditions for the existence of six solutions for (1.1) and our main tool is based on the fixed point index theory. At the end of this paper, we will give an example which illustrates that our work is true. Our method stems from the paper [18].

2. Preliminaries and a Lemma

In the section, we shall give some preliminaries and a lemma which are fundamental to prove our main result.

Let 𝐸 be an ordered Banach space in which the partial ordering ≤ is induced by a cone 𝑃⊂𝐸. A cone 𝑃 is said to be normal if there exists a constant 𝑁>0, such that 𝜃≤𝑥≤𝑦 implies ‖𝑥‖≤𝑁‖𝑦‖, the smallest 𝑁 is called the normal constant of 𝑃. 𝑃 is called solid, if int𝑃≠∅, that is, 𝑃 has nonempty interior. 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∈ int𝑃, we write 𝑥≪𝑦. 𝑃 is called total if 𝐸=𝑃−𝑃. Let 𝐵∶𝐸→𝐸 be a bounded linear operator. 𝐵 is said to be positive if 𝐵(𝑃)⊂𝑃. An operator 𝐴 is strongly increasing, that is, 𝑥<𝑦 implies 𝐴𝑥≪𝐴𝑦. If 𝐴 is a linear operator, 𝐴 is strongly increasing implying 𝐴 is strongly positive.

Let 𝐸 be an ordered Banach space, 𝑃 a total cone in 𝐸, the partial ordering ≤ induced by 𝑃. 𝐵∶𝐸→𝐸 is a positive completely continuous linear operator. Let 𝑟(𝐵)>0 the spectral radius of 𝐵, 𝐵^∗ the conjugated operator of 𝐵, and 𝑃^∗ the conjugated cone of 𝑃. Since 𝑃⊂𝐸 is a total cone (i.e., 𝐸=𝑃−𝑃), according to the famous Krein-Rutman theorem (see [26]), we infer that if 𝑟(𝐵)≠0, then there exist 𝜑∈𝑃⧵{𝜃} and 𝑔^∗∈𝑃^∗⧵{𝜃}, such that 𝐵𝜑=𝑟(𝐵)𝐵𝜑,_∗𝑔^∗=𝑟(𝐵)𝑔^∗. (2.1) Fixed 𝜑∈𝑃⧵{𝜃},𝑔^∗∈𝑃^∗⧵{𝜃} such that (2.1) holds. For 𝛿>0, let𝑃𝑔^∗,𝛿=𝑥∈𝑃,𝑔^∗(𝑥)≥𝛿‖𝑥‖, (2.2) then 𝑃(𝑔^∗,𝛿) is also a cone in 𝐸. One can refer [26–28] for definition and properties about the cones.

Definition 2.1 (see [29]). Let 𝐵 be a positive linear operator. The operator 𝐵 is said to satisfy condition 𝐇, if there exist 𝜑∈𝑃⧵{𝜃},𝑔^∗∈𝑃^∗⧵{𝜃}, and 𝛿>0 such that (2.1) holds, and 𝐵 maps 𝑃 into 𝑃(𝑔^∗,𝛿).

Lemma 2.2 (see [10]). Suppose that 𝑑=1−∑^𝑚−2_𝑖=1𝛼_𝑖≠0. Then, the BVP −𝑢^′′(𝑡)(),𝑢=0,𝑡∈0,1_′(0)=0,𝑢(1)=^𝑚−2_𝑖=1𝛼_𝑖𝑢𝜉_𝑖, (2.3) has Green's function 𝐺()=𝐺()+∑𝑡,𝑠𝑡,𝑠^{𝑚−2𝑖=1}𝛼_𝑖𝐺𝜉_𝑖,𝑠1−∑^𝑚−2_𝑖=1𝛼_𝑖, (2.4) where 𝐺(𝑡,𝑠)=1−𝑡,0≤𝑠≤𝑡≤1,1−𝑠,0≤𝑡≤𝑠≤1. (2.5)

For convenience, we list the following hypotheses which will be used in our main result.(𝐻₁)   𝑓_𝑖,𝑔_𝑖(𝑖=1,2) are strictly increasing;(𝐻₂) there exist constants 𝑘>0,𝑙>0 and 𝐷>0 such that (i)|𝑓₁(±𝑘)±𝑙|<𝑁⁻¹𝑘,(ii)|𝑔₁(±𝑘)±𝐷|<𝑁⁻¹𝑘, (iii)|𝑓₂(±𝑘)|≤𝑙, and (iv) |𝑔₂(±𝑘)|≤𝐷, where 𝑁=max_𝑡∈[0,1]∫¹₀𝐺(𝑡,𝑠)𝑑𝑠;(𝐻₃) there exist constants 0<𝑐₁<𝑘,−𝑘<𝑐₂<0,0<𝑐₃<𝑘,−𝑘<𝑐₄<0, such that, for all 𝑡∈[0,1], we have (i)𝑐₁<∫¹₀𝐺(𝑡,𝑠)𝑓₁(𝑐₁)𝑑𝑠−𝑁𝑙, (ii)𝑐₂>∫¹₀𝐺(𝑡,𝑠)𝑔₁(𝑐₂)𝑑𝑠+𝑁𝐷,(iii)𝑐₃<∫¹₀𝐺(𝑡,𝑠)𝑔₁(𝑐₃)𝑑𝑠−𝑁𝐷, and(iv)𝑐₄>∫¹₀𝐺(𝑡,𝑠)𝑓₁(𝑐₄)𝑑𝑠+𝑁𝑙;(𝐻₄)   (i)lim_{|𝜑|→+∞}(𝑓₁(𝜑)+𝑓₂(𝜓))/𝜑≥2𝜆₁ uniformly for 𝜓∈ℝ, (ii)lim_|Ψ|→+∞(𝑔₁(𝜓)+𝑔₂(𝜑))/𝜓≥2𝜆₁ uniformly for 𝜑∈ℝ, where 𝜆₁ is the first eigenvalue of the following boundary value problem:−𝑢^′′(𝑡)(),𝑢=𝜆𝑢,𝑡∈0,1_′(0)=0,𝑢(1)=^𝑚−2_𝑖=1𝛼_𝑖𝑢𝜉_𝑖. (2.6)

It is well known that 𝜆₁=𝑟⁻¹(𝐻), where linear operator 𝐻∶𝐶[0,1]→𝐶[0,1] is defined as Hu(𝑡)=∫¹₀𝐺(𝑡,𝑠)𝑢(𝑠)𝑑𝑠.

3. Main Results

Theorem 3.1. Assume (𝐻₀), (𝐻₁)−(𝐻₄) hold, then BVP (1.1) has at least six distinct continuous solutions.

Proof. It is easy to check that BVP (1.1) is equivalent to the following integral equation systems: 𝜑(𝑡)=¹₀𝐺(𝑡,𝑠)𝑓₁𝜑(𝑠)+𝑓₂𝜓(𝑠)𝜓(𝑡)=𝑑𝑠,₁₀𝐺(𝑡,𝑠)𝑔₁𝜓(𝑠)+𝑔₂𝜑(𝑠)𝑑𝑠, (3.1) where 𝐺(𝑡,𝑠) is defined as in Lemma 2.2. By (𝐻₀), we know that 𝐺(𝑡,𝑠)≥0,forall𝑡,𝑠∈[0,1].
Let 𝐸=𝐶[0,1]×𝐶[0,1], define the norm in 𝐸 as ‖(𝜑,𝜓)‖=‖𝜑‖+‖𝜓‖. Then, 𝐸 is a Banach space with this norm. Let 𝑃={(𝜑,𝜓)∈𝐸∣𝜑(𝑡)≥0,𝜓(𝑡)≥0, forall𝑡∈[0,1]},𝑄={𝜑∈𝐶[0,1]∣𝜑(𝑡)≥0, forall𝑡∈[0,1]}. Then, 𝑃=𝑄×𝑄 is a normal and solid cone. Set 𝑇∶𝐸→𝐸, such that 𝑇𝜑,𝜓=¹₀𝐺(𝑡,𝑠)𝑓₁𝜑(𝑠)+𝑓₂𝜓(𝑠)𝑑𝑠,¹₀𝐺(𝑡,𝑠)𝑔₁𝜓(𝑠)+𝑔₂𝜑(𝑠)𝑑𝑠, (3.2) it is clear that the solutions of (1.1) are equivalent to the fixed points of 𝑇.
Set 0<𝜉₁≤𝜆₁, let 𝑓=𝑓𝜑,𝜓¹𝜑+𝑓₂𝜓,𝑔₁𝜓+𝑔₂𝜑𝜆₁+𝜉₁,𝐾𝜑,𝜓=𝜆₁+𝜉₁𝐻𝜑,𝜆₁+𝜉₁𝐻𝜓=𝐻₁𝜑,𝐻₁𝜓, (3.3) where 𝐻₁=(𝜆₁+𝜉₁)𝐻, then 𝑇=𝐾𝑓. It is easy to see that 𝐻 is a strongly positive completely continuous operator, and it follows from 𝐺(𝑡,𝑠)≥0 and the continuity of 𝐺(𝑡,𝑠) that 𝐾 is a strongly positive completely continuous operator. Since 𝑓_𝑖,𝑔_𝑖∶𝑅¹→𝑅¹(𝑖=1,2) are strictly increasing continuous functions, we know that 𝑓 is a strictly increasing continuous bounded operator. By 𝑇=𝐾𝑓, we can prove that 𝑇 is completely continuous. We infer from the increasing properties of 𝐾 and 𝑓 that 𝑇 is increasing.
Let 𝜑₁≡𝑐₁,𝜓₁≡−𝑘,𝜑₂≡𝑘,𝜓₂≡𝑐₂,𝜑₃≡−𝑘,𝜓₃≡𝑐₃,𝜑₄≡𝑐₄,𝜓₄≡𝑘, then (𝜑_𝑖,𝜓_𝑖)(𝑖=1,2,3,4) satisfy 𝜑₁,𝜓₁<𝜑₂,𝜓₂,𝜑₃,𝜓₃<𝜑₄,𝜓₄,𝜑₁,𝜓₁≰𝜑₄,𝜓₄,𝜑₃,𝜓₃≰𝜑₂,𝜓₂. (3.4) By (𝐻₂)(iii) and (𝐻₃)(i), we have ¹₀𝐺(𝑡,𝑠)𝑓₁𝜑₁+𝑓₂𝜓₁𝑑𝑠≥¹₀𝐺(𝑡,𝑠)𝑓₁𝑐₁𝑑𝑠−𝑁𝑙>𝑐₁=𝜑₁. (3.5) It follows from (𝐻₂)(ii),(iv), and the increasing property of 𝑔₂ that ¹₀𝐺(𝑡,𝑠)𝑔₁𝜓₁+𝑔₂𝜑₁𝑑𝑠>(−𝑘)𝑁⁻¹¹₀𝐺(𝑡,𝑠)𝑑𝑠≥(−𝑘)𝑁⁻¹𝑁=−𝑘=𝜓₁. (3.6) Equations (3.2), (3.5), and (3.6) imply that 𝜑₁,𝜓₁<𝑇𝜑₁,𝜓₁. (3.7) Similarly, by (𝐻₁)-(𝐻₃), we obtain 𝑇𝜑₂,𝜓₂<𝜑₂,𝜓₂,𝜑₃,𝜓₃<𝑇𝜑₃,𝜓₃,𝑇𝜑₄,𝜓₄<𝜑₄,𝜓₄. (3.8)
By [20, Lemma 3], we get that 𝐻₁ satisfies condition H. Therefore, there exist 𝑗^∗₀∈𝑄^∗⧵{𝜃},𝛿>0, such that 𝐻^∗₁𝑗^∗₀=𝑟𝐻₁𝑗^∗₀,𝑗^∗₀𝐻₁𝜑≥𝛿𝐻₁𝜑,∀𝜑∈𝑄. (3.9)(3.10) By the definition of spectral radius of completely continuous operator, we have 𝑟(𝐾)=𝑟(𝐻₁), and combining (3.9), we infer that 𝐻^∗₁𝑗^∗₀=𝑟(𝐾)𝑗^∗₀. (3.11) Let 𝑗^∗((𝜑,𝜓))=𝑗^∗₀(𝜑)+𝑗^∗₀(𝜓),forall(𝜑,𝜓)∈𝐸, then 𝑗^∗∈𝑃^∗⧵{𝜃}. According to the proof in [18], we can get that 𝐾 satisfies condition 𝐇.
By condition (𝐻₄), we obtain that there exists 𝐶>0, such that 𝑓₁𝜑+𝑓₂𝜓≥𝜆₁+𝜉₁𝑓𝜑,𝜑≥𝐶,𝜓∈ℝ,₁𝜑+𝑓₂𝜓≤𝜆₁+𝜉₁𝑔𝜑,𝜑≤−𝐶,𝜓∈ℝ,₁𝜓+𝑔₂𝜑≥𝜆₁+𝜉₁𝑔𝜓,𝜓≥𝐶,𝜑∈ℝ,₁𝜓+𝑔₂𝜑≤𝜆₁+𝜉₁𝜓,𝜓≤−𝐶,𝜑∈ℝ. (3.12)(3.13)(3.14)(3.15) Equations (3.12)–(3.15) imply 𝑓₁𝜑+𝑓₂𝜓,𝑔₁𝜓+𝑔₂𝜑≥𝜆₁+𝜉₁𝑓𝜑,𝜓,∀𝜑,𝜓≥𝐶,₁𝜑+𝑓₂𝜓,𝑔₁𝜓+𝑔₂𝜑≤𝜆₁+𝜉₁𝜑,𝜓,∀𝜑,𝜓≤−𝐶. (3.16)(3.17) Since 𝑓₁(𝜑),𝑔₁(𝜓) are continuous in [−(𝑐₁+𝑘),𝐶], so they are bounded, then there exists ℎ>0 such that 𝑓₁𝜑≥−ℎ,𝑔₁𝜓≥−ℎ,−𝜑₁,𝜓₁=−𝑐₁+𝑘≤𝜑,𝜓≤𝐶. (3.18) By virtue of (3.18) and the increasing properties of 𝑓₂ and 𝑔₂, one shows 𝑓₁𝜑+𝑓₂𝜓,𝑔₁𝜓+𝑔₂𝜑≥−ℎ−𝑓₂−𝜑₁,𝜓₁,−ℎ−𝑔₂−𝜑₁,𝜓₁,−𝜑₁,𝜓₁≤𝜑,𝜓≤𝐶. (3.19)
In addition, if 𝜑,𝜓 satisfy 𝜑≥𝐶,−‖(𝜑₁,𝜓₁)‖≤𝜓≤𝐶, then it follows from (3.12), (3.18), and the increasing property of 𝑔₂ that 𝑓₁𝜑+𝑓₂𝜓,𝑔₁𝜓+𝑔₂𝜑≥𝜆₁+𝜉₁𝜑,−ℎ+𝑔₂−𝜑₁,𝜓₁≥𝜆₁+𝜉₁𝜑,𝜓−𝜆₁+𝜉₁𝐶+ℎ+𝑔₂−𝜑₁,𝜓₁,𝜆₁+𝜉₁𝐶+ℎ+𝑔₂−𝜑₁,𝜓₁=𝜆₁+𝜉₁𝜑,𝜓−𝑑₁,𝑑₁, (3.20) where 𝑑₁=𝜆₁+𝜉₁𝐶+ℎ+𝑔₂−𝜑₁,𝜓₁. (3.21) Similarly, if 𝜑,𝜓 satisfy 𝜓≥𝐶,−‖(𝜑₁,𝜓₁)‖≤𝜑≤𝐶, then combining the increasing property of 𝑓₂ with (3.14) and (3.18), we know that 𝑓₁𝜑+𝑓₂𝜓,𝑔₁𝜓+𝑔₂𝜑≥𝜆₁+𝜉₁𝜑,𝜓−𝑑₂,𝑑₂, (3.22) where 𝑑₂=(𝜆₁+𝜉₁)𝐶+ℎ+|𝑓₂(−‖(𝜑₁,𝜓₁)‖)|. Let 𝑑=max{𝑑₁,𝑑₂}. By (3.21) and (3.22), we get that if 𝜑≥𝐶,−‖(𝜑₁,𝜓₁)‖≤𝜓≤𝐶 or 𝜓≥𝐶,−‖(𝜑₁,𝜓₁)‖≤𝜑≤𝐶, it is obvious that 𝑓₁𝜑+𝑓₂𝜓,𝑔₁𝜓+𝑔₂𝜑≥𝜆₁+𝜉₁𝜑,𝜓−(𝑑,𝑑). (3.23) It follows from (3.16), (3.19), and (3.23) that 𝑓₁𝜑+𝑓₂𝜓,𝑔₁𝜓+𝑔₂𝜑≥𝜆₁+𝜉₁𝜑,𝜓−(𝑑,𝑑),𝜑,𝜓≥−𝜑₁,𝜓₁. (3.24) In a similar way, from (3.12) and (3.14), we can show that there exists 𝑒>0 such that 𝑓₁𝜑+𝑓₂𝜓,𝑔₁𝜓+𝑔₂𝜑≥𝜆₁+𝜉₁𝜑,𝜓−(𝑒,𝑒),𝜑,𝜓≥−𝜑₃,𝜓₃. (3.25) Let 𝑎=max{𝑑,𝑒}. It follows from (3.24) and (3.25) that if 𝜑,𝜓≥−‖(𝜑₁,𝜓₁)‖ or 𝜑,𝜓≥−‖(𝜑₃,𝜓₃)‖, then 𝑓₁𝜑+𝑓₂𝜓,𝑔₁𝜓+𝑔₂𝜑≥𝜆₁+𝜉₁𝜑,𝜓−(𝑎,𝑎). (3.26) In a similar way, from (3.13) and (3.15), we can prove that there exists constant 𝑎 such that if 𝜑,𝜓≤‖(𝜑₂,𝜓₂)‖ or 𝜑,𝜓≤‖(𝜑₄,𝜓₄)‖, then 𝑓₁𝜑+𝑓₂𝜓,𝑔₁𝜓+𝑔₂𝜑≤𝜆₁+𝜉₁𝜑,𝜓−(𝑎,𝑎). (3.27) Let 𝑃((𝜑₁,𝜓₁))={(𝑢,𝑣)∈𝐸∣(𝑢,𝑣)≥(𝜑₁,𝜓₁)}, 𝑃((𝜑₃,𝜓₃))={(𝑢,𝑣)∈𝐸∣(𝑢,𝑣)≥(𝜑₃,𝜓₃)}, 𝑃((𝜑₂,𝜓₂))={(𝑢,𝑣)∈𝐸∣(𝑢,𝑣)≤(𝜑₂,𝜓₂)}, 𝑃((𝜑₄,𝜓₄))={(𝑢,𝑣)∈𝐸∣(𝑢,𝑣)≤(𝜑₄,𝜓₄)}, forall(𝜑,𝜓)∈𝑃((𝜑₁,𝜓₁))∪𝑃((𝜑₃,𝜓₃)), then 𝜑(𝑡),𝜓(𝑡)≥−‖(𝜑₁,𝜓₁)‖ or 𝜑(𝑡),𝜓(𝑡)≥−‖(𝜑₃,𝜓₃)‖, forall𝑡∈[0,1]; therefore, in virtue of expression of 𝐵,𝑓, and (3.26), we have 𝜑(𝑡)(𝑡)𝜑(𝑡)(𝑡)−𝜆𝐾𝑓,𝜓≥𝐾,𝜓¹+𝜉₁𝑎𝜆₁+𝜉₁¹₀𝐺(𝑡,𝑠)𝑑𝑠,¹₀𝐺()=𝑟𝑡,𝑠𝑑𝑠₋₁(𝐾)+𝜖𝐾𝜑(𝑡),𝜓(𝑡)−𝑢₁(𝑡),𝑣₁(𝑡), (3.28) where 𝜖=1−𝑟⁻¹(𝐾),(𝑢₁(𝑡),𝑣₁(𝑡))=𝑎(∫¹₀𝐺(𝑡,𝑠)𝑑𝑠,∫¹₀𝐺(𝑡,𝑠)𝑑𝑠). Since 0<𝜉₁≤𝜆₁, one can show 𝜆0<𝜖=1−¹𝜆₁+𝜉₁=𝜉₁𝜆₁+𝜉₁≤𝜆₁𝜆₁+𝜉₁=𝑟⁻¹(𝐾). (3.29)
This implies that there exist (𝑢₁,𝑣₁)∈𝐸 and 0<𝜉₂≤𝑟⁻¹(𝐾) such that 𝐾𝑓𝜑,𝜓≥𝑟⁻¹(𝐾)+𝜉₂𝐾𝜑,𝜓−𝑢₁,𝑣₁,∀𝜑,𝜓∈𝑃𝜑₁,𝜓₁∪𝑃𝜑₃,𝜓₃. (3.30) Similarly, we get by (3.27) that there exist (𝑢₂,𝑣₂)∈𝐸 and 0<𝜉₃≤𝑟⁻¹(𝐾) such that 𝐾𝑓𝜑,𝜓≤𝑟⁻¹(𝐾)+𝜉₃𝐾𝜑,𝜓−𝑢₂,𝑣₂,∀𝜑,𝜓∈𝑃𝜑₂,𝜓₂∪𝑃𝜑₄,𝜓₄. (3.31) We get by (3.7) that 𝑇∶𝑃𝜑₁,𝜓₁⟶𝑃𝜑₁,𝜓₁. (3.32) Let 𝐸₁={(𝑢,𝑣)∈𝐸∣(𝜑₁,𝜓₁)≤(𝑢,𝑣)≤(𝜑₂,𝜓₂)}. Since 𝑃 is normal, then 𝐸₁ is bounded (see [28]). Choose 𝑀₁>0 such that 𝑀₁1>max𝜉₂𝛿𝜉₂𝛿𝜑₁,𝜓₁+𝜉₂𝛿𝑇𝜑₁,𝜓₁+𝑟⁻¹(𝐾)𝑔^∗𝑢₁,𝑣₁−𝜉₂𝑔^∗𝑇𝜑₁,𝜓₁,sup_{(𝑢,𝑣)∈𝐸1}‖(𝑢,𝑣)‖+𝜑₁,𝜓₁. (3.33) Let Ω₁={(𝑢,𝑣)∈𝑃((𝜑₁,𝜓₁))∣‖(𝑢,𝑣)−(𝜑₁,𝜓₁)‖<𝑀₁,(𝑢,𝑣)≱(𝜑₃,𝜓₃)}, then 𝐸₁⊂Ω₁ and Ω₁ is a bounded open set. By the proof of Theorem 2.1 in [18], we can show that (𝑢,𝑣)−𝑇(𝑢,𝑣)≠𝜆𝐾(𝑢,𝑣),∀𝜆≥0,(𝑢,𝑣)∈𝜕Ω₁, (3.34) where (𝑢,𝑣) satisfies 𝐾(𝑢,𝑣)=𝑟(𝐾)(𝑢,𝑣).
Equation (3.34) implies that 𝑇 has no fixed point on 𝜕Ω₁. It is easy to prove that 𝑃((𝜑₁,𝜓₁)) is a retract of 𝐸, which together with (3.32) implies that the fixed point index 𝑖(𝑇,Ω₁,𝑃((𝜑₁,𝜓₁))) over Ω₁ with respect to 𝑃((𝜑₁,𝜓₁)) is well defined, and a standard proof yields 𝑖𝑇+𝑠𝐾(𝑢,𝑣),Ω₁,𝑃𝜑₁,𝜓₁=0. (3.35) Set 𝐻(𝑡,(𝑢,𝑣))=(1−𝑡)𝑇(𝑢,𝑣)+𝑡(𝑇(𝑢,𝑣)+𝑠𝐾(𝑢,𝑣)),(𝑡,(𝑢,𝑣))∈[0,1]×Ω₁, then for any (𝑢,𝑣)∈Ω₁,𝑡∈[0,1], we have 𝐻(𝑡,(𝑢,𝑣))∈𝑃((𝜑₁,𝜓₁)). It follows from (3.34) that 𝐻(𝑡,(𝑢,𝑣))≠(𝑢,𝑣),forall(𝑡,(𝑢,𝑣))∈[0,1]×𝜕Ω₁, and by (3.35) and the homotopy invariance of the fixed point index, we get 𝑖𝑇,Ω₁,𝑃𝜑₁,𝜓₁=𝑖𝑇+𝑠𝐾(𝑢,𝑣),Ω₁,𝑃𝜑₁,𝜓₁=0. (3.36) Let 𝑊₁={(𝑢,𝑣)∈𝑃((𝜑₁,𝜓₁))∣(𝑢,𝑣)≪(𝜑₂,𝜓₂)}. By means of usual method (see [30]), we get that 𝑖𝑇,𝑊₁,𝑃𝜑₁,𝜓₁=1. (3.37) It is evident that 𝐴 has no fixed point on 𝜕𝑊₁, by (3.36), (3.37), and the additivity of the fixed point index, we have 𝑖𝑇,Ω₁⧵𝑊₁,𝑃𝜑₁,𝜓₁=𝑖𝑇,Ω₁,𝑃𝜑₁,𝜓₁−𝑖𝑇,𝑊₁,𝑃𝜑₁,𝜓₁=0−1=−1. (3.38) Set 𝐸₂={(𝑢,𝑣)∈𝐸∣(𝜑₃,𝜓₃)≤(𝑢,𝑣)≤(𝜑₄,𝜓₄)}, and choose 𝑀₂>0 such that 𝑀₂1>max𝜉₂𝛿𝜉₂𝛿𝜑₃,𝜓₃+𝜉₂𝛿𝑇𝜑₃,𝜓₃+𝑟⁻¹(𝐾)𝑔^∗𝑢₁,𝑣₁−𝜉₂𝑔^∗𝑇𝜑₃,𝜓₃,sup_{(𝑢,𝑣)∈𝐸2}‖(𝑢,𝑣)‖+𝜑₃,𝜓₃. (3.39) Let Ω₂=(𝑢,𝑣)∈𝑃𝜑₃,𝜓₃∣(𝑢,𝑣)−𝜑₃,𝜓₃<𝑀₂,(𝑢,𝑣)≱𝜑₁,𝜓₁,𝑊₂=(𝑢,𝑣)∈𝑃𝜑₃,𝜓₃∣(𝑢,𝑣)≪𝜑₄,𝜓₄. (3.40) Similarly to the proof of (3.37) and (3.38), we get that 𝑖𝑇,𝑊₂,𝑃𝜑₃,𝜓₃𝑖=1,𝑇,Ω₂⧵𝑊₂,𝑃𝜑₃,𝜓₃=−1. (3.41) Choose 𝑀₃,𝑀₄>0 such that 𝑀₃1>max𝜉₃𝛿𝜉₃𝛿𝜑₂,𝜓₂+𝜉₃𝛿𝑇𝜑₂,𝜓₂+𝑟⁻¹(𝐾)𝑔^∗𝑢₂,𝑣₂−𝜉₃𝑔^∗𝑇𝜑₂,𝜓₂sup_{(𝑢,𝑣)∈𝐸1}‖(𝑢,𝑣)‖+𝜑₂,𝜓₂,𝑀₄1>max𝜉₃𝛿𝜉₃𝛿𝜑₄,𝜓₄+𝜉₃𝛿𝑇𝜑₄,𝜓₄+𝑟⁻¹(𝐾)𝑔^∗𝑢₂,𝑣₂−𝜉₃𝑔^∗𝑇𝜑₄,𝜓₄,sup_{(𝑢,𝑣)∈𝐸2}‖(𝑢,𝑣)‖+𝜑₄,𝜓₄. (3.42) Set Ω₃=(𝑢,𝑣)∈𝑃𝜑₂,𝜓₂∣(𝑢,𝑣)−𝜑₂,𝜓₂<𝑀₃,(𝑢,𝑣)≰𝜑₄,𝜓₄,Ω₄=(𝑢,𝑣)∈𝑃𝜑₄,𝜓₄∣(𝑢,𝑣)−𝜑₄,𝜓₄<𝑀₄,(𝑢,𝑣)≰𝜑₂,𝜓₂,𝑊₃=(𝑢,𝑣)∈𝑃𝜑₂,𝜓₂∣𝜑₁,𝜓₁≪(,𝑊𝑢,𝑣)₄=(𝑢,𝑣)∈𝑃𝜑₄,𝜓₄∣𝜑₂,𝜓₂≪(𝑢,𝑣). (3.43) By virtue of (3.31) and the same method as that for (3.34), we have (𝑢,𝑣)−𝑇(𝑢,𝑣)≠−𝜇𝐾(𝑢,𝑣),∀𝜇≥0,(𝑢,𝑣)∈𝜕Ω₃∪𝜕Ω₄. (3.44) By (3.44), similarly to the proof of (3.38), we can prove that 𝑖𝑇,Ω₃⧵𝑊₃,𝑃𝜑₂,𝜓₂=𝑖𝑇,Ω₄⧵𝑊₄,𝑃𝜑₄,𝜓₄=−1. (3.45) Equations (3.37)–(3.41), (3.45) imply that 𝑇 has at least six distinct fixed points, that is, the system of differential equations (1.1) has at least six solution in 𝐶[0,1]×𝐶[0,1].

4. An Example

In this section, we present a simple example to explain our results.

Consider the following second-order three-point BVP for nonlinear equations system: −𝜑^′′(𝑡)=𝑓₁𝜑(𝑡)+𝑓₂𝜓(𝑡),𝑡∈[0,1],−𝜓^′′(𝑡)=𝑔₁𝜓(𝑡)+𝑔₂𝜑(𝑡)[],𝜑,𝑡∈0,1_′(0)(1)=1=0,𝜑2𝜑14,𝜓^′(0)(1)=1=0,𝜓2𝜓14, (4.1) where 𝑓₁(𝜑)=(𝜑³/30000)+5𝜑^1/3, 𝑔₁(𝜓)=(𝜓³/30000)+4𝜓^1/5, 𝑓₂(𝜓)=(1/6)⋅√⁵100+1/12arctan6𝜓, 𝑔₂(𝜑)=(1/16)⋅√³100+(1/30)arctan7𝜑, 𝛼₁=1/2,𝜉₁=1/4,𝑁=max_𝑡∈[0,1]∫¹₀𝐺(𝑡,𝑠)𝑑𝑠=31/32, 𝑓_𝑖,𝑔_𝑖∶ℝ¹→ℝ¹ are strictly increasing continuous functions, and condition (𝐻₁) is satisfied. Choose 𝑘=100,𝑙=5/12+𝜋/24,𝐷=5/16+𝜋/60. Some direct calculations show 𝑓₁(±100)±𝑙≤10⁶3⋅10⁴+5⋅√³5100++𝜋12<243231⋅100=𝑁⁻¹𝑔𝑘,₁(±100)±𝐷≤1003+4⋅√⁵5100++𝜋16560<34+4⋅2<3231⋅100=𝑁⁻¹𝑓𝑘,₂(≤1±100)6⋅√⁵1100+<112arctan6⋅1006⋅52+1⋅𝜋122=5+𝜋12𝑔24=𝑙,₂(≤1±100)⋅16³√100+1<130arctan7⋅100116⋅5+⋅𝜋302=5+𝜋1660=𝐷, (4.2) Therefore, condition (𝐻₂) is satisfied.

Choosing 𝑐₁=1/8,𝑐₂=−1,𝑐₃=1/4,𝑐₄=−1, it is easy to check that ¹₀𝐺(𝑡,𝑠)𝑓₁𝑐₁𝑑𝑠−𝑁𝑙≥151328³⋅3⋅10⁴+52−31532+𝜋12≥2415⋅5322−5+1126=75−764>1128=𝑐₁,₁₀𝐺(𝑡,𝑠)𝑔₁𝑐₂𝑑𝑠+𝑁𝐷≤1532−13⋅10⁴−4+31532+𝜋16≤6015⋅()+632−4316=−2<−1=𝑐₂,₁₀𝐺(𝑡,𝑠)𝑔₁𝑐₃𝑑𝑠−𝑁𝐷≥151324³⋅3⋅10⁴1+44^1/5−31532+𝜋16≥601532⋅4⁵14−38≥15332⋅4⋅4−38=23>1324=𝑐₃,₁₀𝐺(𝑡,𝑠)𝑓₁𝑐₄𝑑𝑠+𝑁𝑙≤1532−13⋅10⁴−5+31532+𝜋12≤2415()+732−5=12−75+73212<−1=𝑐₄. (4.3) Therefore, condition (𝐻₃) is satisfied. At last, we will check condition (𝐻₄), by the method of [9, 31], and we consider the linear eigenvalue problem 𝑢^′′𝑢+𝜆𝑢=0,0<𝑡<1,_′(0)(1)=1=0,𝑢2𝑢14. (4.4) Let Γ(𝑠)=cos𝑠−(1/2)cos(𝑠/4). By the paper [31], we know that the sequence of positive eigenvalue of (4.4) is exactly given by 𝜆_𝑛=𝑠²_𝑛,𝑛=1,2,…, where 𝑠_𝑛 is the sequence of positive solutions of Γ(𝑠)=0. In [31], Han obtained that 𝑠₁=1.0675, moreover 𝜆₁=𝑠²₁=1.1396. It is easy to know that lim_|𝑥|→∞𝑓₁(𝑥)+𝑓₂𝑦𝑥=lim_|𝑥|→∞𝑥³/30000+5𝑥^1/3+(1/6)⋅√⁵100+(1/12)arctan6𝑦𝑥=+∞≥2⋅1.1396=2𝜆₁, (4.5) uniformly for 𝑦∈ℝ, lim_|𝑦|→∞𝑔₁𝑦+𝑔₂(𝑥)𝑦=lim_|𝑦|→∞𝑦³/30000+4𝑦^1/5+(1/16)⋅√³100+(1/30)arctan7𝑥𝑦=+∞≥2⋅1.1396=2𝜆₁, (4.6) uniformly for 𝑥∈ℝ. Therefore, condition (𝐻₄) is also satisfied. Consequently, all conditions of Theorem 3.1 are satisfied, and we get the system of differential equations (4.1) has at least six solutions in 𝐶[0,1]×𝐶[0,1].

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This project is supported by the National Natural Science Foundation of China (10971046), the University Science and Technology Foundation of Shandong Provincial Education Department (J10LA62), the Natural Science Foundation of Shandong Province (ZR2009AM004, ZR2010AL014), and the Doctor of Scientific Startup Foundation for Shandong University of Finance (08BSJJ32).