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)=f1(φ(t))+f2(ψ(t)),t∈[0,1],-ψ′′(t)=g1(ψ(t))+g2(φ(t)),t∈[0,1],φ′(0)=0,φ(1)=∑i=1m-2αiφ(ξi),ψ′(0)=0,ψ(1)=∑i=1m-2αiψ(ξi),
where fi,gi:ℝ1→ℝ1(i=1,2) are continuous and αi,ξi satisfying

∑i=1m-2αi∈(0,1) with αi∈(0,+∞) for i=1,2,…,m-2 and 0<ξ1<ξ2<⋯<ξm-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)+f(t,y)=0,t∈[0,1],y(0)=0,y(1)-αy(η)=0,
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 f(t,x2)-f(t,x1)≥-M(x2-x1),t∈[0,1],x2≥x1.
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,
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),(cosγ0)u(0)-(sinγ0)u′(0)=∫01u(τ)dα(τ),(cosγ1)u(1)+(sinγ1)u′(1)=∫01u(τ)dβ(τ),
where γ0∈[0,π/2] and γ1∈[0,π/2], ∫01u(τ)dα(τ) and ∫01u(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)+λu(t)=0,0<t<1,u(0)=0,u(1)=∑i=1m-2αiu(ηi),
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 {λn}n=0∞ with eigenfunctions ϕn(t)=sin(λnt).

Very recently, Kong et al. [24] were concerned with the general boundary value problem with a variable wu′′(t)+w(t)f(u)=0,t∈(a,b),cosαu(a)-sinαu′(a)=0,α∈[0,π),u(b)=∑i=1m-2kiu(ηi)
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)φ¯,B*g*=r(B)g*.
Fixed φ¯∈P∖{θ},g*∈P*∖{θ} such that (2.1) holds. For δ>0, letP(g*,δ)={x∈P,g*(x)≥δ‖x‖},
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 [<xref ref-type="bibr" rid="B21">29</xref>]).

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 [<xref ref-type="bibr" rid="B14">10</xref>]).

Suppose that d=1-∑i=1m-2αi≠0. Then, the BVP
-u′′(t)=0,t∈(0,1),u′(0)=0,u(1)=∑i=1m-2αiu(ξi),
has Green's function
G(t,s)=G̃(t,s)+∑i=1m-2αiG̃(ξi,s)1-∑i=1m-2αi,
where
G̃(t,s)={1-t,0≤s≤t≤1,1-s,0≤t≤s≤1.

For convenience, we list the following hypotheses which will be used in our main result.

fi,gi(i=1,2) are strictly increasing;

there exist constants k>0,l>0 and D>0 such that

|f1(±k)±l|<N-1k,

|g1(±k)±D|<N-1k,

|f2(±k)|≤l, and (iv) |g2(±k)|≤D, where N=maxt∈[0,1]∫01G(t,s)ds;

there exist constants 0<c1<k,-k<c2<0,0<c3<k,-k<c4<0, such that, for all t∈[0,1], we have

c1<∫01G(t,s)f1(c1)ds-Nl,

c2>∫01G(t,s)g1(c2)ds+ND,

c3<∫01G(t,s)g1(c3)ds-ND, and

c4>∫01G(t,s)f1(c4)ds+Nl;

lim|φ|→+∞(f1(φ)+f2(ψ))/φ≥2λ1 uniformly for ψ∈ℝ,

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)=λu,t∈(0,1),u′(0)=0,u(1)=∑i=1m-2αiu(ξi).

It is well known that λ1=r-1(H), where linear operator H:C[0,1]→C[0,1] is defined as Hu(t)=∫01G(t,s)u(s)ds.

3. Main ResultsTheorem 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)=∫01G(t,s)[f1(φ(s))+f2(ψ(s))]ds,ψ(t)=∫01G(t,s)[g1(ψ(s))+g2(φ(s))]ds,
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
T(φ,ψ)=(∫01G(t,s)[f1(φ(s))+f2(ψ(s))]ds,∫01G(t,s)[g1(ψ(s))+g2(φ(s))]ds),
it is clear that the solutions of (1.1) are equivalent to the fixed points of T.

Set 0<ξ1≤λ1, let
f(φ,ψ)=(f1(φ)+f2(ψ),g1(ψ)+g2(φ))λ1+ξ1,K(φ,ψ)=((λ1+ξ1)Hφ,(λ1+ξ1)Hψ)=(H1φ,H1ψ),
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 fi,gi: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 (φi,ψi)(i=1,2,3,4) satisfy
(φ1,ψ1)<(φ2,ψ2),(φ3,ψ3)<(φ4,ψ4),(φ1,ψ1)≰(φ4,ψ4),(φ3,ψ3)≰(φ2,ψ2).
By (H2)(iii) and (H3)(i), we have
∫01G(t,s)[f1(φ1)+f2(ψ1)]ds≥∫01G(t,s)f1(c1)ds-Nl>c1=φ1.
It follows from (H2)(ii),(iv), and the increasing property of g2 that
∫01G(t,s)[g1(ψ1)+g2(φ1)]ds>(-k)N-1∫01G(t,s)ds≥(-k)N-1N=-k=ψ1.
Equations (3.2), (3.5), and (3.6) imply that
(φ1,ψ1)<T(φ1,ψ1).
Similarly, by (H1)–(H3), we obtain
T(φ2,ψ2)<(φ2,ψ2),(φ3,ψ3)<T(φ3,ψ3),T(φ4,ψ4)<(φ4,ψ4).

By [20, Lemma 3], we get that H1 satisfies condition H. Therefore, there exist j0*∈Q*∖{θ},δ>0, such that
H1*j0*=r(H1)j0*,j0*(H1φ)≥δ‖H1φ‖,∀φ∈Q.
By the definition of spectral radius of completely continuous operator, we have r(K)=r(H1), and combining (3.9), we infer that
H1*j0*=r(K)j0*.
Let j*((φ,ψ))=j0*(φ)+j0*(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
f1(φ)+f2(ψ)≥(λ1+ξ1)φ,φ≥C,ψ∈R,f1(φ)+f2(ψ)≤(λ1+ξ1)φ,φ≤-C,ψ∈R,g1(ψ)+g2(φ)≥(λ1+ξ1)ψ,ψ≥C,φ∈R,g1(ψ)+g2(φ)≤(λ1+ξ1)ψ,ψ≤-C,φ∈R.
Equations (3.12)–(3.15) imply
(f1(φ)+f2(ψ),g1(ψ)+g2(φ))≥(λ1+ξ1)(φ,ψ),∀φ,ψ≥C,(f1(φ)+f2(ψ),g1(ψ)+g2(φ))≤(λ1+ξ1)(φ,ψ),∀φ,ψ≤-C.
Since f1(φ),g1(ψ) are continuous in [-(c1+k),C], so they are bounded, then there exists h>0 such that
f1(φ)≥-h,g1(ψ)≥-h,-‖(φ1,ψ1)‖=-(c1+k)≤φ,ψ≤C.
By virtue of (3.18) and the increasing properties of f2 and g2, one shows
(f1(φ)+f2(ψ),g1(ψ)+g2(φ))≥(-h-|f2(-‖(φ1,ψ1)‖)|,-h-|g2(-‖(φ1,ψ1)‖)|),-‖(φ1,ψ1)‖≤φ,ψ≤C.

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
(f1(φ)+f2(ψ),g1(ψ)+g2(φ))≥((λ1+ξ1)φ,-h+g2(-‖(φ1,ψ1)‖))≥(λ1+ξ1)(φ,ψ)-((λ1+ξ1)C+h+|g2(-‖(φ1,ψ1)‖)|,(λ1+ξ1)C+h+|g2(-‖(φ1,ψ1)‖)|)=(λ1+ξ1)(φ,ψ)-(d1,d1),
where
d1=(λ1+ξ1)C+h+|g2(-‖(φ1,ψ1)‖)|.
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
(f1(φ)+f2(ψ),g1(ψ)+g2(φ))≥(λ1+ξ1)(φ,ψ)-(d2,d2),
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
(f1(φ)+f2(ψ),g1(ψ)+g2(φ))≥(λ1+ξ1)(φ,ψ)-(d,d).
It follows from (3.16), (3.19), and (3.23) that
(f1(φ)+f2(ψ),g1(ψ)+g2(φ))≥(λ1+ξ1)(φ,ψ)-(d,d),φ,ψ≥-‖(φ1,ψ1)‖.
In a similar way, from (3.12) and (3.14), we can show that there exists e>0 such that
(f1(φ)+f2(ψ),g1(ψ)+g2(φ))≥(λ1+ξ1)(φ,ψ)-(e,e),φ,ψ≥-‖(φ3,ψ3)‖.
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
(f1(φ)+f2(ψ),g1(ψ)+g2(φ))≥(λ1+ξ1)(φ,ψ)-(a,a).
In a similar way, from (3.13) and (3.15), we can prove that there exists constant ã such that if φ,ψ≤∥(φ2,ψ2)∥ or φ,ψ≤∥(φ4,ψ4)∥, then
(f1(φ)+f2(ψ),g1(ψ)+g2(φ))≤(λ1+ξ1)(φ,ψ)-(ã,ã).
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
Kf(φ(t),ψ(t))≥K(φ(t),ψ(t))-(λ1+ξ1)a(λ1+ξ1)(∫01G(t,s)ds,∫01G(t,s)ds)=(r-1(K)+ϵ)K(φ(t),ψ(t))-(u1(t),v1(t)),
where ϵ=1-r-1(K),(u1(t),v1(t))=a(∫01G(t,s)ds,∫01G(t,s)ds). Since 0<ξ1≤λ1, one can show
0<ϵ=1-λ1λ1+ξ1=ξ1λ1+ξ1≤λ1λ1+ξ1=r-1(K).

This implies that there exist (u1,v1)∈E and 0<ξ2≤r-1(K) such that
Kf(φ,ψ)≥(r-1(K)+ξ2)K(φ,ψ)-(u1,v1),∀(φ,ψ)∈P((φ1,ψ1))∪P((φ3,ψ3)).
Similarly, we get by (3.27) that there exist (u2,v2)∈E and 0<ξ3≤r-1(K) such that
Kf(φ,ψ)≤(r-1(K)+ξ3)K(φ,ψ)-(u2,v2),∀(φ,ψ)∈P((φ2,ψ2))∪P((φ4,ψ4)).
We get by (3.7) that
T:P((φ1,ψ1))⟶P((φ1,ψ1)).
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
M1>max{sup(u,v)∈E1sup(u,v)∈E11ξ2δ(ξ2δ‖(φ1,ψ1)‖+ξ2δ‖T(φ1,ψ1)‖+r-1(K)g*((u1,v1))-ξ2g*(T(φ1,ψ1))),sup(u,v)∈E1‖(u,v)‖+‖(φ1,ψ1)‖}.
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,
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
i(T+sK(u¯,v¯),Ω1,P((φ1,ψ1)))=0.
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
i(T,Ω1,P((φ1,ψ1)))=i(T+sK(u¯,v¯),Ω1,P((φ1,ψ1)))=0.
Let W1={(u,v)∈P((φ1,ψ1))∣(u,v)≪(φ2,ψ2)}. By means of usual method (see [30]), we get that
i(T,W1,P((φ1,ψ1)))=1.
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
i(T,Ω1∖W1¯,P((φ1,ψ1)))=i(T,Ω1,P((φ1,ψ1)))-i(T,W1,P((φ1,ψ1)))=0-1=-1.
Set E2={(u,v)∈E∣(φ3,ψ3)≤(u,v)≤(φ4,ψ4)}, and choose M2>0 such that
M2>max{sup(u,v)∈E1sup(u,v)∈E11ξ2δ(ξ2δ‖(φ3,ψ3)‖+ξ2δ‖T(φ3,ψ3)‖+r-1(K)g*((u1,v1))-ξ2g*(T(φ3,ψ3))),sup(u,v)∈E2‖(u,v)‖+‖(φ3,ψ3)‖}.
Let
Ω2={(u,v)∈P((φ3,ψ3))∣‖(u,v)-(φ3,ψ3)‖<M2,(u,v)≱(φ1,ψ1)},W2={(u,v)∈P((φ3,ψ3))∣(u,v)≪(φ4,ψ4)}.
Similarly to the proof of (3.37) and (3.38), we get that
i(T,W2,P((φ3,ψ3)))=1,i(T,Ω2∖W2¯,P((φ3,ψ3)))=-1.
Choose M3,M4>0 such that
M3>max{sup(u,v)∈E11ξ3δ(ξ3δ‖(φ2,ψ2)‖+ξ3δ‖T(φ2,ψ2)‖+r-1(K)g*((u2,v2))-ξ3g*(T(φ2,ψ2)))sup(u,v)∈E1‖(u,v)‖+‖(φ2,ψ2)‖},M4>max{sup(u,v)∈E11ξ3δ(ξ3δ‖(φ4,ψ4)‖+ξ3δ‖T(φ4,ψ4)‖+r-1(K)g*((u2,v2))-ξ3g*(T(φ4,ψ4))),sup(u,v)∈E2‖(u,v)‖+‖(φ4,ψ4)‖}.
Set
Ω3={(u,v)∈P((φ2,ψ2))∣‖(u,v)-(φ2,ψ2)‖<M3,(u,v)≰(φ4,ψ4)},Ω4={(u,v)∈P((φ4,ψ4))∣‖(u,v)-(φ4,ψ4)‖<M4,(u,v)≰(φ2,ψ2)},W3={(u,v)∈P((φ2,ψ2))∣(φ1,ψ1)≪(u,v)},W4={(u,v)∈P((φ4,ψ4))∣(φ2,ψ2)≪(u,v)}.
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.
By (3.44), similarly to the proof of (3.38), we can prove that
i(T,Ω3∖W3¯,P((φ2,ψ2)))=i(T,Ω4∖W4¯,P((φ4,ψ4)))=-1.
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)=f1(φ(t))+f2(ψ(t)),t∈[0,1],-ψ′′(t)=g1(ψ(t))+g2(φ(t)),t∈[0,1],φ′(0)=0,φ(1)=12φ(14),ψ′(0)=0,ψ(1)=12ψ(14),
where f1(φ)=(φ3/30000)+5φ1/3, g1(ψ)=(ψ3/30000)+4ψ1/5, f2(ψ)=(1/6)·1005+1/12arctan6ψ, g2(φ)=(1/16)·1003+(1/30)arctan7φ, α1=1/2,ξ1=1/4,N=maxt∈[0,1]∫01G(t,s)ds=31/32, fi,gi: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⋅1003+512+π24<3231⋅100=N-1k,|g1(±100)±D|≤1003+4⋅1005+516+π60<34+4⋅52<3231⋅100=N-1k,|f2(±100)|≤16⋅1005+112arctan6⋅100<16⋅52+112⋅π2=512+π24=l,|g2(±100)|≤116⋅1003+130arctan7⋅100<116⋅5+130⋅π2=516+π60=D,
Therefore, condition (H2) is satisfied.

Choosing c1=1/8,c2=-1,c3=1/4,c4=-1, it is easy to check that
∫01G(t,s)f1(c1)ds-Nl≥1532(183⋅3⋅104+52)-3132(512+π24)≥1532⋅52-(512+16)=7564-712>18=c1,∫01G(t,s)g1(c2)ds+ND≤1532(-13⋅104-4)+3132(516+π60)≤1532⋅(-4)+616=-32<-1=c2,∫01G(t,s)g1(c3)ds-ND≥1532[143⋅3⋅104+4(14)1/5]-3132(516+π60)≥1532⋅4145-38≥1532⋅4⋅34-38=2332>14=c3,∫01G(t,s)f1(c4)ds+Nl≤1532(-13⋅104-5)+3132(512+π24)≤1532(-5)+712=-7532+712<-1=c4.
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′′+λu=0,0<t<1,u′(0)=0,u(1)=12u(14).
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 λn=sn2,n=1,2,…, where sn is the sequence of positive solutions of Γ(s)=0. In [31], Han obtained that s1=1.0675, moreover λ1=s12=1.1396. It is easy to know that
lim|x|→∞f1(x)+f2(y)x=lim|x|→∞(x3/30000)+5x1/3+(1/6)⋅1005+(1/12)arctan6yx=+∞≥2⋅1.1396=2λ1,
uniformly for y∈ℝ,
lim|y|→∞g1(y)+g2(x)y=lim|y|→∞(y3/30000)+4y1/5+(1/16)⋅1003+(1/30)arctan7xy=+∞≥2⋅1.1396=2λ1,
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).

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