# Positive Solutions for Nonlinear Singular Differential Systems Involving Parameter on the Half-Line

Authors: Lishan Liu; Ying Wang; Xinan Hao; Yonghong Wu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

By using the upper-lower solutions method and the fixed-point theorem on cone in a special space, we study the singular boundary value problem for systems of nonlinear second-order differential equations involving two parameters on the half-line. Some results for the existence, nonexistence and multiplicity of positive solutions for the problem are obtained.


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## Body

## 1. Introduction

In this paper, we are concerned with the following boundary value problem for systems of nonlinear singular second-order ordinary differential equations on the half-line:(Pa,b)-(p1(t)u′(t))′=λϕ1(t)f1(t,u(t),v(t),a,b),0<t<+∞,-(p2(t)v′(t))′=λϕ2(t)f2(t,u(t),v(t),a,b),0<t<+∞,α11u(0)-β11lim⁡t→0+p1(t)u′(t)=0,α12lim⁡t→+∞u(t)+β12lim⁡t→+∞p1(t)u′(t)=0,α21v(0)-β21lim⁡t→0+p2(t)v′(t)=0,α22lim⁡t→+∞v(t)+β22lim⁡t→+∞p2(t)v′(t)=0,
where λ>0 is a parameter, a≥0,b≥0 are constants; f1,f2:ℝ+5→ℝ+ are continuous, ϕ1,ϕ2:(0,+∞)→ℝ+ are continuous and may have singularity at t=0; pi∈C(ℝ+)∩C1(0,+∞) with pi(t)>0 on (0,+∞) and ∫0+∞‍(1/pi(s))ds<+∞(i=1,2); αij,βij≥0(i,j=1,2) with ρi=αi2βi1+αi1βi2+αi1αi2∫0+∞‍(1/pi(s))ds>0(i=1,2), in which ℝ+=[0,+∞) is the set of nonnegative real numbers.Boundary value problems (BVP for short) on infinite interval arise in many applications (see [1, 2] and the references therein). Over the last couple of decades, a great deal of results have been developed for differential, difference, and integral BVPs on the infinite interval, including those by Agarwal and O’Regan [1], O’Regan [2], and many others (see [3–17]). For the study of boundary value problems, Agarwal and O’Regan [1] adopted mainly the method of the nonlinear alternative theorem together with a wonderful diagonalization process and the fixed-point theorem in the Frechet space.Boundary value problems on the half-line arise naturally in the study of radially symmetric solutions of nonlinear elliptic equations (see, [18–20]). Recently, by using the Krasnosel’skii fixed-point theorem, Lian and Ge [6] obtained the criteria for the existence of at least one positive solution, a unique positive solution, and multiple positive solutions of the following BVP
(1.1)(p(t)x′(t))′+λϕ(t)f(t,x(t))=0,0<t<+∞,α1x(0)-β1lim⁡t→0+p(t)x′(t)=0,α2lim⁡t→+∞x(t)+β2lim⁡t→+∞p(t)x′(t)=0,
where λ>0 is a parameter, f:ℝ+×ℝ+→ℝ=(-∞,+∞) and ϕ(t):(0,+∞)→(0,+∞) are continuous. More recently, by employing the method of varying in translation together with the fixed-point theorem in cone, Zhang et al. [14] established the existence of positive solution for the following semipositone singular Sturm-Liouville boundary value problem on the half-line
(1.2)(p(t)x′(t))′+f(t,x)+q(t)=0,0<t<+∞,α1x(0)-β1lim⁡t→0+p(t)x′(t)=0,α2lim⁡t→+∞x(t)+β2lim⁡t→+∞p(t)x′(t)=0,
where f:(0,+∞)×ℝ+→ℝ+ is continuous and allows the nonlinearity to have singularity at t=0, q:(0,+∞)→ℝ is a Lebesgue integrable function. As far as we know, there is very few work concerning the systems of BVPs on the half-line, although the study for the systems of BVPs (Pa,b) on the half-line is very important.Using the fixed-point theorem of cone expansion and compression type, the upper-lower solutions method, and degree arguments, do Ó et al. [21] studied the existence, nonexistence, and multiplicity of positive solutions for the following class of systems of second-order ordinary differential equations on the finite interval [0,1]:
(1.3)-u′′(t)=f(t,u,v,a,b),0<t<1,-v′′(t)=g(t,u,v,a,b),0<t<1,u(0)=u(1)=0,v(0)=v(1)=0,
where f,g:[0,1]×ℝ+4→ℝ+ are continuous and nondecreasing with respect to the last four variables.Motivated by the above works, in this paper, we extend the results of [6, 14, 21, 22] to (Pa,b) and also expand the domain from finite intervals to the infinite interval—the half line.There are two aims in this paper. The first aim is to obtain the existence of positive solutions for the system(Pa,b). For this purpose, we solve the fixed point of an operator F instead of the positive solutions for the system (Pa,b). The main difficulty for this is to testify that the operator F is completely continuous, as the Ascoli-Arzela theorem cannot be used in infinite interval ℝ+. Some modification of the compactness criterion in infinite interval ℝ+ (Lemma 2.4) has thus been made to resolve this problem. The second aim is to show that there exists a continuous curve Γ which splits the positive quadrant of the (a,b)-plane into two disjoint sets 𝒬1 and 𝒬2 such that the system (Pa,b) has at least two positive solutions in 𝒬1, at least one positive solution on the boundary of 𝒬1, and no positive solutions in 𝒬2.The rest of the paper is organized as follows. In Section2, we present some necessary definitions and lemmas that will be used to prove our main results. In Section 3, first, we give Lemma 3.1, which is a result of completely continuous operator, then we discuss our main results.


## 2. Preliminaries and Lemmas

In this section, we present some notations and lemmas that will be used in the proof of our main results.Throughout this paper, the spaceX=E×E will be the basic space to study (Pa,b), where the Banach space E is denoted by E={u∈C(ℝ+):limt→+∞u(t)exists} with the supremum norm ∥u∥∞=supt∈ℝ+|u(t)|. Clearly, (X,∥·∥) is a Banach space with the norm ∥(u,v)∥=∥u∥∞+∥v∥∞ for (u,v)∈X. For convenience, let
(2.1)a¯i=βi1+αi1∫0+∞‍1pi(s)ds,b¯i=βi2+αi2∫0+∞‍1pi(s)ds,i=1,2,a¯i(t)=βi1+αi1∫0t‍1pi(s)ds,b¯i(t)=βi2+αi2∫t+∞‍1pi(s)ds,i=1,2.
Then, it is obvious that αi2a¯i(t)+αi1b¯i(t)=ρi(i=1,2) is a constant for any t∈ℝ+ and a¯i(t) is increasing on t∈ℝ+, b¯i(t) is decreasing on t∈ℝ+ for i=1,2.Lemma 2.12.1(see [6]).
Under the condition∫0+∞‍(1/pi(s))ds<+∞ and ρi>0 for i=1,2, the linear boundary value problem (2.2)(pi(t)x′(t))′+v(t)=0,0<t<+∞,αi1x(0)-βi1lim⁡t→0+pi(t)x′(t)=0,αi2lim⁡t→+∞x(t)+βi2lim⁡t→+∞pi(t)x′(t)=0,i=1,2
has a unique solution for any v∈L1(0,+∞). Moreover, this unique solution can be expressed in the form
(2.3)xi(t)=∫0+∞‍Gi(t,s)v(s)ds,i=1,2,
where the Green function Gi(t,s) is defined by
(2.4)Gi(t,s)=1ρi{a¯i(t)b¯i(s),0≤t≤s<+∞,a¯i(s)b¯i(t),0≤s≤t<+∞,i=1,2.Remark 2.2.
From (2.4), we can get the properties of Gi(t,s) as follows.(1)
Gi(t,s) is continuous and nonnegative on ℝ+×ℝ+.(2)
For eachs∈ℝ+, Gi(t,s) is continuously differentiable on ℝ+ except for t=s.(3)
∂Gi(t,s)/∂t|t=s+-∂Gi(t,s)/∂t|t=s-=-1/pi(s).(4)
For eachs∈ℝ+, Gi(t,s) satisfies the corresponding homogeneous BVP (i.e., the BVP (2.2) with v(t)≡0) on ℝ+ except for t=s. In other words, Gi(t,s) is the Green function of BVP (2.2) on the half-line.(5)
Gi(t,s)≤Gi(s,s)≤(1/ρi)a¯ib¯i<+∞,i=1,2.(6)
G¯i(s)=limt→+∞Gi(t,s)=(1/ρi)βi2a¯i(s)≤Gi(s,s)<+∞,i=1,2.(7)
For anyt∈[δ,1/δ] and s∈ℝ+,Gi(t,s)≥ωiGi(s,s), where
(2.5)δ∈(0,1),ωi=min⁡{b¯i(1/δ)b¯i,a¯i(δ)a¯i},i=1,2,0<ω=min⁡{ω1,ω2}<1.
For any(x1,x2,…,xm)∈ℝm, define |x1,x2,…,xm|=|x1|+|x2|+⋯|xm|. In what follows, we list some conditions for convenience.(H1)
The functionfi:ℝ+5→ℝ+ is continuous and nondecreasing with respect to the last four variables. In other words,
(2.6)fi(t,x1,y1,a1,b1)≤fi(t,x2,y2,a2,b2),i=1,2,
for any t∈ℝ+,(x1,y1,a1,b1)≤(x2,y2,a2,b2), where the order is understood to apply to every component. And there exists (a0,b0) such that fi(t,x,y,a,b) is bounded for any (a,b) satisfying (0,0)≤(a,b)≤(a0,b0), t∈ℝ+, x  and  y in any bounded set of ℝ+.(H2)
The functionϕi(t):(0,+∞)→ℝ+ is continuous and singular at t=0, ϕi(t)≢0 on ℝ+ satisfying 0<∫0+∞‍Gi(s,s)ϕi(s)ds<+∞,i=1,2.(H3)
For the abovea0 and b0 in (H1),
(2.7)fi0=limsup⁡x+y→0+sup⁡t∈R+fi(t,x,y,a0,b0)x+y<L,fi∞=liminf⁡x+y→+∞inf⁡t∈[δ,1/δ]fi(t,x,y,0,0)x+y>l,i=1,2,
where
(2.8)L=12(max⁡{∫0+∞‍G1(s,s)ϕ1(s)ds,∫0+∞‍G2(s,s)ϕ2(s)ds})-1,l=12(min⁡{min⁡t∈[δ,1/δ]∫δ1/δ‍G1(t,s)ϕ1(s)ds,min⁡t∈[δ,1/δ]∫δ1/δ‍G2(t,s)ϕ2(s)ds})-1.(H4)
(2.9)lim⁡|(a,b)|→+∞fi(t,x,y,a,b)=+∞,uniformlyfort∈[δ,1δ],x≥0,y≥0,i=1,2.From the above assumptions(H1) and (H2), it is not difficult to show that the pair (u,v)∈X is a solution of the system (Pa,b) if and only if (u,v)∈X is a solution of the following system of nonlinear integral equations:(Sa,b)u(t)=λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,u(s),v(s),a,b)ds,v(t)=λ∫0+∞‍G2(t,s)ϕ2(s)f2(s,u(s),v(s),a,b)ds,t∈R+.
Define operators Ai:X→E and F:X→X as follows:
(2.10)Ai(u,v)(t)=λ∫0+∞‍Gi(t,s)ϕi(s)fi(s,u(s),v(s),a,b)ds,i=1,2,(2.11)F(u,v)=(A1(u,v),A2(u,v)).
Then, the solution of the system (Pa,b) is equivalent to the fixed point of the operator F. Define a cone K in the Banach space X as follows:
(2.12)K={(u,v)∈X:u(t)≥0,v(t)≥0,t∈R+,u(t)≥ω‖u‖∞,v(t)≥ω‖v‖∞,t∈[δ,1δ]},
which induces a partial order “≤”: (u1,v1)≤(u2,v2) if and only if u1(t)≤u2(t),v1(t)≤v2(t) for any t∈ℝ+.Consider the following system:(S)-(p1(t)u′(t))′=λϕ1(t)g1(t,u(t),v(t)),-(p2(t)v′(t))′=λϕ2(t)g2(t,u(t),v(t)),0<t<+∞,α11u(0)-β11lim⁡t→0+p1(t)u′(t)=0,α12lim⁡t→+∞u(t)+β12lim⁡t→+∞p1(t)u′(t)=0,α21v(0)-β21lim⁡t→0+p2(t)v′(t)=0,α22lim⁡t→+∞v(t)+β22lim⁡t→+∞p2(t)v′(t)=0,
where gi:ℝ+3→ℝ+(i=1,2) are nonnegative continuous functions and are nondecreasing with respect to the last two variables.Definition 2.3.
The pair(u,v) is said to be a lower solution for the system (S) if the pair (u,v) satisfies the following inequality system:
(T)-(p1(t)u′(t))′≤λϕ1(t)g1(t,u(t),v(t)),-(p2(t)v′(t))′≤λϕ2(t)g2(t,u(t),v(t)),0<t<+∞,α11u(0)-β11lim⁡t→0+p1(t)u′(t)≤0,α12lim⁡t→+∞u(t)+β12lim⁡t→+∞p1(t)u′(t)≤0,α21v(0)-β21lim⁡t→0+p2(t)v′(t)≤0,α22lim⁡t→+∞v(t)+β22lim⁡t→+∞p2(t)v′(t)≤0.
Similarly, we define the upper solution for the system (S) by replacing the ≤ in (T) by ≥.Lemma 2.4 (see [1, 23]).
LetE be defined as above and M⊂E. Then, M is relatively compact in E if the following conditions hold.(1)
M is uniformly bounded in E.(2)
The functions inM are equicontinuous on any bounded interval of ℝ+.(3)
The functions inM are equiconvergent at +∞, that is, for any given ε>0, there exists a T=T(ε)>0 such that |f(t)-f(+∞)|<ε, for any t>T,f∈M.Lemma 2.5 (see [24, 25]).
LetP be a positive cone in a real Banach space (E,∥·∥), Pr={x∈P:∥x∥<r}, P¯r,R={x∈P:r≤∥x∥≤R}(0<r<R<+∞), and let A:P¯r,R→P be a completely continuous operator. If the following conditions are satisfied,(1)
∥Ax∥≤∥x∥, for all x∈∂PR,(2)
there exists ae∈∂P1 such that x≠Ax+me for any x∈∂Pr and m>0, then A has fixed points in P¯r,R.Remark 2.6.
If (1) and (2) are satisfied forx∈∂Pr and x∈∂PR, respectively, then Lemma 2.5 still holds.Lemma 2.72.7 (see [25]).
Let(E,∥·∥) be a Banach space, K a cone in E, and let Ω be a bounded open set in E with θ∈Ω. Suppose that A:K∩Ω¯→K is a completely continuous operator. If Ax≠μx for x∈K∩∂Ω and μ≥1, then the fixed-point index (2.13)i(A,K∩Ω,K)=1.Lemma 2.82.8 (see [25]).
Let(E,∥·∥) be a Banach space, K be a cone in E. For r>0, define Kr={x∈K:∥x∥<r}. Suppose that A:K¯r→K is a completely continuous operator such that Ax≠x for x∈∂Kr.(i)
If∥Ax∥≥∥x∥ for x∈∂Kr, then i(A,Kr,K)=0.(ii)
If∥Ax∥≤∥x∥ for x∈∂Kr, then i(A,Kr,K)=1.


## 3. Main Results

### 3.1. The Complete Continuity of the OperatorF

Before presenting the main results, we give a lemma.Lemma 3.1.
Assume that(H1) and (H2) hold. Then, for any (a,b) satisfying (0,0)≤(a,b)≤(a0,b0), F:X→X is a completely continuous operator and F(K)⊆K.Proof.
We divide the proof into four steps.
(i) Firstly, we show thatF:X→X is well defined. For any fixed (u,v)∈X, there exists r1>0 such that |u(t)|≤r1 and |v(t)|≤r1 for any t∈ℝ+. It follows from (H1) and the property (1) of the Green function Gi(t,s) that Ai(u,v)≥0 and
(3.1)Bfir1=sup⁡{fi(t,x,y,a,b):t∈R+,x∈[0,r1],y∈[0,r1],a∈[0,a0],b∈[0,b0]}<+∞.
Thus, by(H1) and (H2), for any t∈ℝ+,a∈[0,a0] and b∈[0,b0], we obtain
(3.2)0≤λ∫0+∞‍Gi(t,s)ϕi(s)fi(s,u(s),v(s),a,b)ds≤λBfir1∫0+∞‍Gi(s,s)ϕi(s)ds<+∞.
Hence, the operator F(u,v)=(A1(u,v),A2(u,v)) is well defined for any (u,v)∈X.
For anyt1,t2,s∈ℝ+, by the property (5) of the Green function Gi(t,s), we have
(3.3)|Gi(t1,s)-Gi(t2,s)|ϕi(s)≤2Gi(s,s)ϕi(s),i=1,2.
So, by (H2), the Lebesgue dominated convergence theorem and the continuity of Gi(t,s), for any t1,t2∈ℝ+, a∈[0,a0],b∈[0,b0], we get
(3.4)|Ai(u,v)(t1)-Ai(u,v)(t2)|≤λ∫0+∞‍|Gi(t1,s)-Gi(t2,s)|ϕi(s)fi(s,u(s),v(s),a,b)ds≤λBfir1∫0+∞‍|Gi(t1,s)-Gi(t2,s)|ϕi(s)ds⟶0,t1⟶t2.
Therefore, Ai(u,v)∈C(ℝ+), and so F(u,v)∈C(ℝ+)×C(ℝ+). By the property (6) of Gi(t,s) and the Lebesgue dominated convergence theorem, we obtain
(3.5)lim⁡t→+∞Ai(u,v)(t)=λ∫0+∞‍G¯i(s)ϕi(s)fi(s,u(s),v(s),a,b)ds<+∞.
Hence, F:X→X is well defined.
(ii) Next we show thatF:X→X is continuous. Let (un,vn),(u,v)∈X,n∈ℕ, and ∥(un,vn)-(u,v)∥→0(n→+∞), we will prove that ∥F(un,vn)-F(u,v)∥→0 (n→+∞). By (2.10), (H1), and (H2), for any t∈ℝ+, a∈[0,a0],b∈[0,b0] and any natural number n, we have
(3.6)|Ai(un,vn)(t)-Ai(u,v)(t)|≤λ∫0+∞‍Gi(s,s)ϕi(s)(∫0+∞‍|fi(s,un(s),vn(s),a,b)|+|fi(s,u(s),v(s),a,b)|)ds≤2λBfir2∫0+∞‍Gi(s,s)ϕi(s)ds<+∞,i=1,2,
where Bfir2=sup{fi(t,x,y,a,b):t∈ℝ+,x∈[0,r2],y∈[0,r2],a∈[0,a0],b∈[0,b0]}<+∞ by (H1), r2 is a real number such that r2≥maxn∈ℕ{∥(u,v)∥,∥(un,vn)∥}, in which ℕ is the natural number set.
For anyε>0, by (H2), there exists a sufficiently large T0>0 such that
(3.7)2λBfir2∫T0+∞‍Gi(s,s)ϕi(s)ds<ε2,i=1,2.
On the other hand, by the continuity of fi(t,x,y,a,b) on [0,T0]×[0,r2]×[0,r2]×[0,a0]×[0,b0], for the above ε>0, there exists a δ>0 such that for any s∈[0,T0], a∈[0,a0],b∈[0,b0] and x,x′,y,y′∈[0,r2], when |x-x′|<δ, |y-y′|<δ, we have
(3.8)|fi(s,x,y,a,b)-fi(s,x′,y′,a,b)|<ε2(λ∫0+∞‍Gi(s,s)ϕi(s)ds)-1,i=1,2.
From ∥(un,vn)-(u,v)∥→0 (n→+∞) and the definition of the norm ∥·∥ in the space X, we can easily conclude that ∥un-u∥∞→0, ∥vn-v∥∞→0(n→+∞). So, for the above δ>0, there exists a sufficiently large natural number N0 such that, when n>N0, for any s∈[0,T0], we have
(3.9)|un(s)-u(s)|≤‖un-u‖∞<δ,|vn(s)-v(s)|≤‖vn-v‖∞<δ.
Hence, by (3.8), when n>N0, for any s∈[0,T0], a∈[0,a0],b∈[0,b0], we get
(3.10)|fi(s,un(s),vn(s),a,b)-fi(s,u(s),v(s),a,b)|<ε2(λ∫0+∞‍Gi(s,s)ϕi(s)ds)-1.
Therefore, by (2.10), (3.7), and (3.10), when n>N0, for any t∈ℝ+, a∈[0,a0] and b∈[0,b0], we obtain
(3.11)|Ai(un,vn)(t)-Ai(u,v)(t)|≤λ∫0+∞‍Gi(s,s)ϕi(s)|fi(s,un(s),vn(s),a,b)-fi(s,u(s),v(s),a,b)|ds≤λ∫0T0‍Gi(s,s)ϕi(s)|fi(s,un(s),vn(s),a,b)-fi(s,u(s),v(s),a,b)|ds+2λBfir2∫T0+∞‍Gi(s,s)ϕi(s)ds<ε,i=1,2.
This implies that the operator Ai:X→E is continuous. Therefore, the operator F:X→X is continuous.
(iii) We need to prove that the operatorF:X→X is compact. Let D be any bounded subset of X. Then, there exists a constant r3>0 such that ∥(u,v)∥≤r3 for any (u,v)∈D. So ∥u∥∞≤r3, ∥v∥∞≤r3 for any (u,v)∈D. By (2.10), (H1), and (H2), for any (u,v)∈D and t∈ℝ+, we have
(3.12)|F(u,v)(t)|=|A1(u,v)(t)|+|A2(u,v)(t)|=|λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,u(s),v(s),a,b)ds|+|λ∫0+∞‍G2(t,s)ϕ2(s)f2(s,u(s),v(s),a,b)ds|≤λBf1r3∫0+∞‍G1(s,s)ϕ1(s)ds+λBf2r3∫0+∞‍G2(s,s)ϕ2(s)ds<+∞,
where Bfir3=sup{fi(t,x,y,a,b):t∈ℝ+,x∈[0,r3],y∈[0,r3],a∈[0,a0],b∈[0,b0]}<+∞ by (H1). Hence, F(D) is uniformly bounded. By the similar proof as for (3.4), we can conclude that Ai(D) is equicontinuous, and so F(D) is also equicontinuous.
From(H2) and the property (6) of the Green function Gi(t,s), for any t∈ℝ+, we have
(3.13)∫0+∞‍|Gi(t,s)-G¯i(s)|ϕi(s)ds≤2∫0+∞‍Gi(s,s)ϕi(s)ds<+∞,i=1,2.
By (2.10), (3.5), and the Lebesgue dominated convergence theorem, for any (u,v)∈D, t∈ℝ+, a∈[0,a0], and b∈[0,b0], we obtain
(3.14)|Ai(u,v)(t)-Ai(u,v)(+∞)|=|λ∫0+∞‍Gi(t,s)ϕi(s)fi(s,u(s),v(s),a,b)ds-λ∫0+∞‍G¯i(s)ϕi(s)fi(s,u(s),v(s),a,b)ds|≤λ∫0+∞‍|Gi(t,s)-G¯i(s)|ϕi(s)fi(s,u(s),v(s),a,b)ds≤λBfir3∫0+∞‍|Gi(t,s)-G¯i(s)|ϕi(s)ds⟶0,t⟶+∞,i=1,2.
This implies that Ai(D) is equiconvergent at +∞. Hence, F(D) is equiconvergent at +∞. Therefore, the above discussion and Lemma 2.4 imply that F:X→X is completely continuous.
(iv) Finally, we proveF(K)⊆K. By the property (1) of Gi(t,s), (H1), and (H2), it is easy to see that, for any (u,v)∈K and t∈ℝ+, Ai(u,v)(t)≥0 and
(3.15)Ai(u,v)(t)=λ∫0+∞‍Gi(t,s)ϕi(s)fi(s,u(s),v(s),a,b)ds≤λ∫0+∞‍Gi(s,s)ϕi(s)fi(s,u(s),v(s),a,b)ds.
So
(3.16)‖Ai(u,v)‖∞≤λ∫0+∞‍Gi(s,s)ϕi(s)fi(s,u(s),v(s),a,b)ds,i=1,2.
On the other hand, by the property (7) of Gi(t,s), we have
(3.17)Ai(u,v)(t)≥ωλ∫0+∞‍Gi(s,s)ϕi(s)fi(s,u(s),v(s),a,b)ds,t∈[δ,1δ],i=1,2.
It follows from (3.16) and (3.17) that
(3.18)Ai(u,v)(t)≥ω‖Ai(u,v)‖∞,t∈[δ,1δ],i=1,2.
Therefore F(K)⊆K. The proof of Lemma 3.1 is completed.


### 3.2. The Positive Solution for System(Pa0,b0)

Theorem 3.2.
Assume that(H1)–(H3) hold, then the system (Pa0,b0) has at least one positive solution for any
(3.19)λ∈(lmin⁡(f1∞,f2∞),Lmax⁡(f10,f20)).Proof.
From (3.19), there exists ε0>0 such that
(3.20)lmin⁡(f1∞,f2∞)-ε0≤λ≤Lmax⁡(f10,f20)+ε0.
By the first inequality of (H3), there exists r>0 such that
(3.21)fi(t,x,y,a0,b0)≤(fi0+ε0)r,0≤x+y≤r,t∈R+,i=1,2.
Setting Kr1={(u,v)∈K:∥(u,v)∥<r1}(r1≤r), by the definition of ∥·∥, we know that
(3.22)u(t)+v(t)≤‖u‖∞+‖v‖∞=‖(u,v)‖=r1≤r,∀(u,v)∈∂Kr1,t∈R+.
Then, for any (u,v)∈∂Kr1,
(3.23)‖Ai(u,v)‖∞=λsup⁡t∈R+|∫0+∞‍Gi(t,s)ϕi(s)fi(s,u(s),v(s),a0,b0)ds|≤λsup⁡t∈R+∫0+∞‍Gi(t,s)ϕi(s)(fi0+ε0)r1ds≤λ(fi0+ε0)r1∫0+∞‍Gi(s,s)ϕi(s)ds≤r1L∫0+∞‍Gi(s,s)ϕi(s)ds≤12r1=12‖(u,v)‖,i=1,2.
Thus,
(3.24)‖F(u,v)‖=‖A1(u,v)‖∞+‖A2(u,v)‖∞≤‖(u,v)‖,(u,v)∈∂Kr1.
On the other hand, by the second inequality of(H3), there exists r0>ωr1>0 such that
(3.25)fi(t,x,y,a0,b0)≥fi(t,x,y,0,0)≥(fi∞-ε0)(x+y),x+y≥r0,t∈[δ,1δ],i=1,2.
Take r2=r0/ω>r1, and let Kr2={(u,v)∈K:∥(u,v)∥<r2}, (u0,v0)=(1/2,1/2)∈∂K1. Then,
(3.26)(u,v)≠F(u,v)+μ(u0,v0),∀(u,v)∈∂Kr2,∀μ>0.
Suppose that (3.26) is false, then there exist (u2,v2)∈∂Kr2 and μ2>0 such that (u2,v2)=F(u2,v2)+μ2(u0,v0). From (3.25) and the fact that
(3.27)u2(t)+v2(t)≥ω‖u2‖∞+ω‖v2‖∞=ω‖(u2,v2)‖=ωr2=r0,t∈[δ,1δ],
we have
(3.28)fi(t,u2(t),v2(t),a0,b0)≥(fi∞-ε0)(u2(t)+v2(t)),t∈[δ,1δ],i=1,2.
Set
(3.29)ξ=min⁡{u2(t)+v2(t):t∈[δ,1δ]}.
Then, u2(t)+v2(t)≥ξ>0 for any t∈[δ,1/δ]. Hence, for any t∈[δ,1/δ], by (3.28), we have
(3.30)u2(t)+v2(t)=A1(u2(t),v2(t))+A2(u2(t),v2(t))+μ2(u0+v0)=λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,u2(s),v2(s),a0,b0)ds+λ∫0+∞‍G2(t,s)ϕ2(s)f2(s,u2(s),v2(s),a0,b0)ds+μ2(u0+v0)≥λ∫δ1/δ‍G1(t,s)ϕ1(s)(f1∞-ε0)(u2(s)+v2(s))ds+λ∫δ1/δ‍G2(t,s)ϕ2(s)(f2∞-ε0)(u2(s)+v2(s))ds+μ2≥min⁡s∈[δ,1/δ](u2(s)+v2(s))λ(f1∞-ε0)min⁡t∈[δ,1/δ]∫δ1/δ‍G1(t,s)ϕ1(s)ds+min⁡s∈[δ,1/δ](u2(s)+v2(s))λ(f2∞-ε0)min⁡t∈[δ,1/δ]∫δ1/δ‍G2(t,s)ϕ2(s)ds+μ2≥ξ+μ2.
Then, we can obtain that
(3.31)u2(t)+v2(t)>ξ+μ2,t∈[δ,1δ].
It is clearly that (3.31) contradicts (3.29), which implies that (3.26) holds.
It follows from (3.24), (3.26), Lemmas 2.5 and 3.1 that the operator F has fixed-point (u,v)∈Kr2∖K-r1 such that 0<r1<∥(u,v)∥<r2. It is easy to see that (u,v) is a positive solution of the system (Pa0,b0). The proof of Theorem 3.2 is completed.Remark 3.3.
Noticing thatl/f1∞<1, l/f2∞<1 and L/f10>1, L/f20>1, we conclude that Theorem 3.2 also holds for λ=1.Remark 3.4.
From Theorem3.2, we can see that fi(t,x,y,a0,b0)(i=1,2) do not need to be superlinear or sublinear. In fact, Theorem 3.2 still holds, if (H1)-(H2) are satisfied and one of the following conditions is satisfied:(1)
f1∞=f2∞=+∞,L>f10>0, L>f20>0, for each λ∈(0,L/max{f10,f20}),(2)
f1∞=f2∞=+∞, f10=f20=0, for each λ∈(0,+∞),(3)
f1∞>l, f2∞>l, f10=f20=0, for each λ∈(l/min{f1∞,f2∞},+∞).


### 3.3.  Lower and Upper Solutions

Theorem 3.5.
Assume that(H1)-(H2) hold. Let (u_,v_) and (u¯,v¯) be a lower solution and an upper solution, respectively, of the system (S) such that (0,0)≤(u_,v_)≤(u¯,v¯). Then, the system (S) has a nonnegative solution (u,v) satisfying (u_,v_)≤(u,v)≤(u¯,v¯).Proof.
Let(3.32)Mi(u,v)(t)=λ∫0+∞‍Gi(t,s)ϕi(s)gi(s,u(s),v(s))ds,i=1,2,H(u,v)=(M1(u,v),M2(u,v)).
Then, the solutions of the system (S) are equivalent to the fixed points of the operator H in K.
Now, we introduce the following auxiliary operatorℋ defined by
(3.33)H(u,v)=(M1(u,v),M2(u,v)),
where
(3.34)Mi(u,v)(t)=λ∫0+∞‍Gi(t,s)ϕi(s)gi(s,ζ1(s,u),ζ2(s,v))ds,i=1,2,
in which
(3.35)ζ1(t,u)=max⁡{u_(t),min⁡{u(t),u¯(t)}},ζ2(t,v)=max⁡{v_(t),min⁡{v(t),v¯(t)}}.
It is easy to prove that the operator ℋ has the following properties.(1)
ℋ is a completely continuous operator.(2)
If the pair(u,v)∈K is a fixed point of ℋ, then (u,v) is a fixed point of H with (u_,v_)≤(u,v)≤(u¯,v¯).(3)
If(u,v)=μℋ(u,v) with 0≤μ≤1, then ∥(u,v)∥≤C**, where C** does not depend on μ and (u,v)∈K.Therefore, by using the topological degree of Leray-Schauder (see [26, Corollary  8.1, page. 61]), we obtain a fixed point of the operator H. The proof of Theorem 3.5 is completed.Theorem 3.6.
Assume that(H1)–(H3) and (3.19) hold. Then for (0,0)≤(a,b)≤(a0,b0), the system (Pa,b) has at least one positive solution.Proof.
From Theorem3.2, we can see that the system (Pa0,b0) has at least one positive solution (u0,v0). Since the functions fi(i=1,2) are increasing functions with respect to the last four variables, we conclude that (u0,v0) is an upper solution and (0,0) is a lower solution for the system (Pa,b). Hence, by Theorem 3.5, we have that the system (Pa,b) has at least one positive solution. This completes the proof of Theorem 3.6.


### 3.4. A Priori Estimate

Theorem 3.7.
Assume that(H1)–(H3) and (3.19) hold. Then for any (a,b) satisfying (0,0)≤(a,b)≤(a0,b0), there exists a constant C0>0 independent of (a,b), such that ∥(u,v)∥≤C0 for all positive solutions (u,v) of the system (Pa,b).Proof.
Assume by contradiction that there exists a sequence of positive solution(un,vn)∈X of system (Pa,b) such that ∥(un,vn)∥→∞. From (3.19), there exists ε0>0 such that λ>l/(min(f1∞,f2∞)-ε0). From assumption (H3), there exists H>0 such that, for (0,0)≤(a,b)≤(a0,b0),
(3.36)fi(t,x,y,a,b)≥fi(t,x,y,0,0)≥(fi∞-ε0)(x+y),x+y≥H,t∈[δ,1δ],i=1,2.
Since un(t)+vn(t)≥ω∥un∥∞+ω∥vn∥∞=ω∥(un,vn)∥,t∈[δ,1/δ], there exists natural number N0 such that, for n>N0, we have un(t)+vn(t)≥H for t∈[δ,1/δ]. It follows from (3.36) that when n>N0, fi(t,un(t),vn(t),a,b)≥(fi∞-ε0)(un(t)+vn(t)),t∈[δ,1/δ],i=1,2. Thus,
(3.37)un(t)+vn(t)=λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,un(s),vn(s),a,b)ds+λ∫0+∞‍G2(t,s)ϕ2(s)f2(s,un(s),vn(s),a,b)ds≥λ∫δ1/δ‍G1(t,s)ϕ1(s)(f1∞-ε0)(un(s)+vn(s))ds+λ∫δ1/δ‍G2(t,s)ϕ2(s)(f2∞-ε0)(un(s)+vn(s))ds≥min⁡s∈[δ,1/δ](un(s)+vn(s))λ(f1∞-ε0)min⁡t∈[δ,1/δ]∫δ1/δ‍G1(t,s)ϕ1(s)ds+min⁡s∈[δ,1/δ](un(s)+vn(s))λ(f2∞-ε0)min⁡t∈[δ,1/δ]∫δ1/δ‍G2(t,s)ϕ2(s)ds>min⁡s∈[δ,1/δ](un(s)+vn(s)),t∈[δ,1δ]
which yields
(3.38)min⁡t∈[δ,1/δ]un(t)+vn(t)>min⁡t∈[δ,1/δ](un(t)+vn(t)),
which is a contradiction. This completes the proof of Theorem 3.7.Remark 3.8.
By the discussions in Sections3.1, 3.2, 3.3 and 3.4, we can conclude that, for any (a,b) satisfying |(0,0)|≤|(a,b)|≤|(a0,b0)|, the system (Pa,b) has at least one positive solution (u,v) with ∥(u,v)∥≤C0. In the following section, we will establish the nonexistence result for the system (Pa,b).


### 3.5.  Non-Existence

Theorem 3.9.
Assume that(H1)–(H4) and (3.19) hold. Then, there exist a_>0,b_>0 such that, for all (a,b) with |(a,b)|>|(a_,b_)|, the system (Pa,b) has no solution.Proof.
Suppose by contradiction that there exists a sequence(an,bn) with |(an,bn)|→+∞(n→+∞) such that, for each nature number n, the system (Pan,bn) has a positive solution (un,vn) in K. From assumption (H4), for any M>0, there exists a constant C>0 such that, for any |(a,b)|>C,
(3.39)fi(t,x,y,a,b)≥M,x,y≥0,t∈[δ,1δ],i=1,2.
By |(an,bn)|→+∞(n→+∞), for the above C>0, there exists a natural number n0 such that, for n>n0,|(an,bn)|>C, then, for n>n0 and t∈[δ,1/δ],
(3.40)un(t)=λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,un(s),vn(s),an,bn)ds>λ∫δ1/δ‍G1(t,s)ϕ1(s)f1(s,un(s),vn(s),an,bn)ds≥Mλω∫δ1/δ‍G1(s,s)ϕ1(s)ds.
By the same way, we can obtain
(3.41)vn(t)≥Mλω∫δ1/δ‍G2(s,s)ϕ2(s)ds.
Since we can choose M arbitrarily large, we conclude that un and vn are unbounded sequences in K, then limt→+∞un(t) and limt→+∞vn(t) do not exist, which contradicts the fact that un,vn∈K. So Theorem 3.9 holds.We next define the set(3.42)A={a>0:thesystem(Pa,b)hasapositivesolutionforsomeb>0}.
From Theorems 3.2 and 3.9, we conclude that A is nonempty and bounded. Thus, 0<a~=supA<+∞. Using the upper-lower solutions method, we see that, for all a∈(0,a~), there exists b>0 such that the system (Pa,b) has a positive solution. We now define the function Γ:[0,a~]→ℝ+ by
(3.43)Γ(a)=sup⁡{b>0:system(Pa,b)hasapositivesolution}.
By Theorem 3.6, the function Γ is continuous and nonincreasing. We thus claim that Γ(a) is attained. In fact, it suffices to use Theorem 3.7 and the compactness of the operator F. Finally, it follows from the definition of the function Γ that the system (Pa,b) has at least one positive solution for 0≤b≤Γ(a) and has no positive solutions for b>Γ(a).


### 3.6.  Existence of Two Positive Solutions

In this section,we will assume that the nonlinearitiesf1 and f2 are strict-increasing with respect to the fifth variable. Fix a∈[0,a~], and let (ϕ¯,ψ¯) be the solution of the problem (Pa,Γ(a)) which is obtained using Theorem 3.6. Our next result allows us to establish another solution of the system (Pa,b)  for 0<b<Γ(a).Lemma 3.10.
For each0<b<Γ(a), there exists ϵ0>0 so that, for all 0<ϵ≤ϵ0 and all t∈ℝ+, one has (3.44)ϕ¯ϵ(t)>λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,ϕ¯ϵ(s),ψ¯ϵ(s),a,b)ds,ψ¯ϵ(t)>λ∫0+∞‍G2(t,s)ϕ2(s)f2(s,ϕ¯ϵ(s),ψ¯ϵ(s),a,b)ds,
where ϕ¯ϵ(t)=ϕ¯(t)+ϵ,ψ¯ϵ(t)=ψ¯(t)+ϵ.Proof.
Fixδ∈(0,1). Since f1 is strict-increasing with respect to the fifth variable, we have that for each 0<b<Γ(a) we may find a positive constant I=I(b) so that, for all s∈[δ,1/δ], we have
(3.45)f1(s,ϕ¯(s),ψ¯(s),a,Γ(a))-f1(s,ϕ¯(s),ψ¯(s),a,b)≥I>0.
By the uniform continuity of f1, there exists ϵ0>0 so that, for all s∈[δ,1/δ] and all 0<ϵ≤ϵ0, we have
(3.46)|f1(s,ϕ¯(s)+ϵ,ψ¯(s)+ϵ,a,b)-f1(s,ϕ¯(s),ψ¯(s),a,b)|≤I2.
Next, we define
(3.47)Φϵ(t,s)=G1(t,s)ϕ1(s)[f1(s,ϕ¯ϵ(s),ψ¯ϵ(s),a,b)-f1(s,ϕ¯(s),ψ¯(s),a,b)],Ψ(t,s)=G1(t,s)ϕ1(s)[f1(s,ϕ¯(s),ψ¯(s),a,Γ(a))-f1(s,ϕ¯(s),ψ¯(s),a,b)].
Assume 0<ϵ≤ϵ0. Then,
(3.48)ϕ¯ϵ(t)>ϕ¯(t)=λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,ϕ¯(s),ψ¯(s),a,Γ(a))ds=λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,ϕ¯ϵ(s),ψ¯ε(s),a,b)ds-λ∫0+∞‍Φϵ(t,s)ds+λ∫0+∞‍Ψ(t,s)ds.
Since Ψ(t,s) is positive and Ψ(t,s)-Φϵ(t,s)>(I/2)G1(t,s)ϕ1(s) for t∈[δ,1/δ], we have
(3.49)ϕ¯ϵ(t)>λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,ϕ¯ϵ(s),ψ¯ϵ(s),a,b)ds-λ∫0δ‍Φϵ(t,s)ds-λ∫1/δ+∞‍Φϵ(t,s)ds+I2λ∫δ1/δ‍G1(t,s)ϕ1(s)ds.
It is not difficult to show that the Lebesgue dominated convergence theorem implies thatλ∫0δ‍Φϵ(t,s)ds+λ∫1/δ+∞‍Φϵ(t,s)ds converges to zero, uniformly in t, as ϵ tends to zero. Thus, for ϵ sufficiently small, we have
(3.50)ϕ¯ϵ(t)>λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,ϕ¯ϵ(s),ψ¯ϵ(s),a,b)ds
uniformly in t∈ℝ+.
A similar computation holds forψ¯ϵ.We are now in a position to show existence of two positive solutions of the system(Pa,b) for 0<b<Γ(a), where a∈[0,a~] is fixed.Theorem 3.11.
Assume that(H1)–(H4) and (3.19) hold. Then, for all a∈[0,a~], the system (Pa,b) has at least two positive solutions for 0<b<Γ(a).Proof.
Consider the set(3.51)Ω={(ϕ,ψ)∈X:-ϵ<ϕ(t)<ϕ¯ϵ(t),-ϵ<ψ(t)<ψ¯ϵ(t),fort∈R+},
where ϕ¯ϵ and ψ¯ϵ are the functions of Lemma 3.10. It is not hard to see that Ω is bounded and open in X and that (0,0)∈Ω. Note that one of the solutions of the system (Pa,b) belongs to K∩Ω¯, we also know that F:K∩Ω¯→K is a completely continuous operator.
Let(ϕ,ψ)∈K∩∂Ω. It follows that there exists t0∈(0,+∞) so that one of the following two cases holds: ϕ(t0)=ϕ¯ϵ(t0) or ψ(t0)=ψ¯ϵ(t0). In the case ϕ(t0)=ϕ¯ϵ(t0), it follows from Lemma 3.10 that, for all μ≥1, we have(3.52)A1(ϕ,ψ)(t0)=λ∫0+∞‍G1(t0,s)ϕ1(s)f1(s,ϕ(s),ψ(s),a,b)ds≤λ∫0+∞‍G1(t0,s)ϕ1(s)f1(s,ϕ¯ϵ(s),ψ¯ϵ(s),a,b)ds<ϕ¯ϵ(t0)=ϕ(t0)≤μϕ(t0).
Similarly, A2(ϕ,ψ)(t0)<μψ(t0) in the case ψ(t0)=ψ¯ϵ(t0). Hence, F(ϕ,ψ)≠μ(ϕ,ψ), for all (ϕ,ψ)∈K∩∂Ω and all μ≥1. Now, according to Lemma 2.7, we have
(3.53)i(F,K∩Ω,K)=1.
On the other hand, a slight change in the proof of Theorem3.7 shows the existence of an r3>0 sufficiently large, say r3>r2, where r2 is as in Theorem 3.2, so that(3.54)‖F(ϕ,ψ)‖>‖(ϕ,ψ)‖
for every ∥(ϕ,ψ)∥=r3 and every (ϕ,ψ)∈K.
Letr4=max{C0+1,r3,∥(ϕ¯ϵ,ψ¯ϵ)∥}, where C0 is as that in Theorem 3.7. Set
(3.55)Kr4={(ϕ,ψ)∈K:‖(ϕ,ψ)‖<r4}.
Then Theorem 3.7 implies that F(ϕ,ψ)≠(ϕ,ψ), for (ϕ,ψ)∈∂Kr4. Consequently, Lemma 2.8 implies i(F,Kr4,K)=0.
Now, by the additivity property of the fixed-point index, we obtain(3.56)i(F,Kr4∖K∩Ω¯,K)=-1.
Therefore, F has another fixed point in Kr4∖K∩Ω¯.Then, from the above discussion, it is easy to obtain the following conclusion.


## 3.1. The Complete Continuity of the OperatorF

Before presenting the main results, we give a lemma.Lemma 3.1.
Assume that(H1) and (H2) hold. Then, for any (a,b) satisfying (0,0)≤(a,b)≤(a0,b0), F:X→X is a completely continuous operator and F(K)⊆K.Proof.
We divide the proof into four steps.
(i) Firstly, we show thatF:X→X is well defined. For any fixed (u,v)∈X, there exists r1>0 such that |u(t)|≤r1 and |v(t)|≤r1 for any t∈ℝ+. It follows from (H1) and the property (1) of the Green function Gi(t,s) that Ai(u,v)≥0 and
(3.1)Bfir1=sup⁡{fi(t,x,y,a,b):t∈R+,x∈[0,r1],y∈[0,r1],a∈[0,a0],b∈[0,b0]}<+∞.
Thus, by(H1) and (H2), for any t∈ℝ+,a∈[0,a0] and b∈[0,b0], we obtain
(3.2)0≤λ∫0+∞‍Gi(t,s)ϕi(s)fi(s,u(s),v(s),a,b)ds≤λBfir1∫0+∞‍Gi(s,s)ϕi(s)ds<+∞.
Hence, the operator F(u,v)=(A1(u,v),A2(u,v)) is well defined for any (u,v)∈X.
For anyt1,t2,s∈ℝ+, by the property (5) of the Green function Gi(t,s), we have
(3.3)|Gi(t1,s)-Gi(t2,s)|ϕi(s)≤2Gi(s,s)ϕi(s),i=1,2.
So, by (H2), the Lebesgue dominated convergence theorem and the continuity of Gi(t,s), for any t1,t2∈ℝ+, a∈[0,a0],b∈[0,b0], we get
(3.4)|Ai(u,v)(t1)-Ai(u,v)(t2)|≤λ∫0+∞‍|Gi(t1,s)-Gi(t2,s)|ϕi(s)fi(s,u(s),v(s),a,b)ds≤λBfir1∫0+∞‍|Gi(t1,s)-Gi(t2,s)|ϕi(s)ds⟶0,t1⟶t2.
Therefore, Ai(u,v)∈C(ℝ+), and so F(u,v)∈C(ℝ+)×C(ℝ+). By the property (6) of Gi(t,s) and the Lebesgue dominated convergence theorem, we obtain
(3.5)lim⁡t→+∞Ai(u,v)(t)=λ∫0+∞‍G¯i(s)ϕi(s)fi(s,u(s),v(s),a,b)ds<+∞.
Hence, F:X→X is well defined.
(ii) Next we show thatF:X→X is continuous. Let (un,vn),(u,v)∈X,n∈ℕ, and ∥(un,vn)-(u,v)∥→0(n→+∞), we will prove that ∥F(un,vn)-F(u,v)∥→0 (n→+∞). By (2.10), (H1), and (H2), for any t∈ℝ+, a∈[0,a0],b∈[0,b0] and any natural number n, we have
(3.6)|Ai(un,vn)(t)-Ai(u,v)(t)|≤λ∫0+∞‍Gi(s,s)ϕi(s)(∫0+∞‍|fi(s,un(s),vn(s),a,b)|+|fi(s,u(s),v(s),a,b)|)ds≤2λBfir2∫0+∞‍Gi(s,s)ϕi(s)ds<+∞,i=1,2,
where Bfir2=sup{fi(t,x,y,a,b):t∈ℝ+,x∈[0,r2],y∈[0,r2],a∈[0,a0],b∈[0,b0]}<+∞ by (H1), r2 is a real number such that r2≥maxn∈ℕ{∥(u,v)∥,∥(un,vn)∥}, in which ℕ is the natural number set.
For anyε>0, by (H2), there exists a sufficiently large T0>0 such that
(3.7)2λBfir2∫T0+∞‍Gi(s,s)ϕi(s)ds<ε2,i=1,2.
On the other hand, by the continuity of fi(t,x,y,a,b) on [0,T0]×[0,r2]×[0,r2]×[0,a0]×[0,b0], for the above ε>0, there exists a δ>0 such that for any s∈[0,T0], a∈[0,a0],b∈[0,b0] and x,x′,y,y′∈[0,r2], when |x-x′|<δ, |y-y′|<δ, we have
(3.8)|fi(s,x,y,a,b)-fi(s,x′,y′,a,b)|<ε2(λ∫0+∞‍Gi(s,s)ϕi(s)ds)-1,i=1,2.
From ∥(un,vn)-(u,v)∥→0 (n→+∞) and the definition of the norm ∥·∥ in the space X, we can easily conclude that ∥un-u∥∞→0, ∥vn-v∥∞→0(n→+∞). So, for the above δ>0, there exists a sufficiently large natural number N0 such that, when n>N0, for any s∈[0,T0], we have
(3.9)|un(s)-u(s)|≤‖un-u‖∞<δ,|vn(s)-v(s)|≤‖vn-v‖∞<δ.
Hence, by (3.8), when n>N0, for any s∈[0,T0], a∈[0,a0],b∈[0,b0], we get
(3.10)|fi(s,un(s),vn(s),a,b)-fi(s,u(s),v(s),a,b)|<ε2(λ∫0+∞‍Gi(s,s)ϕi(s)ds)-1.
Therefore, by (2.10), (3.7), and (3.10), when n>N0, for any t∈ℝ+, a∈[0,a0] and b∈[0,b0], we obtain
(3.11)|Ai(un,vn)(t)-Ai(u,v)(t)|≤λ∫0+∞‍Gi(s,s)ϕi(s)|fi(s,un(s),vn(s),a,b)-fi(s,u(s),v(s),a,b)|ds≤λ∫0T0‍Gi(s,s)ϕi(s)|fi(s,un(s),vn(s),a,b)-fi(s,u(s),v(s),a,b)|ds+2λBfir2∫T0+∞‍Gi(s,s)ϕi(s)ds<ε,i=1,2.
This implies that the operator Ai:X→E is continuous. Therefore, the operator F:X→X is continuous.
(iii) We need to prove that the operatorF:X→X is compact. Let D be any bounded subset of X. Then, there exists a constant r3>0 such that ∥(u,v)∥≤r3 for any (u,v)∈D. So ∥u∥∞≤r3, ∥v∥∞≤r3 for any (u,v)∈D. By (2.10), (H1), and (H2), for any (u,v)∈D and t∈ℝ+, we have
(3.12)|F(u,v)(t)|=|A1(u,v)(t)|+|A2(u,v)(t)|=|λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,u(s),v(s),a,b)ds|+|λ∫0+∞‍G2(t,s)ϕ2(s)f2(s,u(s),v(s),a,b)ds|≤λBf1r3∫0+∞‍G1(s,s)ϕ1(s)ds+λBf2r3∫0+∞‍G2(s,s)ϕ2(s)ds<+∞,
where Bfir3=sup{fi(t,x,y,a,b):t∈ℝ+,x∈[0,r3],y∈[0,r3],a∈[0,a0],b∈[0,b0]}<+∞ by (H1). Hence, F(D) is uniformly bounded. By the similar proof as for (3.4), we can conclude that Ai(D) is equicontinuous, and so F(D) is also equicontinuous.
From(H2) and the property (6) of the Green function Gi(t,s), for any t∈ℝ+, we have
(3.13)∫0+∞‍|Gi(t,s)-G¯i(s)|ϕi(s)ds≤2∫0+∞‍Gi(s,s)ϕi(s)ds<+∞,i=1,2.
By (2.10), (3.5), and the Lebesgue dominated convergence theorem, for any (u,v)∈D, t∈ℝ+, a∈[0,a0], and b∈[0,b0], we obtain
(3.14)|Ai(u,v)(t)-Ai(u,v)(+∞)|=|λ∫0+∞‍Gi(t,s)ϕi(s)fi(s,u(s),v(s),a,b)ds-λ∫0+∞‍G¯i(s)ϕi(s)fi(s,u(s),v(s),a,b)ds|≤λ∫0+∞‍|Gi(t,s)-G¯i(s)|ϕi(s)fi(s,u(s),v(s),a,b)ds≤λBfir3∫0+∞‍|Gi(t,s)-G¯i(s)|ϕi(s)ds⟶0,t⟶+∞,i=1,2.
This implies that Ai(D) is equiconvergent at +∞. Hence, F(D) is equiconvergent at +∞. Therefore, the above discussion and Lemma 2.4 imply that F:X→X is completely continuous.
(iv) Finally, we proveF(K)⊆K. By the property (1) of Gi(t,s), (H1), and (H2), it is easy to see that, for any (u,v)∈K and t∈ℝ+, Ai(u,v)(t)≥0 and
(3.15)Ai(u,v)(t)=λ∫0+∞‍Gi(t,s)ϕi(s)fi(s,u(s),v(s),a,b)ds≤λ∫0+∞‍Gi(s,s)ϕi(s)fi(s,u(s),v(s),a,b)ds.
So
(3.16)‖Ai(u,v)‖∞≤λ∫0+∞‍Gi(s,s)ϕi(s)fi(s,u(s),v(s),a,b)ds,i=1,2.
On the other hand, by the property (7) of Gi(t,s), we have
(3.17)Ai(u,v)(t)≥ωλ∫0+∞‍Gi(s,s)ϕi(s)fi(s,u(s),v(s),a,b)ds,t∈[δ,1δ],i=1,2.
It follows from (3.16) and (3.17) that
(3.18)Ai(u,v)(t)≥ω‖Ai(u,v)‖∞,t∈[δ,1δ],i=1,2.
Therefore F(K)⊆K. The proof of Lemma 3.1 is completed.


## 3.2. The Positive Solution for System(Pa0,b0)

Theorem 3.2.
Assume that(H1)–(H3) hold, then the system (Pa0,b0) has at least one positive solution for any
(3.19)λ∈(lmin⁡(f1∞,f2∞),Lmax⁡(f10,f20)).Proof.
From (3.19), there exists ε0>0 such that
(3.20)lmin⁡(f1∞,f2∞)-ε0≤λ≤Lmax⁡(f10,f20)+ε0.
By the first inequality of (H3), there exists r>0 such that
(3.21)fi(t,x,y,a0,b0)≤(fi0+ε0)r,0≤x+y≤r,t∈R+,i=1,2.
Setting Kr1={(u,v)∈K:∥(u,v)∥<r1}(r1≤r), by the definition of ∥·∥, we know that
(3.22)u(t)+v(t)≤‖u‖∞+‖v‖∞=‖(u,v)‖=r1≤r,∀(u,v)∈∂Kr1,t∈R+.
Then, for any (u,v)∈∂Kr1,
(3.23)‖Ai(u,v)‖∞=λsup⁡t∈R+|∫0+∞‍Gi(t,s)ϕi(s)fi(s,u(s),v(s),a0,b0)ds|≤λsup⁡t∈R+∫0+∞‍Gi(t,s)ϕi(s)(fi0+ε0)r1ds≤λ(fi0+ε0)r1∫0+∞‍Gi(s,s)ϕi(s)ds≤r1L∫0+∞‍Gi(s,s)ϕi(s)ds≤12r1=12‖(u,v)‖,i=1,2.
Thus,
(3.24)‖F(u,v)‖=‖A1(u,v)‖∞+‖A2(u,v)‖∞≤‖(u,v)‖,(u,v)∈∂Kr1.
On the other hand, by the second inequality of(H3), there exists r0>ωr1>0 such that
(3.25)fi(t,x,y,a0,b0)≥fi(t,x,y,0,0)≥(fi∞-ε0)(x+y),x+y≥r0,t∈[δ,1δ],i=1,2.
Take r2=r0/ω>r1, and let Kr2={(u,v)∈K:∥(u,v)∥<r2}, (u0,v0)=(1/2,1/2)∈∂K1. Then,
(3.26)(u,v)≠F(u,v)+μ(u0,v0),∀(u,v)∈∂Kr2,∀μ>0.
Suppose that (3.26) is false, then there exist (u2,v2)∈∂Kr2 and μ2>0 such that (u2,v2)=F(u2,v2)+μ2(u0,v0). From (3.25) and the fact that
(3.27)u2(t)+v2(t)≥ω‖u2‖∞+ω‖v2‖∞=ω‖(u2,v2)‖=ωr2=r0,t∈[δ,1δ],
we have
(3.28)fi(t,u2(t),v2(t),a0,b0)≥(fi∞-ε0)(u2(t)+v2(t)),t∈[δ,1δ],i=1,2.
Set
(3.29)ξ=min⁡{u2(t)+v2(t):t∈[δ,1δ]}.
Then, u2(t)+v2(t)≥ξ>0 for any t∈[δ,1/δ]. Hence, for any t∈[δ,1/δ], by (3.28), we have
(3.30)u2(t)+v2(t)=A1(u2(t),v2(t))+A2(u2(t),v2(t))+μ2(u0+v0)=λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,u2(s),v2(s),a0,b0)ds+λ∫0+∞‍G2(t,s)ϕ2(s)f2(s,u2(s),v2(s),a0,b0)ds+μ2(u0+v0)≥λ∫δ1/δ‍G1(t,s)ϕ1(s)(f1∞-ε0)(u2(s)+v2(s))ds+λ∫δ1/δ‍G2(t,s)ϕ2(s)(f2∞-ε0)(u2(s)+v2(s))ds+μ2≥min⁡s∈[δ,1/δ](u2(s)+v2(s))λ(f1∞-ε0)min⁡t∈[δ,1/δ]∫δ1/δ‍G1(t,s)ϕ1(s)ds+min⁡s∈[δ,1/δ](u2(s)+v2(s))λ(f2∞-ε0)min⁡t∈[δ,1/δ]∫δ1/δ‍G2(t,s)ϕ2(s)ds+μ2≥ξ+μ2.
Then, we can obtain that
(3.31)u2(t)+v2(t)>ξ+μ2,t∈[δ,1δ].
It is clearly that (3.31) contradicts (3.29), which implies that (3.26) holds.
It follows from (3.24), (3.26), Lemmas 2.5 and 3.1 that the operator F has fixed-point (u,v)∈Kr2∖K-r1 such that 0<r1<∥(u,v)∥<r2. It is easy to see that (u,v) is a positive solution of the system (Pa0,b0). The proof of Theorem 3.2 is completed.Remark 3.3.
Noticing thatl/f1∞<1, l/f2∞<1 and L/f10>1, L/f20>1, we conclude that Theorem 3.2 also holds for λ=1.Remark 3.4.
From Theorem3.2, we can see that fi(t,x,y,a0,b0)(i=1,2) do not need to be superlinear or sublinear. In fact, Theorem 3.2 still holds, if (H1)-(H2) are satisfied and one of the following conditions is satisfied:(1)
f1∞=f2∞=+∞,L>f10>0, L>f20>0, for each λ∈(0,L/max{f10,f20}),(2)
f1∞=f2∞=+∞, f10=f20=0, for each λ∈(0,+∞),(3)
f1∞>l, f2∞>l, f10=f20=0, for each λ∈(l/min{f1∞,f2∞},+∞).


## 3.3.  Lower and Upper Solutions

Theorem 3.5.
Assume that(H1)-(H2) hold. Let (u_,v_) and (u¯,v¯) be a lower solution and an upper solution, respectively, of the system (S) such that (0,0)≤(u_,v_)≤(u¯,v¯). Then, the system (S) has a nonnegative solution (u,v) satisfying (u_,v_)≤(u,v)≤(u¯,v¯).Proof.
Let(3.32)Mi(u,v)(t)=λ∫0+∞‍Gi(t,s)ϕi(s)gi(s,u(s),v(s))ds,i=1,2,H(u,v)=(M1(u,v),M2(u,v)).
Then, the solutions of the system (S) are equivalent to the fixed points of the operator H in K.
Now, we introduce the following auxiliary operatorℋ defined by
(3.33)H(u,v)=(M1(u,v),M2(u,v)),
where
(3.34)Mi(u,v)(t)=λ∫0+∞‍Gi(t,s)ϕi(s)gi(s,ζ1(s,u),ζ2(s,v))ds,i=1,2,
in which
(3.35)ζ1(t,u)=max⁡{u_(t),min⁡{u(t),u¯(t)}},ζ2(t,v)=max⁡{v_(t),min⁡{v(t),v¯(t)}}.
It is easy to prove that the operator ℋ has the following properties.(1)
ℋ is a completely continuous operator.(2)
If the pair(u,v)∈K is a fixed point of ℋ, then (u,v) is a fixed point of H with (u_,v_)≤(u,v)≤(u¯,v¯).(3)
If(u,v)=μℋ(u,v) with 0≤μ≤1, then ∥(u,v)∥≤C**, where C** does not depend on μ and (u,v)∈K.Therefore, by using the topological degree of Leray-Schauder (see [26, Corollary  8.1, page. 61]), we obtain a fixed point of the operator H. The proof of Theorem 3.5 is completed.Theorem 3.6.
Assume that(H1)–(H3) and (3.19) hold. Then for (0,0)≤(a,b)≤(a0,b0), the system (Pa,b) has at least one positive solution.Proof.
From Theorem3.2, we can see that the system (Pa0,b0) has at least one positive solution (u0,v0). Since the functions fi(i=1,2) are increasing functions with respect to the last four variables, we conclude that (u0,v0) is an upper solution and (0,0) is a lower solution for the system (Pa,b). Hence, by Theorem 3.5, we have that the system (Pa,b) has at least one positive solution. This completes the proof of Theorem 3.6.


## 3.4. A Priori Estimate

Theorem 3.7.
Assume that(H1)–(H3) and (3.19) hold. Then for any (a,b) satisfying (0,0)≤(a,b)≤(a0,b0), there exists a constant C0>0 independent of (a,b), such that ∥(u,v)∥≤C0 for all positive solutions (u,v) of the system (Pa,b).Proof.
Assume by contradiction that there exists a sequence of positive solution(un,vn)∈X of system (Pa,b) such that ∥(un,vn)∥→∞. From (3.19), there exists ε0>0 such that λ>l/(min(f1∞,f2∞)-ε0). From assumption (H3), there exists H>0 such that, for (0,0)≤(a,b)≤(a0,b0),
(3.36)fi(t,x,y,a,b)≥fi(t,x,y,0,0)≥(fi∞-ε0)(x+y),x+y≥H,t∈[δ,1δ],i=1,2.
Since un(t)+vn(t)≥ω∥un∥∞+ω∥vn∥∞=ω∥(un,vn)∥,t∈[δ,1/δ], there exists natural number N0 such that, for n>N0, we have un(t)+vn(t)≥H for t∈[δ,1/δ]. It follows from (3.36) that when n>N0, fi(t,un(t),vn(t),a,b)≥(fi∞-ε0)(un(t)+vn(t)),t∈[δ,1/δ],i=1,2. Thus,
(3.37)un(t)+vn(t)=λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,un(s),vn(s),a,b)ds+λ∫0+∞‍G2(t,s)ϕ2(s)f2(s,un(s),vn(s),a,b)ds≥λ∫δ1/δ‍G1(t,s)ϕ1(s)(f1∞-ε0)(un(s)+vn(s))ds+λ∫δ1/δ‍G2(t,s)ϕ2(s)(f2∞-ε0)(un(s)+vn(s))ds≥min⁡s∈[δ,1/δ](un(s)+vn(s))λ(f1∞-ε0)min⁡t∈[δ,1/δ]∫δ1/δ‍G1(t,s)ϕ1(s)ds+min⁡s∈[δ,1/δ](un(s)+vn(s))λ(f2∞-ε0)min⁡t∈[δ,1/δ]∫δ1/δ‍G2(t,s)ϕ2(s)ds>min⁡s∈[δ,1/δ](un(s)+vn(s)),t∈[δ,1δ]
which yields
(3.38)min⁡t∈[δ,1/δ]un(t)+vn(t)>min⁡t∈[δ,1/δ](un(t)+vn(t)),
which is a contradiction. This completes the proof of Theorem 3.7.Remark 3.8.
By the discussions in Sections3.1, 3.2, 3.3 and 3.4, we can conclude that, for any (a,b) satisfying |(0,0)|≤|(a,b)|≤|(a0,b0)|, the system (Pa,b) has at least one positive solution (u,v) with ∥(u,v)∥≤C0. In the following section, we will establish the nonexistence result for the system (Pa,b).


## 3.5.  Non-Existence

Theorem 3.9.
Assume that(H1)–(H4) and (3.19) hold. Then, there exist a_>0,b_>0 such that, for all (a,b) with |(a,b)|>|(a_,b_)|, the system (Pa,b) has no solution.Proof.
Suppose by contradiction that there exists a sequence(an,bn) with |(an,bn)|→+∞(n→+∞) such that, for each nature number n, the system (Pan,bn) has a positive solution (un,vn) in K. From assumption (H4), for any M>0, there exists a constant C>0 such that, for any |(a,b)|>C,
(3.39)fi(t,x,y,a,b)≥M,x,y≥0,t∈[δ,1δ],i=1,2.
By |(an,bn)|→+∞(n→+∞), for the above C>0, there exists a natural number n0 such that, for n>n0,|(an,bn)|>C, then, for n>n0 and t∈[δ,1/δ],
(3.40)un(t)=λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,un(s),vn(s),an,bn)ds>λ∫δ1/δ‍G1(t,s)ϕ1(s)f1(s,un(s),vn(s),an,bn)ds≥Mλω∫δ1/δ‍G1(s,s)ϕ1(s)ds.
By the same way, we can obtain
(3.41)vn(t)≥Mλω∫δ1/δ‍G2(s,s)ϕ2(s)ds.
Since we can choose M arbitrarily large, we conclude that un and vn are unbounded sequences in K, then limt→+∞un(t) and limt→+∞vn(t) do not exist, which contradicts the fact that un,vn∈K. So Theorem 3.9 holds.We next define the set(3.42)A={a>0:thesystem(Pa,b)hasapositivesolutionforsomeb>0}.
From Theorems 3.2 and 3.9, we conclude that A is nonempty and bounded. Thus, 0<a~=supA<+∞. Using the upper-lower solutions method, we see that, for all a∈(0,a~), there exists b>0 such that the system (Pa,b) has a positive solution. We now define the function Γ:[0,a~]→ℝ+ by
(3.43)Γ(a)=sup⁡{b>0:system(Pa,b)hasapositivesolution}.
By Theorem 3.6, the function Γ is continuous and nonincreasing. We thus claim that Γ(a) is attained. In fact, it suffices to use Theorem 3.7 and the compactness of the operator F. Finally, it follows from the definition of the function Γ that the system (Pa,b) has at least one positive solution for 0≤b≤Γ(a) and has no positive solutions for b>Γ(a).


## 3.6.  Existence of Two Positive Solutions

In this section,we will assume that the nonlinearitiesf1 and f2 are strict-increasing with respect to the fifth variable. Fix a∈[0,a~], and let (ϕ¯,ψ¯) be the solution of the problem (Pa,Γ(a)) which is obtained using Theorem 3.6. Our next result allows us to establish another solution of the system (Pa,b)  for 0<b<Γ(a).Lemma 3.10.
For each0<b<Γ(a), there exists ϵ0>0 so that, for all 0<ϵ≤ϵ0 and all t∈ℝ+, one has (3.44)ϕ¯ϵ(t)>λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,ϕ¯ϵ(s),ψ¯ϵ(s),a,b)ds,ψ¯ϵ(t)>λ∫0+∞‍G2(t,s)ϕ2(s)f2(s,ϕ¯ϵ(s),ψ¯ϵ(s),a,b)ds,
where ϕ¯ϵ(t)=ϕ¯(t)+ϵ,ψ¯ϵ(t)=ψ¯(t)+ϵ.Proof.
Fixδ∈(0,1). Since f1 is strict-increasing with respect to the fifth variable, we have that for each 0<b<Γ(a) we may find a positive constant I=I(b) so that, for all s∈[δ,1/δ], we have
(3.45)f1(s,ϕ¯(s),ψ¯(s),a,Γ(a))-f1(s,ϕ¯(s),ψ¯(s),a,b)≥I>0.
By the uniform continuity of f1, there exists ϵ0>0 so that, for all s∈[δ,1/δ] and all 0<ϵ≤ϵ0, we have
(3.46)|f1(s,ϕ¯(s)+ϵ,ψ¯(s)+ϵ,a,b)-f1(s,ϕ¯(s),ψ¯(s),a,b)|≤I2.
Next, we define
(3.47)Φϵ(t,s)=G1(t,s)ϕ1(s)[f1(s,ϕ¯ϵ(s),ψ¯ϵ(s),a,b)-f1(s,ϕ¯(s),ψ¯(s),a,b)],Ψ(t,s)=G1(t,s)ϕ1(s)[f1(s,ϕ¯(s),ψ¯(s),a,Γ(a))-f1(s,ϕ¯(s),ψ¯(s),a,b)].
Assume 0<ϵ≤ϵ0. Then,
(3.48)ϕ¯ϵ(t)>ϕ¯(t)=λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,ϕ¯(s),ψ¯(s),a,Γ(a))ds=λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,ϕ¯ϵ(s),ψ¯ε(s),a,b)ds-λ∫0+∞‍Φϵ(t,s)ds+λ∫0+∞‍Ψ(t,s)ds.
Since Ψ(t,s) is positive and Ψ(t,s)-Φϵ(t,s)>(I/2)G1(t,s)ϕ1(s) for t∈[δ,1/δ], we have
(3.49)ϕ¯ϵ(t)>λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,ϕ¯ϵ(s),ψ¯ϵ(s),a,b)ds-λ∫0δ‍Φϵ(t,s)ds-λ∫1/δ+∞‍Φϵ(t,s)ds+I2λ∫δ1/δ‍G1(t,s)ϕ1(s)ds.
It is not difficult to show that the Lebesgue dominated convergence theorem implies thatλ∫0δ‍Φϵ(t,s)ds+λ∫1/δ+∞‍Φϵ(t,s)ds converges to zero, uniformly in t, as ϵ tends to zero. Thus, for ϵ sufficiently small, we have
(3.50)ϕ¯ϵ(t)>λ∫0+∞‍G1(t,s)ϕ1(s)f1(s,ϕ¯ϵ(s),ψ¯ϵ(s),a,b)ds
uniformly in t∈ℝ+.
A similar computation holds forψ¯ϵ.We are now in a position to show existence of two positive solutions of the system(Pa,b) for 0<b<Γ(a), where a∈[0,a~] is fixed.Theorem 3.11.
Assume that(H1)–(H4) and (3.19) hold. Then, for all a∈[0,a~], the system (Pa,b) has at least two positive solutions for 0<b<Γ(a).Proof.
Consider the set(3.51)Ω={(ϕ,ψ)∈X:-ϵ<ϕ(t)<ϕ¯ϵ(t),-ϵ<ψ(t)<ψ¯ϵ(t),fort∈R+},
where ϕ¯ϵ and ψ¯ϵ are the functions of Lemma 3.10. It is not hard to see that Ω is bounded and open in X and that (0,0)∈Ω. Note that one of the solutions of the system (Pa,b) belongs to K∩Ω¯, we also know that F:K∩Ω¯→K is a completely continuous operator.
Let(ϕ,ψ)∈K∩∂Ω. It follows that there exists t0∈(0,+∞) so that one of the following two cases holds: ϕ(t0)=ϕ¯ϵ(t0) or ψ(t0)=ψ¯ϵ(t0). In the case ϕ(t0)=ϕ¯ϵ(t0), it follows from Lemma 3.10 that, for all μ≥1, we have(3.52)A1(ϕ,ψ)(t0)=λ∫0+∞‍G1(t0,s)ϕ1(s)f1(s,ϕ(s),ψ(s),a,b)ds≤λ∫0+∞‍G1(t0,s)ϕ1(s)f1(s,ϕ¯ϵ(s),ψ¯ϵ(s),a,b)ds<ϕ¯ϵ(t0)=ϕ(t0)≤μϕ(t0).
Similarly, A2(ϕ,ψ)(t0)<μψ(t0) in the case ψ(t0)=ψ¯ϵ(t0). Hence, F(ϕ,ψ)≠μ(ϕ,ψ), for all (ϕ,ψ)∈K∩∂Ω and all μ≥1. Now, according to Lemma 2.7, we have
(3.53)i(F,K∩Ω,K)=1.
On the other hand, a slight change in the proof of Theorem3.7 shows the existence of an r3>0 sufficiently large, say r3>r2, where r2 is as in Theorem 3.2, so that(3.54)‖F(ϕ,ψ)‖>‖(ϕ,ψ)‖
for every ∥(ϕ,ψ)∥=r3 and every (ϕ,ψ)∈K.
Letr4=max{C0+1,r3,∥(ϕ¯ϵ,ψ¯ϵ)∥}, where C0 is as that in Theorem 3.7. Set
(3.55)Kr4={(ϕ,ψ)∈K:‖(ϕ,ψ)‖<r4}.
Then Theorem 3.7 implies that F(ϕ,ψ)≠(ϕ,ψ), for (ϕ,ψ)∈∂Kr4. Consequently, Lemma 2.8 implies i(F,Kr4,K)=0.
Now, by the additivity property of the fixed-point index, we obtain(3.56)i(F,Kr4∖K∩Ω¯,K)=-1.
Therefore, F has another fixed point in Kr4∖K∩Ω¯.Then, from the above discussion, it is easy to obtain the following conclusion.


## 4.  Conclusion

Assume that (H1)–(H4) and (3.19) hold. Then, there exist a constant a~>0 and a non-increasing continuous function Γ:[0,a~]→ℝ+ so that, for all a∈[0,a~], the system (Pa,b) has at least one positive solution for 0≤b≤Γ(a), has no positive solutions for b>Γ(a), and has at least two positive solutions for 0<b<Γ(a) when f1 and f2 are strict-increasing with respect to the fifth variable.



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*Source: 161925-2012-08-15.xml*