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

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.

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: −𝑝₁(𝑡)𝑢^′(𝑡)^′=𝜆𝜙₁(𝑡)𝑓₁((𝑡)(𝑡))−𝑝𝑡,𝑢,𝑣,𝑎,𝑏,0<𝑡<+∞,₂(𝑡)𝑣^′(𝑡)^′=𝜆𝜙₂(𝑡)𝑓₂((𝑡)(𝑡))𝛼𝑡,𝑢,𝑣,𝑎,𝑏,0<𝑡<+∞,₁₁𝑢(0)−𝛽₁₁lim_𝑡→0+𝑝₁(𝑡)𝑢^′(𝑡)𝛼=0,₁₂lim_𝑡→+∞𝑢(𝑡)+𝛽₁₂lim_𝑡→+∞𝑝₁(𝑡)𝑢^′(𝑡)𝛼=0,₂₁𝑣(0)−𝛽₂₁lim_𝑡→0+𝑝₂(𝑡)𝑣^′(𝑡)𝛼=0,₂₂lim_𝑡→+∞𝑣(𝑡)+𝛽₂₂lim_𝑡→+∞𝑝₂(𝑡)𝑣^′(𝑡)=0, 𝑃_𝑎,𝑏 where 𝜆>0 is a parameter, 𝑎≥0,𝑏≥0 are constants; 𝑓₁,𝑓₂∶ℝ⁵₊→ℝ₊ are continuous, 𝜙₁,𝜙₂∶(0,+∞)→ℝ₊ are continuous and may have singularity at 𝑡=0; 𝑝_𝑖∈𝐶(ℝ₊)∩𝐶¹(0,+∞) with 𝑝_𝑖(𝑡)>0 on (0,+∞) and ∫₀^+∞(1/𝑝_𝑖(𝑠))𝑑𝑠<+∞(𝑖=1,2); 𝛼_𝑖𝑗,𝛽_𝑖𝑗≥0(𝑖,𝑗=1,2) with 𝜌_𝑖=𝛼_𝑖2𝛽_𝑖1+𝛼_𝑖1𝛽_𝑖2+𝛼_𝑖1𝛼_𝑖2∫₀^+∞(1/𝑝_𝑖(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 𝑝(𝑡)𝑥^′(𝑡)^′(𝑡)𝑓((𝑡𝛼+𝜆𝜙𝑡,𝑥))=0,0<𝑡<+∞,₁𝑥(0)−𝛽₁lim_𝑡→0+𝑝(𝑡)𝑥^′(𝑡)𝛼=0,₂lim_𝑡→+∞𝑥(𝑡)+𝛽₂lim_𝑡→+∞𝑝(𝑡)𝑥^′(𝑡)=0, (1.1) 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 𝑝(𝑡)𝑥^′(𝑡)^′()(𝑡)𝛼+𝑓𝑡,𝑥+𝑞=0,0<𝑡<+∞,₁𝑥(0)−𝛽₁lim_𝑡→0+𝑝(𝑡)𝑥^′(𝑡)𝛼=0,₂lim_𝑡→+∞𝑥(𝑡)+𝛽₂lim_𝑡→+∞𝑝(𝑡)𝑥^′(𝑡)=0, (1.2) 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 𝑃_𝑎,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]: −𝑢^′′(𝑡)=𝑓(𝑡,𝑢,𝑣,𝑎,𝑏),0<𝑡<1,−𝑣^′′(𝑡)()𝑢(0)(1)𝑣(0)(1)=𝑔𝑡,𝑢,𝑣,𝑎,𝑏,0<𝑡<1,=𝑢=0,=𝑣=0, (1.3) where 𝑓,𝑔∶[0,1]×ℝ⁴₊→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 𝑃_𝑎,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 𝑃_𝑎,𝑏. For this purpose, we solve the fixed point of an operator 𝐹 instead of the positive solutions for the system 𝑃_𝑎,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 𝑃_𝑎,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 Section 2, 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 space 𝑋=𝐸×𝐸 will be the basic space to study 𝑃_𝑎,𝑏, where the Banach space 𝐸 is denoted by 𝐸={𝑢∈𝐶(ℝ₊)∶lim_𝑡→+∞𝑢(𝑡)exists} with the supremum norm ‖𝑢‖_∞=sup_𝑡∈ℝ+|𝑢(𝑡)|. Clearly, (𝑋,‖⋅‖) is a Banach space with the norm ‖(𝑢,𝑣)‖=‖𝑢‖_∞+‖𝑣‖_∞ for (𝑢,𝑣)∈𝑋. For convenience, let 𝑎_𝑖=𝛽_𝑖1+𝛼_𝑖10^+∞1𝑝_𝑖(𝑠)𝑑𝑠,𝑏_𝑖=𝛽_𝑖2+𝛼_𝑖20^+∞1𝑝_𝑖(𝑠)𝑑𝑠,𝑖=1,2,𝑎_𝑖(𝑡)=𝛽_𝑖1+𝛼_𝑖1^𝑡₀1𝑝_𝑖(𝑠)𝑑𝑠,𝑏_𝑖(𝑡)=𝛽_𝑖2+𝛼_𝑖2𝑡^+∞1𝑝_i(𝑠)𝑑𝑠,𝑖=1,2. (2.1) Then, it is obvious that 𝛼_𝑖2𝑎_𝑖(𝑡)+𝛼_𝑖1𝑏_𝑖(𝑡)=𝜌_𝑖(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 2.1 2.1(see [6]). Under the condition ∫₀^+∞(1/𝑝_𝑖(𝑠))𝑑𝑠<+∞ and 𝜌_𝑖>0 for 𝑖=1,2, the linear boundary value problem 𝑝_𝑖(𝑡)𝑥^′(𝑡)^′(𝑡)𝛼+𝑣=0,0<𝑡<+∞,_𝑖1𝑥(0)−𝛽_𝑖1lim_𝑡→0+𝑝_𝑖(𝑡)𝑥^′(𝑡)𝛼=0,_𝑖2lim_𝑡→+∞𝑥(𝑡)+𝛽_𝑖2lim_𝑡→+∞𝑝_𝑖(𝑡)𝑥^′(𝑡)=0,𝑖=1,2 (2.2) has a unique solution for any 𝑣∈𝐿¹(0,+∞). Moreover, this unique solution can be expressed in the form 𝑥_𝑖(𝑡)=₀^+∞𝐺_𝑖(𝑡,𝑠)𝑣(𝑠)𝑑𝑠,𝑖=1,2, (2.3) where the Green function 𝐺_𝑖(𝑡,𝑠) is defined by 𝐺_𝑖()=1𝑡,𝑠𝜌_𝑖𝑎_𝑖(𝑡)𝑏_𝑖(𝑠),0≤𝑡≤𝑠<+∞,𝑎_𝑖(𝑠)𝑏_𝑖(𝑡),0≤𝑠≤𝑡<+∞,𝑖=1,2. (2.4)

Remark 2.2. From (2.4), we can get the properties of 𝐺_𝑖(𝑡,𝑠) as follows.(1)𝐺_𝑖(𝑡,𝑠) is continuous and nonnegative on ℝ₊×ℝ₊.(2)For each 𝑠∈ℝ₊, 𝐺_𝑖(𝑡,𝑠) is continuously differentiable on ℝ₊ except for 𝑡=𝑠.(3)𝜕𝐺_𝑖(𝑡,𝑠)/𝜕𝑡|_𝑡=𝑠+−𝜕𝐺_𝑖(𝑡,𝑠)/𝜕𝑡|_𝑡=𝑠−=−1/𝑝_𝑖(𝑠).(4)For each 𝑠∈ℝ₊, 𝐺_𝑖(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, 𝐺_𝑖(𝑡,𝑠) is the Green function of BVP (2.2) on the half-line.(5)𝐺_𝑖(𝑡,𝑠)≤𝐺_𝑖(𝑠,𝑠)≤(1/𝜌_𝑖)𝑎_𝑖𝑏_𝑖<+∞,𝑖=1,2.(6)𝐺_𝑖(𝑠)=lim_𝑡→+∞𝐺_𝑖(𝑡,𝑠)=(1/𝜌_𝑖)𝛽_𝑖2𝑎_𝑖(𝑠)≤𝐺_𝑖(𝑠,𝑠)<+∞,𝑖=1,2.(7)For any 𝑡∈[𝛿,1/𝛿] and 𝑠∈ℝ₊,𝐺_𝑖(𝑡,𝑠)≥𝜔_𝑖𝐺_𝑖(𝑠,𝑠), where 𝛿∈(0,1),𝜔_𝑖=min𝑏_𝑖(1/𝛿)𝑏_𝑖,𝑎_𝑖(𝛿)𝑎_𝑖,𝑖=1,2,0<𝜔=min𝜔₁,𝜔₂<1. (2.5)
For any (𝑥₁,𝑥₂,…,𝑥_𝑚)∈ℝ^𝑚, define |𝑥₁,𝑥₂,…,𝑥_𝑚|=|𝑥₁|+|𝑥₂|+⋯|𝑥_𝑚|. In what follows, we list some conditions for convenience.(H₁) The function 𝑓_𝑖∶ℝ⁵₊→ℝ₊ is continuous and nondecreasing with respect to the last four variables. In other words, 𝑓_𝑖𝑡,𝑥₁,𝑦₁,𝑎₁,𝑏₁≤𝑓_𝑖𝑡,𝑥₂,𝑦₂,𝑎₂,𝑏₂,𝑖=1,2, (2.6) 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 𝑓_𝑖(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 function 𝜙_𝑖(𝑡)∶(0,+∞)→ℝ₊ is continuous and singular at 𝑡=0, 𝜙_𝑖(𝑡)≢0 on ℝ₊ satisfying 0<∫₀^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑑𝑠<+∞,𝑖=1,2.(H₃) For the above 𝑎₀ and 𝑏₀ in (H₁), 𝑓⁰_𝑖=limsup_{𝑥+𝑦→0+}sup_𝑡∈ℝ+𝑓_𝑖𝑡,𝑥,𝑦,𝑎₀,𝑏₀𝑓𝑥+𝑦<𝐿,_𝑖∞=liminf_{𝑥+𝑦→+∞}inf_{[]𝑡∈𝛿,1/𝛿}𝑓_𝑖𝑡,𝑥,𝑦,0,0𝑥+𝑦>𝑙,𝑖=1,2, (2.7) where 1𝐿=2max₀^+∞𝐺₁(𝑠,𝑠)𝜙₁(𝑠)𝑑𝑠,₀^+∞𝐺₂(𝑠,𝑠)𝜙₂(𝑠)𝑑𝑠⁻¹,1𝑙=2minmin_{[]𝑡∈𝛿,1/𝛿𝛿}^1/𝛿𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑑𝑠,min_{[]𝑡∈𝛿,1/𝛿𝛿}^1/𝛿𝐺₂(𝑡,𝑠)𝜙₂(𝑠)𝑑𝑠⁻¹. (2.8)(H₄)lim_{|(𝑎,𝑏)|→+∞}𝑓_𝑖1𝑡,𝑥,𝑦,𝑎,𝑏=+∞,uniformlyfor𝑡∈𝛿,𝛿,𝑥≥0,𝑦≥0,𝑖=1,2. (2.9)

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 𝑃_𝑎,𝑏 if and only if (𝑢,𝑣)∈𝑋 is a solution of the following system of nonlinear integral equations:𝑢(𝑡)=𝜆₀^+∞𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁((𝑠)(𝑠))𝑣(𝑡)𝑠,𝑢,𝑣,𝑎,𝑏𝑑𝑠,=𝜆₀^+∞𝐺₂(𝑡,𝑠)𝜙₂(𝑠)𝑓₂(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠,𝑡∈ℝ₊. 𝑆_𝑎,𝑏 Define operators 𝐴_𝑖∶𝑋→𝐸 and 𝐹∶𝑋→𝑋 as follows: 𝐴_𝑖(𝑢,𝑣)(𝑡)=𝜆₀^+∞𝐺_𝑖(𝑡,𝑠)𝜙_𝑖(𝑠)𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠,𝑖=1,2, (2.10)𝐹(𝑢,𝑣)=𝐴₁(𝑢,𝑣),𝐴₂(𝑢,𝑣). (2.11) Then, the solution of the system 𝑃_𝑎,𝑏 is equivalent to the fixed point of the operator 𝐹. Define a cone 𝐾 in the Banach space 𝑋 as follows: 𝐾=(𝑢,𝑣)∈𝑋∶𝑢(𝑡)≥0,𝑣(𝑡)≥0,𝑡∈ℝ₊,𝑢(𝑡)≥𝜔‖𝑢‖_∞,𝑣(𝑡)≥𝜔‖𝑣‖_∞1,𝑡∈𝛿,𝛿, (2.12) 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: −𝑝₁(𝑡)𝑢^′(𝑡)^′=𝜆𝜙₁(𝑡)𝑔₁((𝑡)(𝑡,−𝑝𝑡,𝑢,𝑣))₂(𝑡)𝑣^′(𝑡)^′=𝜆𝜙₂(𝑡)𝑔₂((𝑡)(𝑡𝛼𝑡,𝑢,𝑣)),0<𝑡<+∞,₁₁𝑢(0)−𝛽₁₁lim_𝑡→0+𝑝₁(𝑡)(𝑡)𝛼𝑢′=0,₁₂lim_𝑡→+∞𝑢(𝑡)+𝛽₁₂lim_𝑡→+∞𝑝₁(𝑡)(𝑡)𝛼𝑢′=0,₂₁𝑣(0)−𝛽₂₁lim_𝑡→0+𝑝₂(𝑡)(𝑡)𝛼𝑣′=0,₂₂lim_𝑡→+∞𝑣(𝑡)+𝛽₂₂lim_𝑡→+∞𝑝₂(𝑡)𝑣′(𝑡)=0, (𝑆) where 𝑔_𝑖∶ℝ³₊→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: −𝑝₁(𝑡)𝑢^′(𝑡)^′≤𝜆𝜙₁(𝑡)𝑔₁((𝑡)(𝑡,−𝑝𝑡,𝑢,𝑣))₂(𝑡)𝑣^′(𝑡)^′≤𝜆𝜙₂(𝑡)𝑔₂((𝑡)(𝑡𝛼𝑡,𝑢,𝑣)),0<𝑡<+∞,₁₁𝑢(0)−𝛽₁₁lim_𝑡→0+𝑝₁(𝑡)(𝑡)𝛼𝑢′≤0,₁₂lim_𝑡→+∞𝑢(𝑡)+𝛽₁₂lim_𝑡→+∞𝑝₁(𝑡)(𝑡)𝛼𝑢′≤0,₂₁𝑣(0)−𝛽₂₁lim_𝑡→0+𝑝₂(𝑡)(𝑡)𝛼𝑣′≤0,₂₂lim_𝑡→+∞𝑣(𝑡)+𝛽₂₂lim_𝑡→+∞𝑝₂(𝑡)𝑣′(𝑡)≤0. (𝑇) Similarly, we define the upper solution for the system (𝑆) by replacing the ≤ in (𝑇) by ≥.

Lemma 2.4 (see [1, 23]). Let 𝐸 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 in 𝑀 are equicontinuous on any bounded interval of ℝ₊.(3)The functions in 𝑀 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]). Let 𝑃 be a positive cone in a real Banach space (𝐸,‖⋅‖), 𝑃_𝑟={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∈𝜕𝑃_𝑅,(2)there exists a 𝑒∈𝜕𝑃₁ such that 𝑥≠𝐴𝑥+𝑚𝑒 for any 𝑥∈𝜕𝑃_𝑟 and 𝑚>0, then 𝐴 has fixed points in 𝑃_𝑟,𝑅.

Remark 2.6. If (1) and (2) are satisfied for 𝑥∈𝜕𝑃_𝑟 and 𝑥∈𝜕𝑃_𝑅, respectively, then Lemma 2.5 still holds.

Lemma 2.7 2.7 (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 𝑖(𝐴,𝐾∩Ω,𝐾)=1. (2.13)

Lemma 2.8 2.8 (see [25]). Let (𝐸,‖⋅‖) be a Banach space, 𝐾 be a cone in 𝐸. For 𝑟>0, define 𝐾_𝑟={𝑥∈𝐾∶‖𝑥‖<𝑟}. Suppose that 𝐴∶𝐾_𝑟→𝐾 is a completely continuous operator such that 𝐴𝑥≠𝑥 for 𝑥∈𝜕𝐾_𝑟.(i)If ‖𝐴𝑥‖≥‖𝑥‖ for 𝑥∈𝜕𝐾_𝑟, then 𝑖(𝐴,𝐾_𝑟,𝐾)=0.(ii)If ‖𝐴𝑥‖≤‖𝑥‖ for 𝑥∈𝜕𝐾_𝑟, then 𝑖(𝐴,𝐾_𝑟,𝐾)=1.

3. Main Results

3.1. The Complete Continuity of the Operator 𝐹

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 that 𝐹∶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 𝐺_𝑖(𝑡,𝑠) that 𝐴_𝑖(𝑢,𝑣)≥0 and 𝐵^𝑟1_𝑓𝑖=sup𝑓_𝑖𝑡,𝑥,𝑦,𝑎,𝑏∶𝑡∈ℝ₊,𝑥∈0,𝑟₁,𝑦∈0,𝑟₁,𝑎∈0,𝑎₀,𝑏∈0,𝑏₀<+∞. (3.1)
Thus, by (H₁) and (H_𝟐), for any 𝑡∈ℝ₊,𝑎∈[0,𝑎₀] and 𝑏∈[0,𝑏₀], we obtain 0≤𝜆₀^+∞𝐺_𝑖(𝑡,𝑠)𝜙_𝑖(𝑠)𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠≤𝜆𝐵^𝑟1_𝑓𝑖0^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑑𝑠<+∞. (3.2) Hence, the operator 𝐹(𝑢,𝑣)=(𝐴₁(𝑢,𝑣),𝐴₂(𝑢,𝑣)) is well defined for any (𝑢,𝑣)∈𝑋.
For any 𝑡₁,𝑡₂,𝑠∈ℝ₊, by the property (5) of the Green function 𝐺_𝑖(𝑡,𝑠), we have 𝐺_𝑖𝑡₁,𝑠−𝐺_𝑖𝑡₂,𝑠𝜙_𝑖(𝑠)≤2𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠),𝑖=1,2. (3.3) So, by (H₂), the Lebesgue dominated convergence theorem and the continuity of 𝐺_𝑖(𝑡,𝑠), for any 𝑡₁,𝑡₂∈ℝ₊, 𝑎∈[0,𝑎₀],𝑏∈[0,𝑏₀], we get 𝐴_𝑖(𝑢,𝑣)𝑡₁−𝐴_𝑖(𝑢,𝑣)𝑡₂≤𝜆₀^+∞𝐺_𝑖𝑡₁,𝑠−𝐺_𝑖𝑡₂,𝑠𝜙_𝑖(𝑠)𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠≤𝜆𝐵^𝑟1_𝑓𝑖0^+∞𝐺_𝑖𝑡₁,𝑠−𝐺_𝑖𝑡₂,𝑠𝜙_𝑖(𝑠)𝑑𝑠⟶0,𝑡₁⟶𝑡₂. (3.4) Therefore, 𝐴_𝑖(𝑢,𝑣)∈𝐶(ℝ₊), and so 𝐹(𝑢,𝑣)∈𝐶(ℝ₊)×𝐶(ℝ₊). By the property (6) of 𝐺_𝑖(𝑡,𝑠) and the Lebesgue dominated convergence theorem, we obtain lim_𝑡→+∞𝐴_𝑖(𝑢,𝑣)(𝑡)=𝜆₀^+∞𝐺_𝑖(𝑠)𝜙_𝑖(𝑠)𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠<+∞. (3.5) Hence, 𝐹∶𝑋→𝑋 is well defined.
(ii) Next we show that 𝐹∶𝑋→𝑋 is continuous. Let (𝑢_𝑛,𝑣_𝑛),(𝑢,𝑣)∈𝑋,𝑛∈ℕ, and ‖(𝑢_𝑛,𝑣_𝑛)−(𝑢,𝑣)‖→0(𝑛→+∞), we will prove that ‖𝐹(𝑢_𝑛,𝑣_𝑛)−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 𝐴_𝑖𝑢_𝑛,𝑣_𝑛(𝑡)−𝐴_𝑖(𝑢,𝑣)(𝑡)≤𝜆₀^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑓_𝑖𝑠,𝑢_𝑛(𝑠),𝑣_𝑛(𝑠),𝑎,𝑏+𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠≤2𝜆𝐵^𝑟2_𝑓𝑖0^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑑𝑠<+∞,𝑖=1,2, (3.6) where 𝐵^𝑟2_𝑓𝑖=sup{𝑓_𝑖(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≥max_𝑛∈ℕ{‖(𝑢,𝑣)‖,‖(𝑢_𝑛,𝑣_𝑛)‖}, in which ℕ is the natural number set.
For any 𝜀>0, by (H₂), there exists a sufficiently large 𝑇₀>0 such that 2𝜆𝐵^𝑟2_𝑓𝑖𝑇^+∞₀𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝜀𝑑𝑠<2,𝑖=1,2. (3.7) On the other hand, by the continuity of 𝑓_𝑖(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 𝑓_𝑖𝑠,𝑥,𝑦,𝑎,𝑏−𝑓_𝑖𝑠,𝑥^′,𝑦^′<𝜀,𝑎,𝑏2𝜆₀^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑑𝑠⁻¹,𝑖=1,2. (3.8) From ‖(𝑢_𝑛,𝑣_𝑛)−(𝑢,𝑣)‖→0 (𝑛→+∞) and the definition of the norm ‖⋅‖ in the space 𝑋, we can easily conclude that ‖𝑢_𝑛−𝑢‖_∞→0, ‖𝑣_𝑛−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 𝑢_𝑛(𝑠)−𝑢(𝑠)≤𝑢_𝑛−𝑢_∞<𝛿,𝑣_𝑛(𝑠)−𝑣(𝑠)≤𝑣_𝑛−𝑣_∞<𝛿. (3.9) Hence, by (3.8), when 𝑛>𝑁₀, for any 𝑠∈[0,𝑇₀], 𝑎∈[0,𝑎₀],𝑏∈[0,𝑏₀], we get 𝑓_𝑖𝑠,𝑢_𝑛(𝑠),𝑣_𝑛(𝑠),𝑎,𝑏−𝑓_𝑖((𝑠)(𝑠)<𝜀𝑠,𝑢,𝑣,𝑎,𝑏)2𝜆₀^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑑𝑠⁻¹. (3.10) 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 𝐴_𝑖𝑢_𝑛,𝑣_𝑛(𝑡)−𝐴_𝑖(𝑢,𝑣)(𝑡)≤𝜆₀^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑓_𝑖𝑠,𝑢_𝑛(𝑠),𝑣_𝑛(𝑠),𝑎,𝑏−𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠≤𝜆^𝑇0₀𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑓_𝑖𝑠,𝑢_𝑛(𝑠),𝑣_𝑛(𝑠),𝑎,𝑏−𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠+2𝜆𝐵^𝑟2_𝑓𝑖𝑇^+∞₀𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑑𝑠<𝜀,𝑖=1,2. (3.11) This implies that the operator 𝐴_𝑖∶𝑋→𝐸 is continuous. Therefore, the operator 𝐹∶𝑋→𝑋 is continuous.
(iii) We need to prove that the operator 𝐹∶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 |𝐹(𝑢,𝑣)(𝑡)|=𝐴₁(𝑢,𝑣)(𝑡)+𝐴₂(𝑡=𝜆𝑢,𝑣)()_0+∞𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁((𝑠)(𝑠))+𝜆𝑠,𝑢,𝑣,𝑎,𝑏𝑑𝑠_0+∞𝐺₂(𝑡,𝑠)𝜙₂(𝑠)𝑓₂(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠≤𝜆𝐵^𝑟3_𝑓10^+∞𝐺₁(𝑠,𝑠)𝜙₁(𝑠)𝑑𝑠+𝜆𝐵^𝑟3_𝑓20^+∞𝐺₂(𝑠,𝑠)𝜙₂(𝑠)𝑑𝑠<+∞, (3.12) where 𝐵^𝑟3_𝑓𝑖=sup{𝑓_𝑖(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 𝐴_𝑖(𝐷) is equicontinuous, and so 𝐹(𝐷) is also equicontinuous.
From (H₂) and the property (6) of the Green function 𝐺_𝑖(𝑡,𝑠), for any 𝑡∈ℝ₊, we have ₀^+∞𝐺_𝑖(𝑡,𝑠)−𝐺_𝑖(𝑠)𝜙_𝑖(𝑠)𝑑𝑠≤2₀^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑑𝑠<+∞,𝑖=1,2. (3.13) 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 𝐴_𝑖(𝑢,𝑣)(𝑡)−𝐴_𝑖(𝑢,𝑣)(+∞)=𝜆₀^+∞𝐺_𝑖(𝑡,𝑠)𝜙_𝑖(𝑠)𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠−𝜆₀^+∞𝐺_𝑖(𝑠)𝜙_𝑖(𝑠)𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠≤𝜆₀^+∞𝐺_𝑖(𝑡,𝑠)−𝐺_𝑖(𝑠)𝜙_𝑖(𝑠)𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠≤𝜆𝐵^𝑟3_𝑓𝑖0^+∞𝐺_𝑖(𝑡,𝑠)−𝐺_𝑖(𝑠)𝜙_𝑖(𝑠)𝑑𝑠⟶0,𝑡⟶+∞,𝑖=1,2. (3.14) This implies that 𝐴_𝑖(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 prove 𝐹(𝐾)⊆𝐾. By the property (1) of 𝐺_𝑖(𝑡,𝑠), (H₁), and (H₂), it is easy to see that, for any (𝑢,𝑣)∈𝐾 and 𝑡∈ℝ₊, 𝐴_𝑖(𝑢,𝑣)(𝑡)≥0 and 𝐴_𝑖(𝑢,𝑣)(𝑡)=𝜆₀^+∞𝐺_𝑖(𝑡,𝑠)𝜙_𝑖(𝑠)𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠≤𝜆₀^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠. (3.15) So 𝐴_𝑖(𝑢,𝑣)_∞≤𝜆₀^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑓_𝑖(𝑠,𝑢(𝑠),𝑣(𝑠),𝑎,𝑏)𝑑𝑠,𝑖=1,2. (3.16) On the other hand, by the property (7) of 𝐺_𝑖(𝑡,𝑠), we have 𝐴_𝑖(𝑢,𝑣)(𝑡)≥𝜔𝜆₀^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑓_𝑖((𝑠)(𝑠))1𝑠,𝑢,𝑣,𝑎,𝑏𝑑𝑠,𝑡∈𝛿,𝛿,𝑖=1,2. (3.17) It follows from (3.16) and (3.17) that 𝐴_𝑖(𝑢,𝑣)(𝑡)≥𝜔𝐴_𝑖(𝑢,𝑣)_∞1,𝑡∈𝛿,𝛿,𝑖=1,2. (3.18) Therefore 𝐹(𝐾)⊆𝐾. The proof of Lemma 3.1 is completed.

3.2. The Positive Solution for System (𝑃_𝑎0,𝑏0)

Theorem 3.2. Assume that (H₁)–(H₃) hold, then the system (𝑃_𝑎0,𝑏0) has at least one positive solution for any 𝑙𝜆∈min𝑓_1∞,𝑓_2∞,𝐿max𝑓⁰₁,𝑓⁰₂. (3.19)

Proof. From (3.19), there exists 𝜀₀>0 such that 𝑙min𝑓_1∞,𝑓_2∞−𝜀₀𝐿≤𝜆≤max𝑓⁰₁,𝑓⁰₂+𝜀₀. (3.20) By the first inequality of (H₃), there exists 𝑟>0 such that 𝑓_𝑖𝑡,𝑥,𝑦,𝑎₀,𝑏₀≤𝑓⁰_𝑖+𝜀₀𝑟,0≤𝑥+𝑦≤𝑟,𝑡∈ℝ₊,𝑖=1,2. (3.21) Setting 𝐾_𝑟1={(𝑢,𝑣)∈𝐾∶‖(𝑢,𝑣)‖<𝑟₁}(𝑟₁≤𝑟), by the definition of ‖⋅‖, we know that 𝑢(𝑡)+𝑣(𝑡)≤‖𝑢‖_∞+‖𝑣‖_∞=‖(𝑢,𝑣)‖=𝑟₁≤𝑟,∀(𝑢,𝑣)∈𝜕𝐾_𝑟1,𝑡∈ℝ₊. (3.22) Then, for any (𝑢,𝑣)∈𝜕𝐾_𝑟1, 𝐴_𝑖(𝑢,𝑣)_∞=𝜆sup_𝑡∈ℝ+0^+∞𝐺_𝑖(𝑡,𝑠)𝜙_𝑖(𝑠)𝑓_𝑖𝑠,𝑢(𝑠),𝑣(𝑠),𝑎₀,𝑏₀𝑑𝑠≤𝜆sup_𝑡∈ℝ+0^+∞𝐺_𝑖(𝑡,𝑠)𝜙_𝑖(𝑠)𝑓⁰_𝑖+𝜀₀𝑟₁𝑓𝑑𝑠≤𝜆⁰_𝑖+𝜀₀𝑟₁₀^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)𝑑𝑠≤𝑟₁𝐿₀^+∞𝐺_𝑖(𝑠,𝑠)𝜙_𝑖(𝑠)≤1𝑑𝑠2𝑟₁=12‖(𝑢,𝑣)‖,𝑖=1,2. (3.23) Thus, ‖𝐹(𝑢,𝑣)‖=𝐴₁(𝑢,𝑣)_∞+𝐴₂(𝑢,𝑣)_∞≤‖(𝑢,𝑣)‖,(𝑢,𝑣)∈𝜕𝐾_𝑟1. (3.24)
On the other hand, by the second inequality of (H₃), there exists 𝑟₀>𝜔𝑟₁>0 such that 𝑓_𝑖𝑡,𝑥,𝑦,𝑎₀,𝑏₀≥𝑓_𝑖𝑡,𝑥,𝑦,0,0≥𝑓_𝑖∞−𝜀₀𝑥+𝑦,𝑥+𝑦≥𝑟₀1,𝑡∈𝛿,𝛿,𝑖=1,2. (3.25) Take 𝑟₂=𝑟₀/𝜔>𝑟₁, and let 𝐾_𝑟2={(𝑢,𝑣)∈𝐾∶‖(𝑢,𝑣)‖<𝑟₂}, (𝑢₀,𝑣₀)=(1/2,1/2)∈𝜕𝐾₁. Then, (𝑢,𝑣)≠𝐹(𝑢,𝑣)+𝜇𝑢₀,𝑣₀,∀(𝑢,𝑣)∈𝜕𝐾_𝑟2,∀𝜇>0. (3.26) Suppose that (3.26) is false, then there exist (𝑢₂,𝑣₂)∈𝜕𝐾_𝑟2 and 𝜇₂>0 such that (𝑢₂,𝑣₂)=𝐹(𝑢₂,𝑣₂)+𝜇₂(𝑢₀,𝑣₀). From (3.25) and the fact that 𝑢₂(𝑡)+𝑣₂(𝑡)≥𝜔𝑢_2∞+𝜔𝑣_2∞=𝜔𝑢₂,𝑣₂=𝜔𝑟₂=𝑟₀1,𝑡∈𝛿,𝛿, (3.27) we have 𝑓_𝑖𝑡,𝑢₂(𝑡),𝑣₂(𝑡),𝑎₀,𝑏₀≥𝑓_𝑖∞−𝜀₀𝑢₂(𝑡)+𝑣₂(𝑡1),𝑡∈𝛿,𝛿,𝑖=1,2. (3.28) Set 𝜉=min𝑢₂(𝑡)+𝑣₂(𝑡)1∶𝑡∈𝛿,𝛿. (3.29) 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 𝑢₂(𝑡)+𝑣₂(𝑡)=𝐴₁𝑢₂(𝑡),𝑣₂(𝑡)+𝐴₂𝑢₂(𝑡),𝑣₂(𝑡)+𝜇₂𝑢₀+𝑣₀=𝜆₀^+∞𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁𝑠,𝑢₂(𝑠),𝑣₂(𝑠),𝑎₀,𝑏₀𝑑𝑠+𝜆₀^+∞𝐺₂(𝑡,𝑠)𝜙₂(𝑠)𝑓₂𝑠,𝑢₂(𝑠),𝑣₂(𝑠),𝑎₀,𝑏₀𝑑𝑠+𝜇₂𝑢₀+𝑣₀≥𝜆_𝛿^1/𝛿𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓_1∞−𝜀₀𝑢₂(𝑠)+𝑣₂(𝑠)𝑑𝑠+𝜆_𝛿^1/𝛿𝐺₂(𝑡,𝑠)𝜙₂(𝑠)𝑓_2∞−𝜀₀𝑢₂(𝑠)+𝑣₂(𝑠)𝑑𝑠+𝜇₂≥min_{[]𝑠∈𝛿,1/𝛿}𝑢₂(𝑠)+𝑣₂(𝑠)𝜆𝑓_1∞−𝜀₀min_{[]𝑡∈𝛿,1/𝛿𝛿}^1/𝛿𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑑𝑠+min_{[]𝑠∈𝛿,1/𝛿}𝑢₂(𝑠)+𝑣₂(𝑠)𝜆𝑓_2∞−𝜀₀min_{[]𝑡∈𝛿,1/𝛿𝛿}^1/𝛿𝐺₂(𝑡,𝑠)𝜙₂(𝑠)𝑑𝑠+𝜇₂≥𝜉+𝜇₂. (3.30) Then, we can obtain that 𝑢₂(𝑡)+𝑣₂(𝑡)>𝜉+𝜇₂1,𝑡∈𝛿,𝛿. (3.31) 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)∈𝐾_𝑟2⧵𝐾_𝑟1 such that 0<𝑟₁<‖(𝑢,𝑣)‖<𝑟₂. It is easy to see that (𝑢,𝑣) is a positive solution of the system (𝑃_𝑎0,𝑏0). The proof of Theorem 3.2 is completed.

Remark 3.3. Noticing that 𝑙/𝑓_1∞<1, 𝑙/𝑓_2∞<1 and 𝐿/𝑓⁰₁>1, 𝐿/𝑓⁰₂>1, we conclude that Theorem 3.2 also holds for 𝜆=1.

Remark 3.4. From Theorem 3.2, we can see that 𝑓_𝑖(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)𝑓_1∞=𝑓_2∞=+∞,𝐿>𝑓⁰₁>0, 𝐿>𝑓⁰₂>0, for each 𝜆∈(0,𝐿/max{𝑓⁰₁,𝑓⁰₂}),(2)𝑓_1∞=𝑓_2∞=+∞, 𝑓⁰₁=𝑓⁰₂=0, for each 𝜆∈(0,+∞),(3)𝑓_1∞>𝑙, 𝑓_2∞>𝑙, 𝑓⁰₁=𝑓⁰₂=0, for each 𝜆∈(𝑙/min{𝑓_1∞,𝑓_2∞},+∞).

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 𝑀_𝑖(𝑢,𝑣)(𝑡)=𝜆₀^+∞𝐺_𝑖(𝑡,𝑠)𝜙_𝑖(𝑠)𝑔_𝑖((𝑠)(𝑠𝐻()=𝑀𝑠,𝑢,𝑣))𝑑𝑠,𝑖=1,2,𝑢,𝑣₁(𝑢,𝑣),𝑀₂(𝑢,𝑣). (3.32) 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 operator ℋ defined by ℋ(𝑢,𝑣)=ℳ₁(𝑢,𝑣),ℳ₂(𝑢,𝑣), (3.33) where ℳ_𝑖(𝑢,𝑣)(𝑡)=𝜆₀^+∞𝐺_𝑖(𝑡,𝑠)𝜙_𝑖(𝑠)𝑔_𝑖𝑠,𝜁₁(𝑠,𝑢),𝜁₂(𝑠,𝑣)𝑑𝑠,𝑖=1,2, (3.34) in which 𝜁₁(𝑡,𝑢)=max𝑢(𝑡),min{𝑢(𝑡),𝑢(𝑡)},𝜁₂(𝑡,𝑣)=max𝑣(𝑡),min{𝑣(𝑡),𝑣(𝑡)}. (3.35) 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 𝑃_𝑎,𝑏 has at least one positive solution.

Proof. From Theorem 3.2, we can see that the system (𝑃_𝑎0,𝑏0) has at least one positive solution (𝑢₀,𝑣₀). Since the functions 𝑓_𝑖(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 𝑃_𝑎,𝑏. Hence, by Theorem 3.5, we have that the system 𝑃_𝑎,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 𝑃_𝑎,𝑏.

Proof. Assume by contradiction that there exists a sequence of positive solution (𝑢_𝑛,𝑣_𝑛)∈𝑋 of system 𝑃_𝑎,𝑏 such that ‖(𝑢_𝑛,𝑣_𝑛)‖→∞. From (3.19), there exists 𝜀₀>0 such that 𝜆>𝑙/(min(𝑓_1∞,𝑓_2∞)−𝜀₀). From assumption (H₃), there exists 𝐻>0 such that, for (0,0)≤(𝑎,𝑏)≤(𝑎₀,𝑏₀), 𝑓_𝑖𝑡,𝑥,𝑦,𝑎,𝑏≥𝑓_𝑖𝑡,𝑥,𝑦,0,0≥𝑓_𝑖∞−𝜀₀1𝑥+𝑦,𝑥+𝑦≥𝐻,𝑡∈𝛿,𝛿,𝑖=1,2. (3.36) Since 𝑢_𝑛(𝑡)+𝑣_𝑛(𝑡)≥𝜔‖𝑢_𝑛‖_∞+𝜔‖𝑣_𝑛‖_∞=𝜔‖(𝑢_𝑛,𝑣_𝑛)‖,𝑡∈[𝛿,1/𝛿], there exists natural number 𝑁₀ such that, for 𝑛>𝑁₀, we have 𝑢_𝑛(𝑡)+𝑣_𝑛(𝑡)≥𝐻 for 𝑡∈[𝛿,1/𝛿]. It follows from (3.36) that when 𝑛>𝑁₀, 𝑓_𝑖(𝑡,𝑢_𝑛(𝑡),𝑣_𝑛(𝑡),𝑎,𝑏)≥(𝑓_𝑖∞−𝜀₀)(𝑢_𝑛(𝑡)+𝑣_𝑛(𝑡)),𝑡∈[𝛿,1/𝛿],𝑖=1,2. Thus, 𝑢_𝑛(𝑡)+𝑣_𝑛(𝑡)=𝜆₀^+∞𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁𝑠,𝑢_𝑛(𝑠),𝑣_𝑛(𝑠),𝑎,𝑏𝑑𝑠+𝜆₀^+∞𝐺₂(𝑡,𝑠)𝜙₂(𝑠)𝑓₂𝑠,𝑢_𝑛(𝑠),𝑣_𝑛(𝑠),𝑎,𝑏𝑑𝑠≥𝜆_𝛿^1/𝛿𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓_1∞−𝜀₀𝑢_𝑛(𝑠)+𝑣_𝑛(𝑠)𝑑𝑠+𝜆_𝛿^1/𝛿𝐺₂(𝑡,𝑠)𝜙₂(𝑠)𝑓_2∞−𝜀₀𝑢_𝑛(𝑠)+𝑣_𝑛(𝑠)𝑑𝑠≥min_{[]𝑠∈𝛿,1/𝛿}𝑢_𝑛(𝑠)+𝑣_𝑛(𝑠)𝜆𝑓_1∞−𝜀₀min_{[]𝑡∈𝛿,1/𝛿𝛿}^1/𝛿𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑑𝑠+min_{[]𝑠∈𝛿,1/𝛿}𝑢_𝑛(𝑠)+𝑣_𝑛(𝑠)𝜆𝑓_2∞−𝜀₀min_{[]𝑡∈𝛿,1/𝛿𝛿}^1/𝛿𝐺₂(𝑡,𝑠)𝜙₂(𝑠)𝑑𝑠>min_{[]𝑠∈𝛿,1/𝛿}𝑢_𝑛(𝑠)+𝑣_𝑛(𝑠)1,𝑡∈𝛿,𝛿 (3.37) which yields min_{[]𝑡∈𝛿,1/𝛿}𝑢_𝑛(𝑡)+𝑣_𝑛(𝑡)>min_{[]𝑡∈𝛿,1/𝛿}𝑢_𝑛(𝑡)+𝑣_𝑛(𝑡), (3.38) which is a contradiction. This completes the proof of Theorem 3.7.

Remark 3.8. By the discussions in Sections 3.1, 3.2, 3.3 and 3.4, we can conclude that, for any (𝑎,𝑏) satisfying |(0,0)|≤|(𝑎,𝑏)|≤|(𝑎₀,𝑏₀)|, the system 𝑃_𝑎,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 𝑃_𝑎,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 𝑃_𝑎,𝑏 has no solution.

Proof. Suppose by contradiction that there exists a sequence (𝑎_𝑛,𝑏_𝑛) with |(𝑎_𝑛,𝑏_𝑛)|→+∞(𝑛→+∞) such that, for each nature number 𝑛, the system (𝑃_{𝑎𝑛,𝑏𝑛}) has a positive solution (𝑢_𝑛,𝑣_𝑛) in 𝐾. From assumption (H₄), for any 𝑀>0, there exists a constant 𝐶>0 such that, for any |(𝑎,𝑏)|>𝐶, 𝑓_𝑖1𝑡,𝑥,𝑦,𝑎,𝑏≥𝑀,𝑥,𝑦≥0,𝑡∈𝛿,𝛿,𝑖=1,2. (3.39) By |(𝑎_𝑛,𝑏_𝑛)|→+∞(𝑛→+∞), for the above 𝐶>0, there exists a natural number 𝑛₀ such that, for 𝑛>𝑛₀,|(𝑎_𝑛,𝑏_𝑛)|>𝐶, then, for 𝑛>𝑛₀ and 𝑡∈[𝛿,1/𝛿], 𝑢_𝑛(𝑡)=𝜆₀^+∞𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁𝑠,𝑢_𝑛(𝑠),𝑣_𝑛(𝑠),𝑎_𝑛,𝑏_𝑛𝑑𝑠>𝜆_𝛿^1/𝛿𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁𝑠,𝑢_𝑛(𝑠),𝑣_𝑛(𝑠),𝑎_𝑛,𝑏_𝑛𝑑𝑠≥𝑀𝜆𝜔_𝛿^1/𝛿𝐺₁(𝑠,𝑠)𝜙₁(𝑠)𝑑𝑠. (3.40) By the same way, we can obtain 𝑣_𝑛(𝑡)≥𝑀𝜆𝜔_𝛿^1/𝛿𝐺₂(𝑠,𝑠)𝜙₂(𝑠)𝑑𝑠. (3.41) Since we can choose 𝑀 arbitrarily large, we conclude that 𝑢_𝑛 and 𝑣_𝑛 are unbounded sequences in 𝐾, then lim_𝑡→+∞𝑢_𝑛(𝑡) and lim_𝑡→+∞𝑣_𝑛(𝑡) do not exist, which contradicts the fact that 𝑢_𝑛,𝑣_𝑛∈𝐾. So Theorem 3.9 holds.

We next define the set 𝐴=𝑎>0∶thesystem𝑃_𝑎,𝑏hasapositivesolutionforsome𝑏>0. (3.42) 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 𝑃_𝑎,𝑏 has a positive solution. We now define the function Γ∶[0,𝑎]→ℝ₊ by Γ(𝑎)=sup𝑏>0∶system𝑃_𝑎,𝑏hasapositivesolution. (3.43) 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 𝑃_𝑎,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 nonlinearities 𝑓₁ 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 (𝑃_{𝑎,Γ(𝑎)}) which is obtained using Theorem 3.6. Our next result allows us to establish another solution of the system 𝑃_𝑎,𝑏 for 0<𝑏<Γ(𝑎).

Lemma 3.10. For each 0<𝑏<Γ(𝑎), there exists 𝜖₀>0 so that, for all 0<𝜖≤𝜖₀ and all 𝑡∈ℝ₊, one has 𝜙_𝜖(𝑡)>𝜆₀^+∞𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁𝑠,𝜙_𝜖(𝑠),𝜓_𝜖(𝑠),𝑎,𝑏𝑑𝑠,𝜓_𝜖(𝑡)>𝜆₀^+∞𝐺₂(𝑡,𝑠)𝜙₂(𝑠)𝑓₂𝑠,𝜙_𝜖(𝑠),𝜓_𝜖(𝑠),𝑎,𝑏𝑑𝑠, (3.44) where 𝜙_𝜖(𝑡)=𝜙(𝑡)+𝜖,𝜓_𝜖(𝑡)=𝜓(𝑡)+𝜖.

Proof. Fix 𝛿∈(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 𝑓₁𝑠,𝜙(𝑠),𝜓(𝑠),𝑎,Γ(𝑎)−𝑓₁𝑠,𝜙(𝑠),𝜓(𝑠),𝑎,𝑏≥𝐼>0. (3.45) 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 𝑓₁𝑠,𝜙(𝑠)+𝜖,𝜓(𝑠)+𝜖,𝑎,𝑏−𝑓₁𝑠,𝜙(𝑠),𝜓(𝑠)≤𝐼,𝑎,𝑏2. (3.46) Next, we define Φ_𝜖(𝑡,𝑠)=𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁𝑠,𝜙_𝜖(𝑠),𝜓_𝜖(𝑠),𝑎,𝑏−𝑓₁𝑠,𝜙(𝑠),𝜓(𝑠),Ψ(),𝑎,𝑏𝑡,𝑠=𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁𝑠,𝜙(𝑠),𝜓(𝑠),𝑎,Γ(𝑎)−𝑓₁𝑠,𝜙(𝑠),𝜓(𝑠),𝑎,𝑏. (3.47) Assume 0<𝜖≤𝜖₀. Then, 𝜙_𝜖(𝑡)>𝜙(𝑡)=𝜆₀^+∞𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁𝑠,𝜙(𝑠),𝜓(𝑠),𝑎,Γ(𝑎)𝑑𝑠=𝜆₀^+∞𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁𝑠,𝜙_𝜖(𝑠),𝜓_𝜀(𝑠),𝑎,𝑏𝑑𝑠−𝜆₀^+∞Φ_𝜖(𝑡,𝑠)𝑑𝑠+𝜆₀^+∞Ψ(𝑡,𝑠)𝑑𝑠. (3.48) Since Ψ(𝑡,𝑠) is positive and Ψ(𝑡,𝑠)−Φ_𝜖(𝑡,𝑠)>(𝐼/2)𝐺₁(𝑡,𝑠)𝜙₁(𝑠) for 𝑡∈[𝛿,1/𝛿], we have 𝜙_𝜖(𝑡)>𝜆₀^+∞𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁𝑠,𝜙_𝜖(𝑠),𝜓_𝜖(𝑠),𝑎,𝑏𝑑𝑠−𝜆^𝛿₀Φ_𝜖(𝑡,𝑠)𝑑𝑠−𝜆^+∞_1/𝛿Φ_𝜖()𝐼𝑡,𝑠𝑑𝑠+2𝜆_𝛿^1/𝛿𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑑𝑠. (3.49)
It is not difficult to show that the Lebesgue dominated convergence theorem implies that 𝜆∫^𝛿₀Φ_𝜖(𝑡,𝑠)𝑑𝑠+𝜆∫^+∞_1/𝛿Φ_𝜖(𝑡,𝑠)𝑑𝑠 converges to zero, uniformly in 𝑡, as 𝜖 tends to zero. Thus, for 𝜖 sufficiently small, we have 𝜙_𝜖(𝑡)>𝜆₀^+∞𝐺₁(𝑡,𝑠)𝜙₁(𝑠)𝑓₁𝑠,𝜙_𝜖(𝑠),𝜓_𝜖(𝑠),𝑎,𝑏𝑑𝑠 (3.50) uniformly in 𝑡∈ℝ₊.
A similar computation holds for 𝜓_𝜖.

We are now in a position to show existence of two positive solutions of the system 𝑃_𝑎,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 𝑃_𝑎,𝑏 has at least two positive solutions for 0<𝑏<Γ(𝑎).

Proof. Consider the set Ω=𝜙,𝜓∈𝑋∶−𝜖<𝜙(𝑡)<𝜙_𝜖(𝑡),−𝜖<𝜓(𝑡)<𝜓_𝜖(𝑡),for𝑡∈ℝ₊, (3.51) 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 𝑃_𝑎,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 have𝐴₁𝜙,𝜓𝑡₀=𝜆₀^+∞𝐺₁𝑡₀,𝑠𝜙₁(𝑠)𝑓₁𝑠,𝜙(𝑠),𝜓(𝑠),𝑎,𝑏𝑑𝑠≤𝜆₀^+∞𝐺₁𝑡₀,𝑠𝜙₁(𝑠)𝑓₁𝑠,𝜙_𝜖(𝑠),𝜓_𝜖(𝑠)<,𝑎,𝑏𝑑𝑠𝜙_𝜖𝑡₀=𝜙𝑡₀≤𝜇𝜙𝑡₀. (3.52) 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 𝑖(𝐹,𝐾∩Ω,𝐾)=1. (3.53)
On the other hand, a slight change in the proof of Theorem 3.7 shows the existence of an 𝑟₃>0 sufficiently large, say 𝑟₃>𝑟₂, where 𝑟₂ is as in Theorem 3.2, so that𝐹𝜙,𝜓>𝜙,𝜓 (3.54) for every ‖(𝜙,𝜓)‖=𝑟₃ and every (𝜙,𝜓)∈𝐾.
Let 𝑟₄=max{𝐶₀+1,𝑟₃,‖(𝜙_𝜖,𝜓_𝜖)‖}, where 𝐶₀ is as that in Theorem 3.7. Set 𝐾_𝑟4=𝜙,𝜓∈𝐾∶𝜙,𝜓<𝑟₄. (3.55) Then Theorem 3.7 implies that 𝐹(𝜙,𝜓)≠(𝜙,𝜓), for (𝜙,𝜓)∈𝜕𝐾_𝑟4. Consequently, Lemma 2.8 implies 𝑖(𝐹,𝐾_𝑟4,K)=0.
Now, by the additivity property of the fixed-point index, we obtain 𝑖𝐹,𝐾_𝑟4⧵𝐾∩Ω,𝐾=−1. (3.56) Therefore, 𝐹 has another fixed point in 𝐾_𝑟4⧵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 𝑃_𝑎,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.

Acknowledgments

L. Liu, Y. Wang, and X. Hao authors were supported financially by the National Natural Science Foundation of China (11071141 and 11126231), the Natural Science Foundation of Shandong Province of China (ZR2011AQ008) and a Project of Shandong Province Higher Educational Science and Technology Program (J11LA06). The Y. Wu author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.