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

and Applied Analysis 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 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 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 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 ai βi1 αi1 ∫ ∞ 0 1 pi s ds, bi βi2 αi2 ∫ ∞ 0 1 pi s ds, i 1, 2,

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 λφ t f t, x t 0, 0 < t < ∞, 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 f t, x q t 0, 0 < t < ∞, 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 upperlower solutions method, and degree arguments, doÓ 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 : Abstract and Applied Analysis 3 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 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 Q 1 and Q 2 such that the system P a,b has at least two positive solutions in Q 1 , at least one positive solution on the boundary of Q 1 , and no positive solutions in Q 2 .
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.

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

2.1
Then, it is obvious that α i2 a i t α i1 b 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.
has a unique solution for any v ∈ L 1 0, ∞ . Moreover, this unique solution can be expressed in the form 7 For any t ∈ δ, 1/δ and s ∈ R , G i t, s ≥ ω i G i s, s , where For any In what follows, we list some conditions for convenience.
H 1 The function f i : R 5 → R is continuous and nondecreasing with respect to the last four variables. In other words, where the order is understood to apply to every component. And there x and y in any bounded set of R .

2.7
Abstract and Applied Analysis From the above assumptions H 1 and H 2 , 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: Define operators A i : X → E and F : X → X as follows: 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: which induces a partial order "≤": Abstract and Applied Analysis Consider the following system: T Similarly, we define the upper solution for the system S by replacing the ≤ in T by ≥. 3 The functions in M are equiconvergent at ∞, that is, for any given 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, Abstract and Applied Analysis 7 2 there exists a e ∈ ∂P 1 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.
ii If Ax ≤ x for x ∈ ∂K r , then i A, K r , K 1.

The Complete Continuity of the Operator F
Before presenting the main results, we give a lemma. 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 r 1 > 0 such that |u t | ≤ r 1 and |v t | ≤ r 1 for any t ∈ R . It follows from H 1 and the property 1 of the Green function G i t, s that A i u, v ≥ 0 and Thus, by H 1 and H 2 , for any t ∈ R , a ∈ 0, a 0 and b ∈ 0, b 0 , we obtain Hence, the operator F u, v A 1 u, v , A 2 u, v is well defined for any u, v ∈ X. For any t 1 , t 2 , s ∈ R , by the property 5 of the Green function G i t, s , we have 3.3 8

Abstract and Applied Analysis
So, by H 2 , the Lebesgue dominated convergence theorem and the continuity of G i t, s , for any t 1 , t 2 ∈ R , a ∈ 0, a 0 , b ∈ 0, b 0 , we get 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 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 , H 1 , and H 2 , for any t ∈ R , a ∈ 0, a 0 , b ∈ 0, b 0 and any natural number n, we have where B r 2 f i sup{f i t, x, y, a, b : t ∈ R , x ∈ 0, r 2 , y ∈ 0, r 2 , a ∈ 0, a 0 , b ∈ 0, b 0 } < ∞ by H 1 , r 2 is a real number such that r 2 ≥ max n∈N { u, v , u n , v n }, in which N is the natural number set.
For any ε > 0, by H 2 , there exists a sufficiently large T 0 > 0 such that On the other hand, by the continuity of f i t, x, y, a, b on 0, T 0 × 0, r 2 × 0, r 2 × 0, a 0 × 0, b 0 , for the above ε > 0, there exists a δ > 0 such that for any s ∈ 0, T 0 , a ∈ 0, a 0 , b ∈ 0, b 0 and x, x , y, y ∈ 0, r 2 , when |x − x | < δ, |y − y | < δ, we have 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 N 0 such that, when n > N 0 , for any s ∈ 0, T 0 , we have Hence, by 3.8 , when n > N 0 , for any s ∈ 0, T 0 , a ∈ 0, a 0 , b ∈ 0, b 0 , we get

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 r 3 > 0 such that u, v ≤ r 3 for any u, v ∈ D. So u ∞ ≤ r 3 , v ∞ ≤ r 3 for any u, v ∈ D. By 2.10 , H 1 , and H 2 , for any u, v ∈ D and t ∈ R , we have , x, y, a, b : t ∈ R , x ∈ 0, r 3 , y ∈ 0, r 3 , a ∈ 0, a 0 , b ∈ 0, b 0 } < ∞ by H 1 . 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 H 2 and the property 6 of the Green function G i t, s , for any t ∈ R , we have 3.13 By 2.10 , 3.5 , and the Lebesgue dominated convergence theorem, for any u, v ∈ D, t ∈ R , a ∈ 0, a 0 , and b ∈ 0, b 0 , we obtain 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 , H 1 , and H 2 , it is easy to see that, for any u, v ∈ K and t ∈ R , A i u, v t ≥ 0 and

3.15
So On the other hand, by the property 7 of G i t, s , we have It follows from 3.16 and 3.17 that Therefore F K ⊆ K. The proof of Lemma 3.1 is completed.

The Positive Solution for System P a 0 ,b 0
Theorem 3.2. Assume that H 1 − H 3 hold, then the system (P a 0 ,b 0 ) has at least one positive solution for any

3.19
Proof. From 3.19 , there exists ε 0 > 0 such that By the first inequality of H 3 , there exists r > 0 such that Then, for any u, v ∈ ∂K r 1 ,

3.23
Thus, On the other hand, by the second inequality of H 3 , there exists r 0 > ωr 1 > 0 such that 3.25

3.30
Then, we can obtain that It is clearly that 3.31 contradicts 3.29 , which implies that 3.26 holds. 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 | ≤ | a 0 , b 0 |, the system P a,b has at least one positive solution u, v with u, v ≤ C 0 . In the following section, we will establish the nonexistence result for the system P a,b . 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 a n ,b n has a positive solution u n , v n in K. From assumption H 4 , for any M > 0, there exists a constant C > 0 such that, for any | a, b | > C,

Non-Existence
By | a n , b n | → ∞ n → ∞ , for the above C > 0, there exists a natural number n 0 such that, for n > n 0 , | a n , b n | > C, then, for n > n 0 and t ∈ δ, 1/δ ,

3.40
By the same way, we can obtain v n t ≥ Mλω 1/δ δ G 2 s, s φ 2 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 : the system P a,b has a positive solution for some b > 0 .

3.42
From Theorems 3.2 and 3.9, we conclude that A is nonempty and bounded. Thus, 0 < a sup A < ∞. 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 sup b > 0 : system P a,b has a positive solution .

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 .

Existence of Two Positive Solutions
In this section,we will assume that the nonlinearities f 1 and f 2 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 .  f 1 s, φ s , ψ s , a, Γ a − f 1 s, φ s , ψ s , a, b . 3.47

3.49
It is not difficult to show that the Lebesgue dominated convergence theorem implies that λ δ 0 Φ t, s ds λ ∞ 1/δ Φ t, s ds converges to zero, uniformly in t, as tends to zero. Thus, for sufficiently small, we have , s φ 1 s f 1 s, φ s , ψ s , a, b ds 3.50 A similar computation holds for ψ .
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 H 1 -H 4 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 : − < φ t < φ t , − < ψ t < ψ t , for t ∈ R , 3.51 where φ and ψ 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 t 0 ∈ 0, ∞ so that one of the following two cases holds: φ t 0 φ t 0 or ψ t 0 ψ t 0 . In the case φ t 0 φ t 0 , it follows from Lemma 3.10 that, for all μ ≥ 1, we have

3.56
Therefore, F has another fixed point in K r 4 \ K ∩ Ω.
Then, from the above discussion, it is easy to obtain the following conclusion.

Conclusion
Assume that H 1 -H 4 and 3.19 hold. Then, there exist a constant a > 0 and a nonincreasing 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 f 1 and f 2 are strict-increasing with respect to the fifth variable.