Research Article | Open Access
Positive Solutions for Nonlinear Singular Differential Systems Involving Parameter on the Half-Line
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.
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: where is a parameter, are constants; are continuous, are continuous and may have singularity at ; with on and ; with , in which 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 , O’Regan , and many others (see [3–17]). For the study of boundary value problems, Agarwal and O’Regan  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  obtained the criteria for the existence of at least one positive solution, a unique positive solution, and multiple positive solutions of the following BVP where is a parameter, and are continuous. More recently, by employing the method of varying in translation together with the fixed-point theorem in cone, Zhang et al.  established the existence of positive solution for the following semipositone singular Sturm-Liouville boundary value problem on the half-line where is continuous and allows the nonlinearity to have singularity at , 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 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.  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 : where are continuous and nondecreasing with respect to the last four variables.
There are two aims in this paper. The first aim is to obtain the existence of positive solutions for the system . For this purpose, we solve the fixed point of an operator instead of the positive solutions for the system . The main difficulty for this is to testify that the operator is completely continuous, as the Ascoli-Arzela theorem cannot be used in infinite interval . Some modification of the compactness criterion in infinite interval (Lemma 2.4) has thus been made to resolve this problem. The second aim is to show that there exists a continuous curve which splits the positive quadrant of the -plane into two disjoint sets and such that the system has at least two positive solutions in , at least one positive solution on the boundary of , and no positive solutions in .
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 will be the basic space to study , where the Banach space is denoted by with the supremum norm . Clearly, is a Banach space with the norm for . For convenience, let Then, it is obvious that is a constant for any and is increasing on , is decreasing on for .
Lemma 2.1 2.1(see ). Under the condition and for , the linear boundary value problem has a unique solution for any . Moreover, this unique solution can be expressed in the form where the Green function is defined by
Remark 2.2. From (2.4), we can get the properties of as follows.(1) is continuous and nonnegative on .(2)For each , is continuously differentiable on except for .(3).(4)For each , satisfies the corresponding homogeneous BVP (i.e., the BVP (2.2) with ) on except for . In other words, is the Green function of BVP (2.2) on the half-line.(5).(6).(7)For any and , where
For any , define . In what follows, we list some conditions for convenience. The function is continuous and nondecreasing with respect to the last four variables. In other words, for any , where the order is understood to apply to every component. And there exists such that is bounded for any satisfying , , and in any bounded set of . The function is continuous and singular at , on satisfying . For the above and in , where
From the above assumptions and , it is not difficult to show that the pair is a solution of the system if and only if is a solution of the following system of nonlinear integral equations: Define operators and as follows: Then, the solution of the system is equivalent to the fixed point of the operator . Define a cone in the Banach space as follows: which induces a partial order “≤”: if and only if for any .
Consider the following system: where are nonnegative continuous functions and are nondecreasing with respect to the last two variables.
Definition 2.3. The pair is said to be a lower solution for the system if the pair satisfies the following inequality system: Similarly, we define the upper solution for the system by replacing the in by .
Lemma 2.4 (see [1, 23]). Let be defined as above and . Then, is relatively compact in if the following conditions hold.(1) is uniformly bounded in .(2)The functions in are equicontinuous on any bounded interval of .(3)The functions in are equiconvergent at , that is, for any given , there exists a such that , for any .
Lemma 2.5 (see [24, 25]). Let be a positive cone in a real Banach space , , , and let be a completely continuous operator. If the following conditions are satisfied,(1), for all ,(2)there exists a such that for any and , then has fixed points in .
Remark 2.6. If (1) and (2) are satisfied for and , respectively, then Lemma 2.5 still holds.
3. Main Results
3.1. The Complete Continuity of the Operator
Before presenting the main results, we give a lemma.
Lemma 3.1. Assume that and hold. Then, for any satisfying , is a completely continuous operator and .
Proof. We divide the proof into four steps.
(i) Firstly, we show that is well defined. For any fixed , there exists such that and for any . It follows from and the property (1) of the Green function that and
Thus, by and , for any and , we obtain Hence, the operator is well defined for any .
For any , by the property (5) of the Green function , we have So, by , the Lebesgue dominated convergence theorem and the continuity of , for any , , we get Therefore, , and so . By the property (6) of and the Lebesgue dominated convergence theorem, we obtain Hence, is well defined.
(ii) Next we show that is continuous. Let , and , we will prove that (). By (2.10), , and , for any , and any natural number , we have where by , is a real number such that , in which is the natural number set.
For any , by , there exists a sufficiently large such that On the other hand, by the continuity of on , for the above , there exists a such that for any , and , when , , we have From () and the definition of the norm in the space , we can easily conclude that , . So, for the above , there exists a sufficiently large natural number such that, when , for any , we have Hence, by (3.8), when , for any , , we get Therefore, by (2.10), (3.7), and (3.10), when , for any , and , we obtain This implies that the operator is continuous. Therefore, the operator is continuous.
(iii) We need to prove that the operator is compact. Let be any bounded subset of . Then, there exists a constant such that for any . So , for any . By (2.10), , and , for any and , we have where by . Hence, is uniformly bounded. By the similar proof as for (3.4), we can conclude that is equicontinuous, and so is also equicontinuous.
From and the property (6) of the Green function , for any , we have By (2.10), (3.5), and the Lebesgue dominated convergence theorem, for any , , , and , we obtain This implies that is equiconvergent at . Hence, is equiconvergent at . Therefore, the above discussion and Lemma 2.4 imply that is completely continuous.
(iv) Finally, we prove . By the property (1) of , , and , it is easy to see that, for any and , and So On the other hand, by the property (7) of , we have It follows from (3.16) and (3.17) that Therefore . The proof of Lemma 3.1 is completed.
3.2. The Positive Solution for System
Theorem 3.2. Assume that – hold, then the system () has at least one positive solution for any
Proof. From (3.19), there exists such that
By the first inequality of , there exists such that
Setting , by the definition of , we know that
Then, for any ,
On the other hand, by the second inequality of , there exists such that Take , and let , . Then, Suppose that (3.26) is false, then there exist and such that . From (3.25) and the fact that we have Set Then, for any . Hence, for any , by (3.28), we have Then, we can obtain that 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 has fixed-point such that . It is easy to see that is a positive solution of the system . The proof of Theorem 3.2 is completed.
Remark 3.3. Noticing that , and , , we conclude that Theorem 3.2 also holds for .
Remark 3.4. From Theorem 3.2, we can see that do not need to be superlinear or sublinear. In fact, Theorem 3.2 still holds, if - are satisfied and one of the following conditions is satisfied:(1),, , for each ,(2), , for each ,(3), , , for each .
3.3. Lower and Upper Solutions
Then, the solutions of the system are equivalent to the fixed points of the operator in .
Now, we introduce the following auxiliary operator defined by where in which It is easy to prove that the operator has the following properties.(1) is a completely continuous operator.(2)If the pair is a fixed point of , then is a fixed point of with .(3)If with , then , where does not depend on and .Therefore, by using the topological degree of Leray-Schauder (see [26, Corollary 8.1, page. 61]), we obtain a fixed point of the operator . The proof of Theorem 3.5 is completed.
Theorem 3.6. Assume that – and (3.19) hold. Then for , the system has at least one positive solution.
Proof. From Theorem 3.2, we can see that the system has at least one positive solution . Since the functions are increasing functions with respect to the last four variables, we conclude that is an upper solution and is a lower solution for the system . Hence, by Theorem 3.5, we have that the system has at least one positive solution. This completes the proof of Theorem 3.6.
3.4. A Priori Estimate
Theorem 3.7. Assume that – and (3.19) hold. Then for any satisfying , there exists a constant independent of , such that for all positive solutions of the system .
Proof. Assume by contradiction that there exists a sequence of positive solution of system such that . From (3.19), there exists such that