Abstract
By using fixed-point theorem and under suitable conditions, we investigate the existence and multiplicity positive solutions to the following systems: , where are four positive constants and , , and . We derive two explicit intervals of and , such that the existence and multiplicity of positive solutions for the systems is guaranteed.
1. Introduction
In this paper, we are concerned with the existence and multiplicity of positive solutions to the following boundary value problem of second-order nonlinear differential systems with two parameters: where are four positive constants.
In addition, we assume the following conditions throughout this paper:() does not vanish identically on any subinterval of ;(); ().
The boundary value problem of ordinary differential systems has attracted much attention, see [1–6] and the references therein. Recently, Fink and Gatica [4] and Ma [7] have studied the existence of positive solutions of the following systems: In [4], a multiplicity result has been established when . In [7], a multiplicity result has been given for the more general case .
Also, by using Krasnoselskill fixed-point theorem, Ru and An [8] considered the existence of positive solutions of the following systems: where , .
More recently, Dalbono and Mckenna [5] proved the existence and multiplicity of solutions to a class of asymmetric weakly coupled systems as follows: where is suitably small and the positive numbers , satisfy Applying a classical change of variables, the authors transformed the initial problem into an equivalent problem whose solutions can be characterized by their nodal properties. Meanwhile, in [5], there are two open questions. (1)Can one replace the near-diagonal matrix with something more general and use information on the eigenvalues of matrix?(2)Can one replace the near-diagonal matrix with something more general and use information on the eigenvalues of matrix?
Inspired by the above works and the two open questions, we consider the existence and multiplicity of positive solutions to (1.1). The paper is organized as follows. In Section 2, we state some preliminaries. In Sections 3 and 4, we prove the existence and multiplicity results of (1.1).
2. Preliminaries
In order to prove our results, we state the well-known fixed-point theorem [9]:
Lemma 2.1. Let be a Banach space and let be a cone in . Assume , are two open subsets of with , , and let be a completely continuous operator such that either(i) and ; or(ii) and .Then, has a fixed point in .
To be convenient, we introduce the following notations:
and suppose that .
Let , be function of the corresponding to linear boundary value problem:
Then, the solution of (2.2) is given by
It is well known that can be expressed by
In addition, it can be easily to be checked that
See [6], we have the following lemma.
Lemma 2.2. has the following properties:(i), (ii), (iii). It is obvious that problem (1.1) is equivalent to the equation: and consequently it is equivalent to the fixed-point problem: with given by For convenience, denote It is obvious that is completely continuous. Let , and define a cone in by where .
Lemma 2.3. . Proof. For any , by Lemma 2.2, we have Similarly, for any , we have Then, for any , we have Thus, . Therefore, .
3. Existence Results
We assume the following:; ; .
Theorem 3.1. Assume – hold. Then one has the following:
(1)If , then for each and , (1.1) has at least one positive solution. (2)If , then for each and , (1.1) has at least one positive solution. (3)If , then for each and , (1.1) has at least one positive solution.(4)If , then for each and , (1.1) has at least one positive solution.(5)If and , then for each and , (1.1) has at least one positive solution.(6)If or , then for each and , (1.1) has at least one positive solution. (7)If and , then for each or (1.1) has at least one positive solution.
Proof. We only prove case (1.1). The other cases can be proved similarly. In order to apply the Lemma 2.1, we construct the sets .
Let , and we choose such that
By the definition of and , there exists , such that
Choosing with , we have
namely, . In the same way, we also have
then . Thus, if we set , then
On the other hand, by the definition of , there exists , such that , for . Let and . If we choose with , such that , then we have
Hence
Therefore, it follows from (3.5) to (3.7) and Lemma 2.2, has a fixed point , which is a positive solution of (1.1).
Similarly, we also have the following results.
Theorem 3.2. Assume – hold. Then one has the following:(1)If , then for each and , (1.1) has at least one positive solution.(2)If , then for each and , (1.1) has at least one positive solution.(3)If , then for each and , (1.1) has at least one positive solution.(4)If , then for each and , (1.1) has at least one positive solution.(5)If and , then for each and , (1.1) has at least one positive solution.(6)If or then for each and , (1.1) has at least one positive solution.(7)If and then for each or , (1.1) has at least one positive solution.
Proof. We only prove case . The other cases can be proved similarly.
Let , and we choose such that
by the definition of , there exists , such that
Choosing with , such that , then we have
So, if we set , then
On the other hand, by the definition of and , there exists , such that
Choosing with , we have
In the same way, we also have
Hence, if we set , then
Therefore, it follows from (3.11) to (3.15) and Lemma 2.2 that has a fixed point , which is a positive solution of (1.1).
4. Multiplicity Results
Theorem 4.1. Assume – hold. In addition, assume that there exist three constants , where is sufficient small with such that(i), (ii), for .Then, for any , , or , , the problem (1.1) has at least two positive solutions.
Proof. We only prove the case of , , The other case is similar.
Step 1. Since , there exists such that
Set , then we have
In the same way, we also have
Hence,
Step 2. Since , there exists such that
Similarly, set , then
Step 3. Let , we can see that
Then,
Consequently, by Lemma 2.1 and from (4.4)–(4.8), has two fixed points: and , so (1.1) has at least two positive solutions satisfying .
Theorem 4.2. Assume – hold. In addition, assume that there exist constants , where is sufficient large with such that(i) or ,(ii) or , for ,then, for any , , or , , the problem (1.1) has at least two positive solutions.
Proof. We only prove the case of , , The other case is similar.
Step 1. Since , there exists such that
Set , then we have
Hence,
Step 2. Since , there exists such that
Similarly, set , then
Step 3. Let , we can see that
In the same way, we also have
Hence,
Consequently, by Lemma 2.1 and from (4.11)–(4.16), has two fixed points: and , so (1.1) has at least two positive solutions satisfying .