We use a fixed point theorem in a cone to obtain the existence of positive solutions of the differential equation, , , with some suitable boundary conditions, where is a parameter.
1. Introduction
We consider the existence of positive solutions of the following twopoint boundary value problem:
where and are nonnegative constants, and .
In the last thirty years, there are many mathematician considered the boundary value problem (BVP_{λ}) with , see, for example, Chu et al. [1], Chu et al. [2], Chu and Zhau [3], Chu and Jiang [4], Coffman and Marcus [5], Cohen and Keller [6], Erbe [7], Erbe et al. [8], Erbe and Wang [9], Guo and Lakshmikantham [10], Iffland [11], Njoku and Zanolin [12], Santanilla [13].
In 1993, Wong [14] showed the following excellent result.
Theorem 1 A (see [14]).
Assume that
is an increasing function with respect to . If there exists a constant such that
where for , then, there exists such that the boundary value problem (BVP_{λ}) with has a positive solution in for , while there is no such solution for in which
Seeing such facts, we cannot but ask "whether or not we can obtain a similar conclusion for the boundary value problem (BVP_{λ})." We give a confirm answer to the question.
First, We observe the following statements.
(1)Let
on , then is the Green's function of the differential equation in with respect to the boundary value condition .
(2), is a cone in the Banach space with .
In order to discuss our main result, we need the follo wing useful lemmas which due to Lian et al. [15] and Guo and Lakshmikantham [10], respectively.
Lemma 1 B (see [10]).
Suppose that be defined as in . Then, we have the following results.
for and )
for and )
Lemma 1 C (see [10, Lemmas and ]).
Let be a real Banach space, and let be a cone. Assume that and is completely continuous. Then
(1) if
(2)
where is the fixed point index of a compact map , such that for , with respect to .
2. Main Results
Now, we can state and prove our main result.
Theorem 2.1.
Suppose that there exist two distinct positive constants , and a function with and such that
Then (BVP_{λ}) has a positive solution with between and if
where
Proof.
It is clear that (BVP_{λ}) has a solution if, and only if, is the solution of the operator equation
It follows from the definition of in our observation and Lemma B that
Hence, , which implies . Furthermore, it is easy to check that is completely continuous. If there exists a such that , then we obtain the desired result. Thus, we may assume that
where and . We now separate the rest proof into the following three steps.
Step 1.
It follows from the definitions of and that, for ,
which implies
Hence, by (2.5),
which implies
Hence
We now claim that
In fact, if there exist and such that then, by (2.11),
which gives a contradiction. This proves that (2.13) holds. Thus, by Lemma C,
Step 2.
First, we claim that
Suppose to the contrary that there exist and such that
It is clear that (2.17) is equivalent to
Since and it follows that there exists a such that
Let
Then . From on , we see that on on and on . It follows from
and on that
Hence,
Thus
This contradiction implies
Therefore, by Lemma C,
Step 3.
It follows from Steps (1) and (2) and the property of the fixed point index (see, for example, [10, Theorem ]) that the proof is complete.
Remark 2.2.
It follows from the conclusion of Theorem 2.1 that the positive constant and nonnegative function satisfy
There are many functions and positive constants satisfying (2.27). For example, Suppose that and . Let on , then on and
Remark 2.3.
We now define
A simple calculation shows that
Then, we have the following results.
(i)Suppose that . Taking , there exists ( can be chosen small arbitrarily) such that
Hence,
It follows from Remark 2.2 that the hypothesis (2.2) of Theorem 2.1 is satisfied if .
(ii)Suppose that . Taking , there exists ( can be chosen large arbitrarily) such that
Hence,
which satisfies the hypothesis (2.1) of Theorem 2.1.
(iii)Suppose that . Taking , there exists ( can be chosen small arbitrarily) such that
Hence,
which satisfies the hypothesis (2.1) of Theorem 2.1.
(iv)Suppose that . Taking , there exists a ( can be chosen large arbitrarily) such that
Hence, we have the following two cases.
Case i.
Assume that is bounded, say
for some constant . Taking (since can be chosen large arbitrarily, can be chosen large arbitrarily, too),
Case ii.
Assume that is unbounded, then there exist a ( can be chosen large arbitrarily) and such that
It follows from and (2.37) that
By Cases (i), (ii) and Remark 2.2, we see that the hypothesis (2.2) of Theorem 2.1 is satisfied if .
We immediately conclude the following corollaries.
Corollary 2.4.
(BVP_{λ}) has at least one positive solution for if one of the following conditions holds:
Proof.
It follows from Remark 2.3 and Theorem 2.1 that the desired result holds, immediately.
Corollary 2.5.
Let
on for some and .
Then, for , (BVP_{λ}) has at least two positive solutions and such that
Proof.
It follows from Remark 2.3 that there exist two real numbers satisfying
Hence, by Theorem 2.1 and Remark 2.2, we see that for each , there exist two positive solutions and of (BVP_{λ}) such that
Thus, we complete the proof.
Corollary 2.6.
Let
on , for some .
Then, for , (BVP_{λ}) has at least two positive solutions and such that
Proof.
It follows from Remark 2.3 that there exist two real numbers satisfying
Hence, by Theorem 2.1 and Remark 2.2, we see that, for each , (BVP_{λ}) has two positive solutions and such that
Thus, we completed the proof.
3. Examples
To illustrate the usage of our results, we present the following examples.
Example 3.1.
Consider the following boundary value problem:
Clearly,
If we take , then it follows from of Corollary 2.4 that (BVP.1) has a solution if .
Example 3.2.
Consider the following boundary value problem:
Clearly,
If we take , then it follows from of Corollary 2.4 that (BVP.2) has a solution if .
Example 3.3.
Consider the following boundary value problem:
Clearly, if we take and ,
Hence, it follows from Corollary 2.5 that (BVP.3) has two solutions if .
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