Existence of Positive Solutions for Nonlinear Eigenvalue Problems

We consider the existence of positive solutions of the following two-point boundary value problem:

(BVP λa)

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

(1.1)

is an increasing function with respect to . If there exists a constant such that

(1.2)

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

(1.3)

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

(1.4)

(2)

where is the fixed point index of a compact map , such that for , with respect to .

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

(2.1)

(2.2)

Then (BVP _λ ) has a positive solution with between and if

(2.3)

where

(2.4)

Proof.

It is clear that (BVP _λ ) has a solution if, and only if, is the solution of the operator equation

(2.5)

It follows from the definition of in our observation and Lemma B that

(2.6)

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

(2.7)

where and . We now separate the rest proof into the following three steps.

Step 1.

It follows from the definitions of and that, for ,

(2.8)

which implies

(2.9)

Hence, by (2.5),

(2.10)

which implies

(2.11)

Hence

(2.12)

We now claim that

(2.13)

In fact, if there exist and such that then, by (2.11),

(2.14)

which gives a contradiction. This proves that (2.13) holds. Thus, by Lemma C,

(2.15)

Step 2.

First, we claim that

(2.16)

Suppose to the contrary that there exist and such that

(2.17)

It is clear that (2.17) is equivalent to

(2.18)

Since and it follows that there exists a such that

(2.19)

Let

(2.20)

Then . From on , we see that on on and on . It follows from

(2.21)

and on that

(2.22)

Hence,

(2.23)

Thus

(2.24)

This contradiction implies

(2.25)

Therefore, by Lemma C,

(2.26)

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

(2.27)

There are many functions and positive constants satisfying (2.27). For example, Suppose that and . Let on , then on and

(2.28)

Remark 2.3.

We now define

(2.29)

A simple calculation shows that

(2.30)

Then, we have the following results.

(i)Suppose that . Taking , there exists ( can be chosen small arbitrarily) such that

(2.31)

Hence,

(2.32)

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

(2.33)

Hence,

(2.34)

which satisfies the hypothesis (2.1) of Theorem 2.1.

(iii)Suppose that . Taking , there exists ( can be chosen small arbitrarily) such that

(2.35)

Hence,

(2.36)

which satisfies the hypothesis (2.1) of Theorem 2.1.

(iv)Suppose that . Taking , there exists a ( can be chosen large arbitrarily) such that

(2.37)

Hence, we have the following two cases.

Case i.

Assume that is bounded, say

(2.38)

for some constant . Taking (since can be chosen large arbitrarily, can be chosen large arbitrarily, too),

(2.39)

Case ii.

Assume that is unbounded, then there exist a ( can be chosen large arbitrarily) and such that

(2.40)

It follows from and (2.37) that

(2.41)

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

(2.42)

Proof.

It follows from Remark 2.3 that there exist two real numbers satisfying

(2.43)

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

(2.44)

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

(2.45)

Proof.

It follows from Remark 2.3 that there exist two real numbers satisfying

(2.46)

Hence, by Theorem 2.1 and Remark 2.2, we see that, for each , (BVP _λ ) has two positive solutions and such that

(2.47)

Thus, we completed the proof.

To illustrate the usage of our results, we present the following examples.

Example 3.1.

Consider the following boundary value problem:

(BVP.1)

Clearly,

(3.1)

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:

(BVP.2)

Clearly,

(3.2)

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:

(BVP.3)

Clearly, if we take and ,

(3.3)

Hence, it follows from Corollary 2.5 that (BVP.3) has two solutions if .