Existence of Positive Solutions for Nonlinear Eigenvalue Problems

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.

We consider the existence of positive solutions of the following two-point 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 .

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.

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 .

1. Chu, J, Chen, H, O'Regan, D: Positive periodic solutions and eigenvalue intervals for systems of second order differential equations. Mathematische Nachrichten. 281(11), 1549–1556 (2008). Publisher Full Text

2. Chu, J, O'Regan, D, Zhang, M: Positive solutions and eigenvalue intervals for nonlinear systems. Proceedings of the Indian Academy of Sciences. 117(1), 85–95 (2007). Publisher Full Text

3. Chu, J, Zhou, Z: Positive solutions and eigenvalues of nonlocal boundary-value problems. Electronic Journal of Differential Equations.(86), 1–9 (2005)

4. Chu, J, Jiang, D: Eigenvalues and discrete boundary value problems for the one-dimensional -Laplacian. Journal of Mathematical Analysis and Applications. 305(2), 452–465 (2005). Publisher Full Text

5. Coffman, CV, Marcus, M: Existence and uniqueness results for semi-linear Dirichlet problems in annuli. Archive for Rational Mechanics and Analysis. 108(4), 293–307 (1989)

6. Cohen, DS, Keller, HB: Some positone problems suggested by nonlinear heat generation. Journal of Mathematics and Mechanics. 16, 1361–1376 (1967)

7. Erbe, LH: Existence of solutions to boundary value problems for second order differential equations. Nonlinear Analysis: Theory, Methods & Applications. 6(11), 1155–1162 (1982). PubMed Abstract | Publisher Full Text

8. Erbe, LH, Hu, S, Wang, H: Multiple positive solutions of some boundary value problems. Journal of Mathematical Analysis and Applications. 184(3), 640–648 (1994). Publisher Full Text

9. Erbe, LH, Wang, H: On the existence of positive solutions of ordinary differential equations. Proceedings of the American Mathematical Society. 120(3), 743–748 (1994). Publisher Full Text

10. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering,p. viii+275. Academic Press, Boston, Mass, USA (1988)

11. Iffland, G: Positive solution of a problem of Emden-Fowler type with a free boundary. SIAM Journal on Mathematical Analysis. 18(2), 283–292 (1987). Publisher Full Text

12. Njoku, FI, Zanolin, F: Positive solutions for two-point BVPs: existence and multiplicity results. Nonlinear Analysis: Theory, Methods & Applications. 13(11), 1329–1338 (1989). PubMed Abstract | Publisher Full Text

13. Santanilla, J: Nonnegative solutions to boundary value problems for nonlinear first and second order ordinary differential equations. Journal of Mathematical Analysis and Applications. 126(2), 397–408 (1987). Publisher Full Text

14. Wong, FH: Existence of positive solutions of singular boundary value problems. Nonlinear Analysis: Theory, Methods & Applications. 21(5), 397–406 (1993). PubMed Abstract | Publisher Full Text

15. Lian, W-C, Wong, FH, Yeh, CC: On the existence of positive solutions of nonlinear second order differential equations. Proceedings of the American Mathematical Society. 124(4), 1117–1126 (1996). Publisher Full Text