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Abstract and Applied Analysis

Volume 2012 (2012), Article ID 918082, 17 pages

http://dx.doi.org/10.1155/2012/918082

## Existence and Multiplicity of Positive Solutions of a Nonlinear Discrete Fourth-Order Boundary Value Problem

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 16 September 2011; Revised 14 December 2011; Accepted 25 December 2011

Academic Editor: Yuriy Rogovchenko

Copyright © 2012 Ruyun Ma and Yanqiong Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

we show the existence and multiplicity of positive solutions of the nonlinear discrete fourth-order boundary value problem , , , where , is continuous, and is continuous, , . The main tool is the Dancer's global bifurcation theorem.

#### 1. Introduction

It's well known that the fourth order boundary value problem can describe the stationary states of the deflection of an elastic beam with both ends hinged, (it also models a rotating shaft). The existence and multiplicity of positive solutions of the boundary value problem (1.1) have been considered extensively in the literature, see [1–10]. The existence and multiplicity of positive solutions of the parameterized boundary value problem have also been studied by several authors, see Bai and Wang [11], Cid et al. [12], and the references therein.

However, relatively little is known about the corresponding discrete fourth-order problems. Let

Zhang et al. [13], and He and Yu [14] used the fixed point index theory in cones to study the following discrete analogue where denote the fourth forward difference operator and . It has been pointed out in [13, 14] that (1.4), (1.5) are equivalent to the equation of the form: where

Notice that two distinct Green's functions used in (1.6) make the construction of cones and the verification of strong positivity of become more complex and difficult. Therefore, Ma and Xu [15] considered (1.4) with the boundary condition
and introduced the definition of * generalized positive solutions*:

*Definition 1.1. *A function is called a * generalized positive solution* of (1.4), (1.8), if satisfies (1.4), (1.8), and on and on .

*Remark 1.2. *Notice that the fact is a generalized positive solution of (1.4), (1.8) does not means that on . In fact, satisfies

(1) for ;

(2) ;

(3) .

Ma and Xu [15] also applied the fixed point theorem in cones to obtain some results on the existence of generalized positive solutions.

It is the purpose of this paper to show some new results on the existence and multiplicity of generalized positive solutions of (1.4), (1.8) by Dancer's global bifurcation theorem. To wit, we get the following.

Theorem 1.3. *Let , and**
Assume that there exists such that f is nondecreasing on . Then*(i)*(1.4), (1.8) have at least one generalized positive solution if ;*(ii)* (1.4), (1.8) have at least two generalized positive solutions if
where , is defined as (2.3) and is the first eigenvalue of
*

The “dual” of Theorem 1.3 is as follows.

Theorem 1.4. *Let , and**
Assume that there exists such that is nondecreasing on . Then*

(i)* (1.4), (1.8) have at least a generalized positive solution provided
**where *
(ii)*(1.4), (1.8) have at least two generalized positive solutions provided *

The rest of the paper is organized as follows: in Section 2, we present the form of the Green's function of (1.4), (1.8) and its properties, and we enunciate the Dancer's global bifurcation theorem ([16, Corollary 15.2]). In Section 3, we use the Dancer's bifurcation theorem to prove Theorems 1.3 and 1.4 and in Section 4, we finish the paper presenting a couple of illustrative examples.

*Remark 1.5. *For other results on the existence and multiplicity of positive solutions and nodal solutions for fourth-order boundary value problems based on bifurcation techniques, see [17–21].

#### 2. Preliminaries and Dancer's Global Bifurcation Theorem

Lemma 2.1. *Let . Then the linear boundary value problem**
has a solution**
where
*

*Proof. * By a simple summing computation and , we can obtain

This together with , it follows that
Therefore, (2.2) holds.

*Remark 2.2. * It has been pointed out in [15] that (2.1) is equivalent to the summation equation of the form
It is easy to verify that (2.2) and (2.6) are equivalent.

By a similar method in [9], it follows that satisfies
where
Moreover, we have that
here .

Let be a real Banach space with a cone such that . Let us consider the equation:
under the assumptions:

(A1) The operators are compact. Furthermore, is linear, as , and . (A2) The spectral radius of is positive. Denote . (A3) is strongly positive.Dancer's global bifurcation theorem is the following.

Theorem 2.3 (see [16, Corollary 15.2]). *Let
**
If (A1) and (A2) are satisfied, then is a bifurcation point of (2.11) and has an unbounded solution component which passes through . Additionally, if (A3) is satisfied, then and always implies and .*

#### 3. Proof of the Main Results

Before proving Theorem 1.3, we state some preliminary results and notations. Let Then is a Banach space under the normal: See [22] for the detail.

Let Then is normal and has a nonempty interior and .

Let . Then is a Banach space under the norm: Define by setting

It is easy to check that is compact.

Lemma 3.1. *Let with and for some , and
**
Then .*

* Proof. *It is enough to show that there exist two constants such that
In fact, we have from (3.7) that
This together with the relation implies that
Combining (3.9) with (3.10) and the fact that
for some constants , we conclude that (3.8) is true.

Let be such that clearly Let us consider as a bifurcation problem from the trivial solution .

By (1.4), (1.8), it follows that if is one solution of (1.4), (1.8), then satisfies . So, is a solution of (1.4), (1.8), if and only if, solves the operator equation

Now, let be the linear operator:

Let be the operators: respectively. Then Lemma 3.1 yields that is strongly positive. Moreover, [16, Theorem 7.c] implies .

Now, it follows from Theorem 2.3 that there exists a continuum which joins with infinity in and It is easy to check that

Lemma 3.2. *Let with . Then the eigenvalue problems
**
have the principal eigenvalue , such that . Moreover, the corresponding eigenfunctions are positive in .*

*Proof . *Let be the operator

Then Lemma 3.1 yields that is strongly positive. By Krein-Rutman theorem [16, Theorem 7.c] the spectral radius and there exist with on such that
That is, the eigenvalue problems (3.22) have the principal eigenvalues , and is the corresponding eigenfunctions of , .

Next, we prove . Since , it follows that
Therefore, .

Suppose that is a strict subset of and denote the restriction of on . Consider the linear eigenvalue problems: Then we get the following result.

Lemma 3.3. *Let is the principal eigenvalue of (3.17), then (3.26) has only one principal eigenvalue such that .*

*Proof. *It is not difficult to prove that (3.26) has only one principal eigenvalue by Lemma 3.1, and the corresponding eigenfunction on . So we only to verify that .

Let be the corresponding eigenfunction of , we have that
So
Thus .

*Proof of Theorem 1.3. *We divide the proof into three steps.

Let be such that
Then
*Step 1. *We show that there exists a constant such that for all .

Suppose on the contrary that
Let . Then it follows from (3.30) that
Since
there exists a constant , such that
Let be the principal eigenvalue of the linear eigenvalue problems:
Combining (3.31) and (3.34) with the relation (3.32), using Lemma 3.2, we get
This contradicts (3.31). So for all .*Step 2. *We show that joins with .

Assume that there exist and such that

First, we show that

Suppose on the contrary that
for some (independent on ). Then it follows from (3.30) and that
and subsequently, is bounded. This is a contradiction. So, (3.38) holds.

Next, we show that

In fact,
This together with (2.7) imply that (3.41) is valid.

Finally, we have from the facts that and that
Consider the following linear eigenvalue problems:
By Lemma 3.3 and (3.32), (3.44) has a positive principal eigenvalue , and
which contradicts (3.43). Thus .*Step 3. *Fixed such that
Then there exists such that
We show that there is no such that
In fact, if there exists satisfying (3.48), then
for , and subsequently, . Therefore, no satisfies (3.48).Now, combining the conclusions in Steps 2 and 3, using the fact that no satisfies (3.48), it concludes that for every , (1.4), (1.8) has at least two generalized positive solutions in . For arbitrary , we may find satisfying (3.47). So, for every , (1.4), (1.8) has at least two generalized positive solutions in .

*Proof of Theorem 1.4. *We divide the proof into three steps.*Step 1. *We show that there exists a positive constant such that
Suppose on the contrary that there exists such that
Then we have from (3.32), (3.51), and that
However, this contradicts with the fact that for all . Therefore, (3.50) holds.*Step 2. *We show that for any closed interval , there exists such that
Suppose on the contrary that there exists with
Then by (3.38),
and subsequently
This together with (3.32) and and that
However, this contradicts with the fact that
Therefore, (3.53) holds.*Step 3. *Fixed such that
Then there exists such that
We show that there is no such that
Suppose on the contrary that there exists satisfying (3.61). Then for ,
and subsequently, . Therefore, there is no such that (3.61) holds.

Now, combining the conclusions in Steps 2 and 3, using the fact that no satisfying (3.61), it concludes that for every , (1.4), (1.8) has at least two generalized positive solutions in . For arbitrary , we may find satisfying (3.60). So, for every , (1.4), (1.8) has at least two generalized positive solutions in .

#### 4. Some Examples

In this section, we will apply our results to two examples.

For convenience, set , then , .

*Example 4.1. *Let us consider the boundary value problem
where
Clearly, is nondecreasing, , . Take . By a simple computation, it follows that , and , then
So, Theorem 1.3(i) implies that (4.1) has at least one generalized positive solution for
Theorem 1.3(ii) implies that (4.1) has at least two generalized positive solutions for

*Example 4.2. *Let us consider the boundary value problem:
where
Obviously, is nondecreasing in , so , . By a simple computation, it follows that and , . Take . Since , it follows that
Therefore, (i) of Theorem 1.4 implies that(4.6) has at least one generalized positive solution for
(ii) of Theorem 1.4 implies that (4.6) has at least two generalized positive solutions for

#### Acknowledgments

This paper was written when the first author visited Tabuk University, Tabuk, during May 16-June 13, 2011 and he is very thankful to the administration of Tabuk University for providing him the hospitalities during the stay. The second author gratefully acknowledges the partial financial support from the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah. The authors thank the referees for their valuable comments.

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