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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 918082, 17 pages
http://dx.doi.org/10.1155/2012/918082
Research Article

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 [110]. 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 [1721].

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.

References

  1. R. P. Agarwal, “On fourth order boundary value problems arising in beam analysis,” Differential and Integral Equations, vol. 2, no. 1, pp. 91–110, 1989. View at Zentralblatt MATH
  2. A. Cabada, “The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 185, no. 2, pp. 302–320, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. C. De Coster and L. Sanchez, “Upper and lower solutions, Ambrosetti-Prodi problem and positive solutions for fourth order O.D.E,” Rivista di Matematica Pura ed Applicata, no. 14, pp. 57–82, 1994. View at Zentralblatt MATH
  4. Z. Liu and F. Li, “Multiple positive solutions of nonlinear two-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 203, no. 3, pp. 610–625, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Z.-C. Hao and L. Debnath, “On eigenvalue intervals and eigenfunctions of fourth-order singular boundary value problems,” Applied Mathematics Letters, vol. 18, no. 5, pp. 543–553, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. Guo and Y. Gao, “The method of upper and lower solutions for a Lidstone boundary value problem,” Czechoslovak Mathematical Journal, vol. 55(130), no. 3, pp. 639–652, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Chu and D. O'Regan, “Positive solutions for regular and singular fourth-order boundary value problems,” Communications in Applied Analysis, vol. 10, no. 2-3, pp. 185–199, 2006. View at Zentralblatt MATH
  8. G. Han and Z. Xu, “Multiple solutions of some nonlinear fourth-order beam equations,” Nonlinear Analysis, vol. 68, no. 12, pp. 3646–3656, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J. R. L. Webb, G. Infante, and D. Franco, “Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions,” Proceedings of the Royal Society of Edinburgh A, vol. 138, no. 2, pp. 427–446, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. Ma and H. Wang, “On the existence of positive solutions of fourth-order ordinary differential equations,” Applicable Analysis, vol. 59, no. 1–4, pp. 225–231, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Z. Bai and H. Wang, “On positive solutions of some nonlinear fourth-order beam equations,” Journal of Mathematical Analysis and Applications, vol. 270, no. 2, pp. 357–368, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. A. Cid, D. Franco, and F. Minhós, “Positive fixed points and fourth-order equations,” Bulletin of the London Mathematical Society, vol. 41, no. 1, pp. 72–78, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. B. Zhang, L. Kong, Y. Sun, and X. Deng, “Existence of positive solutions for BVPs of fourth-order difference equations,” Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 583–591, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Z. He and J. Yu, “On the existence of positive solutions of fourth-order difference equations,” Applied Mathematics and Computation, vol. 161, no. 1, pp. 139–148, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. Ma and Y. Xu, “Existence of positive solution for nonlinear fourth-order difference equations,” Computers & Mathematics with Applications, vol. 59, no. 12, pp. 3770–3777, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems, Springer, New York, NY, USA, 1986, Translated from the German by Peter R. Wadsac.
  17. B. P. Rynne, “Infinitely many solutions of superlinear fourth order boundary value problems,” Topological Methods in Nonlinear Analysis, vol. 19, no. 2, pp. 303–312, 2002. View at Zentralblatt MATH
  18. R. Ma, “Existence of positive solutions of a fourth-order boundary value problem,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 1219–1231, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. R. Ma and J. Xu, “Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem,” Nonlinear Analysis, vol. 72, no. 1, pp. 113–122, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. R. Ma, “Nodal solutions of boundary value problems of fourth-order ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 424–434, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. R. Ma, “Nodal solutions for a fourth-order two-point boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 314, no. 1, pp. 254–265, 2006. View at Zentralblatt MATH
  22. Y. Xu, C. Gao, and R. Ma, “Solvability of a nonlinear fourth-order discrete problem at resonance,” Applied Mathematics and Computation, vol. 216, no. 2, pp. 662–670, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH