Abstract and Applied Analysis

Volume 2014 (2014), Article ID 578672, 10 pages

http://dx.doi.org/10.1155/2014/578672

Research Article

The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions

Yanping Guo,¹ Xuefei Lv,² Yude Ji,² and Yongchun Liang¹

¹School of Electrical Engineering, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, China

²College of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, China

Received 10 March 2014; Accepted 13 May 2014; Published 17 July 2014

Academic Editor: Abdul Latif

Copyright © 2014 Yanping Guo et al. 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 consider the fourth-order difference equation: ,   subject to the boundary conditions: , , , where and for ,   is continuous. is nonnegative ; is nonnegative for . Using fixed point theorem of cone expansion and compression of norm type and Hölder’s inequality, various existence, multiplicity, and nonexistence results of positive solutions for above problem are derived, which extends and improves some known recent results.

1. Introduction

Boundary value problems (BVPs) for ordinary differential equations arise in different areas of applied mathematics and so on. The existence of solutions for second order and higher order nonlocal boundary value problems has been studied by several authors; for example, see [1–11] and the references therein. Many authors have also discussed the existence of positive solutions for higher order difference equation BVPs [12, 13], by using fixed point theorem of cone expansion and compression of norm type, sufficient conditions for the existence of positive solutions for fourth-order and third-order difference equation BVPs are established, respectively. Recently, there has been much attention on the existence of positive solutions for the fourth-order differential equations with integral boundary conditions [14–17]. In [17], Zhang and Ge considered the differential equation BVP:

where, is symmetric on , for some , and it is symmetric on the interval,is continuous, andfor all, andare nonnegative, symmetric on. The authors made use of fixed point theorem of cone expansion and compression of norm type and Hölder inequality to prove the existence of positive solutions for the above problem.

Motivated by the above works, we intend to study the existence and nonexistence of positive solutions of the following fourth-order difference BVP with sum form boundary conditions:

Throughout this paper, we make the following assumptions:

 is symmetric onand; there is asuch thatfor;is continuous;is nonnegative onand,is nonnegative onand, where,;,.

In order to establish the existence of positive solutions of the problem (2), we need the following definitions, theorem, and lemma.

Definition 1. A functionis said to be a solution of problem (2) ifsatisfying BVP (2).

Definition 2 (see [18]). Letbe a real Banach space over  . A nonempty closed setis said to be a cone provided that

(i)for alland alland(ii)implies.

Every coneinduces an ordering ingiven byif and only if.

Theorem 3 (see [18]). Letbe a cone in a real Banach space. Assume thatare bounded open sets inwith. If

is completely continuous such that either

(i),and,, or(ii),and,, thenhas at least one fixed point in.

Lemma 4 (Hölder). Suppose thatis a real-valued column; let

, satisfy the conditionwhich are called conjugate exponent, andfor. If, then

which can be recorded as

2. Preliminaries

Let; is a real Banach space with the normdefined by

Letbe a cone of, and

where.

In our main results, we will use the following lemmas and properties.

Lemma 5. Suppose thatandhold and; then, for all, the BVP

has unique solutiongiven by

where

where,.

Proposition 6. Assume that; then

Proposition 7. Assume that; then

where

Proposition 8. Suppose that; then

where

Proof. From Lemma 5 and Proposition 7, we have

On the other hand, using, we get

Lemma 9. Suppose that, for all; the BVP

has unique solutiongiven by

where

Proof. From the properties of difference operator, we can get

then we have

It can imply that

Let; we have

That is

thus,

We can get

From the boundary conditions, we have

thus,

Because

we have

Thus, we get

where

Multiplying the above equation with, summing them fromto, we have

So, we can get

Proposition 10. Assume that; then

Proposition 11. For, one has

where.

Proposition 12. If, then, for all, one has

where, .

We construct a cone onby

where

Obviously,is a closed convex cone of.

Define an operatoras

Let

Then we can obtain the following properties.

Proposition 13. Ifandhold, then

Proposition 14. Ifandhold, then, for all , one has

Lemma 15. Suppose thathold; ifis a solution of the equation

thenis a solution of the BVP (2).

Lemma 16. Assume thathold; thenandis completely continuous.

Proof. From above works, for all, we have

Because

That is to sayfor.

On the other hand, for, we have

Similarly, we can get

So,and. It is easy to see thatis completely continuous.

3. The Existence of One Positive Solution

In this part, we apply Theorem 3 and Lemma 4 to prove the existence of one positive solution for BVP (2). We need consider the following cases for:,, and.

Let

wheredenotesor, and

Remark 17. If we only consider the case, then we can take

Firstly, the following theorem deals with the case .

Theorem 18. Suppose thathold. If there exist two constants, withsuch that

for allandfor all; orfor allandfor all, then BVP (2) has at least one positive solution.

Proof. We only consider condition. For, from the definition ofwe obtain that

So, for, we have, which implies that. Thus, for, fromwe get

that is,implies that

On the other hand, for, we have, which implies that; therefore for, fromwe have

that is,implies that

From the above works, we apply (i) of Theorem 3 to yield thathas a fixed point, and. Thus, it follows that BVP (2) has at least one positive solution.

The following theorem deals with the case.

Theorem 19. Suppose thathold and or holds. Then BVP (2) has at least one positive solution.

Proof. Letreplace and repeat the argument of Theorem 18.

Finally we consider the case.

Theorem 20. Suppose thathold and or holds. Then BVP (2) has at least one positive solution.

Proof. Similar to the proof of Theorem 18, for, we have

So, for, we have. And from the proof of Theorem 18,for. Thus Theorem 20 is proved.

Theorem 21. Assume thathold. If one of the following conditions is satisfied

and(particularly,and);and (particularly,and), then BVP (2) has at least one positive solution.

Proof. We only consider the case. The proof of caseis similar to case. Considering, there existssuch thatfor,, wheresatisfies. Then, for,, fromwe have

that is,implies that.

Next, considering, then there existssuch that

wheresatisfies; assume that

then

Choosing.

Thus, for, we obtain

that is,, we have.

From above works and (ii) of Theorem 3 we know thathas a fixed point, , and, . Thus, it implies that BVP (2) has a positive solution.

Theorem 22. Suppose thathold. If there exist two constantswithsuch that

is nondecreasing onfor all;andfor all, then the BVP (2) has at least one positive solution.

Proof. For, from the definition ofwe have . So, for, we have, from conditions and , we obtain

that is, for, we can imply that

On the other hand, for, we have, , this together withand, we have

that is, for, we can imply that

From above works and Theorem 3, we prove thathas a fixed point,, and,. So the BVP (2) has at least one positive solution.

4. The Existence of Multiple Positive Solutions

Theorem 23. Assume thathold, and the following two conditions hold:

and (particularly,);there existssuch that.

Then BVP (2) has at least two positive solutions, , which satisfy

Proof. We can take two constants, and suppose that. If; from the proof of Theorem 21 we have

If, similarly, we have

On the other hand, from, for, we have

that is,

Applying Theorem 3, we can prove thathas a fixed pointand a fixed point. Thus, we prove that BVP (2) has two positive solutions , . From above formula we obtainand. So.

Remark 24. From the proof of Theorem 23 we obtain that ifholds and(or), then BVP (2) has at least one positive solution. It satisfies(or).

Using a similar method we can obtain the following results.

Theorem 25. Suppose thathold, and the following two conditions hold

and;there existssuch that, then BVP (2) has at least two positive solutions, which satisfy

Remark 26. Ifholds and(or), then BVP (2) has at least one positive solutionsatisfying(or).

Theorem 27. Assume thathold. If there existpositive numbers, withsuch that

, for andfor ,; or, for andfor , , then BVP (2) has at leastpositive solutionssatisfying, .

Theorem 28. Assume thathold. If there existpositive numbers, withsuch that

is nondecreasing onfor all;, and, then BVP (2) has at leastpositive solutionssatisfying.

5. The Nonexistence of Positive Solution

Theorem 29. Suppose thathold, and, for all, ; then BVP (2) has no positive solution.

Proof. Assume thatis a positive solution of BVP (2). Then, , and

which is a contradiction.

Similarly, we can get the following result.

Theorem 30. Suppose thathold and , for all , ; then BVP (2) has no positive solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was supported by the Natural Science Foundation of China (11371120), the Natural Science Foundation of Hebei Province (A2013208147) and the Education Department of Hebei Province Science and Technology Research Project (Z2014095).

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