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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

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 [111] 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 [1417]. 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, and for all , and are 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 on and ; there is a such that for ; is continuous; is nonnegative on and , is nonnegative on and , 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 function is said to be a solution of problem (2) if satisfying BVP (2).

Definition 2 (see [18]). Let be a real Banach space over   . A nonempty closed set is said to be a cone provided that(i) for all and all and(ii) implies .

Every cone induces an ordering in given by if and only if .

Theorem 3 (see [18]). Let be a cone in a real Banach space . Assume that are bounded open sets in with . If is completely continuous such that either(i) , and , ,or(ii) , and , ,then has at least one fixed point in .

Lemma 4 (Hölder). Suppose that is a real-valued column; let , satisfy the condition which are called conjugate exponent, and for . If , then which can be recorded as

2. Preliminaries

Let ; is a real Banach space with the norm defined by

Let be a cone of , and where .

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

Lemma 5. Suppose that and hold and ; then, for all , the BVP has unique solution given 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 solution given 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 from to , 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 on by where Obviously, is a closed convex cone of .

Define an operator as Let Then we can obtain the following properties.

Proposition 13. If and hold, then

Proposition 14. If and hold, then, for all , one has

Lemma 15. Suppose that hold; if is a solution of the equation then is a solution of the BVP (2).

Lemma 16. Assume that hold; then and is completely continuous.

Proof. From above works, for all , we have Because That is to say for .

On the other hand, for , we have Similarly, we can get So, and . It is easy to see that is 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 where denotes or , and

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

Firstly, the following theorem deals with the case .

Theorem 18. Suppose that hold. If there exist two constants , with such that for all and for all ;or for all and for all , then BVP (2) has at least one positive solution.

Proof. We only consider condition . For , from the definition of we obtain that So, for , we have , which implies that . Thus, for , from we get that is, implies that

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

From the above works, we apply (i) of Theorem 3 to yield that has 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 that hold and or holds. Then BVP (2) has at least one positive solution.

Proof. Let replace and repeat the argument of Theorem 18.

Finally we consider the case .

Theorem 20. Suppose that hold 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 that hold. 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 case is similar to case . Considering , there exists such that for , , where satisfies . Then, for , , from we have that is, implies that .

Next, considering , then there exists such that where satisfies ; assume that then Choosing .

Thus, for , we obtain that is, , we have .

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

Theorem 22. Suppose that hold. If there exist two constants with such that is nondecreasing on for all ; and for all ,then the BVP (2) has at least one positive solution.

Proof. For , from the definition of we 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 with and , we have that is, for , we can imply that

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

4. The Existence of Multiple Positive Solutions

Theorem 23. Assume that hold, and the following two conditions hold: and (particularly, ); there exists such 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 that has a fixed point and a fixed point . Thus, we prove that BVP (2) has two positive solutions , . From above formula we obtain and . So .

Remark 24. From the proof of Theorem 23 we obtain that if holds 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 that hold, and the following two conditions hold and ; there exists such that ,then BVP (2) has at least two positive solutions , which satisfy

Remark 26. If holds and (or ), then BVP (2) has at least one positive solution satisfying (or ).

Theorem 27. Assume that hold. If there exist positive numbers , with such that , for and for , ; or , for and for , ,then BVP (2) has at least positive solutions satisfying , .

Theorem 28. Assume that hold. If there exist positive numbers , with such that is nondecreasing on for all ; , and ,then BVP (2) has at least positive solutions satisfying .

5. The Nonexistence of Positive Solution

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

Proof. Assume that is a positive solution of BVP (2). Then , , and which is a contradiction.

Similarly, we can get the following result.

Theorem 30. Suppose that hold and , for all