Positive Solutions for Systems of Second-Order Difference Equations

Abstract

We study the existence and nonexistence of positive solutions of some systems of nonlinear second-order difference equations subject to multipoint boundary conditions which contain some positive constants.

1. Introduction

The mathematical modeling of many nonlinear problems from computer science, economics, mechanical engineering, control systems, biological neural networks, and others leads to the consideration of nonlinear difference equations (see [1, 2]). In the last decades, many authors have investigated such problems by using various methods, such as fixed point theorems, the critical point theory, upper and lower solutions, the fixed point index theory, and the topological degree theory (see, e.g., [3–15]).

In this paper, we consider the system of nonlinear second-order difference equations: with the multipoint boundary conditions where , , is the forward difference operator with stepsize , , , means that for , for all , for all , for all , for all , for all , for all , for all , for all , , , , , and and are positive constants.

Under some assumptions on the functions and , we will prove the existence of positive solutions of problem -. By a positive solution of - we mean a pair of sequences satisfying and with for all and for all , for all . We will also give sufficient conditions for the nonexistence of positive solutions for this problem. System with the multipoint boundary conditions , , , and () has been investigated in [16]. Some systems of difference equations with parameters, subject to multipoint boundary conditions, were studied in [17, 18], by using the Guo-Krasnosel’skii fixed point theorem. We also mention the paper [19], where the authors investigated the existence and multiplicity of positive solutions for the system , , , with the multipoint boundary conditions with , by using some theorems from the fixed point index theory.

In Section 2, we present some auxiliary results which investigate a second-order difference equation subject to multipoint boundary conditions. In Section 3, we will prove our main results, and in Section 4, we will present an example which illustrates the obtained theorems. Our main existence result is based on the Schauder fixed point theorem which we present now.

Theorem 1. Let be a Banach space and a nonempty, bounded, convex, and closed subset. If the operator is completely continuous (continuous and compact, i.e., mapping bounded sets into relatively compact sets), then has at least one fixed point.

2. Auxiliary Results

In this section, we present some auxiliary results from [17] related to the following second-order difference equation with the multipoint boundary conditions:where , , for all , for all , for all , for all , , and .

Lemma 2 (see [17]). If for all , for all , for all , for all , , , , and for all , then the solution of (1) is given by for all , where Green’s function is defined by

Lemma 3 (see [17]). If for all , , for all , , for all , , for all , and , then Green’s function of problem (1) satisfies for all , . Moreover, if for all , then the unique solution , , of problem (1) satisfies for all .

Lemma 4 (see [17]). Assume that for all , , for all , , for all , , for all , and . Then Green’s function of problem (1) satisfies the following inequalities:(a), , where (b)For every , one has  where  and is the largest integer not greater than .

Lemma 5 (see [17]). Assume that for all , , for all , , for all , , for all , , , and for all . Then the solution , of problem (1) satisfies the inequality .

We can also formulate similar results as Lemmas 2–5 above for the discrete boundary value problemwhere , , for all , for all , for all , for all , , , and for all . We denote by , and the corresponding constants and functions for problem (6) defined in a similar manner as , and , respectively.

3. Main Results

We present first the assumptions that we will use in the sequel:() for all , , for all , , for all , , for all , , for all , , for all , , for all , , and for all , .()The constants for all , and there exist such that , .() are continuous functions and there exists such that , for all , where and are defined in Section 2.() are continuous functions and satisfy the conditions

Our first theorem is the following existence result for problem -.

Theorem 6. Assume that assumptions hold. Then problem - has at least one positive solution for and sufficiently small.

Proof. We consider the problemsProblems (7) and (8) have the solutionsrespectively, where and are defined in Section 2. By assumption we obtain for all and for all .
We define the sequences , by where is a solution of -. Then - can be equivalently written aswith the boundary conditionsUsing Green’s functions and from Section 2, a pair is a solution of problem (11)-(12) if and only if it is a solution for the problemwhere , are given in (9).
We consider the Banach space with the norm , , and we define the set .
We also define the operator by For sufficiently small and , by , we deduceThen, by using Lemma 3, we obtain for all and . By Lemma 4, for all , we have Therefore .
Using standard arguments, we deduce that is completely continuous. By Theorem 1, we conclude that has a fixed point . This element together with given by (13) represents a solution for (11)-(12). This shows that our problem - has a positive solution with , ( for all and for all ) for sufficiently small and .

In what follows, we present sufficient conditions for the nonexistence of positive solutions of -.

Theorem 7. Assume that assumptions , , and hold. Then problem - has no positive solution for and sufficiently large.

Proof. We suppose that is a positive solution of -. Then with , is a solution for (11)-(12), where and are the solutions of problems (7) and (8), respectively (given by (9)). By there exists such that and then and . By using Lemma 3 we have for all , and by Lemma 5, we obtain and , where and are defined in Section 2.
Using now (9), we deduce that Therefore, we obtain where and .
We now considerBy using , for defined above, we conclude that there exists such that , for all . We consider and sufficiently large such thatBy , (11), (12), and the above inequalities, we deduce that and .
Now, by using Lemma 4 and the above considerations, we have Therefore, we obtainIn a similar manner, we deduce So, we obtainBy (22) and (24), we obtain , which is a contradiction, because . Then, for and sufficiently large, problem - has no positive solution.

Similar results as Theorems 6 and 7 can be obtained if instead of boundary conditions we have or or where and are positive constants.

For problem –, instead of sequences and from the proof of Theorem 6, the solutions of problemsare respectively. By assumption we obtain , for all , and for all .

For problem –, instead of sequences and from Theorem 6, the solutions of problems (7) and (26) are and , respectively, which satisfy , for all , and for all . For problem –, instead of sequences and from Theorem 6, the solutions of problems (25) and (8) are and , respectively, which satisfy , for all , and for all .

Therefore we also obtain the following results.

Theorem 8. Assume that assumptions hold. Then problem – has at least one positive solution , for all , and for all for and sufficiently small.

Theorem 9. Assume that assumptions , , and hold. Then problem – has no positive solution , for all , and for all for and sufficiently large.

Theorem 10. Assume that assumptions hold. Then problem – has at least one positive solution , for all , and for all for and sufficiently small.

Theorem 11. Assume that assumptions , , and hold. Then problem – has no positive solution , for all , and for all for and sufficiently large.

Theorem 12. Assume that assumptions hold. Then problem – has at least one positive solution , for all , and for all for and sufficiently small.

Theorem 13. Assume that assumptions , , and hold. Then problem – has no positive solution , for all , and for all for and sufficiently large.

4. An Example

We consider , , for all , , , , , , , , , , , , , , , , , , , , , , and . We also consider the functions , , and , for all , with and . We have .

Therefore, we consider the system of second-order difference equations with the multipoint boundary conditions where and are positive constants.

We have , , , and . The functions and are given by

Hence, we deduce that assumptions , , and are satisfied. In addition, by using the above functions and , we obtain , , and then . We choose and if we select , satisfying the conditions , , then we conclude that , for all . For example, if , , then for and the above conditions for and are satisfied. So, assumption is also satisfied. By Theorems 6 and 7 we deduce that problem - has at least one positive solution (here and for all ) for sufficiently small and and no positive solution for sufficiently large and .

By the proofs of Theorems 6 and 7 we can find some intervals for and such that problem - has at least one positive solution, or it has no positive solution. We consider , , , , , (as above), , and . Then , , and the sequences and from (9) are and for all . We also obtain and . If we choose and , then inequalities (15) are satisfied. Because and , for and problem - has at least one positive solution.

Now we choose (the constant from the beginning of the proof of Theorem 7), and then we obtain , , , , , , , and (given by (19)). For , the inequalities and are satisfied for and , respectively. We consider , and then for and , inequalities (20) are satisfied. Therefore, if and , problem - 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 authors thank the referees for their valuable comments and suggestions. The work of R. Luca was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project no. PN-II-ID-PCE-2011-3-0557.