#### Abstract

We present sufficient conditions ensuring the lower and upper functions on the reversed-order for the periodic difference equations. This enables us to obtain the existence of positive periodic solutions of the second-order difference equation where , and are -periodic functions, .

#### 1. Introduction

The theory of nonlinear difference equations has been widely used to study discrete models appearing in many fields such as computer science, economics, neural network, ecology, and cybernetics, see for example, [1]. Recently, there are many papers to study the existence of positive periodic solutions for second-order difference equations, see [2–7] and their references therein. However, there are few techniques for studying the existence of positive solutions of difference equations with singularity, and thus the results in the field are very rare, see [8–13]. The existence of positive periodic solutions for continuous case has been studied by Torres, see [14, 15].

Let denote the integer set, for with , .

In 2012, Lu and Ma are concerned with the existence of positive periodic solutions of the second-order difference equation as follows: where , , and are -periodic functions, , . Special cases of (1) are In the related literature, it is said that (3) has an attractive singularity, whereas (4) has a repulsive singularity. They use the well-order lower and upper functions of (1) to show the existence of positive -periodic solution of (1), (2), and (3), respectively, see [13].

However, the well-order lower and upper solutions lose their effects to deal with case (4). In this paper, we are devoted to constructing lower and upper functions on the reversed-order for (1) and dealing with the problems (1), (2), and (4), respectively.

The structure of the paper is as follows. Section 2 contains the tools needed in the proofs. In Section 3 we state and prove the main results and develop some corollaries for the equation with a singularity of mixed type. To illustrate the results, some examples are given.

#### 2. Auxiliary Results

Let under the norm . Then is a Banach space.

The proofs of our results rely on the method of upper and lower functions. The following lemmas are classical and can be found, for example, in [3]. We introduce them in a form suitable for us.

Lemma 1. *Let there exist positive functions , , such that
**
and for . Then there exists at least one positive -periodic solution to (1).**A function (resp., ) verifying (6) (resp, (7)) is called lower (resp, upper) function (solution) of (1). When the order between the lower and the upper functions is the inverse, an additional hypothesis is needed.*

*Definition 2. *A function and is said to verify the property () if the implication
holds.

Lemma 3. *Let there exist positive functions , satisfying (6), (7), and , . Moreover, there exists with the property (), such that
**
where for . Then there exists at least one positive -periodic solution to (1).*

*Proof. *From the condition (9), it follows that
That is, the nonlinearity is increasing.

Define the operator as the unique solution of problem (1) as follows:
where is the Green’s function of
As satisfies the property (), it follows that . Now we divide the proof into three steps. *Step 1*. We show
where is a nonempty bounded closed subset in .

In fact, for , set . From the definitions of , , and , combining with (9), we have
Using property (), we get .

Analogously, we can prove that . Thus, (13) holds. *Step 2*. Let , , where , satisfy . Then we claim that
In fact, let , it follows from (10) and (11) that
*Step 3*. The sequences and are obtained by recurrence:
From the results of Steps 1 and 2, it follows that
Moreover, from the definition of , we get

This together with (18), we can easily get that there exists depending only on but not on and , such that , so is bounded in . Similarly, is bounded in . Therefore, we can conclude that and converge uniformly to the extremal solutions of the problem (1). Subsequently, there exists at least one positive -periodic solutions of (1).

Property () is just an anti-maximum principle for the linear operator with periodic boundary conditions, and it is equivalent to have a nonnegative Green function. Reference [7] provides sufficient conditions for to verify property (). In particular, we have the following lemma.

Lemma 4 (see [7]). *Let us assume , and the following conditions holds:
**
Then verifies the property ().*

To finish this section, we give a technical bound on the amplitude of oscillation of a periodic function.

Lemma 5 ([13, Lemma 2.2]). *Given , then
**
where
**
Moreover, (21) is fulfilled as an equality if and only if is a constant function.*

#### 3. The Main Results

For the sake of brevity we will use the following notation throughout the paper: The following theorems are the main results of the paper.

Theorem 6. *Let , , let functions , be such that the equalities
**
are fulfilled and let there exist such that
**
where
**
Moreover, define
**
and assume that verifies the property (). Then problem (1) has at least one positive -periodic solution.*

*Proof. *Let be defined by (28). Then and in view of (24) and (25), we have
Moreover, according to (26) and (27)
Now (29) and (30) imply
Consequently, is an upper function to (1).

Further, we can choose such that
and put
Then , and in view of (24) and (25) we have
Moreover, according to (27) and (32)
Now (34) and (35) imply
Consequently, is a lower function to (1) and according to (30) and (35) we have

Furthermore, note that the function
is nondecreasing for . Therefore we have
whenever for , hence, we get
Thus, the assertion follows from Lemma 3.

*Remark 7. * Note that for every such that , the periodic solution of the equation
is given by the Green formula:
where . Therefore, the periodic functions and with properties (24) and (25) exist and, moreover, are unique up to a constant term, the value of which has no influence on the validity of the condition (26). A similar observation can be made in relation to the formulations of the theorems given below.

Theorem 8. *Let , , , , let functions , be such that (24) and (25) are fulfilled and let there exist such that
**
where and are defined by (27). Moreover, assume that verifies the property (), where is given by (28). Then problem (1) has at least one positive -periodic solution.*

*Proof. *Note that the inequality implies
Therefore, analogously to the proof of Theorem 6, one can show that there exist lower and upper functions , satisfying (37). Consequently, the assertion follows from Lemma 3 with .

Corollary 9. *Let , , , and let be such that (24) is fulfilled. Let
**
where is given by (27) and
**
Moreover, let us define
**
and assume that verifies the property (). Then problem (2) has at least one positive solution.*

*Proof. *Put and
Then the assertion follows from Theorem 8.

Corollary 10. *Let , , let be such that (25) is fulfilled, and let
**
where is defined by (27) and
**
Moreover, if
**
holds. Then problem (4) has at least one positive -periodic solution.*

*Proof. *Put ,
and define a function by (28). Let for . Then (51) guaranties that satisfies the property (). Moreover, (51) yields (25). Therefore, the assertion follows from Theorem 6.

Corollary 11. *Let , and
**
Then problem (4) has at least one positive -periodic solution.*

*Proof. *According to Lemma 5,
Then (53) implies (51). Consequently, the assertion follows from Corollary 10.

*Example 12. *Let us consider the boundary value problem:
where is an integer.

Obviously, as . Let , , . Then , and
Thus, from Corollary 11, problem (55) has at least one positive -periodic solution.

#### Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (no. 11061030, no. 11361054), Gansu provincial National Science Foundation of China (no. 1208RJZA258) and SRFDP (no. 20126203110004).