#### Abstract

We will consider the existence of multiple positive periodic solutions for a class of abstract difference equations by using the well-known fixed point theorem (due to Krasnoselskii).

In the past several years, the existence of periodic solutions for first-order functional differential equations has been extensively investigated (see [1–3], and the references therein). In [4–6], the existence of periodic positive solutions for difference equations has been considered. To the best of our knowledge, however, little has been done for the abstract difference equations (see [7–9]). In this note, we will consider this problem. To this end, let be a real Banach space and let be a cone, then a Banach space with a partial ordering = induced by a cone is called an ordered Banach space. On the other hand, we will denote the identity operator defined on by .

In [7–9], the authors considered the existence of periodic solutions for the abstract equation In this note, we will consider the equation where is a -periodic sequence of bounded linear operator defined on and satisfies for , and for any and for any ,?? is an integer valued -periodic sequence, and is a -periodic sequence of bounded functions from to , and is a positive constant.

If (4) has a -periodic solution in , then we have Summing the above equation from to , we have That is, where If (7) has a -periodic solution in , then we have This equation is equivalent to (4). Thus, we have the following result.

Theorem 1. *Assume that and are invertible and . Then is a -periodic solution of (4) if and only if it is a -periodic solution of (7).*

We now assume that for and and that . To obtain our main results, we firstly give a lemma. The proof of that lemma can be found in [10].

Lemma 1. *Let be a Banach space, and let be a cone. Assume are bounded open subsets of such that . Suppose that is a completely continuous operator such that*(1)* for and for or that*(2)* for and for .**Then has a fixed point in .*

Now let be the set of all -periodic sequences in , endowed with the usual linear structure and the norm Then is a Banach space with cone Define a mapping by Then it is easily seen that is completely continuous on (bounded) subset of , and for , so that That is, is contained in .

Lemma 2. *Assume that there exist two positive numbers and such that ,
**
where
**
Then there exists which is a fixed point of and satisfies .*

*Proof. *Let . Assume that , then, for any which satisfies , in view of (15), we have
That is, for . For any which satisfies , we have
That is, we have for . In view of Theorem 1, there exists , which satisfies such that . If , (19) is replaced by in view of (16) and (20) is replaced by in view of (15). The same conclusion is proved. The proof is complete.

Theorem 2. *Suppose (H _{1}), (L_{1}), and (L_{2}) hold. Then for any , (4) has at least two positive periodic solutions, where
*

*Proof. *In view of (H_{1}), we can let . By (L_{1}) and (L_{2}), we see further that . Thus, there exists such that . For any , by the intermediate value theorem, there exist and such that . Thus, we have for and , and for and . On the other hand, in view of (L_{1}) and (L_{2}), we see that there exist and such that for . That is, for and for ). An application of Lemma 2 leads to two distinct solutions of (4).

Theorem 3. *Suppose (H _{2}), (L_{3}), and (L_{4}) hold. Then for any , (4) has at least two positive periodic solutions, where
*

*and is defined by (18).*

*Proof. *Let . Clearly, ,. From (L_{3}) and (L_{4}), we see that . Thus, there exists such that . For any , there exist and such that . Thus we have for and , and for and . On the other hand, in view of (L_{3}), we see that there exists such that for and . Thus we have for and . In view of (L_{4}), we see that there exists such that for and . Let . Then we have for and , where and . An application of Lemma 2 leads to two distinct solutions of (4).

Theorem 4. *Assume that (H _{2}), (L_{5}), and (L_{6}) hold. Then, for each satisfying
*

*or*

*equation (4) has a positive periodic solution.*

*Proof. *Suppose (23) holds. Let such that
Note that , then there exists such that for and . So, for with , we have
Next, since , there exists a such that for and . Let . Then for with ,
In view of Lemma 1, we see that (4) has a positive periodic solution.

The other case is similarly proved.

Our Theorems 1–4 generalize the main results from [5, 6].

If , is a Hilbert space, , and are invertible self-conjugate operator defined on , , ( are self-conjugate operator defined on , then , satisfy conditions of this paper.

As an example, let both and be real bounded sequence, and are also real bounded sequence, where is complete orthonormal set of space . Let for any , then and are both self-conjugate operator, and satisfy all of above conditions.

#### Acknowledgments

The project is partially supported by the Natural Science Foundation of Shanxi Province (2008011002-1) and Shanxi Datong University (2010-B-01, 2009-Y-15) and by High Science and Technology Foundation of Shanxi Province (20101109).