Discrete Dynamics in Nature and SocietyVolume 2011 (2011), Article ID 870164, 7 pagesdoi:10.1155/2011/870164

Research Article

Existence of Periodic Positive Solutions for Abstract Difference Equations

Shugui Kang, Yaqiong Cui, and Jianmin Guo

Institute of Applied Mathematics, Shanxi Datong University Datong, Shanxi 037009, China

Received 30 April 2011; Accepted 9 June 2011

Academic Editor: M. De la Sen

Copyright © 2011 Shugui Kang 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 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 .

For the sake of convenience, the conditions needed for our criteria are listed as follows., and there exists with such that () for and . and for and . for . for . for . for . for and . for and .

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 (H₁), (L₁), and (L₂) hold. Then for any , (4) has at least two positive periodic solutions, where

Proof. In view of (H₁), we can let . By (L₁) and (L₂), 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 (L₁) and (L₂), 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 (H₂), (L₃), and (L₄) hold. Then for any , (4) has at least two positive periodic solutions, where and is defined by (18).

Proof. Let . Clearly, ,. From (L₃) and (L₄), 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 (L₃), we see that there exists such that for and . Thus we have for and . In view of (L₄), 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 (H₂), (L₅), and (L₆) 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).

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