Research Article Existence of Periodic Positive Solutions for Abstract Difference Equations

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1 Discrete Dynamics in Nature and Society Volume 2011, Article ID , 7 pages doi: /2011/ 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 , China Correspondence should be addressed to Shugui Kang, dtangshugui@126.com Received 30 April 2011; Accepted 9 June 2011 Academic Editor: M. De la Sen Copyright q 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 wor is properly cited. We will consider the existence of multiple positive periodic solutions for a class of abstract difference equations by using the well-nown fixed point theorem due to Krasnoselsii. In the past several years, the existence of periodic solutions for first-order functional differential equations y t a t y t f t, y t τ t ) 1 has been extensively investigated see 1 3, and the references therein. In 4 6, the existence of periodic positive solutions for difference equations x n 1 a n x n λh n f x n τ n ) 2 has been considered. To the best of our nowledge, 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 X be a real Banach space and let K X be a cone, then a Banach space X with a partial ordering induced by a cone K is called an ordered Banach space. On the other hand, we will denote the identity operator defined on X by I. In 7 9, the authors considered the existence of periodic solutions for the abstract equation x n 1 A n x n F n x n. 3

2 2 Discrete Dynamics in Nature and Society In this note, we will consider the equation x n 1 A n x n λf n xn τ n ), n Z, 4 where {A n } n Z is a T-periodic sequence of bounded linear operator defined on X and satisfies T 1 0 A 1 I 1 A n T 1 0 A 1 I A n for n Z, T 1 0 A 1 I x K and T 1 0 A 1 I 1 x K for any x K, A x K and x K for any x K 0, 1,...,T 1, {τ n } n Z is an integer valued T-periodic sequence, and {F n } n Z is a T-periodic sequence of bounded functions from X to K, andλ is a positive constant. If 4 has a T-periodic solution in X, then we have n 0 x n 1 n 1 0 x n n 0 λfn xn τ n )). 5 Summing the above equation from n to n T 1, we have n 1 0 n T 1 n I )x n s 0 λfs xs τ s )). 6 That is, ) x n λ G n, s F s xs τ s, n Z, 7 where G n, s T 1 0 I ) 1 s. n 8 If 7 has a T-periodic solution in X, then we have x n 1 x n T 1 0 T 1 0 I ) 1 s n T 1 n 1 ) 1 n T 1 I s n λfs xs τ s )) λfs xs τ s ))

3 Discrete Dynamics in Nature and Society 3 ) T 1 1 ) T 1 I A n I I G n, s )) λf s xs τ s 0 T 1 0 ) 1 n T I n 1 0 I ) λfn xn τ n )) A n x n x n λf n xn τ n ). 9 This equation is equivalent to 4. Thus, we have the following result. Theorem 1. Assume that A 0,A 1,...,A T 1 and T 1 0 A 1 I are invertible and n 1 A 1 n 2 A 1 n T 0 A 1 1 A 1 T 1 n Z. Then {x n} n Z x n X is a T-periodic solution of 4 if and only if it is a T-periodic solution of 7. We now assume that 0 <N G n, s M< for n Z and n s n T 1and that σ N/M. To obtain our main results, we firstly give a lemma. The proof of that lemma can be found in 10. Lemma 1. Let E be a Banach space, and let P E be a cone. Assume Ω 1, Ω 2 are bounded open subsets of E such that 0 Ω 1 Ω 1 Ω 2. Suppose that T : P Ω 2 \ Ω 1 P is a completely continuous operator such that 1 Tu u for u P Ω 1 and Tu u for u P Ω 2 or that 2 Tu u for u P Ω 1 and Tu u for u P Ω 2. Then T has a fixed point in P Ω 2 \ Ω 1. For the sae of convenience, the conditions needed for our criteria are listed as follows. H 1 F n C X, X, and there exists {u } X with u 0 such that F n u > θ u θ for n 1, 2,...,T and 1, 2,... H 2 F n C X, X and F n u >θfor u>θand n 1, 2,...,T. L 1 lim u 0 F n u / u for n 1, 2,...,T. L 2 lim u F n u / u for n 1, 2,...,T. L 3 lim u 0 F n u / u 0forn 1, 2,...,T. L 4 lim u F n u / u 0forn 1, 2,...,T. L 5 lim u 0 F n u / u l for n 1, 2,...,T and 0 <l<. L 6 lim u F n u / u L for n 1, 2,...,T and 0 <L<. Now let Ŷ be the set of all T-periodic sequences in X, endowed with the usual linear structure and the norm u max 0 n T 1 u n. 10

4 4 Discrete Dynamics in Nature and Society Then Ŷ is a Banach space with cone Ω { } u {u n } Ŷ : u n θ, u n σ u, n Z. 11 Define a mapping H : Ŷ Ŷ by Hu n λ G n, s )) F s us τ s, n Z. 12 Then it is easily seen that H is completely continuous on bounded subset of Ω, andfor u Ω, Hu n λ G n, s ) Fs us τ s ) λm Fs us τ s 13 so that Hu n λn ) Fs us τ s σ Hu n T 1 14 That is, HΩ is contained in Ω. Lemma 2. Assume that there exist two positive numbers a and b such that a / b, max F n x a 0 x a,0 n T 1 λa, min F n x b σb x b, 0 n T 1 λb, where A max G n, s, 0 n T 1 B n T 1 min 0 n T 1 G n, s Then there exists u Ω which is a fixed point of H and satisfies min{a, b} u max{a, b}.

5 Discrete Dynamics in Nature and Society 5 Proof. Let Ω ξ {w Ω w <ξ}. Assume that a<b, then, for any u Ω which satisfies u a, inviewof 15, we have { } a Hu n λ G n, s λa λa a a. λa 19 That is, Hu u for u Ω a. For any u Ω which satisfies u b, we have { } b Hu n λ G n, s λb λb b b. λb 20 That is, we have Hu u for u Ω b. In view of Theorem 1, there exists u Ω, which satisfies a u b such that Hu u. Ifa>b, 19 is replaced by Hu n b in view of 16 and 20 is replaced by Hu n a 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 λ 0,λ, 4 has at least two positive periodic solutions, where λ 1 A sup r>0 r max 0 u r,0 n T 1 F n u. 21 Proof. In view of H 1, we can let q r r/ A max 0 u r,0 n T 1 F n u. By L 1 and L 2, we see further that lim r 0 q r lim r q r 0. Thus, there exists r 0 > 0 such that q r 0 max r>0 q r λ. For any λ 0,λ, by the intermediate value theorem, there exist a 1 0,r 0 and a 2 r 0, such that q a 1 q a 2 λ. Thus, we have F n u a 1 / λa for u 0,a 1 and n 0, 1, 2,...,T 1, and F n u a 2 / λa for u 0,a 2 and n 0, 1, 2,...,T 1. On the other hand, in view of L 1 and L 2, we see that there exist b 1 0,a 1 and b 2 a 2, such that F n u / u 1/ λσb for u 0,b 1 b 2 σ,. That is, F n u b 1 / λb for u b 1 σ, b 1 and F n u b 2 / λb for u b 2 σ, b 2. 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 λ>λ, 4 has at least two positive periodic solutions, where λ 1 B inf r r>0 min σr u r,0 n T 1 F n u, 22 and B is defined by 18. Proof. Let p r r/ B min σr u r,0 n T 1 F n u. Clearly, p C 0,, 0,. From L 3 and L 4, we see that lim r 0 p r lim r p r. Thus, there exists r 0 > 0 such that p r 0 min r>0 p r λ. For any λ>λ, there exist b 1 0,r 0 and b 2 r 0, such that p b 1 p b 2 λ. Thus we have F n u b 1 / λb for u σb 1,b 1 and n 0, 1,...,T 1, and F n u b 2 / λb for u σb 2,b 2 and n 0, 1,...,T 1. On the other hand, in view of L 3, we see that there exists a 1 0,b 1 such that F n u / u 1/ λa for u 0,a 1 and

6 6 Discrete Dynamics in Nature and Society n 0, 1,...,T 1. Thus we have F n u a 1 / λa for 0 u a 1 and n 0, 1,...,T 1. In view of L 4, we see that there exists a b 2, such that F n u / u 1/ λa for u a, and n 0, 1,...,T 1. Let δ max 0 u a,0 n T 1 F n u. Then we have F n u a 2 / λa for u 0,a 2 and n 0, 1,...,T 1, where a 2 >aand a 2 λδa. 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 λ satisfying 1 σbl <λ< 1 Al 23 or 1 σbl <λ< 1 AL, 24 equation 4 has a positive periodic solution. Proof. Suppose 23 holds. Let ε>0 such that 1 σb L ε λ 1 A l ε. 25 Note that l>0, then there exists H 1 > 0 such that F n u l ε u for 0 < u H 1 and n 0, 1,...,T 1. So, for u Ω with u H 1, we have Hu n λ l ε G n, s u s τ s λ l ε u G n, s 26 λa l ε u u. Next, since L>0, there exists a H 2 > 0 such that F n u L ε u for u H 2 and n 0, 1,...,T 1. Let H 2 max{2h 1, H 2 }. Then for u Ω with u H 2, Hu n λ L ε G n, s us τ s λ L ε σ u G n, s 27 λ L ε σb u u. In view of Lemma 1, weseethat 4 has a positive periodic solution. The other case is similarly proved.

7 Discrete Dynamics in Nature and Society 7 Our Theorems 1 4 generalize the main results from 5, 6. If T 2, X is a Hilbert space, A 0,A 1,and 0 A 1 1 I are invertible self-conjugate operator defined on X, A 0 A 1, 0 A 1 1 I A 0, 0 A 1 1 I A 1 are self-conjugate operator defined on X, then A 0, A 1 satisfy conditions of this paper. As an example, let both {λ n } and {λ n} be real bounded sequence, {μ n } and {μ n} are also real bounded sequence, where 1, λ n / 0, μ n λ n 0, λ n 0, 1 μ n λ, λ n / 0, n 0, λ n {e n } is complete orthonormal set of space l 2 : e n {0,...,0, n 1, 0,...0} n 1, 2,....Let A 0 x ξ n λ n e n, n 1 A 1 x ξ n λ ne n n 1 29 for any x n 1 ξ ne n, then A 0 and A 1 are both self-conjugate operator, and satisfy all of above conditions. Acnowledgments The project is partially supported by the Natural Science Foundation of Shanxi Province and Shanxi Datong University 2010-B-01, 2009-Y-15 andbyhighscience and Technology Foundation of Shanxi Province References 1 S. Kang, B. Shi, and G. Wang, Existence of maximal and minimal periodic solutions for first-order functional differential equations, Applied Mathematics Letters, vol. 23, no. 1, pp , S. Kang and S. S. Cheng, Existence and uniqueness of periodic solutions of mixed monotone functional differential equations, Abstract and Applied Analysis, vol. 2008, Article ID , 13 pages, S. Kang and G. Zhang, Existence of nontrivial periodic solutions for first order functional differential equations, Applied Mathematics Letters, vol. 18, no. 1, pp , R. Y. Zhang, Z. C. Wang, Y. Chen, and J. Wu, Periodic solutions of a single species discrete population model with periodic harvest/stoc, Computers & Mathematics with Applications, vol. 39, no. 1-2, pp , S. Cheng and G. Zhang, Positive periodic solutions of a discrete population model, Functional Differential Equations, vol. 7, no. 3-4, pp , Y. Gao, G. Zhang, and W. G. Ge, Existence of periodic positive solutions for delay difference equations, Systems Science and Mathematical Sciences, vol. 23, no. 2, pp , M. I. Gil and S. S. Cheng, Periodic solutions of a perturbed difference equation, Applicable Analysis, vol. 76, no. 3-4, pp , M. Gil, Periodic solutions of abstract difference equations, Applied Mathematics E-Notes, vol. 1, pp , M I. Gil, S. Kang, and G. Zhang, Positive periodic solutions of abstract difference equations, Applied Mathematics E-Notes, vol. 4, pp , D. J. Guo and V. Lashmiantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Orlando, Fla, USA, 1988.

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