Existence of Periodic Positive Solutions for Abstract Difference Equations

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).

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In the past several years, the existence of periodic solutions for first-order functional differential equations 𝑦^′(𝑡)=−𝑎(𝑡)𝑦(𝑡)+𝑓𝑡,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_𝑛𝑥_𝑛+𝜆ℎ_𝑛𝑓𝑥_{𝑛−𝜏(𝑛)} (2) 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 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_𝑛𝑥_𝑛+𝐹_𝑛𝑥_𝑛. (3) In this note, we will consider the equation 𝑥_𝑛+1=𝐴_𝑛𝑥_𝑛+𝜆𝐹_𝑛𝑥_{𝑛−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_𝑘=0𝐴_𝑘⁻¹−𝐼)⁻¹𝐴_𝑛(∏^𝑇−1_𝑘=0𝐴_𝑘⁻¹−𝐼)=𝐴_𝑛 for 𝑛∈𝑍, (∏^𝑇−1_𝑘=0𝐴_𝑘⁻¹−𝐼)𝑥∈𝐾 and (∏^𝑇−1_𝑘=0𝐴_𝑘⁻¹−𝐼)⁻¹𝑥∈𝐾 for any 𝑥∈𝐾,𝐴_𝑘𝑥∈𝐾 and 𝐴_𝑘⁻¹𝑥∈𝐾 for any 𝑥∈𝐾(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, and 𝜆 is a positive constant.

If (4) has a 𝑇-periodic solution in 𝑋, then we have ^𝑛_𝑘=0𝐴_𝑘⁻¹𝑥_𝑛+1−^𝑛−1_𝑘=0𝐴_𝑘⁻¹𝑥_𝑛=^𝑛_𝑘=0𝐴_𝑘⁻¹𝜆𝐹_𝑛𝑥_{𝑛−𝜏(𝑛)}. (5) Summing the above equation from 𝑛 to 𝑛+𝑇−1, we have ^𝑛−1_𝑘=0𝐴_𝑘^{−1𝑛+𝑇−1}_𝑘=𝑛𝐴_𝑘⁻¹−𝐼𝑥_𝑛=^{𝑛+𝑇−1𝑠}_{𝑠=𝑛𝑘=0}𝐴_𝑘⁻¹𝜆𝐹_𝑠𝑥_{𝑠−𝜏(𝑠)}. (6) That is, 𝑥_𝑛=𝜆^{𝑛+𝑇−1}_𝑠=𝑛𝐺(𝑛,𝑠)𝐹_𝑠𝑥_{𝑠−𝜏(𝑠)},𝑛∈𝑍, (7) where 𝐺(𝑛,𝑠)=^𝑇−1_𝑘=0𝐴_𝑘⁻¹−𝐼^𝑠−1_𝑘=𝑛𝐴_𝑘⁻¹. (8) If (7) has a 𝑇-periodic solution in 𝑋, then we have 𝑥_𝑛+1−𝑥_𝑛=^𝑇−1_𝑘=0𝐴_𝑘⁻¹−𝐼^{−1𝑛+𝑇𝑠}_{𝑠=𝑛+1𝑘=𝑛+1}𝐴_𝑘⁻¹𝜆𝐹_𝑠𝑥_{𝑠−𝜏(𝑠)}−^𝑇−1_𝑘=0𝐴_𝑘⁻¹−𝐼^{−1𝑛+𝑇−1𝑠}_{𝑠=𝑛𝑘=𝑛}𝐴_𝑘⁻¹𝜆𝐹_𝑠𝑥_{𝑠−𝜏(𝑠)}=^𝑇−1_𝑘=0𝐴_𝑘⁻¹−𝐼⁻¹𝐴_𝑛^𝑇−1_𝑘=0𝐴_𝑘⁻¹−𝐼−𝐼^{𝑛+𝑇−1}_𝑠=𝑛𝐺(𝑛,𝑠)𝜆𝐹_𝑠𝑥_{𝑠−𝜏(𝑠)}+^𝑇−1_𝑘=0𝐴_𝑘⁻¹−𝐼^{−1𝑛+𝑇}_𝑘=𝑛+1𝐴_𝑘⁻¹−𝐼𝜆𝐹_𝑛𝑥_{𝑛−𝜏(𝑛)}=𝐴_𝑛𝑥_𝑛−𝑥_𝑛+𝜆𝐹_𝑛𝑥_{𝑛−𝜏(𝑛)}. (9) This equation is equivalent to (4). Thus, we have the following result.

Theorem 1. Assume that 𝐴₀,𝐴₁,…,𝐴_𝑇−1 and (∏^𝑇−1_𝑘=0𝐴_𝑘⁻¹−𝐼) are invertible and 𝐴⁻¹_𝑛+1𝐴⁻¹_𝑛+2⋯𝐴⁻¹_𝑛+𝑇=𝐴₀⁻¹𝐴₁⁻¹⋯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−1 and 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 Ω₁,Ω₂ are bounded open subsets of 𝐸 such that 0∈Ω₁⊂Ω₁⊂Ω₂. Suppose that 𝑇∶𝑃∩(Ω₂⧵Ω₁)→𝑃 is a completely continuous operator such that(1)‖𝑇𝑢‖≤‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω₁ and ‖𝑇𝑢‖⩾‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω₂ or that(2)‖𝑇𝑢‖⩾‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω₁ and ‖𝑇𝑢‖≤‖𝑢‖ for 𝑢∈𝑃∩𝜕Ω₂.
Then 𝑇 has a fixed point in 𝑃∩(Ω₂⧵Ω₁).

For the sake of convenience, the conditions needed for our criteria are listed as follows.(H₁)𝐹_𝑛∈𝐶(𝑋,𝑋), and there exists {𝑢_𝑘}⊂𝑋 with ‖𝑢_𝑘‖→0 such that 𝐹_𝑛(𝑢_𝑘)>𝜃 (𝑢_𝑘⩾𝜃) for 𝑛=1,2,…,𝑇 and 𝑘=1,2,….(H₂)𝐹_𝑛∈𝐶(𝑋,𝑋) and 𝐹_𝑛(𝑢)>𝜃 for 𝑢>𝜃 and 𝑛=1,2,…,𝑇.(L₁)lim_{‖𝑢‖→0}‖𝐹_𝑛(𝑢)‖/‖𝑢‖=∞ for 𝑛=1,2,…,𝑇.(L₂)lim_{‖𝑢‖→∞}‖𝐹_𝑛(𝑢)‖/‖𝑢‖=∞ for 𝑛=1,2,…,𝑇.(L₃)lim_{‖𝑢‖→0}‖𝐹_𝑛(𝑢)‖/‖𝑢‖=0 for 𝑛=1,2,…,𝑇.(L₄)lim_{‖𝑢‖→∞}‖𝐹_𝑛(𝑢)‖/‖𝑢‖=0 for 𝑛=1,2,…,𝑇.(L₅)lim_{‖𝑢‖→0}‖𝐹_𝑛(𝑢)‖/‖𝑢‖=𝑙 for 𝑛=1,2,…,𝑇 and 0<𝑙<∞.(L₆)lim_{‖𝑢‖→∞}‖𝐹_𝑛(𝑢)‖/‖𝑢‖=𝐿 for 𝑛=1,2,…,𝑇 and 0<𝐿<∞.

Now let 𝑌 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) Then 𝑌 is a Banach space with cone Ω=𝑢=𝑢_𝑛∈𝑌∶𝑢_𝑛⩾𝜃,𝑢_𝑛⩾𝜎‖𝑢‖,𝑛∈𝑍. (11) Define a mapping 𝐻∶𝑌→𝑌 by (𝐻𝑢)_𝑛=𝜆^{𝑛+𝑇−1}_𝑠=𝑛𝐺(𝑛,𝑠)𝐹_𝑠𝑢_{𝑠−𝜏(𝑠)},𝑛∈𝑍. (12) Then it is easily seen that 𝐻 is completely continuous on (bounded) subset of Ω, and for 𝑢∈Ω, (𝐻𝑢)_𝑛≤𝜆^{𝑛+𝑇−1}_𝑠=𝑛‖𝐺(𝑛,𝑠)‖⋅𝐹_𝑠𝑢_{𝑠−𝜏(𝑠)}≤𝜆𝑀^{𝑛+𝑇−1}_𝑠=𝑛𝐹_𝑠𝑢_{𝑠−𝜏(𝑠)} (13) so that (𝐻𝑢)_𝑛⩾𝜆𝑁^{𝑛+𝑇−1}_𝑠=𝑛𝐹_𝑠𝑢_{𝑠−𝜏(𝑠)}⩾𝜎‖𝐻𝑢‖ (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_{‖𝑥‖0≤≤𝑎,0≤𝑛≤𝑇−1}𝐹_𝑛(𝑥≤𝑎)𝜆𝐴,min_{‖𝑥‖𝜎𝑏≤≤𝑏,0≤𝑛≤𝑇−1}𝐹_𝑛(𝑥≥𝑏),𝜆𝐵 (15)(16) where 𝐴=max_{0≤𝑛≤𝑇−1}^{𝑛+𝑇−1}_𝑠=𝑛‖𝐺(𝑛,𝑠)‖,𝐵=min_{0≤𝑛≤𝑇−1}^{𝑛+𝑇−1}_𝑠=𝑛‖𝐺(𝑛,𝑠)‖. (17)(18) Then there exists 𝑢∈Ω which is a fixed point of 𝐻 and satisfies min{𝑎,𝑏}≤‖𝑢‖≤max{𝑎,𝑏}.

Proof. Let Ω_𝜉={𝑤∈Ω∣‖𝑤‖<𝜉}. Assume that 𝑎<𝑏, then, for any 𝑢∈Ω which satisfies ‖𝑢‖=𝑎, in view of (15), we have (𝐻𝑢)_𝑛≤𝜆^{𝑛+𝑇−1}_𝑠=𝑛‖𝐺(⋅𝑎𝑛,𝑠)‖𝑎𝜆𝐴≤𝜆𝐴⋅𝜆𝐴=𝑎. (19) That is, ‖𝐻𝑢‖≤‖𝑢‖ for 𝑢∈𝜕Ω_𝑎. For any 𝑢∈Ω which satisfies ‖𝑢‖=𝑏, we have (𝐻𝑢)_𝑛⩾𝜆^{𝑛+𝑇−1}_𝑠=𝑛‖𝐺(⋅𝑏𝑛,𝑠)‖𝑏𝜆𝐵⩾𝜆𝐵⋅𝜆𝐵=𝑏. (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. If 𝑎>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 (H₁), (L₁), and (L₂) hold. Then for any 𝜆∈(0,𝜆^∗), (4) has at least two positive periodic solutions, where 𝜆^∗=1𝐴sup_𝑟>0𝑟max_{0≤‖𝑢‖≤𝑟,0≤𝑛≤𝑇−1}𝐹_𝑛(𝑢.) (21)

Proof. In view of (H₁), we can let 𝑞(𝑟)=𝑟/(𝐴max_{0≤‖𝑢‖≤𝑟,0≤𝑛≤𝑇−1}‖𝐹_𝑛(𝑢)‖). By (L₁) and (L₂), we see further that lim_𝑟→0𝑞(𝑟)=lim_𝑟→∞𝑞(𝑟)=0. Thus, there exists 𝑟₀>0 such that 𝑞(𝑟₀)=max_𝑟>0𝑞(𝑟)=𝜆^∗. For any 𝜆∈(0,𝜆^∗), by the intermediate value theorem, there exist 𝑎₁∈(0,𝑟₀) and 𝑎₂∈(𝑟₀,∞) such that 𝑞(𝑎₁)=𝑞(𝑎₂)=𝜆. Thus, we have ‖𝐹_𝑛(𝑢)‖≤𝑎₁/(𝜆𝐴) for ‖𝑢‖∈[0,𝑎₁] and 𝑛=0,1,2,…,𝑇−1, and ‖𝐹_𝑛(𝑢)‖≤𝑎₂/(𝜆𝐴) for ‖𝑢‖∈[0,𝑎₂] and 𝑛=0,1,2,…,𝑇−1. On the other hand, in view of (L₁) and (L₂), we see that there exist 𝑏₁∈(0,𝑎₁) and 𝑏₂∈(𝑎₂,∞) such that ‖𝐹_𝑛(𝑢)‖/‖𝑢‖⩾1/(𝜆𝜎𝐵) for ‖𝑢‖∈(0,𝑏₁]∪[𝑏₂𝜎,∞). That is, ‖𝐹_𝑛(𝑢)‖⩾𝑏₁/(𝜆𝐵) for ‖𝑢‖∈[𝑏₁𝜎,𝑏₁] and ‖𝐹_𝑛(𝑢)‖⩾𝑏₂/(𝜆𝐵) for ‖𝑢‖∈[𝑏₂𝜎,𝑏₂]). An application of Lemma 2 leads to two distinct solutions of (4).

Theorem 3. Suppose (H₂), (L₃), and (L₄) hold. Then for any 𝜆>𝜆^∗∗, (4) has at least two positive periodic solutions, where 𝜆^∗∗=1𝐵inf_𝑟>0𝑟min_{𝜎𝑟≤‖𝑢‖≤𝑟,0≤𝑛≤𝑇−1}𝐹_𝑛(𝑢,) (22) and 𝐵 is defined by (18).

Proof. Let 𝑝(𝑟)=𝑟/(𝐵min_{𝜎𝑟≤‖𝑢‖≤𝑟,0≤𝑛≤𝑇−1}‖𝐹_𝑛(𝑢)‖). Clearly, 𝑝∈𝐶((0,∞),(0,∞)). From (L₃) and (L₄), we see that lim_𝑟→0𝑝(𝑟)=lim_𝑟→∞𝑝(𝑟)=∞. Thus, there exists 𝑟₀>0 such that 𝑝(𝑟₀)=min_𝑟>0𝑝(𝑟)=𝜆^∗∗. For any 𝜆>𝜆^∗∗, there exist 𝑏₁∈(0,𝑟₀) and 𝑏₂∈(𝑟₀,∞) such that 𝑝(𝑏₁)=𝑝(𝑏₂)=𝜆. Thus we have ‖𝐹_𝑛(𝑢)‖⩾𝑏₁/(𝜆𝐵) for ‖𝑢‖∈[𝜎𝑏₁,𝑏₁] and 𝑛=0,1,…,𝑇−1, and ‖𝐹_𝑛(𝑢)‖⩾𝑏₂/(𝜆𝐵) for ‖𝑢‖∈[𝜎𝑏₂,𝑏₂] and 𝑛=0,1,…,𝑇−1. On the other hand, in view of (L₃), we see that there exists 𝑎₁∈(0,𝑏₁) such that ‖𝐹_𝑛(𝑢)‖/‖𝑢‖≤1/(𝜆𝐴) for ‖𝑢‖∈(0,𝑎₁] and 𝑛=0,1,…,𝑇−1. Thus we have ‖𝐹_𝑛(𝑢)‖≤𝑎₁/(𝜆𝐴) for 0≤‖𝑢‖≤𝑎₁ and 𝑛=0,1,…,𝑇−1. In view of (L₄), we see that there exists 𝑎∈(𝑏₂,∞) 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)‖≤𝑎₂/(𝜆𝐴) for ‖𝑢‖∈[0,𝑎₂] and 𝑛=0,1,…,𝑇−1, where 𝑎₂>𝑎 and 𝑎₂⩾𝜆𝛿𝐴. An application of Lemma 2 leads to two distinct solutions of (4).

Theorem 4. Assume that (H₂), (L₅), and (L₆) hold. Then, for each 𝜆 satisfying 11𝜎𝐵𝐿<𝜆<𝐴𝑙 (23) or 11𝜎𝐵𝑙<𝜆<,𝐴𝐿 (24)equation (4) has a positive periodic solution.

Proof. Suppose (23) holds. Let 𝜀>0 such that 1()1𝜎𝐵𝐿−𝜀≤𝜆≤𝐴().𝑙+𝜀 (25) Note that 𝑙>0, then there exists 𝐻₁>0 such that ‖𝐹_𝑛(𝑢)‖≤(𝑙+𝜀)‖𝑢‖ for 0<‖𝑢‖≤𝐻₁ and 𝑛=0,1,…,𝑇−1. So, for 𝑢∈Ω with ‖𝑢‖=𝐻₁, we have (𝐻𝑢)_𝑛≤𝜆(𝑙+𝜀)^{𝑛+𝑇−1}_𝑠=𝑛‖𝐺(𝑛,𝑠)‖⋅𝑢_{𝑠−𝜏(𝑠)}≤𝜆(𝑙+𝜀)‖𝑢‖^{𝑛+𝑇−1}_𝑠=𝑛‖𝐺((𝑢‖≤‖𝑢‖.𝑛,𝑠)‖≤𝜆𝐴𝑙+𝜀)‖ (26) Next, since 𝐿>0, there exists a 𝐻₂>0 such that ‖𝐹_𝑛(𝑢)‖⩾(𝐿−𝜀)‖𝑢‖ for ‖𝑢‖⩾𝐻₂ and 𝑛=0,1,…,𝑇−1. Let 𝐻₂=max{2𝐻₁,𝐻₂}. Then for 𝑢∈Ω with ‖𝑢‖=𝐻₂, (𝐻𝑢)_𝑛⩾𝜆(𝐿−𝜀)^{𝑛+𝑇−1}_𝑠=𝑛‖𝐺(𝑛,𝑠)‖⋅𝑢_{𝑠−𝜏(𝑠)}⩾𝜆(𝐿−𝜀)𝜎‖𝑢‖^{𝑛+𝑇−1}_𝑠=𝑛‖𝐺(()‖𝑢‖⩾‖𝑢‖.𝑛,𝑠)‖⩾𝜆𝐿−𝜀𝜎𝐵 (27) 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 𝑇=2, 𝑋 is a Hilbert space, 𝐴₀,𝐴₁, and 𝐴₀⁻¹𝐴₁⁻¹−𝐼 are invertible self-conjugate operator defined on 𝑋, 𝐴₀𝐴₁,(𝐴₀⁻¹𝐴₁⁻¹−𝐼)𝐴₀, (𝐴₀⁻¹𝐴₁⁻¹−𝐼)𝐴₁ are self-conjugate operator defined on 𝑋, then 𝐴₀, 𝐴₁ 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 𝜇_𝑛=1𝜆_𝑛,𝜆_𝑛≠0,0,𝜆_𝑛𝜇=0,^′𝑛=1𝜆^′_𝑛,𝜆^′_𝑛≠0,0,𝜆^′_𝑛=0. (28){𝑒_𝑛} is complete orthonormal set of space 𝑙²∶𝑒_𝑛={0,…,0,1^(𝑛),0,…0}(𝑛=1,2,…). Let 𝐴₀𝑥=^∞_𝑛=1𝜉_𝑛𝜆_𝑛𝑒_𝑛,𝐴₁𝑥=^∞_𝑛=1𝜉_𝑛𝜆^′_𝑛𝑒_𝑛 (29) for any 𝑥=∑^∞_𝑛=1𝜉_𝑛𝑒_𝑛, then 𝐴₀ and 𝐴₁ 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).