Unique Existence Theorem of Solution of Almost Periodic Differential Equations on Time Scales

Abstract

By using the theory of calculus on time scales and 𝑀-matrix theory, the unique existence theorem of solution of almost periodic differential equations on almost periodic time scales is established. The result can be used to a large of dynamic systems.

1. Introduction

It is well known that in Celestial mechanics, almost periodic solutions and stable solutions to differential equations or difference equations are intimately related. In the same way, stable electronic circuits, ecological systems, neural networks, and so on exhibit almost periodic behavior. A vast amount of researches have been directed toward studying these phenomena, we refer the readers to [1–5] and the references therein.

Also, the theory of calculus on time scales (see [6] and references cited therein) was initiated by Stefan Hilger in his Ph.D. thesis in 1988 [7] in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has received much attention since his foundational work (see, e.g., [8–12]). Therefore, it is practicable to study that on time scales which can unify the continuous and discrete situations.

Recently, the conceptions of almost periodic time scales and almost periodic functions on almost periodic time scales have been established, one can see [8]. Consider the following almost periodic system:𝑥^Δ(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡),𝑡∈𝕋, (1.1) where 𝕋 is an almost periodic time scale, 𝐴(𝑡) is an almost periodic matrix function, 𝑓(𝑡) is an almost periodic vector function. The authors in [8] only proved the existence of almost periodic solution for system (1.1) (see Lemma  2.13 in [8]), but the uniqueness has not been considered. However, the unique existence theorem of solution usually plays an important role in applications, so, the theories need to be explored.

The main purpose of this paper is by using the theory of calculus on time scales and 𝑀-matrix theory to establish the unique existence theorem of solution of system (1.1).

2. Preliminaries

The basic theories of calculus on time scales, one can see [6]. In order to obtain the unique existence theorem of solution of system (1.1), we first make the following preparations.

Lemma 2.1 (see [6]). Let 𝑔∶𝕋×ℝ→ℝ be a function with 𝑔𝑡,𝑥₁≤𝑔𝑡,𝑥₂,∀𝑡∈𝕋,𝑥₁≤𝑥₂. (2.1) Let 𝜐,𝜔∶𝕋→ℝ be differentiable with 𝜐^Δ(𝑡)≤𝑔(𝑡,𝜐(𝑡)),𝜔^Δ(𝑡)≥𝑔(𝑡,𝜔(𝑡)),∀𝑡∈𝑡₀,+∞_𝕋. (2.2) Then, 𝜐𝑡₀<𝜔𝑡₀,𝑡₀∈𝕋, (2.3) implies 𝜐(𝑡)<𝜔(𝑡),∀𝑡∈𝑡₀,+∞_𝕋. (2.4)

Theorem 2.2. If the following conditions satisfy:(1)𝐷⁺𝑥^Δ_𝑖(𝑡)≤^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑥_𝑗(𝑡),𝑡∈𝑡₀,+∞_𝕋,𝑖,𝑗=1,2,…,𝑛, (2.5)where 𝑎_𝑖𝑗≥0(𝑖≠𝑗),𝑏_𝑖𝑗≥0,∑^𝑛_𝑖=1𝑥_𝑖(𝑡₀)>0,𝑥_𝑖(𝑡)=sup_{𝑠∈[𝑡−𝜏0,𝑡]𝕋}𝑥_𝑖(𝑠), and 𝜏₀>0 is a constant;(2)𝑀∶=−(𝑎_𝑖𝑗+𝑏_𝑖𝑗)_𝑛×𝑛 is an 𝑀-matrix; then there exists constants 𝛾_𝑖>0 and 𝑎>0, such that the solutions of inequality (1) satisfy 𝑥_𝑖(𝑡)≤𝛾_𝑖^𝑛_𝑗=1𝑥_𝑗𝑡₀𝑒_⊖𝑎𝑡,𝑡₀,∀𝑡∈𝑡₀,+∞_𝕋,𝑖=1,2,…,𝑛. (2.6)

Proof. Assume that 𝐺(𝑡,𝑥(𝑡),𝑥(𝑡))=𝑔₁(𝑡,𝑥(𝑡),𝑥(𝑡)),𝑔₂(𝑡,𝑥(𝑡),𝑥(𝑡)),…,𝑔_𝑛(𝑡,𝑥(𝑡),𝑥(𝑡)), (2.7) where 𝑔_𝑖(𝑡,𝑥(𝑡),𝑥(𝑡))=^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑥_𝑖(𝑡),𝑖=1,2,…,𝑛. (2.8)
By condition (1), then 𝐷⁺𝑥^Δ_𝑖(𝑡)≤𝑔_𝑖(𝑡,𝑥(𝑡),𝑥(𝑡)),∀𝑡∈𝑡₀,+∞_𝕋,𝑖=1,2,…,𝑛. (2.9)
By condition (2), there exist constants 𝜉>0 and 𝑑_𝑖>0(𝑖=1,2,…,𝑛) such that ^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑑_𝑖<−𝜉,𝑖=1,2,…,𝑛. (2.10) Choose 0<𝑎≪1, such that 𝑎𝑑_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗𝑑_𝑖+𝑏_𝑖𝑗𝑑_𝑖𝑒_𝑎𝑡,𝑡−𝜏₀<0,∀𝑡∈𝑡₀,+∞_𝕋,𝑖=1,2,…,𝑛. (2.11)
If 𝑡∈[𝑡₀−𝜏₀,𝑡₀]_𝕋, choose 𝐹≫1, such that 𝐹𝑑_𝑖𝑒_⊖𝑎𝑡,𝑡₀>1,𝑖=1,2,…,𝑛. (2.12)
For any 𝜀>0, let 𝑞_𝑖(𝑡)=𝐹𝑑_𝑖^𝑛_𝑗=1𝑥_𝑗𝑡₀+𝜀𝑒_⊖𝑎𝑡,𝑡₀,𝑖=1,2,…,𝑛. (2.13)
From (2.11), for any 𝑡∈[𝑡₀,+∞)_𝕋, we have 𝐷⁺𝑞^Δ_𝑖(𝑡)=(⊖𝑎)𝐹𝑑_𝑖^𝑛_𝑗=1𝑥_𝑗𝑡₀+𝜀𝑒_⊖𝑎𝑡,𝑡₀≥−𝑎𝐹𝑑_𝑖^𝑛_𝑗=1𝑥_𝑗𝑡₀+𝜀𝑒_⊖𝑎𝑡,𝑡₀>^𝑛_𝑗=1𝑎_𝑖𝑗𝑑_𝑖+𝑏_𝑖𝑗𝑑_𝑖𝑒_𝑎𝑡,𝑡−𝜏₀𝐹^𝑛_𝑗=1𝑥_𝑗𝑡₀+𝜀𝑒_⊖𝑎𝑡,𝑡₀=^𝑛_𝑗=1𝑎_𝑖𝑗𝑑_𝑖𝐹^𝑛_𝑗=1𝑥_𝑗𝑡₀+𝜀𝑒_⊖𝑎𝑡,𝑡₀+^𝑛_𝑗=1𝑏_𝑖𝑗𝑑_𝑖𝐹^𝑛_𝑗=1𝑥_𝑗𝑡₀+𝜀𝑒_⊖𝑎𝑡−𝜏₀,𝑡₀≥^𝑛_𝑗=1𝑎_𝑖𝑗𝑞_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑞_𝑖(𝑡)=𝑔_𝑖𝑡,𝑞(𝑡),𝑞(𝑡),𝑖=1,2,…,𝑛, (2.14) that is, 𝐷⁺𝑞^Δ_𝑖(𝑡)>𝑔_𝑖𝑡,𝑞(𝑡),𝑞(𝑡),∀𝑡∈𝑡₀,+∞_𝕋,𝑖=1,2,…,𝑛. (2.15)
For 𝑡∈[𝑡₀−𝜏₀,𝑡₀]_𝕋, by (2.12), we can get 𝑞_𝑖(𝑡)=𝐹𝑑_𝑖^𝑛_𝑗=1𝑥_𝑗𝑡₀+𝜀𝑒_⊖𝑎𝑡,𝑡₀>^𝑛_𝑗=1𝑥_𝑗𝑡₀+𝜀,𝑖=1,2,…,𝑛. (2.16) Let 𝑥_𝑖(𝑡)≤∑^𝑛_𝑗=1𝑥_𝑗(𝑡₀)+𝜀,𝑡∈[𝑡₀−𝜏₀,𝑡₀]_𝕋, then 𝑞_𝑖𝑡₀>𝑥_𝑖𝑡₀,𝑖=1,2,…,𝑛. (2.17)
Together with (2.9), (2.15), and (2.17), by Lemma 2.1, we can get 𝑥_𝑖(𝑡)<𝑞_𝑖(𝑡)=𝐹𝑑_𝑖^𝑛_𝑗=1𝑥_𝑗𝑡₀+𝜀𝑒_⊖𝑎𝑡,𝑡₀,∀𝑡∈𝑡₀,+∞_𝕋,𝑖=1,2,…,𝑛. (2.18) Let 𝜀→0⁺,𝐹𝑑_𝑖=𝛾_𝑖, then 𝑥_𝑖(𝑡)≤𝛾_𝑖^𝑛_𝑗=1𝑥_𝑗𝑡₀𝑒_⊖𝑎𝑡,𝑡₀,∀𝑡∈𝑡₀,+∞_𝕋,𝑖=1,2,…,𝑛. (2.19) The proof is completed.

Definition 2.3. The almost periodic solution 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^𝑇 of (1.1) is said to be exponentially stable, if there exists a positive 𝛼 such that for any 𝛿∈[𝑡₀−𝜏₀,𝑡₀]_𝕋,𝜏₀>0, there exists 𝑁=𝑁(𝛿)≥1 such that for any solution 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 satisfying 𝑥−𝑥^∗≤𝑁𝜙−𝑥^∗𝑒_⊖𝛼(𝑡,𝛿),𝑡∈𝑡₀,+∞_𝕋, (2.20) where 𝜙(𝑠),𝑠∈[𝑡₀−𝜏₀,𝑡₀]_𝕋, is the initial condition.

3. Unique Existence Theorem

In this section, we will establish the unique existence theorem of solution of system (1.1) based on the theory of calculus on time scales and 𝑀-matrix theory. The conceptions of almost periodic time scales and almost periodic functions on almost periodic time scales, one can see [8].

Definition 3.1 (see [8]). Let 𝑥∈ℝ^𝑛, and let 𝐴(𝑡) be an 𝑛×𝑛 rd-continuous matrix on 𝕋, the linear system 𝑥^Δ(𝑡)=𝐴(𝑡)𝑥(𝑡),𝑡∈𝕋, (3.1) is said to admit an exponential dichotomy on 𝕋 if there exist positive constants 𝑘,𝛼, projection 𝑃, and the fundamental solution matrix 𝑋(𝑡) of (3.1), satisfying 𝑋(𝑡)𝑃𝑋⁻¹(𝜎(𝑠))₀≤𝑘𝑒_⊖𝛼((𝑠(𝑠),𝑋(𝑡)𝑋𝑡,𝜎)),𝑠,𝑡∈𝕋,𝑡≥𝜎)(𝐼−𝑃⁻¹(𝜎(𝑠))₀≤𝑘𝑒_⊖𝛼(𝜎(𝑠),𝑡),𝑠,𝑡∈𝕋,𝑡≤𝜎(𝑠), (3.2) where |⋅|₀ is a matrix norm on 𝕋.

Lemma 3.2 (see [8]). If the linear system (3.1) admits exponential dichotomy, then system (1.1) has a bounded solution 𝑥(𝑡) as follows: 𝑥(𝑡)=^𝑡_−∞𝑋(𝑡)𝑃𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠−_𝑡^+∞𝑋(𝑡)(𝐼−𝑃)𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠, (3.3) where 𝑋(𝑡) is the fundamental solution matrix of (3.1).

Let 𝐴(𝑡)=(𝑎_𝑖𝑗(𝑡))_𝑛×𝑛,𝐴=(sup(𝑎_𝑖𝑗(𝑡)))_𝑛×𝑛,1≤𝑖,𝑗≤𝑛,𝑡∈𝕋.

Lemma 3.3. Assume that the conditions of Lemma 3.2 hold, if −𝐴 is an 𝑀-matrix, then the almost periodic solution of system (1.1) is globally exponentially stable and unique.

Proof. According to Lemma 3.2, we know that (1.1) has an almost periodic solution 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^𝑇. Suppose that 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 be an arbitrary solution of (1.1). Then, system (1.1) can be written as 𝑥(𝑡)−𝑥^∗(𝑡)^Δ=𝐴(𝑡)𝑥(𝑡)−𝐴(𝑡)𝑥^∗(𝑡). (3.4)
Assume that the initial condition of (1.1) is 𝜙(𝑠)=(𝜙₁(𝑠),…,𝜙_𝑛(𝑠))^𝑇,𝑠∈[𝑡₀−𝜏₀,𝑡₀]_𝕋,𝜏₀>0, then the initial condition of (3.4) is 𝜙(𝑠)=𝜙(𝑠)−𝑥^∗(𝑠),𝑠∈[𝑡₀−𝜏₀,𝑡₀]_𝕋.
Let 𝑉(𝑡)=|𝑥(𝑡)−𝑥^∗(𝑡)|, the upper right derivative 𝐷⁺𝑉^Δ(𝑡) along the solutions of system (3.4) is as follows: 𝐷⁺𝑉^Δ(𝑡)=sgn𝑥(𝑡)−𝑥^∗(𝑡)𝑥(𝑡)−𝑥^∗(𝑡)^Δ≤𝐴𝑉(𝑡)+𝟎𝑉(𝑡), (3.5) where 𝟎 is an 𝑛×𝑛-matrix with all its elements are zeros.
Since −(𝐴+𝟎)=−𝐴 is an 𝑀-matrix, according to Theorem 2.2, then there exist constants 𝛼>0,𝛾₀>0, for any 𝛿∈[𝑡₀−𝜏₀,𝑡₀]_𝕋, 𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)≤𝛾₀sup_{𝛿∈[𝑡0−𝜏0,𝑡0]𝕋}𝜙_𝑖(𝛿)−𝑥^∗_𝑖(𝛿)𝑒_⊖𝛼𝑡,𝑡₀≤𝛾₀𝑒_⊖𝛼𝑡₀,𝛿sup_{𝛿∈[𝑡0−𝜏0,𝑡0]𝕋}𝜙_𝑖(𝛿)−𝑥^∗_𝑖(𝛿)𝑒_⊖𝛼(𝑡,𝛿), (3.6) where 𝑡∈[𝑡₀,+∞)_𝕋,𝑖=1,2,…,𝑛.
Then, there exists a positive number 𝜂>𝑒_⊖𝛼(𝑡₀,𝛿)/𝛾₀, such that 𝑥−𝑥^∗≤𝑁𝜙−𝑥^∗𝑒_⊖𝛼(𝑡,𝛿),𝑡∈𝑡₀,+∞_𝕋, (3.7) where 𝑁=𝑁(𝛿)=𝜂𝛾₀/𝑒_⊖𝛼(𝑡₀,𝛿)>1,‖𝑥‖=max_{1≤𝑖≤𝑛}sup_{𝑡∈[𝑡0,+∞)𝕋}|𝑥_𝑖(𝑡)|.
From Definition 2.3, the almost periodic solution 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^𝑇 is globally exponentially stable. Thus, the almost periodic solution of system (1.1) is globally exponentially stable.
In (3.7), let 𝑡→+∞, then 𝑒_⊖𝛼(𝑡,𝛿)→0, so, we can get 𝑥=𝑥^∗. Hence, the almost periodic system (1.1) has a unique almost periodic solution. The proof is completed.

Together with Lemmas 3.2 and 3.3, we can get the following theorem.

Theorem 3.4. If the linear system (3.1) admits an exponential dichotomy, −𝐴 is an 𝑀-matrix, then system (1.1) has a unique almost periodic solution 𝑥(𝑡), and 𝑥(𝑡)=^𝑡_−∞𝑋(𝑡)𝑃𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠−_𝑡^+∞𝑋(𝑡)(𝐼−𝑃)𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠, (3.8) where 𝑋(𝑡) is the fundamental solution matrix of (3.1).

Lemma 3.5 (see [8]). Let 𝑐_𝑖(𝑡) be an almost periodic function on 𝕋, where 𝑐_𝑖(𝑡)>0, −𝑐_𝑖(𝑡)∈ℛ⁺, for all 𝑡∈𝕋, and min_{1≤𝑖≤𝑛}inf_𝑡∈𝕋𝑐_𝑖(𝑡)=𝑚>0, (3.9) then the linear system 𝑥^Δ(𝑡)=diag−𝑐₁(𝑡),−𝑐₂(𝑡),…,−𝑐_𝑛(𝑡)𝑥(𝑡) (3.10) admits an exponential dichotomy on 𝕋.

Corollary 3.6. In system (1.1), if 𝐴(𝑡)=diag(−𝑎₁₁(𝑡),−𝑎₂₂(𝑡),…,−𝑎_𝑛𝑛(𝑡)),𝑡∈𝕋, and min_{1≤𝑖≤𝑛}{inf_𝑡∈𝕋𝑎_𝑖𝑖(𝑡)}=𝑎>0, then system (1.1) has a unique almost periodic solution 𝑥(𝑡), and 𝑥(𝑡)=^𝑡_−∞𝑋(𝑡)𝑃𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠−_𝑡^+∞𝑋(𝑡)(𝐼−𝑃)𝑋⁻¹(𝜎(𝑠))𝑓(𝑠)Δ𝑠, (3.11) where 𝑋(𝑡) is the fundamental solution matrix of (3.1).

Proof. Obviously, −𝐴 is an 𝑀-matrix, since 𝐴(𝑡)=diag(−𝑎₁₁(𝑡),−𝑎₂₂(𝑡),…,−𝑎_𝑛𝑛(𝑡)),𝑡∈𝕋, and min_{1≤𝑖≤𝑛}{inf_𝑡∈𝕋𝑎_𝑖𝑖(𝑡)}=𝑎>0. By Lemma 3.5, the linear system (3.1) admits an exponential dichotomy. According to Theorem 3.4, it is easy to see that the almost periodic system (1.1) has exactly one almost periodic solution. The proof is completed.

Remark 3.7. As an application, consider system (1.1) in paper [8], by using fixed-point theorem, the authors in [8] proved (1.1) has a unique almost periodic solution, one can see Theorem  3.2 in [8] for more detail. However, from the proof of Theorem  3.2 in paper [8], one can see that (3.5) is a solution of system (3.4), but the uniqueness cannot be determined, so, the proof of Theorem  3.2 in paper [8] is questionable. Our results obtained in this paper can solve the problem. By Corollary 3.6, one can get system (3.4) has exactly one solution as (3.5) in [8], then by the same method in [8], under the conditions of Theorem  3.2, system (1.1) has a unique almost periodic solution. Also, the results can be used to other neural networks.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 61073065.