Asymptotic Behavior of Solutions to a Linear Volterra Integrodifferential System

Abstract

We investigate the asymptotic behavior of solutions to a linear Volterra integrodifferential system 𝑥^′_𝑖(𝑡)=𝑎_𝑖(𝑡)+𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)+∑^𝑛_𝑗=1∫^𝑡₀𝐾_𝑖𝑗(𝑡,𝑠)𝑥_𝑗(𝑠)𝑑𝑠,𝑡∈ℝ⁺, 𝑖=1,2,…,𝑛. We show that under some suitable conditions, there exists a solution for the above integrodifferential system, which is asymptotically equivalent to some given functions. Two examples are given to illustrate our theorem.

1. Introduction

Throughout this paper, we denote by ℕ the set of positive integers, by ℝ the set of all real numbers, by ℝ⁺ the set of all nonnegative real numbers, and by ℝ^𝑛 the set of all 𝑛-dimensional real vectors. Moreover, 𝐵𝐶(ℝ⁺,ℝ^𝑛) denotes the Banach space of all bounded and continuous functions 𝑓∶ℝ⁺→ℝ^𝑛 with the norm 𝑓=sup_𝑡∈ℝ+max_{1≤𝑗≤𝑛}𝑓_𝑗(𝑡), (1) where 𝑓(𝑡)=(𝑓₁(𝑡),…,𝑓_𝑛(𝑡))^𝑇 for 𝑡∈ℝ⁺.

The aim of this paper is to study some asymptotic behavior of solutions to the following linear Volterra integrodifferential system: 𝑥^′_𝑖(𝑡)=𝑎_𝑖(𝑡)+𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1^𝑡₀𝐾_𝑖𝑗(𝑡,𝑠)𝑥_𝑗(𝑠)𝑑𝑠,𝑡∈ℝ⁺,𝑖=1,2,…,𝑛, (2) where 𝑎_𝑖,𝑏_𝑖∶ℝ⁺→ℝ and 𝐾_𝑖𝑗∶ℝ⁺×ℝ⁺→ℝ, 𝑖,𝑗=1,2,…,𝑛 are all continuous functions.

Definition 1. We call 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∶ℝ⁺→ℝ^𝑛 a solution of system (2) if 𝑥 is continuously differentiable and satisfies (2).

The asymptotic behavior of solutions has been an important and interesting topic in the qualitative theory of differential and difference equations. Especially, recently, many authors have made interesting and important contributions on the asymptotic behavior of solutions for Volterra type difference equations (e.g., we refer the reader to [1–10] and references therein).

Very recently, Diblík and Schmeidel [6] obtained a very interesting result concerning the asymptotic behavior of solutions for the following linear Volterra difference equation: 𝑥(𝑛+1)=𝑎(𝑛)+𝑏(𝑛)𝑥(𝑛)+^𝑛_𝑖=0𝐾(𝑛,𝑖)𝑥(𝑖). (3) More specifically, they proved that for every admissible constant 𝑐∈ℝ, there exists a solution 𝑥=𝑥(𝑛) of (3) such that 𝑥(𝑛)∼𝑐+^𝑛−1_𝑖=0𝑎(𝑖)𝛽()𝛽(𝑛)𝑖+1,𝑛⟶∞, (4) where 𝛽(𝑛)=∏^𝑛−1_𝑖=0𝑏(𝑖). However, to the best of our knowledge, it seems that there is no literature concerning such asymptotic behavior of solutions for Volterra type differential equations. That is the main motivation of this paper. In this paper, we will adopt the idea in the proof of [6] to investigate some asymptotic behaviors of solutions for Volterra differential system (2).

2. Main Result

Before establishing our main result, we first give an “Arzela-Ascoli” type theorem for the subsets of 𝐵𝐶(ℝ⁺,ℝ^𝑛).

Lemma 2. Let ℱ⊂𝐵𝐶(ℝ⁺,ℝ^𝑛), satisfying (i) ℱ is uniformly bounded; (ii) ℱ is equiuniformly continuous on every compact subset of ℝ⁺; (iii) for every 𝜀>0, there exist 𝑓_𝜀∈𝐵𝐶(ℝ⁺,ℝ^𝑛) and 𝑇_𝜀>0 such that ‖𝑓(𝑡)−𝑓_𝜀(𝑡)‖<𝜀 for all 𝑓∈ℱ and 𝑡≥𝑇_𝜀. Then ℱ is precompact in 𝐵𝐶(ℝ⁺,ℝ^𝑛).

Proof. By the condition (iii), for every 𝑘∈ℕ, there exist 𝑇_𝑘>0 such that 𝐹₁(𝑡)−𝐹₂(𝑡<1)𝑘 (5) for all 𝐹₁,𝐹₂∈ℱ and 𝑡≥𝑇_𝑘.
Let {𝑓_𝑛} be a sequence in ℱ. By (i) and (ii), it follows from Arzela-Ascoli theorem that for every 𝑘∈ℕ, there exists a subsequence {𝑓^𝑘_𝑛}⊂{𝑓_𝑛} such that {𝑓^𝑘_𝑛} is uniformly convergent on [0,𝑇_𝑘]. Then, by choosing the diagonal sequence, we can get a subsequence {𝑓_𝑚}⊂{𝑓_𝑛} such that, for every 𝑘∈ℕ, {𝑓_𝑚} is uniformly convergent on [0,𝑇_𝑘].
It remains to show that {𝑓_𝑚} is uniformly convergent on ℝ⁺. For every 𝜀>0, choose 𝑘₀∈ℕ with 1/𝑘₀<𝜀. Since {𝑓_𝑚} is uniformly convergent on [0,𝑇_𝑘0], for the above 𝜀>0, there exists 𝑁∈ℕ such that sup_{𝑡∈[0,𝑇𝑘0]}𝑓_𝑚1(𝑡)−𝑓_𝑚2(𝑡)≤𝜀 (6) for all 𝑚₁,𝑚₂≥𝑁; Combining this with (5), we conclude that sup_𝑡∈ℝ+𝑓_𝑚1(𝑡)−𝑓_𝑚2(𝑡)≤𝜀 (7) for all 𝑚₁,𝑚₂≥𝑁, that is, {𝑓_𝑚} is uniformly convergent on ℝ⁺. This completes the proof.

Throughout the rest of this paper, for every 𝑖∈{1,2,…,𝑛}, we assume that 𝐴_𝑖=sup_𝑡∈ℝ+𝐴_𝑖(𝑡)<+∞, (8) where 𝐴_𝑖(𝑡)=^𝑡₀𝑎_𝑖(𝑠)𝛽_𝑖(𝑠)𝛽𝑑𝑠,_𝑖(𝑡)=exp^𝑡₀𝑏_𝑖(𝑠)𝑑𝑠,𝑡∈ℝ⁺,0≤𝑀_𝑖∶=^𝑛_𝑗=10^+∞𝑡₀𝐾_𝑖𝑗()𝛽𝑡,𝑠^𝑗(𝑠)𝛽_𝑖(𝑡)𝑑𝑠𝑑𝑡<1. (9)

Theorem 3. Assume that 0<liminf_𝑡→+∞𝛽_𝑖(𝑡)≤limsup_𝑡→+∞𝛽_𝑖(𝑡)<+∞,𝑖=1,2,…,𝑛. (10) Let 𝑐=(𝑐₁,𝑐₂,…,𝑐_𝑛)^𝑇∈ℝ^𝑛 with 𝑐_𝑖+𝐴_𝑖(𝑡)≥0 for all 𝑖∈{1,2,…,𝑛} and 𝑡∈ℝ⁺. Then, there exists a solution 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∶ℝ⁺→ℝ^𝑛 of system (2) such that 𝑥_𝑖(𝑡)∼𝑐_𝑖+𝐴_𝑖(𝑡)𝛽_𝑖(𝑡),𝑡⟶+∞,𝑖=1,2,…,𝑛, (11) provided that liminf_𝑡→+∞(𝑐_𝑖+𝐴_𝑖(𝑡))>0.

Proof. We define that 𝛼_𝑖(0)=𝑐_𝑖+𝐴_𝑖𝑀_𝑖1−𝑀_𝑖,𝛼_𝑖(𝑡)=𝑐_𝑖+𝐴_𝑖+𝛼_𝑖(0×_{𝑛𝑗=1𝑡+∞𝑠0})𝐾_𝑖𝑗𝛽𝑠,𝑝^𝑗𝑝𝛽_𝑖(𝑠)𝑑𝑝𝑑𝑠, (12) for all 𝑖∈{1,2,…,𝑛} and 𝑡∈ℝ⁺. Moreover, we define an operator 𝜌=(𝜌₁,𝜌₂,…,𝜌_𝑛)^𝑇 on 𝑆=𝑧∈𝐵𝐶ℝ⁺,ℝ^𝑛∶𝑐_𝑖+𝐴_𝑖(𝑡)−𝛼_𝑖(0)≤𝑧_𝑖(𝑡)≤𝑐_𝑖+𝐴_𝑖(𝑡)+𝛼_𝑖(0),𝑖=1,2,…,𝑛,𝑡∈ℝ⁺ (13) by 𝜌_𝑖𝑧(𝑡)=𝑐_𝑖+𝐴_𝑖(𝑡)−^𝑛_𝑗=1𝑡^+∞𝑠₀𝐾_𝑖𝑗𝑠,𝑝𝑧_𝑗𝑝𝛽_𝑗𝑝𝛽_𝑖(𝑠)𝑑𝑝𝑑𝑠,𝑖=1,2…,𝑛, (14) for 𝑡∈ℝ⁺ and 𝑧∈𝑆. It is easy to see that 𝑆 is a nonempty, closed, and convex set in 𝐵𝐶(ℝ⁺,ℝ^𝑛). Next, we divide the remaining proof into two steps.
Step  1. 𝜌(𝑆)⊂𝑆, 𝜌 is continuous, and 𝜌(𝑆) is compact.
Let 𝑧∈𝑆. We have 𝜌_𝑖𝑧(𝑡)≤𝑐_𝑖+𝐴_𝑖+𝑀_𝑖⋅‖𝑧‖<+∞,𝑡∈ℝ⁺,𝑖=1,2,…,𝑛. (15) In addition, since 𝑧∈𝑆, we have 𝑧_𝑖(𝑡)≤𝑐_𝑖+𝐴_𝑖+𝛼_𝑖(0),𝑡∈ℝ⁺,𝑖=1,2,…,𝑛. (16) Then, it follows that 𝜌_𝑖𝑧(𝑡)−𝑐_𝑖+𝐴_𝑖(𝑡)≤^𝑛_𝑗=1𝑡^+∞𝑠₀𝐾_𝑖𝑗𝑠,𝑝𝑧_𝑗𝑝𝛽_𝑗𝑝𝛽_𝑖(𝑠)≤𝑐𝑑𝑝𝑑𝑠_𝑖+𝐴_𝑖+𝛼_𝑖(0)⋅𝑀_𝑖=𝛼_𝑖(0) (17) for all 𝑖∈{1,2,…,𝑛} and 𝑡∈ℝ⁺. Thus, we conclude that 𝜌(𝑆)⊂𝑆.
For every 𝜀>0, there exists a constant 𝛿=𝜀/max_{1≤𝑖≤𝑛}𝑀_𝑖 such that for all 𝑧,𝑦∈𝑆 with ‖𝑧−𝑦‖<𝛿, we have 𝜌_𝑖𝑧(𝑡)−𝜌_𝑖𝑦(𝑡=_{𝑛𝑗=1𝑡+∞𝑠0})𝐾_𝑖𝑗𝑠,𝑝𝑧_𝑗𝑝𝛽_𝑗𝑝𝛽_𝑖(𝑠)−𝑑𝑝𝑑𝑠_{𝑛𝑗=1𝑡+∞𝑠0}𝐾_𝑖𝑗𝑠,𝑝𝑦_𝑗𝑝𝛽_𝑗𝑝𝛽_𝑖(𝑠)𝑑𝑝𝑑𝑠≤𝛿⋅𝑀_𝑖≤𝜀,𝑖∈{1,2,…,𝑛},𝑡∈ℝ⁺, (18) which means that 𝜌 is continuous.
Next, we show that 𝜌(𝑆) is precompact. Firstly, for every 𝑥∈𝑆, we have 𝜌𝑥=sup_𝑡∈ℝ+max_{1≤𝑖≤𝑛}𝜌_𝑖𝑥(𝑡)≤max_{1≤𝑖≤𝑛}𝑐_𝑖+𝐴_𝑖+𝛼_𝑖(0), (19) which means that 𝜌(𝑆) is uniformly bounded. Secondly, for every 𝑧∈𝑆, 𝑡₁,𝑡₂∈ℝ⁺ and 𝑖=1,2,…,𝑛, we have 𝜌_𝑖𝑧𝑡₁−𝜌_𝑖𝑧𝑡₂=^𝑛_𝑗=1𝑡^+∞₁^𝑠₀𝐾_𝑖𝑗𝑠,𝑝𝑧_𝑗𝑝𝛽_𝑗𝑝𝛽_𝑖(𝑠)−𝑑𝑝𝑑𝑠_{𝑛𝑗=1𝑡+∞2𝑠0}𝐾_𝑖𝑗𝑠,𝑝𝑧_𝑗𝑝𝛽_𝑗𝑝𝛽_𝑖(𝑠)𝑑𝑝𝑑𝑠≤max_{1≤𝑖≤𝑛}𝑐_𝑖+𝐴_𝑖+𝛼_𝑖(0⋅_{𝑛𝑗=1𝑡2𝑡1𝑠0})𝐾_𝑖𝑗𝛽𝑠,𝑝^𝑗𝑝𝛽_𝑖(𝑠)𝑑𝑝𝑑𝑠, (20) which yields that 𝜌(𝑆) is equiuniformly continuous on every compact subsets of ℝ⁺. Thirdly, by the definition of 𝑀_𝑖, for every 𝜀>0, there exists 𝑇>0 such that for all 𝑡≥𝑇 and 𝑧∈𝑆, we have ^𝑛_𝑗=1𝑡^+∞𝑠₀𝐾_𝑖𝑗𝛽𝑠,𝑝^𝑗𝑝𝛽_𝑖(𝑠)<𝜀𝑑𝑝𝑑𝑠max_{1≤𝑖≤𝑛}𝑐_𝑖+𝐴_𝑖+𝛼_𝑖(0),𝑖=1,2,…,𝑛, (21) which yields that 𝜌_𝑖𝑧−𝜌_𝑖0<𝜀,𝑖=1,2,…,𝑛, (22) and thus ‖𝜌𝑧−𝜌0‖<𝜀. Then, by Lemma 2, we know that 𝜌(𝑆) is precompact.
Step  2. By Step  1 and Schauder's fixed-point theorem, 𝜌 has a fixed point in 𝑆; that is, there exists 𝑧⁰=(𝑧⁰₁,𝑧⁰₂,…,𝑧⁰_𝑛)^𝑇∈𝑆 such that 𝑧⁰_𝑖(𝑡)=𝑐_𝑖+𝐴_𝑖(𝑡)−^𝑛_𝑗=1𝑡^+∞𝑠₀𝐾_𝑖𝑗𝑠,𝑝𝑧⁰_𝑗𝑝𝛽_𝑗𝑝𝛽_𝑖(𝑠)𝑑𝑝𝑑𝑠, (23) for all 𝑖∈{1,2,…,𝑛} and 𝑡∈ℝ⁺. Noting that sup_𝑧∈𝑆‖𝑧‖≤max_{1≤𝑖≤𝑛}𝑐_𝑖+𝐴_𝑖+𝛼_𝑖(0), (24) we have 𝑧⁰_𝑖(𝑡)−𝑐_𝑖+𝐴_𝑖(𝑡)≤max_{1≤𝑖≤𝑛}𝑐_𝑖+𝐴_𝑖+𝛼_𝑖(0⋅_{𝑛𝑗=1𝑡+∞𝑠0})𝐾_𝑖𝑗𝛽𝑠,𝑝^𝑗𝑝𝛽_𝑖(𝑠)𝑑𝑝𝑑𝑠, (25) for all 𝑖∈{1,2,…,𝑛} and 𝑡∈ℝ⁺. Then, it is easy to see that lim_𝑡→+∞𝑧⁰_𝑖(𝑡)−𝑐_𝑖+𝐴_𝑖(𝑡)=0,𝑖=1,2,…,𝑛, (26) Combining this with liminf_𝑡→+∞𝑐_𝑖+𝐴_𝑖(𝑡)>0, (27) we have lim_𝑡→+∞𝑧⁰_𝑖(𝑡)𝑐_𝑖+𝐴_𝑖(𝑡)=1,𝑖=1,2,…,𝑛, (28) that is, 𝑧⁰_𝑖(𝑡)∼𝑐_𝑖+𝐴_𝑖(𝑡),𝑡⟶+∞,𝑖=1,2,…,𝑛. (29)
Now, define a function 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∶ℝ⁺→ℝ^𝑛 by 𝑥_𝑖(𝑡)=𝑧⁰_𝑖(𝑡)𝛽_𝑖(𝑡),𝑖=1,2,…,𝑛,𝑡∈ℝ⁺. (30) It follows from (23) that 𝑑𝑧𝑑𝑡^0𝑖(𝑡)=𝑎_𝑖(𝑡)𝛽_𝑖(𝑡)+^𝑛_𝑗=1^𝑡₀𝐾_𝑖𝑗𝑡,𝑝𝑧⁰_𝑗𝑝𝛽_𝑗𝑝𝛽_𝑖(𝑡)𝑑𝑝,𝑖=1,2,…,𝑛,𝑡∈ℝ⁺, (31) which yields that 𝑥^′_𝑖(𝑡)𝛽_𝑖(𝑡)−𝑥_𝑖(𝑡)𝛽^′_𝑖(𝑡)𝛽²_𝑖(𝑡)=𝑎_𝑖(𝑡)𝛽_𝑖(𝑡)+^𝑛_𝑗=1^𝑡₀𝐾_𝑖𝑗()𝑥𝑡,𝑠^𝑗(𝑠)𝛽_𝑖(𝑡)𝑑𝑠,𝑖=1,2,…,𝑛,𝑡∈ℝ⁺. (32) Then, we get 𝑥^′_𝑖(𝑡)=𝑎_𝑖(𝑡)+𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1^𝑡₀𝐾_𝑖𝑗(𝑡,𝑠)𝑥_𝑗(𝑠)𝑑𝑠,𝑖=1,2,…,𝑛,𝑡∈ℝ⁺, (33) which means that 𝑥 is a solution to system (2). In addition, combining (28) with the assumption 0<liminf_𝑡→+∞𝛽_𝑖(𝑡)≤limsup_𝑡→+∞𝛽_𝑖(𝑡)<+∞,𝑖=1,2,…,𝑛, (34) we get lim_𝑡→+∞𝑥_𝑖(𝑡)𝑐_𝑖+𝐴_𝑖(𝑡)𝛽_𝑖(𝑡)=1,𝑖=1,2,…,𝑛, (35) which yields (11).

Example 4. Let 𝑛=1, and for all 𝑡,𝑠∈ℝ⁺, 𝑎₁(𝑡)()𝑏=expsin𝜋𝑡cos𝑡,₁(𝑡)𝐾=𝜋cos𝜋𝑡,₁₁()=()𝑡,𝑠expsin𝜋𝑡(1+𝑡+𝑠)³().expsin𝜋𝑠 (36) Then, for all 𝑡∈ℝ⁺, we have 𝛽₁(𝑡)=exp(sin𝜋𝑡), 𝐴₁(𝑡)=^𝑡₀𝑎₁(𝑠)𝛽₁(𝑠)𝐴𝑑𝑠=sin𝑡,₁=sup_𝑡∈ℝ+𝐴₁(𝑡(),𝑀)=1∈0,+∞₁=₀^+∞𝑡₀𝐾₁₁()𝛽(𝑠)𝑡,𝑠𝛽(𝑡)𝑑𝑠𝑑𝑡=₀^+∞𝑡₀1(1+𝑡+𝑠)³=𝑑𝑠𝑑𝑡_0+∞121(1+𝑡)²−1(1+2𝑡)²=1𝑑𝑡2₀^+∞1(1+𝑡)²1𝑑𝑡−2₀^+∞1(1+2𝑡)²=1𝑑𝑡2−14=14∈(0,1). (37) In addition, it is easy to see that 0<𝑒⁻¹=liminf_𝑡→+∞𝛽₁(𝑡)≤limsup_𝑡→+∞𝛽₁(𝑡)=𝑒<+∞. (38)
Thus, by Theorem 3, we conclude that for every 𝑐>1, there exists a solution 𝑥∶ℝ⁺→ℝ for (2) such that 𝑥(𝑡)∼(𝑐+sin𝑡)exp(sin𝜋𝑡),𝑡⟶+∞. (39)

Remark 5. It is needed to note that in the above example, (𝑐+sin𝑡)exp(sin𝜋𝑡) is not a solution to (2).

Example 6. Consider the following system: 𝑥^′_𝑖(𝑡)=𝑎_𝑖(𝑡)+𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)+²_𝑗=1^𝑡₀𝐾_𝑖𝑗(𝑡,𝑠)𝑥_𝑗(𝑠)𝑑𝑠,𝑖=1,2, (40) where 𝑎₁(𝑡)()𝑎=expsin𝜋𝑡cos𝑡,₂(𝑡)()𝑏=−expcos𝜋𝑡sin𝑡,₁(𝑡)𝑏=𝜋cos𝜋𝑡,₂(𝑡)𝐾=−𝜋sin𝜋𝑡,_𝑖𝑗()=()𝑡,𝑠−1^𝑖+𝑗exp(sin𝜋𝑡)16(1+𝑡+𝑠)³exp(sin𝜋𝑠) (41) for all 𝑖,𝑗=1,2, and 𝑡,𝑠∈ℝ⁺. By a direct calculation, we get 𝛽₁(𝑡)(),𝛽=expsin𝜋𝑡₂(𝑡)(),𝐴=exp−1+cos𝜋𝑡₁(𝑡)=^𝑡₀𝑎₁(𝑠)𝛽₁(𝑠)𝑑𝑠=sin𝑡,𝐴₁𝐴=1,₂(𝑡)=^𝑡₀𝑎₂(𝑠)𝛽₂(𝑠)𝑑𝑠=(−1+cos𝑡)𝑒,𝑡∈ℝ⁺,𝐴₂𝑀=2𝑒,₁=²_𝑗=10^+∞𝑡₀𝐾_1𝑗()𝛽𝑡,𝑠^𝑗(𝑠)𝛽₁(𝑡)≤𝑑𝑠𝑑𝑡1+𝑒𝑀64<1,₂=²_𝑗=10^+∞𝑡₀𝐾_2𝑗()𝛽𝑡,𝑠^𝑗(𝑠)𝛽₂(𝑡)≤𝑑𝑠𝑑𝑡𝑒+2𝑒³64<1. (42) Moreover, we have 0<𝑒⁻¹=liminf_𝑡→+∞𝛽₁(𝑡)≤limsup_𝑡→+∞𝛽₁(𝑡)=𝑒<+∞,0<𝑒⁻²=liminf_𝑡→+∞𝛽₂(𝑡)≤limsup_𝑡→+∞𝛽₂(𝑡)=1<+∞. (43) Then, by Theorem 3, for every 𝑐=(𝑐₁,𝑐₂)^𝑇∈ℝ² with 𝑐₁>1 and 𝑐₂>2𝑒, there exists a solution 𝑥=(𝑥₁,𝑥₂)∶ℝ⁺→ℝ² of system (40) such that 𝑥₁(𝑡)∼𝑐₁()𝑥+sin𝑡expsin𝜋𝑡,𝑡⟶+∞,₂(𝑡)∼𝑐₂+(−1+cos𝑡)𝑒exp(−1+cos𝜋𝑡),𝑡⟶+∞. (44)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to Professor Diblík and two anonymous referees for their valuable suggestions and comments. In addition, Hui-Sheng Ding acknowledges support from the NSF of China (11101192), the Program for Cultivating Young Scientist of Jiangxi Province (20133BCB23009), and the NSF of Jiangxi Province; Yue-Wen Cheng acknowledges support from the Graduate Innovation Fund of Jiangxi Province.