# Asymptotic Behavior of Solutions to a Linear Volterra Integrodifferential System

Authors: Yue-Wen Cheng; Hui-Sheng Ding
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate the asymptotic behavior of solutions to a linear Volterra integrodifferential systemxi′(t)=ai(t)+bi(t)xi(t)+∑j=1n‍∫0t‍Kij(t,s)xj(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.


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## Body

## 1. Introduction

Throughout this paper, we denote byℕ the set of positive integers, by ℝ the set of all real numbers, by ℝ+ the set of all nonnegative real numbers, and by ℝn the set of all n-dimensional real vectors. Moreover, BC(ℝ+,ℝn) denotes the Banach space of all bounded and continuous functions f:ℝ+→ℝn with the norm
(1)∥f∥=supt∈ℝ+max1≤j≤n|fj(t)|,
where f(t)=(f1(t),…,fn(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:(2)xi′(t)=ai(t)+bi(t)xi(t)+∑j=1n‍∫0t‍Kij(t,s)xj(s)ds,xi′(t)=ai(t)+bi(t)xiiii(t)+t∈ℝ+,i=1,2,…,n,
where ai,bi:ℝ+→ℝ and Kij:ℝ+×ℝ+→ℝ, i,j=1,2,…,n are all continuous functions.Definition 1.
We callx=(x1,x2,…,xn)T:ℝ+→ℝn 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:
(3)x(n+1)=a(n)+b(n)x(n)+∑i=0n‍K(n,i)x(i).
More specifically, they proved that for every admissible constant c∈ℝ, there exists a solution x=x(n) of (3) such that
(4)x(n)~(c+∑i=0n-1‍a(i)β(i+1))β(n),n⟶∞,
where β(n)=∏i=0n-1b(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 ofBC(ℝ+,ℝn).Lemma 2.
Letℱ⊂BC(ℝ+,ℝn), satisfying (i) ℱ is uniformly bounded; (ii) ℱ is equiuniformly continuous on every compact subset of ℝ+; (iii) for every ε>0, there exist fε∈BC(ℝ+,ℝn) and Tε>0 such that ∥f(t)-fε(t)∥<ε for all f∈ℱ and t≥Tε. Then ℱ is precompact in BC(ℝ+,ℝn).Proof.
By the condition (iii), for everyk∈ℕ, there exist Tk>0 such that
(5)∥F1(t)-F2(t)∥<1k
for all F1,F2∈ℱ and t≥Tk.
Let{fn} be a sequence in ℱ. By (i) and (ii), it follows from Arzela-Ascoli theorem that for every k∈ℕ, there exists a subsequence {fnk}⊂{fn} such that {fnk} is uniformly convergent on [0,Tk]. Then, by choosing the diagonal sequence, we can get a subsequence {fm}⊂{fn} such that, for every k∈ℕ, {fm} is uniformly convergent on [0,Tk].
It remains to show that{fm} is uniformly convergent on ℝ+. For every ε>0, choose k0∈ℕ with 1/k0<ε. Since {fm} is uniformly convergent on [0,Tk0], for the above ε>0, there exists N∈ℕ such that
(6)supt∈[0,Tk0]∥fm1(t)-fm2(t)∥≤ε
for all m1,m2≥N; Combining this with (5), we conclude that
(7)supt∈ℝ+∥fm1(t)-fm2(t)∥≤ε
for all m1,m2≥N, that is, {fm} is uniformly convergent on ℝ+. This completes the proof.Throughout the rest of this paper, for everyi∈{1,2,…,n}, we assume that
(8)Ai=supt∈ℝ+|Ai(t)|<+∞,
where
(9)Ai(t)=∫0t‍ai(s)βi(s)ds,βi(t)=exp(∫0t‍bi(s)ds),t∈ℝ+,0≤Mi∶=∑j=1n‍∫0+∞‍(∫0t‍|Kij(t,s)βj(s)βi(t)|ds)dt<1.Theorem 3.
Assume that(10)0<liminft→+∞βi(t)≤limsupt→+∞βi(t)<+∞,mmmmmmmmmmmmiiii=1,2,…,n.
Let c=(c1,c2,…,cn)T∈ℝn with ci+Ai(t)≥0 for all i∈{1,2,…,n} and t∈ℝ+. Then, there exists a solution x=(x1,x2,…,xn)T:ℝ+→ℝn of system (2) such that
(11)xi(t)~(ci+Ai(t))βi(t),t⟶+∞,i=1,2,…,n,
provided that liminft→+∞(ci+Ai(t))>0.Proof.
We define that(12)αi(0)=(ci+Ai)Mi1-Mi,αi(t)=(ci+Ai+αi(0))×∑j=1n‍∫t+∞‍(∫0s‍|Kij(s,p)βj(p)βi(s)|dp)ds,
for all i∈{1,2,…,n} and t∈ℝ+. Moreover, we define an operator ρ=(ρ1,ρ2,…,ρn)T on
(13)S={z∈BC(ℝ+,ℝn):ci+Ai(t)-αi(0)≤zi(t)≤ci+Ai(t)+αi(0),i=1,2,…,n,t∈ℝ+}
by
(14)(ρiz)(t)=ci+Ai(t)-∑j=1n‍∫t+∞‍(∫0s‍Kij(s,p)zj(p)βj(p)βi(s)dp)ds,ciciciciciciciciciciicicicicicicicicicicii=1,2…,n,
for t∈ℝ+ and z∈S. It is easy to see that S is a nonempty, closed, and convex set in BC(ℝ+,ℝn). Next, we divide the remaining proof into two steps.
Step  1.
ρ
(
S
)
⊂
S, ρ is continuous, and ρ(S)¯ is compact.
Letz∈S. We have
(15)|(ρiz)(t)|≤|ci|+Ai+Mi·∥z∥<+∞,mmmmmiiiiiiimit∈ℝ+,i=1,2,…,n.
In addition, since z∈S, we have
(16)|zi(t)|≤ci+Ai+αi(0),t∈ℝ+,i=1,2,…,n.
Then, it follows that
(17)|(ρiz)(t)-(ci+Ai(t))|≤∑j=1n‍∫t+∞‍(∫0s‍|Kij(s,p)zj(p)βj(p)βi(s)|dp)ds≤[ci+Ai+αi(0)]·Mi=αi(0)
for all i∈{1,2,…,n} and t∈ℝ+. Thus, we conclude that ρ(S)⊂S.
For everyε>0, there exists a constant δ=ε/max1≤i≤nMi such that for all z,y∈S with ∥z-y∥<δ, we have
(18)|(ρiz)(t)-(ρiy)(t)|=|∑j=1n‍∫t+∞‍(∫0s‍Kij(s,p)zj(p)βj(p)βi(s)dp)ds-∑j=1n‍∫t+∞‍(∫0s‍Kij(s,p)yj(p)βj(p)βi(s)dp)ds|≤δ·Mi≤ε,i∈{1,2,…,n},t∈ℝ+,
which means that ρ is continuous.
Next, we show thatρ(S) is precompact. Firstly, for every x∈S, we have
(19)∥ρx∥=supt∈ℝ+max1≤i≤n|(ρix)(t)|≤max1≤i≤n[ci+Ai+αi(0)],
which means that ρ(S) is uniformly bounded. Secondly, for every z∈S, t1,t2∈ℝ+ and i=1,2,…,n, we have
(20)|(ρiz)(t1)-(ρiz)(t2)|=|∑j=1n‍∫t1+∞‍(∫0s‍Kij(s,p)zj(p)βj(p)βi(s)dp)ds-∑j=1n‍∫t2+∞‍(∫0s‍Kij(s,p)zj(p)βj(p)βi(s)dp)ds|≤max1≤i≤n[ci+Ai+αi(0)]·|∑j=1n‍∫t1t2‍(∫0s‍|Kij(s,p)βj(p)βi(s)|dp)ds|,
which yields that ρ(S) is equiuniformly continuous on every compact subsets of ℝ+. Thirdly, by the definition of Mi, for every ε>0, there exists T>0 such that for all t≥T and z∈S, we have
(21)∑j=1n‍∫t+∞‍(∫0s‍|Kij(s,p)βj(p)βi(s)|dp)ds<εmax1≤i≤n[ci+Ai+αi(0)],i=1,2,…,n,
which yields that
(22)∥ρiz-ρi0∥<ε,i=1,2,…,n,
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=(z10,z20,…,zn0)T∈S such that
(23)zi0(t)=ci+Ai(t)-∑j=1n‍∫t+∞‍(∫0s‍Kij(s,p)zj0(p)βj(p)βi(s)dp)ds,
for all i∈{1,2,…,n} and t∈ℝ+. Noting that
(24)supz∈S∥z∥≤max1≤i≤n[ci+Ai+αi(0)],
we have
(25)|zi0(t)-(ci+Ai(t))|≤max1≤i≤n[ci+Ai+αi(0)]·∑j=1n‍∫t+∞‍(∫0s‍|Kij(s,p)βj(p)βi(s)|dp)ds,
for all i∈{1,2,…,n} and t∈ℝ+. Then, it is easy to see that
(26)limt→+∞|zi0(t)-(ci+Ai(t))|=0,i=1,2,…,n,
Combining this with
(27)liminft→+∞(ci+Ai(t))>0,
we have
(28)limt→+∞zi0(t)ci+Ai(t)=1,i=1,2,…,n,
that is,
(29)zi0(t)~ci+Ai(t),t⟶+∞,i=1,2,…,n.
Now, define a functionx=(x1,x2,…,xn)T:ℝ+→ℝn by
(30)xi(t)=zi0(t)βi(t),i=1,2,…,n,t∈ℝ+.
It follows from (23) that
(31)ddtzi0(t)=ai(t)βi(t)+∑j=1n‍∫0t‍Kij(t,p)zj0(p)βj(p)βi(t)dp,βjβjβjβjβjβjβjβjβjβji=1,2,…,n,t∈ℝ+,
which yields that
(32)xi′(t)βi(t)-xi(t)βi′(t)βi2(t)=ai(t)βi(t)+∑j=1n‍∫0t‍Kij(t,s)xj(s)βi(t)ds,βiβiβiβiβiβiβiii=1,2,…,n,t∈ℝ+.
Then, we get
(33)xi′(t)=ai(t)+bi(t)xi(t)+∑j=1n‍∫0t‍Kij(t,s)xj(s)ds,222i=1,2,…,n,t∈ℝ+,
which means that x is a solution to system (2). In addition, combining (28) with the assumption
(34)0<liminft→+∞βi(t)≤limsupt→+∞βi(t)<+∞,jjjjjjjjjjjjjjkkkkkkkkllli=1,2,…,n,
we get
(35)limt→+∞xi(t)(ci+Ai(t))βi(t)=1,i=1,2,…,n,
which yields (11).Example 4.
Letn=1, and for all t,s∈ℝ+,
(36)a1(t)=exp(sinπt)cost,b1(t)=πcosπt,K11(t,s)=exp(sinπt)(1+t+s)3exp(sinπs).
Then, for all t∈ℝ+, we have β1(t)=exp(sinπt),
(37)A1(t)=∫0t‍a1(s)β1(s)ds=sint,A1=supt∈ℝ+{|A1(t)|}=1∈(0,+∞),M1=∫0+∞‍(∫0t‍|K11(t,s)β(s)β(t)|ds)dt=∫0+∞‍(∫0t‍1(1+t+s)3ds)dt=∫0+∞‍12[1(1+t)2-1(1+2t)2]dt=12∫0+∞‍1(1+t)2dt-12∫0+∞‍1(1+2t)2dt=12-14=14∈(0,1).
In addition, it is easy to see that
(38)0<e-1=liminft→+∞β1(t)≤limsupt→+∞β1(t)=e<+∞.
Thus, by Theorem3, we conclude that for every c>1, there exists a solution x:ℝ+→ℝ for (2) such that
(39)x(t)~(c+sint)exp(sinπt),t⟶+∞.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:(40)xi′(t)=ai(t)+bi(t)xi(t)+∑j=12‍∫0t‍Kij(t,s)xj(s)ds,i=1,2,
where
(41)a1(t)=exp(sinπt)cost,a2(t)=-exp(cosπt)sint,b1(t)=πcosπt,b2(t)=-πsinπt,Kij(t,s)=(-1)i+jexp(sinπt)16(1+t+s)3exp(sinπs)
for all i,j=1,2, and t,s∈ℝ+. By a direct calculation, we get
(42)β1(t)=exp(sinπt),β2(t)=exp(-1+cosπt),A1(t)=∫0t‍a1(s)β1(s)ds=sint,A1=1,A2(t)=∫0t‍a2(s)β2(s)ds=(-1+cost)e,t∈ℝ+,A2=2e,M1=∑j=12‍∫0+∞‍(∫0t‍|K1j(t,s)βj(s)β1(t)|ds)dt≤1+e64<1,M2=∑j=12‍∫0+∞‍(∫0t‍|K2j(t,s)βj(s)β2(t)|ds)dt≤e+2e364<1.
Moreover, we have
(43)0<e-1=liminft→+∞β1(t)≤limsupt→+∞β1(t)=e<+∞,0<e-2=liminft→+∞β2(t)≤limsupt→+∞β2(t)=1<+∞.
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
(44)x1(t)~(c1+sint)exp(sinπt),t⟶+∞,x2(t)~[c2+(-1+cost)e]exp(-1+cosπt),t⟶+∞.



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*Source: 245905-2013-10-28.xml*