AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 245905 10.1155/2013/245905 245905 Research Article Asymptotic Behavior of Solutions to a Linear Volterra Integrodifferential System Cheng Yue-Wen Ding Hui-Sheng Diblík Josef College of Mathematics and Information Science Jiangxi Normal University Nanchang, Jiangxi 330022 China jxnu.edu.cn 2013 28 10 2013 2013 24 07 2013 18 09 2013 2013 Copyright © 2013 Yue-Wen Cheng and Hui-Sheng Ding. 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 work is properly cited.

We investigate the asymptotic behavior of solutions to a linear Volterra integrodifferential system xi(t)=ai(t)+bi(t)xi(t)+j=1n0tKij(t,s)xj(s)ds,t+, 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 the set of positive integers, by the set of all real numbers, by + the set of all nonnegative real numbers, and by n the set of all n-dimensional real vectors. Moreover, BC(+,n) denotes the Banach space of all bounded and continuous functions f:+n with the norm (1)f=supt+max1jn|fj(t)|, where f(t)=(f1(t),,fn(t))T for t+.

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=1n0tKij(t,s)xj(s)ds,xi(t)=ai(t)+bi(t)xiiii(t)+t+,i=1,2,,n, where ai,bi:+ and Kij:+×+, i,j=1,2,,n are all continuous functions.

Definition 1.

We call x=(x1,x2,,xn)T:+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  and references therein).

Very recently, Diblík and Schmeidel  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, 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  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(+,n).

Lemma 2.

Let BC(+,n), satisfying (i) is uniformly bounded; (ii) is equiuniformly continuous on every compact subset of +; (iii) for every ε>0, there exist fεBC(+,n) and Tε>0 such that f(t)-fε(t)<ε for all f and tTε. Then is precompact in BC(+,n).

Proof.

By the condition (iii), for every k, there exist Tk>0 such that (5)F1(t)-F2(t)<1k for all F1,F2 and tTk.

Let {fn} be a sequence in . By (i) and (ii), it follows from Arzela-Ascoli theorem that for every k, 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, {fm} is uniformly convergent on [0,Tk].

It remains to show that {fm} is uniformly convergent on +. For every ε>0, choose k0 with 1/k0<ε. Since {fm} is uniformly convergent on [0,Tk0], for the above ε>0, there exists N such that (6)supt[0,Tk0]fm1(t)-fm2(t)ε for all m1,m2N; Combining this with (5), we conclude that (7)supt+fm1(t)-fm2(t)ε for all m1,m2N, that is, {fm} is uniformly convergent on +. This completes the proof.

Throughout the rest of this paper, for every i{1,2,,n}, we assume that (8)Ai=supt+|Ai(t)|<+, where (9)Ai(t)=0tai(s)βi(s)ds,βi(t)=exp(0tbi(s)ds),t+,0Mi=j=1n0+(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)Tn with ci+Ai(t)0 for all i{1,2,,n} and t+. Then, there exists a solution x=(x1,x2,,xn)T:+n 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=1nt+(0s|Kij(s,p)βj(p)βi(s)|dp)ds, for all i{1,2,,n} and t+. Moreover, we define an operator ρ=(ρ1,ρ2,,ρn)T on (13)S={zBC(+,n):ci+Ai(t)-αi(0)zi(t)ci+Ai(t)+αi(0),i=1,2,,n,t+} by (14)(ρiz)(t)=ci+Ai(t)-j=1nt+(0sKij(s,p)zj(p)βj(p)βi(s)dp)ds,ciciciciciciciciciciicicicicicicicicicicii=1,2,n, for t+ and zS. It is easy to see that S is a nonempty, closed, and convex set in BC(+,n). Next, we divide the remaining proof into two steps.

Step  1. ρ ( S ) S , ρ is continuous, and ρ(S)¯ is compact.

Let zS. We have (15)|(ρiz)(t)||ci|+Ai+Mi·z<+,mmmmmiiiiiiimit+,i=1,2,,n. In addition, since zS, we have (16)|zi(t)|ci+Ai+αi(0),t+,i=1,2,,n. Then, it follows that (17)|(ρiz)(t)-(ci+Ai(t))|j=1nt+(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+. Thus, we conclude that ρ(S)S.

For every ε>0, there exists a constant δ=ε/max1inMi such that for all z,yS with z-y<δ, we have (18)|(ρiz)(t)-(ρiy)(t)|=|j=1nt+(0sKij(s,p)zj(p)βj(p)βi(s)dp)ds-j=1nt+(0sKij(s,p)yj(p)βj(p)βi(s)dp)ds|δ·Miε,i{1,2,,n},t+, which means that ρ is continuous.

Next, we show that ρ(S) is precompact. Firstly, for every xS, we have (19)ρx=supt+max1in|(ρix)(t)|max1in[ci+Ai+αi(0)], which means that ρ(S) is uniformly bounded. Secondly, for every zS, t1,t2+ and i=1,2,,n, we have (20)|(ρiz)(t1)-(ρiz)(t2)|=|j=1nt1+(0sKij(s,p)zj(p)βj(p)βi(s)dp)ds-j=1nt2+(0sKij(s,p)zj(p)βj(p)βi(s)dp)ds|max1in[ci+Ai+αi(0)]·|j=1nt1t2(0s|Kij(s,p)βj(p)βi(s)|dp)ds|, which yields that ρ(S) is equiuniformly continuous on every compact subsets of +. Thirdly, by the definition of Mi, for every ε>0, there exists T>0 such that for all tT and zS, we have (21)j=1nt+(0s|Kij(s,p)βj(p)βi(s)|dp)ds<εmax1in[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)TS such that (23)zi0(t)=ci+Ai(t)-j=1nt+(0sKij(s,p)zj0(p)βj(p)βi(s)dp)ds, for all i{1,2,,n} and t+. Noting that (24)supzSzmax1in[ci+Ai+αi(0)], we have (25)|zi0(t)-(ci+Ai(t))|max1in[ci+Ai+αi(0)]·j=1nt+(0s|Kij(s,p)βj(p)βi(s)|dp)ds, for all i{1,2,,n} and t+. 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 function x=(x1,x2,,xn)T:+n by (30)xi(t)=zi0(t)βi(t),i=1,2,,n,t+. It follows from (23) that (31)ddtzi0(t)=ai(t)βi(t)+j=1n0tKij(t,p)zj0(p)βj(p)βi(t)dp,βjβjβjβjβjβjβjβjβjβji=1,2,,n,t+, which yields that (32)xi(t)βi(t)-xi(t)βi(t)βi2(t)=ai(t)βi(t)+j=1n0tKij(t,s)xj(s)βi(t)ds,βiβiβiβiβiβiβiii=1,2,,n,t+. Then, we get (33)xi(t)=ai(t)+bi(t)xi(t)+j=1n0tKij(t,s)xj(s)ds,222i=1,2,,n,t+, 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.

Let n=1, and for all t,s+, (36)a1(t)=exp(sinπt)cost,b1(t)=πcosπt,K11(t,s)=exp(sinπt)(1+t+s)3exp(sinπs). Then, for all t+, we have β1(t)=exp(sinπt), (37)A1(t)=0ta1(s)β1(s)ds=sint,A1=supt+{|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=120+1(1+t)2dt-120+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 Theorem 3, we conclude that for every c>1, there exists a solution x:+ 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=120tKij(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+. 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+,A2=2e,M1=j=120+(0t|K1j(t,s)βj(s)β1(t)|ds)dt1+e64<1,M2=j=120+(0t|K2j(t,s)βj(s)β2(t)|ds)dte+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)T2 with c1>1 and c2>2e, there exists a solution x=(x1,x2):+2 of system (40) such that (44)x1(t)~(c1+sint)exp(sinπt),t+,x2(t)~[c2+(-1+cost)e]exp(-1+cosπt),t+.

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.

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