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

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.

Let z∈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 every ε>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 that ρ(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 function x=(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).