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Zhang et al. Advances in Difference Equations (2017) 2017:123 Page 4 of 18
(i) for all φ,ψ ∈BC([–τ ,];R
d
) and t ∈[t
,T],
δ(t,φ)–δ(t,ψ)
∨
σ (t,φ)–σ (t,ψ)
≤Kφ – ψ
,(.)
(ii) for all (φ,t) ∈BC([–τ ,];R
d
) ×[t
,T],
δ(t,)
∨
σ (t,)
≤K,(.)
then system (.) with initial value (.) has a unique solution X(t) with X(t) ∈ M
([t
–
τ ,T];R
d
).
Firstly, let us prove a conclusion which will be used in the proof of Theorem ..
Lemma . Let (.) and (.) hold. If X(t) is the solution of (.) with initial value (.),
then
E
sup
t
–τ <s≤T
X(s)
≤Eξ
+
Eξ
+E
(T –t
)ξ
+
K(T – t
)
q–
(T –t
+)
q((q))
×E
q
K(T – t
)
q–
(T –t
+)
(q)
(.)
and X(t) ∈M
([t
– τ ,T];R
d
).
Proof Define the stopping time τ
n
= T ∧inf{t ∈[t
,T]:X
t
≥n} for every integer n ≥.
Obviously, τ
n
↑T a.s. Let X
n
(t)=X(t ∧τ
n
)fort ∈[t
,T]. Therefore,
X
n
(t)=ξ() + ξ
()(t ∧τ
n
– t
)+
(q)
t∧τ
n
t
(t ∧τ
n
– s)
q–
δ
s,X
n
s
ds
+
(q)
t∧τ
n
t
(t ∧τ
n
– s)
q–
σ
s,X
n
s
dB(s)fort
≤t ≤ T,(.)
X
n
(t)
≤
ξ()
+
ξ
()(t ∧τ
n
– t
)
+
((q))
t∧τ
n
t
(t ∧τ
n
– s)
q–
δ
s,X
n
s
ds
+
((q))
t∧τ
n
t
(t ∧τ
n
– s)
q–
σ
s,X
n
s
dB(s)
for t
≤t ≤ T.(.)
Taking the expectation on both sides of (.), we obtain
E
X
n
(t)
≤ E
ξ()
+E
ξ
()(t ∧τ
n
– t
)
+
((q))
E
t∧τ
n
t
(t ∧τ
n
– s)
q–
δ
s,X
n
s
ds
+
((q))
E
t∧τ
n
t
(t ∧τ
n
– s)
q–
σ
s,X
n
s
dB(s)
.(.)