NONLINEAR INTEGRAL INEQUALITIES INVOLVING ... - Ele-Math

NONLINEAR INTEGRAL INEQUALITIES 819
Then for t0 t t6 , the inequality
u(t) k(t)ψ −1
Ψ −1
W −1
W Ψ(1) + A1(t)
(2.26)
holds, where the functions Ψ and W are defined by (2.4) and (2.5), respectively, and
A1(t) =
n αi(t) p m
fi(s) k(s) ds+
β j(t) p g j(s) k(s) ds, (2.27)
∑
i=1
αi(t0)
∑
j=1
β j(t0)
t6 = sup τ ∈ [t0,T) : W Ψ(1) + A1(t) ∈ Dom W −1 ,
W −1
W Ψ(1) + A1(t) ∈ Dom Ψ −1 and
Ψ −1
W −1
W Ψ(1)
+ A1(t) ∈ Dom ψ −1
for t ∈ [t0,τ] .
Proof. From (2.24), (2.25), (B2 ), (B6 ), and 0 1
k(t) 1, we obtain
n
u(t)
αi(t)
ψ 1+
k(t) ∑ fi(s)u
i=1 αi(t0)
p
u(s)
(s)ωi ds (2.28)
k(s)
m β j(t)
+ ∑ g j(s)u p
maxξ∈[s−h,s] u(ξ)
(s) ˜ω j
ds, t ∈ [t0,T),
k(s)
j=1
β j(t0)
u(t) φ(t)
k(t0) k(t0) 1, t ∈ [J − h,t0]. (2.29)
Let s ∈ [β j(t0),βj(T)), where 1 j m is arbitrary. From the monotonicity of the
function k in [t0,T), we obtain the inequality
max ξ ∈[s−h,s] u(ξ)
k(s)
= u(ξ1)
k(s)
u(ξ1)
max
k(ξ1)
ξ ∈[s−h,s]
where ξ1 ∈ [s − h,s]. Define a function v ∈ C([J − h,T),R+) by
⎧
⎪⎨
u(t)
for t ∈ [t0,T)
k(t)
v(t) =
⎪⎩
u(t)
for t ∈ [J − h,t0].
k(t0)
u(ξ)
k(ξ) ,
Then inequalities (2.28) and (2.29) can be rewritten as
n
ψ v(t) 1+
αi(t)
fi(s)k p (s)v p
(s)ωi v(s) ds (2.30)
∑
i=1 αi(t0)
m β j(t)
+
∑
j=1
β j(t0)
g j(s)k p (s)v p (s) ˜ω j
max v(ξ)
ξ ∈[s−h,s]
ds, t ∈ [t0,T),
v(t) 1, t ∈ [J − h,t0]. (2.31)