Explicit bounds of unknown function of some new weakly singular retarded integral inequalities for discontinuous functions and their applications
Li and Wang Journal of Inequalities and Applications
Explicit bounds of unknown function of some new weakly singular retarded integral inequalities for discontinuous functions and their applications
Zizun Li 0 1 2 3 4 6 7
WuSheng Wang 0 1 3 5
0 Guangxi 533000 , P.R. China
1 Statistics, Baise University , Baise
2 School of Mathematics
3 University , Chengdu, Sichuan
4 School of Mathematics , Sichuan
5 School of Mathematics and Statistics, Hechi University , Yizhou, Guangxi 546300 , P.R. China
6 School of Mathematics and Statistics, Baise University , Baise, Guangxi 533000 , P.R. China
7 School of Mathematics, Sichuan University , Chengdu, Sichuan 610064 , P.R. China
The purpose of the present paper is to establish some new retarded weakly singular integral inequalities of GronwallBellman type for discontinuous functions, which generalize some known weakly singular and impulsive integral inequalities. The inequalities given here can be used in the analysis of the qualitative properties of certain classes of singular differential equations and singular impulsive equations.
integral inequality for discontinuous function; retarded; weakly singular

explicit bounds; singular integral equation
1 Introduction
Being an important tool in the study of qualitative properties of solutions of differential
equations and integral equations, various generalizations of GronwallBellman integral
inequality and their applications have attracted great interest of many mathematicians
(such as [–] and the references therein). Gronwall [] and Bellman [] established the
integral inequality
for some constant c ≥ , obtained the estimation of an unknown function,
Abdeldaim [] discussed the following nonlinear integral inequality:
u(t) ≤ c +
f (s)u(s) ds, t ∈ [a, b],
u(t) ≤ c exp
f (s) ds , t ∈ [a, b].
α(t)
s
s
f (s) u–p(s) +
g(τ )uq(τ ) dτ
p ∈ [, ),
f (s) u(s) +
g(τ )u(τ ) dτ
ds, p ∈ [, ).
p
p
ds,
Usually, this type integral inequalities have regular or continuous integral kernels, but
some problems of theory and practicality require us to solve integral inequalities with
singular kernels. For example, to prove a global existence and an exponential decay result for
a parabolic Cauchy problem. Henry [] investigated the following linear singular integral
inequality:
t
u(t) ≤ a + b
(t – s)β–u(s) ds.
Sano and Kunimatsu[] generalized Henry’s type inequality to
t
t
t
t
t
≤ u(t) ≤ c + ctα– + c
u(s) ds + c
(t – s)β–u(s) ds,
u(t) ≤ a(t) + b(t)
(t – s)β–u(s) ds,
and gave a sufficient condition for stabilization of semilinear parabolic distributed
systems. Ye et al. [] discussed the linear singular integral inequality
and they used it to study the dependence of the solution and the initial condition to a
certain fractional differential equation with RiemannLiouville fractional derivatives. All
inequalities of this type are proved by an iteration argument and the estimation formulas are
expressed by a complicated power series which is sometimes not very convenient for
applications. To avoid the weakness, Medveď [] presented a new method to solve integral
inequalities of HenryGronwall type, then he got the explicit bounds with a quite simple
formula, similar to the classic GronwallBellman inequalities. Furthermore, he also obtained
global solutions of the semilinear evolutions in []. In , Ma and Pečarić [] used the
modification of Medveď ’s method to study a new weakly singular integral inequality,
t tβ – sβ γ –sξ–f (s)uq(s) ds, t ∈ [, +∞).
Besides the results mentioned above, various investigators have discovered many useful
and new weakly singular integral inequalities, mainly inspired by their applications in
various branches of fractional differential equations (see [, –] and the references
therein).
In analyzing the impulsive phenomenon of a physical system governed by certain
differential and integral equations, by estimating the unknown function in the integral
inequality of the discontinuous functions, Some properties of the solution of some impulsive
differential equations can be studied. These inequalities and their various linear and
nonlinear generalizations are crucial in the discussion of the existence, uniqueness,
boundedness, stability, and other qualitative properties of solutions of differential and integral
equations (see [, , –] and the references therein). Tatar [] discussed the
following class of integral inequalities:
<tk<t
u(t) ≤ ϕ(t), t ∈ [–τ , ], τ > ,
where ki(t, s) = (t – s)βi–sγi Fi(s), i = , . Iovane [] studied the following discontinuous
function integral inequality:
t<ti<t
where w(u) is monotone decreasing continuous function defined on [, ∞), and w(u) >
when u > . Liu et al. [] investigated the impulsive integral inequality with delay
f (s)uq(s) + h(s)ur σ (s) ds +
βium(ti – ),
∀t ≥ t,
up(t) ≤ a(t) +
+
where u(t), a(t) and gi(t), bj(t), cj(t) ( ≤ i ≤ N , ≤ j ≤ L) are positive and continuous
functions on [t, ∞), and cj(t) are nondecreasing functions on [t, ∞), and φi(t), wj(t) are
continuous functions on [t, ∞) and t ≤ φi(t) ≤ t, t ≤ wj(t) ≤ t.
However, in certain situations, such as some classes of delay impulsive differential
equations and delay impulsive integral equations, it is desirable to find some new delay
impulsive inequalities, in order to achieve a diversity of desired goals. In this paper, we discuss
a class of retarded integral inequalities with weak singularity for discontinuous functions,
Let α(t) be a continuous, differentiable and increasing function on [t, +∞) with α(t) ≤
t, α(t) = t, then
which generalize the inequality () in [] to the weakly singular integral inequality, and ()
in [] to the retarded inequality. We use the modification of Medveď ’s method to obtain
the explicit estimations of the unknown function in the inequality (), and we use the
analysis technique to get the explicit estimations of the unknown function in the inequalities
() and (). Finally, we give two examples to illustrate applications of our results.
2 Main results
Throughout this paper, R denotes the set of real numbers and R+ = [, ∞) is the given
subset of R, and C(M, S) denotes the class of all continuous functions defined on set M
with range in the set S.
The following lemmas are very useful in the procedures of our proof in our main results.
Lemma Suppose that f (x) and g(x) are nonnegative and continuous functions on [c, d].
Let p > , q + p = . Then
c
d
α(t)
α(t)
f (s)g(s) ds ≤
c
d
α(t)
α(t)
()
Proof We prove the inequality (). Using the inequality (), we obtain
f (s)g(s) ds =
f α(s) g α(s) α (s) ds =
f α(s) α (s) /pg α(s) α (s) /q ds
≤
=
t
t
t
t
α(t)
α(t)
f p α(s) α (s) ds
Lemma ([, ]) Let β, γ , ξ and p be positive constants. Then
t tβ – sβ p(γ –)sp(ξ–) ds = tβθ B
p(ξ –β) + , p(γ – ) + , t ∈ [, +∞).
Let α(t) be a continuous, differentiable and increasing function on [t, +∞) with α(t) ≤ t,
α(t) = t, then
α(t)
α(t)
αβ (t) – sβ p(γ –)sp(ξ–) ds ≤ αθβ(t) B
p(ξ –β) + , p(γ – ) + , t ∈ [, +∞),
where B[x, y] = sx–( – s)y– ds (x > , y > ) is the wellknown betafunction and θ =
p[β(γ – ) + ξ – ] + . Suppose that the positive constants β, γ , ξ , p and p satisfy
conditions:
() if β ∈ (, ], γ ∈ (/, ) and ξ ≥ / – γ , p = /γ ;
() if β ∈ (, ], γ ∈ (, /] and ξ > ( – γ )/( – γ ), p = ( + γ )/( + γ ), then
B
pi(ξ –β) + , pi(γ – ) + ∈ [, +∞),
and θi = pi[β(γ – ) + ξ – ] + ≥ are valid for i = , .
Lemma Let u(t), a(t), b(t), h(t) ∈ C(R+, R+), α(t) be a continuous, differentiable and
increasing function on R+ with α(t) ≤ t, α() = . If u(t) satisfies the following inequality:
Then
where
Proof Define a function v(t) on R+ by
v(t) = e α(t)
h(s)u(s) ds,
we have v() = . Differentiating v(t) with respect to t and using () and (), we have
Integrating both sides of the inequality () from to t, since v() = we get
v(t) ≤
t
From () and (), we obtain
α (s)h α(s) a α(s) e α(s) ds =
h(s)a(s)e(s) ds.
α(t)
α(t)
h(s)u(s) ds ≤ e(α(t))
α(t)
h(s)a(s)e(s) ds.
Substituting the inequality () into () we get the required estimation (). The proof is
completed.
Lemma Let a ≥ , p ≥ q ≥ and p = , then
a pq ≤ pq a + p –p q .
a pq ≤ pq K (q–p)/pa + p –p q K q/p,
for any K > . Let K = , we get ().
Proof If q = , the inequality above is obviously valid. On the other hand, if q > , let δ =
q/p, then δ ≤ , by [], [] (Lemma .), we obtain
Theorem Let a(t), f(t), f(t), f(t) ∈ C(R+, R+), and a(t) is a nondecreasing function, and
let α(t) be a continuous, differentiable and increasing function on R+ with α(t) ≤ t, α() = .
Let β, γ , ξ be positive constants. Suppose that u(t) satisfies the inequality ().
() If β ∈ (, ], γ ∈ (/, ) and ξ ≥ / – γ , we have
() If β ∈ (, ], γ ∈ (, /] and ξ > ( – γ )/( – γ ), we have
w(t) ≤
a˜ (t) +
b˜(t)
e˜(α(t))
α(t)
h˜ (s)a˜ (s)e˜(s) ds
,
t ∈ R+,
+γγ
()
γ
a˜ (t) = –γ a –γ (t),
,
t
,
where
where
e˜(t) = exp –
Proof If β ∈ (, ], γ ∈ (/, ) and ξ ≥ / – γ , let
if β ∈ (, ], γ ∈ (, /] and ξ > ( – γ )/( – γ ), let
( + γ )
p = ( + γ ) ,
q =
( + γ )
γ ,
then
+ = ,
pi qi
i = , .
Using Hölder’s inequality in Lemma applied to (), we have
u(t) ≤ a(t) + α(t) αβ (t) – sβ pi(γ –)spi(ξ –) ds /pi
α(t) fqi (s)uqi (s) ds /qi
α(t) αβ (t) – sβ pi(γ –)spi(ξ –) ds /pi
α(t)
s
f(s)
f(τ )u(τ ) dτ
qi
ds
/qi
.
()
qi
/qi
ds
α(t) αβ (t) – sβ pi(γ –)spi(ξ –) ds /pi
α(t) αβ (t) – sβ pi(γ –)spi(ξ –) ds /pi
α(t) fqi (s)uqi (s) ds /qi
× α(t) f(s) s f(τ )u(τ ) dτ qi ds /qi .
Then z(t) is a nondecreasing function, and u(t) ≤ z(t), from (), we have
z(t) ≤ a(t) + α(t) αβ (t) – sβ pi(γ –)spi(ξ –) ds /pi
α(t) fqi (s)zqi (s) ds /qi
α(t) αβ (t) – sβ pi(γ –)spi(ξ –) ds /pi
α(t) αβ (t) – sβ pi(γ –)spi(ξ –) ds /pi
α(t) αβ (t) – sβ pi(γ –)spi(ξ –) ds /pi
f(s) s f(τ )z(τ ) dτ
α(t) fqi (s)zqi (s) ds /qi
Set
z(t) = a(t) +
+
+
×
+
+
α(t) fqi (s)zqi (s) ds
+ qi–
α(t) αβ (t) – sβ pi(γ –)spi(ξ –) ds qi/pi
× α(t) f(s) s f(τ ) dτ qi zqi (s) ds.
Using Lemma , the inequality () can be restated as
zqi (t) ≤ qi–aqi (t) + qi– Miαθi (t) qi/pi
× α(t) fqi (s) + f(s) s
for t ∈ R+, where
Theorem Let u(t) is a nonnegative piecewise continuous function with discontinuous
of the first kind in the points ti (t < t < t < · · · , limi→∞ ti = ∞), a(t), f (t) ∈ C(R+, R+),
a(t) ≥ , and let α(t) be a continuous, differentiable and increasing function on [t, +∞)
with α(t) ≤ t, α(ti) = ti, i = , , , . . . . Let p, β , γ be positive constants, βi ∈ [, ∞). If u(t)
satisfies the inequality (), then we have
,
s
t
,
+
βju(tj – ),
Proof Firstly, we consider the case t ∈ [t, t), denoting
v(t) = a(t) +
α(t)
t
then v(t) is a nonnegative and nondecreasing continuous function, and
u(t) ≤ v(t),
v(t) = a(t).
Differentiating () with respect to t, we have
v (t) = a (t) + α (t) tβ – αβ (t) γ –f α(t) u α(t)
u α(t) +
≤ a (t) + α (t) tβ – αβ (t) γ –f α(t) v α(t) v α(t) +
α(t)
t
p
then (t) is a nonnegative and nondecreasing function, and (t) = a(t), since a(t) ≥ ,
we can conclude that v(t) ≤ (t), differentiating (), from (), we obtain
+ α (t)g α(t)
(t) = v α(t) v α(t) α (t) + α (t)g α(t) v α(t)
≤ α(t) α (t) a (t) + α (t) tβ – αβ (t) γ –f α(t)
α(t)
p(t)
≤ (t)α (t) a (t) + α (t) tβ – αβ (t) γ –f α(t)
(t) p(t)
From (), we have
–(p+) (t) ≤
–(p+)(t) α (t)a (t) + α (t)g α(t)
+ α (t) tβ – αβ (t) γ –f α(t) .
Let η(t) =
–(p+)(t), then η (t) = –(p + ) –(p+) (t), () can be restated as
η (t) + (p + )η(t) α (t)a (t) + α (t)g α(t)
≥ –(p + ) α (t) tβ – αβ (t) γ –f α(t) .
()
()
()
()
()
()
()
η(t) exp (p + )
a α–(s) + g(s) ds
≥ –(p + ) α (t) tβ – αβ (t) γ –
f α(t)
× exp (p + )
a α–(s) + g(s) ds ,
integrating both sides of () from t to t, we obtain
η(t) exp (p + )
a α–(s) + g(s) ds
– η(t)
≥ –(p + ) α (t) tβ – αβ (t) γ –
f α(t)
× exp (p + )
a α–(s) + g(s) ds
≥
α(t)
t
–(p + ) tβ – sβ (t)
γ –
f (s)
× exp (p + )
a α–(τ ) + g(τ ) dτ
ds,
p(t) ≤
(t) =
α(t)
Multiplying by exp((p + ) tα(t)(a (α–(s)) + g(s)) ds) on both sides of (), we have
()
()
()
()
()
()
()
since η(t) =
–(p+)(t) = a–(p+)(t), denoting
g(τ )) dτ ), from (), we have
(t) = exp((p + ) tα(s)(a (α–(τ )) +
η(t) ≥
α(t)(tβ – sβ )γ –f (s) (s)
– a(p+)(t)(p + ) t
a(p+)(t) (t)
,
by η(t) =
–(p+)(t), from (), we have
p
p+
,
p
p+
,
a(p+)(t) (t)
α(t)(tβ – sβ )γ –f (s) (s) ds
– a(p+)(t)(p + ) t
α(t)(tβ – sβ )γ –f (s) ds > , setting
where – a(p+)(t)(p + ) t
a(p+)(t) (t)
α(t)(tβ – sβ )γ –f (s) (s) ds
– a(p+)(t)(p + ) t
from (), (), () and (), we have
v (t) ≤ a (t) + α (t) tβ – αβ (t)
γ –f α(t) v α(t)
(t).
Integrating both side of () from t to t, we get
t
t
v(t) ≤ a(t) +
α (s) tβ – αβ (s)
γ –f α(s) v α(s)
(s) ds
= a(t) +
tβ – s
β γ –
f (s)v(s)
α–(s)
α (s)
ds.
Equation () has the same form as Lemma , and the functions of () satisfy the
conditions of Theorem . Consequently, by using a similar procedure to Lemma and
Theorem , we can get the desired estimations () for t ∈ [t, t).
Next, let us consider the interval [t, t), when t ∈ [t, t), () can be restated as
α(t) tβ – sβ γ –f (s)u(s) u(s) +
()
()
then
(t) is a nonnegative and nondecreasing function, and
Differentiating with respect to t both sides of (), we obtain
(t) = A(t) + α (t) tβ – α(t)β γ –f α(t) u α(t) u α(t) +
≤ A(t) + α (t) tβ – α(t)β γ –f α(t)
α(t)
×
α(t) +
α(t)
t
p
g(s) (s) ds ,
α(t)
t
p
() has the same form of (), and using a similar procedure for t ∈ [t, t), we can get the
desired estimations () for t ∈ [t, t).
Consequently, by using a similar procedure for t ∈ [ti, ti+), we can get the desired
estimations () for t ∈ [ti, ti+). Thus we complete the proof of Theorem .
t ∈ [ti, ti+), i = , , , . . . ,
b˜(t)
e˜i(α(t)) ti
α(t)
where M, θ are the same as in Theorem , and
E(t) = a(t),
t ∈ [t, t),
Ei(t) = a(t) + b(t)
i
j=
tj
α(ti)
β
α (t) – s
β γ – ξ –
s
m
f (s) u (s) +
t ∈ [ti, ti+), i = , , , . . . ,
b˜ (t)
e˜i(α(t))
α(t)
h˜ i(s)a˜ i(s)e˜i(s) ds
γ
+γ /p
,
()
where M, θ are the same as in Theorem and Ei, Ai, Bi, hi, i = , , , . . . , are the same in
() of Theorem ,
a˜ i(t) =
+γ
γ
A
i
+γ
γ
θ
b˜ (t) = Mα (t)
(t),
Proof When t ∈ [t, t), () can be restated as
ti
t
α(t)
t
p
u (t) ≤ a(t) + b(t)
β
α (t) – s
β γ – ξ –
s
m
f (s) u (s) +
g(τ )un(τ ) dτ
q
ds,
()
s
t
by Lemma , we obtain
s
t
um(s) +
+ b(t)
+ b(t)
α(t)
t
t
α(t)
t
α(t)
α(t)
α(t)
t
t
from () and (), we have
By Lemma and (), we obtain
up(t) ≤ a(t) + w(t)
or
u(t) ≤ a(t) + w(t) /p.
um(t) ≤ a(t) + w(t) m/p
un(t) ≤ a(t) + w(t) n/p
≤ mp a(t) + w(t) + p –p m ,
≤ np a(t) + w(t) + p –p n .
Substituting the inequality () and () into () we have
w(t) ≤ b(t)
αβ (t) – sβ γ –sξ –( – q)f (s) ds
αβ (t) – sβ γ –sξ –qf (s)um(s) ds
αβ (t) – sβ γ –sξ –qf (s)
g(τ )un(τ ) dτ ds,
s
t
s
s
t
t
αβ (t) – sβ γ –sξ –qf (s) mp a(s) + w(s) + p –p m
ds
αβ (t) – sβ γ –sξ –qf (s)
g(τ ) np a(τ ) + w(τ ) + p –p n
dτ ds
αβ (t) – sβ γ –sξ –f (s) ( – q) + q
αβ (t) – sβ γ –sξ –qf (s)
αβ (t) – sβ γ –sξ – mq f (s)w(s) ds
p
mp a(s) + p –p m
g(τ ) np a(τ ) + p –p n
ds
dτ ds
α(t)
where
A(t) = b(t)
α(t) tβ – sβ γ –sξ–B(s) ds,
g(t) = mpq f (t),
g(t) = np g(t).
≤ b(t)
αβ (t) – sβ γ –sξ–B(s) ds + b(t)
α(t) tβ – sβ γ –sξ–g(s)w(s) ds
αβ (t) – sβ γ –sξ–g(s)
g(τ )w(τ ) dτ ds,
s
t
s
t
s
t
q
q
Since () have the same form as () and the functions of () satisfy the conditions of
Theorem , applying Theorem to (), considering equation (), we can get the desired
estimations () and () for t ∈ [t, t).
Then, when t ∈ [t, t), () can be restated as
ds
α(t)
t
+ βup(t – ) + b(t)
αβ (t) – sβ γ –sξ–f (s)
× um(s) +
then we have
t
+ βup(t – ),
up(t) ≤ E(t) + b(t)
From (), we can conclude that the estimates ()and () are valid for t ∈ [t, t).
Consequently, by using a similar procedure for t ∈ [ti, ti+), we complete the proof of
theorem.
3 Some applications
Example Consider the following Volterra type retarded weakly singular integral
equations:
α(t)
t
yp(t) –
which arises very often in various problems, especial describing physical processes with
aftereffects. Ma and Pečarić [] discussed the case α(t) = t, g(t) ≡ in ().
t
t
where
γ
a˜ (t) = –γ
θ = γ β(γ – ) + ξ – + ,
A(t) = ( – q) + q p h(t) + p p–
+ qK q–
t
g(τ ) h(τ ) + p – dτ ,
p p
A(t) = pq ,
A(t) = qK q–,
A(t) = p g(t) .
() If β ∈ (, ], γ ∈ (, /] and ξ > ( – γ )/( – γ ), we have
y(t) ≤
h(t) + a˜ (t) +
b˜ (t)
e˜(α(t)) t
α(t)
+γγ /p
h˜(s)a˜ (s)e˜(s) ds
t
α(t) +γ
A γ (s) ds,
Applying Theorem for t ∈ [t, t) (with m = n = , a(t) = h(t), b(t) = λt–βδ/ (γ ), ξ =
β( + δ)) to (), we obtain the desired estimations () and ().
Example Consider the following impulsive differential system:
d(xd(tt)) = F(t, x), t = ti, t ∈ [t, ∞),
(x)t=ti = βix(ti – ),
x(t) = x,
F(s, x) ≤ tβ – sβ γ –f (s) x(s),
t
t
x(t) = x +
F s, x(s) ds +
βix(ti – ).
By using the condition (), from (), we have
x(t) ≤ x +
t tβ – sβ γ –f (s) x(s) ds +
t
Let u(t) = x(t), from (), we get
u(t) ≤ x +
t tβ – sβ γ –f (s) u(s) ds +
t
βiu(ti – ).
where f (t) ∈ C(R+, R+), β ∈ (, ], γ ∈ (/, ).
Then the impulsive differential system () and () are equivalent to the integral
equation
()
()
()
()
()
()
By Lemma , we have
tt tβ – sβ γ –f (s) u(s) + ds +
tt tβ – sβ γ – f (s) u(s) ds +
tt tβ – sβ γ – f (s) ds +
t<ti<t
βiu(ti – )
tt tβ – sβ γ – f (s) u(s) ds +
,
where M, θ are the same as in Theorem , and
αβ (t) – sβ γ –f (s) u(s) ds
E(t) = a(t), t ∈ [t, t),
Ei(t) = a(t) +
i
ti
j= tj
+
i
βju(tj – ), t ∈ [ti, ti+), i = , , . . . ,
j=
γ
a˜ i(t) = –γ Ai–γ (t), i = , , , . . . ,
t tβ – sβ γ –Bi(s) ds, i = , , , . . . ,
Ai(t) =
ti
Bi(t) = f (t) + Ei(t) , i = , , , . . . ,
γ
b˜ (t) = Mαθ (t) –γ ,
g(t) = f (t).
4 Conclusion
In this paper, we generalized the weakly singular integral inequality. The first inequality
was a generally weak singular type, the second inequality was a likeweakly singular type
with discontinuous functions, the third inequality was a type of weakly singular integral
inequality with impulsive. We used analytical methods, reducing the inequality with the
known results in the lemma, and the estimations of the upper bound of the unknown
functions were given. The results were applied to the weakly singular integral equation
and the impulsive differential system.
Acknowledgements
The authors are very grateful to the editor and the referees for their careful comments and valuable suggestions on this
paper. This work is supported by the Natural Science Foundation of China (11561019), Guangxi Natural Science
Foundation (2016GXNSFAA380090) and (2016GXNSFAA380125).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
LZZ organized and wrote this paper. WWS examined all the steps of the proofs in this research and gave some advice. All
authors read and approved the final manuscript.
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1. Abdeldaim , A , Yakout, M: On some new integral inequalities of GronwallBellmanPachpatte type . Appl. Math. Comput . 217 , 7887  7899 ( 2011 )
2. Agarwal , RP: Difference Equations and Inequalities. Dekker, New York ( 1993 )
3. Agarwal , RP , Deng, SF , Zhang, WN: Generalization of a retarded Gronwalllike inequality and its applications . Appl. Math. Comput . 165 , 599  612 ( 2005 )
4. Bainov , DD , Simeonov, P : Integral Inequalities and Applications. Kluwer Academic, Dordrecht ( 1992 )
5. Bellman , R: The stability of solutions of linear differential equations . Duke Math. J . 10 , 643  647 ( 1943 )
6. Cheng, KL, Guo, C , Tang, M: Some nonlinear GronwallBellmanGamidov integral inequalities and their weakly singular analogues with applications . Abstr. Appl. Anal . 2014 , Article ID 562691 ( 2014 )
7. Cheung , WS: Some new nonlinear inequalities and applications to boundary value problems . Nonlinear Anal . 64 , 2112  2128 ( 2006 )
8. Deng , SF , Prather, C: Generalization of an impulsive nonlinear singular GronwallBihari inequality with delay . J. Inequal. Pure Appl. Math. 9, Article 34 ( 2008 )
9. ElOwaidy , H , Ragab, AA , Abuelela, W , ElDeeb, AA: On some new nonlinear integral inequalities of GronwallBellman type . Kyungpook Math. J . 54 , 555  575 ( 2014 )
10. Gllo , A , Piccirilo, AM : About some new generalizations of BellmanBihari results for integrofunctional inequalities with discontinuous functions and applications . Nonlinear Anal . 71 , e2276  e2287 ( 2009 )
11. Gronwall , TH: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations . Ann. Math. 20 , 292  296 ( 1919 )
12. Abdeldaim , A : Nonlinear retarded integral inequalities of GronwallBellman type and applications . J. Math. Inequal . 10 ( 1 ), 285  299 ( 2016 )
13. Henry , D : Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. , vol. 840 . Springer, Berlin ( 1981 )
14. Sano , H , Kunimatsu, N: Modified Gronwall's inequality and its application to stabilization problem for semilinear parabolic systems . Syst. Control Lett . 22 , 145  156 ( 1994 )
15. Ye , HP , Gao, JM , Ding, YS: A generalized Gronwall inequality and its application to a fractional differential equation . J. Math. Anal. Appl . 328 , 1075  1081 ( 2007 )
16. Medved', M: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions . J. Math. Anal. Appl . 214 , 349  366 ( 1997 )
17. Medved', M: Integral inequalities and global solutions of semilinear evolution equations . J. Math. Anal. Appl . 267 , 643  650 ( 2002 )
18. Ma, QH, Pecˇaric´, J: Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations . J. Math. Anal. Appl . 341 ( 2 ), 894  905 ( 2008 )
19. Li , WN, Han, MA , Meng, FW: Some new delay integral inequalities and their applications . J. Comput. Appl . Math. 180 , 191  200 ( 2005 )
20. Lipovan , O: A retarded Gronwalllike inequality and its applications . J. Math. Anal. Appl . 252 , 389  401 ( 2000 )
21. Ma, QH , Yang, EH: Estimations on solutions of some weakly singular Volterra integral inequalities . Acta Math. Appl. Sin . 25 , 505  515 ( 2002 )
22. Mazouzi , S , Tatar, N: New bounds for solutions of a singular integrodifferential inequality . Math. Inequal. Appl . 13 ( 2 ), 427  435 ( 2010 )
23. Medved', M: Nonlinear singular integral inequalities for functions in two and n independent variables . J. Inequal. Appl . 5 ( 3 ), 287  308 ( 2000 )
24. Pachpatte , BG: Inequalities for Differential and Integral Equations . Academic Press, New York ( 1998 )
25. Tatar , NE: An impulsive nonlinear singular version of the GronwallBihari inequality . J. Inequal. Appl . 2006 , Article ID 84561 ( 2006 )
26. Wang , H , Zheng, KL: Some nonlinear weakly singular integral inequalities with two variables and applications . J. Inequal. Appl . 2010 , Article ID 345701 ( 2010 )
27. Willett , D: Nonlinear vector integral equations as contraction mappings . Arch. Ration. Mech. Anal . 15 , 79  86 ( 1964 )
28. Iovane , G: Some new integral inequalities of BellmanBihari type with delay for discontinuous functions . Nonlinear Anal . 66 , 498  508 ( 2007 )
29. Liu, XH , Zhang, LH, Agarwal, P , Wang, GT : On some new integral inequalities of GronwallBellmanBihari type with delay for discontinuous functions and their applications . Indag. Math. 27 , 1  10 ( 2016 )
30. Mi, YZ , Zhong, JY : Generalization of the BellmanBihari type integral inequality with delay for discontinuous functions . J. Sichuan Univ. Natur. Sci. Ed . 52 , 33  38 ( 2015 )
31. Mitropolskiy , YA , Iovane, G , Borysenko, SD : About a generalization of BellmanBihari type inequalities for discontinuous functions and their applications . Nonlinear Anal . 66 , 2140  2165 ( 2007 )
32. Yan , Y: Some new GronwallBellman type impulsive integral inequality and its application . J. Sichuan Normal Univ. Nat. Sci . 36 ( 4 ), 603  609 ( 2013 )
33. Zheng , B: Explicit bounds derived by some new inequalities and applications in fractional integral equations . J. Inequal. Appl . 2014 , 4 ( 2014 )
34. Zheng , ZW , Gao , X , Shao , J: Some new generalized retarded inequalities for discontinuous functions and their applications . J. Inequal. Appl . 2016 , 7 ( 2016 )
35. Kuczma , M: An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality . University of Katowice, Katowice ( 1985 )
36. Jiang , FC , Meng, FW : Explicit bounds on some new nonlinear integral inequalities with delay . J. Comput. Appl . Math. 205 , 479  486 ( 2007 )