1. Introduction
The Gronwall type integral inequalities provide a necessary tool for the study of the theory of differential equations, integral equations and inequalities of the various types (please, see Gronwall [1] and Guiliano [2] ). Some applications of this result to the study of stability of the solution of linear and nonlinear differential equations may be found in Bellman [3] . Numerous applications to the existence and uniqueness theory of differential equations may be found in Nemyckii-Stepanov [4] , Bihari [5] , and Langenhop [6] . During the past few years several authors (see references below and some of the references cited therein) have established several Gronwall type integral inequalities in one or two independent real variables [1] -[14] . Of course, such results have application in the theory of partial differential equations and Volterra integral equations.
In [14] , Pachpatte investigated the following inequality:
Lemma: Let 
be nonnegative continuous functions defined on 
and 
be nonnegative constant. Let 
be a nonnegative continuous function defined for
, and monotonic nondecreasing with respect to 
for ant fixed
. If
For
, then
where 
is the maximal solution of
For
.
2. Main Results
Theorem 2.1: Let 
and 
be nonnegative continuous functions defined on
. Let 
be a positive continuous and nondecreasing functions in both variables and defined for
. If
(2.1)
Then
where 
and
, (2.2)
and 
is the maximal solution of
(2.3)
Proof: Define a function 
by the right-hand side of (2.1).Then
(2.4)
By using (2.4) in (2.1), we get
(2.5)
Since 
is a positive continuous and nondecreasing function, then
(2.6)
Let
(2.7)
From (2.6) and (2.7), we observe that
(2.8)
And
(2.9)
Differentiating both sides of (2.8) with respect to 
and
, we get
(2.10)
By keeping first 
fixed in (2.10) and set 
and integrate from 0 to 
then again keeping 
fixed, set 
and integrate from 0 to 
respectively and using (2.9), we get
(2.11)
From (2.7) and (2.11), it is clear that
From (2.4), it can be restated as
where 
is the maximal solution of
This completes the proof.
Theorem 2.2: Let 
be defined as in Theorem 2.1. If
(2.12)
Then
where 
and
, (2.13)
where 
is the maximal solution of
Proof: Define a function 
by the right-hand side of (2.12).Then
(2.14)
By using (2.14) in (2.12), we get
(2.15)
By following the same steps of Theorem 2.1 from (2.5)-(2.11), we get
(2.16)
From (2.7), (2.14) and (2.16), we observe that
where 
is the maximal solution of
This completes the proof.
Theorem 2.3: Let 
be defined as in Theorem 2.1. If
(2.17)
Then
where 
and
, (2.18)
where 
is the maximal solution of
Proof: Define a function 
by the right-hand side of (2.17).Then
(2.19)
where 
By using (2.19) in the above equation, we get
(2.20)
and
(2.21)
Since 
is a positive continuous and nondecreasing function, then from (2.20),
(2.22)
Let
(2.23)
where
(2.24)
And
(2.25)
Differentiating both sides of (2.24) with respect to 
and
, and from (2.23), we get
(2.26)
By keeping first x fixed in (2.26) and set 
and integrate from 0 to y then again keeping y fixed, set 
and integrate from 0 to x respectively and using (2.25), we get
(2.27)
From (2.19), (2.23) and (2.27), it is clear that
where 
is the maximal solution of
This completes the proof.
Application: As an application, let us consider the bound on the solution of a nonlinear hyperbolic partial differential equation of the form
(2.28)
with the given boundary conditions
(2.29)
where 
and 
such that
(2.30)
(2.31)
where 
and 
be nonnegative continuous functions defined on a domain D
. The Equation (2.28) with (2.29) is equivalent to the integral equation
(2.32)
Let 
be any solution of (2.28) with (2.29) and taking absolute values of both sides, we get
(2.33)
Using (2.30)-(2.32) in (2.33) and assuming that
, where 
be a positive continuous and nondecreasing function defined in the respective domain, we have
Then 
where
, and 
is the maximal solution of
The remaining proof will be the same as the proof of Theorem 2.1 with suitable modifications.
We note that Theorem 2.1 can be used to study the stability, boundedness and continuous dependence of the solutions of (2.28).