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. Some applications of this result can be used to the study of existence, uniqueness theory of differential equations and the stability of the solution of linear and nonlinear differential equations. During the past few years, several authors have established several Gronwall type integral inequalities in one or two independent real variables [1] -[15] . Of course, such results have application in the theory of partial differential equations and Volterra integral equations.

Closely related to the foregoing first-order ordinary differential operators is the following result of Bellman [11] : If the functions and are nonnegative for, and if, the inequality

implies that

Our aim in this paper is to establish new explicit bounds on some basic integral inequalities of one independent variable which will be equally important in handling the inequality (1.1). Given application in this paper is also illustrating the usefulness of our result.

Lemma 2.1: Let and be nonnegative continuous functions defined for. Let defined for and also be nonnegative continuous functions defined for. If

Then

Proof: Define a function by the right-hand side of (2.1), such that

where

Then. From (2.1) and (2.3), we observe that

Differentiating both sides of (2.3) with respect to t, we get

By using (2.5) and since, the above equation can be restated as

Integrating both sides of (2.6) from 0 to t and also using (2.4), we observe that

From (2.5) and (2.7), we get the required inequality (2.2).

Theorem 2.2: Let, and be nonnegative continuous functions defined for. Let defined for and also be nonnegative continuous functions defined for. If

Then

Proof: Define a function by the right-hand side of (2.8), such that

where

Then. From (2.9) and (2.10), we observe that

Differentiating both sides of (2.10) with respect to t, we get

By using (2.12), the above equation can be restated as

where

and

Again differentiating both sides of (2.14) with respect to x and using (2.13) and using the fact that, we get

By applying Lemma 2.1 implies the estimation of as

By substituting (2.17) in (2.13), we have

Integrating both sides of the above inequality from 0 to t and also using (2.11), we observe that

From (2.12) and (2.18), we get the required inequality (2.9). This completes the proof.

Theorem 2.3: Let, and, and be defined as in Theorem 2.2. If

Then

Proof: The proof of Theorem 2.3 is the same as the proof of Theorem 2.2 and by applying the Lemma 2.1 with suitable modifications.

As an application, let us consider the bound for the solution of Volterra integral equation of the form

where x, f and g are the elements of Rⁿ, is a n × n matrix, and and T be a continuous operator such that T maps into.

Define

and

Also let, where (3.4)

Then

Proof: Taking absolute value of the both sides of (3.1), we get

By substituting from (3.2), (3.3), (3.4) and (3.5) in (3.6), we have

The remaining proof will be the same as the proof of Theorem 2.2 with suitable modifications. We note that Theorem 2.2 can be used to study the stability, boundedness and continuous dependence of the solutions of (3.1).

We finally mention that the integral inequalities obtained in this paper allow us to study the stability, boundedness and asymptotic behavior of the solutions of a class of more general partial differential and integral equations.