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The goal of the present paper is to establish some new approach on the basic integral inequality of Gronwall-Bellman type and its generalizations involving function of one independent variable which provides explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain partial differential and integral equations.

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 [

Closely related to the foregoing first-order ordinary differential operators is the following result of Bellman [

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

Then

Proof: Define a function

where

Then

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

By using (2.5) and since

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

Then

Proof: Define a function

where

Then

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

By applying Lemma 2.1 implies the estimation of

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

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 R^{n},

Define

and

Also let

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.