The differential and integral inequalities occupy a very privileged position in the theory of differential and integral equations. In recent years, these inequalities have been greatly enriched by the recognition of their potential and intrinsic worth in many applications of the applied sciences. The integrodifferential inequalities recently established by Gronwall and others [1] -[12] have attracted considerable attention in the theory of differential and integral equations. This fact encourages us to find the explicit bounds on some fundamental integrodifferential inequalities which can be applied fairly well to achieve a diversity of desired goals. In [3] , Pachpatte (1977) gave the following useful integrodifferential inequality:

Let, and be nonnegative continuous functions defined on and is constant. If

for and is defined by

then

for where

.

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

Theorem 2.1: Let, and be nonnegative continuous functions defined on for which the inequality

holds, where is positive constant and. If

and

then

, and

also

Proof: Define a function by the right-hand side of (2.1). Then

where

Then from (2.1) and (2.7), we have

Integrating both sides of (2.9) from 0 to t, we observe that

Differentiating both sides of (2.7) with respect to and using (2.9) and (2.10), we get

Define a function by the right-hand side of (2.11), then

where

It is clear that

By using (2.12) in (2.11), we have

Differentiating both sides of (2.12) with respect to, we get

By using (2.14) and (2.15) in the above equation, we observe that

Let

where

and

Using (2.17) in (2.16), we get

Differentiating both sides of (2.17) with respect to, we get

Inequality (2.21) by using (2.19) and (2.20), and since if takes the form

Let

where

Differentiating both sides of (2.23) with respect to, we get

Inequality (2.22) by using (2.23) and (2.25), takes the form

Multiplying both sides of (2.26) by and integrating the resulting inequality from 0 to, and using (2.24), we have

By using (2.23) in the above inequality, it can be seen that

which can be rewritten as

Using (2.27) in (2.20), we observe that

Let

where

Differentiating both sides of (2.29) with respect to, we get

Inequality (2.28) by using (2.29) and (2.31), takes the form

Multiplying both sides of (2.32) by and integrating the resulting inequality from 0 to, and using (2.29) and (2.30), we have

which can be rewritten as

From (2.15) and (2.33), we get

Integrating both sides of the above inequality from 0 to, and from (2.8), we observe that

From (2.9) and (2.34), we have

Application: As an application we obtain the bound on the solution of the differential equation of the formulation of the form

with the given initial conditions

where is a continuous function and are real constants.,. Here we assume that the solution of (2.35) and (2.36) exists on Assume that the function in (2.35) satisfies the condition

where is a real valued nonnegative continuous function defined on. If

and

then the bounds on the solution (2.35) takes the form

, where, , and

Also

Proof: Integrating both sides of (2.35) from 0 to, and using (2.36), we observe that

Taking absolute values of both sides of the above equation and using (2.37), we get

The remaining proof is the same as Theorem 2.1 by following the same steps from (2.7)-(2.35) in (2.39) with suitable modifications, we get the required bound of (2.35).

We note that many generalizations, extensions, variants and applications of the inequality given in this paper are possible and we hope that the result given here will assure greater importance in near future.