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The aim of this present paper is to establish some new integrodifferential inequalities of Gronwall type involving functions of one independent variable which provide explicit bounds on unknown functions. The inequalities given here can be used in the analysis of a class of differential equations as handy tools.

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 [

Let

for

then

for

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

holds, where

and

then

also

Proof: Define a function

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

Define a function

where

It is clear that

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

Differentiating both sides of (2.12) with respect to

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

Inequality (2.21) by using (2.19) and (2.20), and since

Let

where

Differentiating both sides of (2.23) with respect to

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

Multiplying both sides of (2.26) by

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

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

Multiplying both sides of (2.32) by

which can be rewritten as

From (2.15) and (2.33), we get

Integrating both sides of the above inequality from 0 to

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

where

and

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

Also

Proof: Integrating both sides of (2.35) from 0 to

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.