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.