International Journal of Analysis and Applications
ISSN 2291-8639
Volume 6, Number 2 (2014), 164-169
http://www.etamaths.com

BOUNDS OF CERTAIN DYNAMIC INEQUALITIES ON TIME

SCALES

DEEPAK B. PACHPATTE

Abstract. In this paper we study explicit bounds of certain dynamic integral

inequalities on time scales. These estimates give the bounds on unknown

functions which can be used in studying the qualitative aspects of certain
dynamic equations. Using these inequalities we prove the uniqueness of some

partial integro-differential equations on time scales.

1. Introduction

In 1989 German Mathematician Stefan Hilger [4] initiated the study of time scale
in his Ph.D dissertation. Dynamic inequalities on time scales has applications in
various fields. During past few years many authors have studied various types of
dynamic equations and inequalities on time scales, its properties and applications
[1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Motivated by need for the diverse applications
and to widen the scope of such inequalities in this paper we obtain some explicit
bounds of certain dynamic inequalities on time scales.

2. Main Results

Let R denotes the real number Z the set of integers and T denotes the arbitrary
time scales and IT = I ∩ T, I = [t0,∞). Let Crd be the set of all rd continuous
function. We assume here basic understaing of time scale calculus. The research
monograph [2, 3] gives the basic information on time scales calculus.

Now we give here our main results.
Theorem 2.1 Let p(t,s),q(t,s) ∈ Crd(IT × IT,R+) and be nondecreasing for

t ∈ IT for each s ∈ IT and

(2.1) u (t) ≤ c +
t∫

t0

p (t,τ)u (τ) ∆τ +

α∫
t0

q (t,τ)u (τ) ∆τ,

for t ∈ IT where c ≥ 0 is a constant. If

(2.2) k (t) =

α∫
t0

q (t,τ) ep (τ,s)∆τ < 1,

2010 Mathematics Subject Classification. 26D10, 26E70, 34N05.
Key words and phrases. explicit bounds; integral inequality; Dynamic equations; time scales.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

164



BOUNDS OF CERTAIN DYNAMIC INEQUALITIES ON TIME SCALES 165

then

(2.3) u (t) ≤
c

1 −k (t)
ep (t,τ) .

Proof. For fixed X, t0 ≤ X ≤ α then for t0 ≤ t ≤ X we have

(2.4) u (t) ≤ c +
t∫

t0

p (X,τ)u (τ) ∆τ +

α∫
t0

q (X,τ)u (τ) ∆τ.

Define a function w(t,X), t0 ≤ t ≤ X by right hand side of (2.4) we get

(2.5) u(t) ≤ w(t,X), t0 ≤ t ≤ X,

(2.6) w (t0,X) = c +

α∫
t0

q (X,τ)u (τ) ∆τ,

and

(2.7) w∆ (t,X) = p (X,t) u (t) ≤ p (X,t) w (t) ,

for t0 ≤ T.
By taking t = s and integrating it with respect to s from t0 to X we have

(2.8) w (X,X) ≤ w (t0,X) ep (t0,X) .

Since X is arbitrary in (2.5) and (2.8) X replaced by t and u(t) ≤ w(t,t) we get

(2.9) u (t) ≤ w (t0, t) ep (t0,s) ,

where

(2.10) w (t0, t) = c +

α∫
t0

q (t,τ)u (τ) ∆τ.

Using (2.9) on the right hand side of (2.10) and by (2.2) we have

(2.11) w (t0, t) ≤
c

1 −k (t)
.

Using (2.11) in (2.9) we get the result
Theorem 2.2 Let f(t,τ),g(t,τ),h(t,τ) ∈ Crd(IT × IT,R+), f(t,τ),g(t,τ) are

nondecreasing in t for each τ ∈ IT and if

(2.12) u (t) ≤ c +
t∫

t0

f (t,τ)


u (τ) + τ∫

t0

h (τ,s) u (s) ∆s


 ∆τ + α∫

t0

g (t,τ)u (τ) ∆τ.

for t ∈ IT. Then

(2.13) u (t) ≤
c

1 −k1 (t)
ef (t0,τ) ,

where

(2.14) k1 (t) =

τ∫
t0

g (t,τ)eF (t0,τ) ∆τ < 1,



166 PACHPATTE

(2.15) F (t,τ) = f (t,τ)


1 + τ∫

t0

h (τ,ξ) ∆ξ


 .

Proof. Let c > 0 and for any fix X ∈ IT then for t0 ≤ t ≤ X. From (2.13) we have

(2.16) u (t) ≤ c+
t∫

t0

f (X,τ)


u (τ) + τ∫

t0

h (τ,s) u (s) ∆s


 ∆τ + α∫

t0

g (X,τ)u (τ) ∆τ.

Define a function w(t,X), t ∈ [t0,X], u(t) ≤ w(t,X),w(t,x) > 0

(2.17) w (t0,X) = c +

α∫
t0

g (X,τ)u (τ) ∆τ,

and

w∆ (t,X) = f (X,t)


u (τ) + t∫

t0

h (t,s) u (s) ∆s




≤ f (X,t)


w (t) + t∫

t0

h (t,s) w (s,X) ∆s


 .(2.18)

From (2.17) and since w(t,X) is nondecreasing in t we have

(2.19)
w∆ (t,X)

w (t,X)
≤ f (X,t)


1 + t∫

t0

h (t,s) ∆s


 ,

for t ∈ [t0,X].
Now taking t = ξ and integrating with respect to ξ from t0 to X we get

(2.20) w (X,X) ≤ w (t0,X) eF (t0,τ)

Since X is arbitrary with X replaced by t we have for t ∈ IT,

(2.21) w (t,t) ≤ w (t0, t) eF (t0,τ) ,

and

(2.22) w (t0, t) = c +

α∫
t0

g (t,τ)u (τ) ∆τ,

for t ∈ IT.
Since u(t) ≤ w(t,t) we get from (2.21)

(2.23) u (t) ≤ w (t0, t) eF (t0,τ) .

Now from (2.23), (2.22) and from (2.14) we have

(2.24) w (t0, t) ≤
c

1 −k1 (t)
.

Using (2.24) in (2.25) we get

(2.25) u (t) ≤
c

1 −k1 (t)
eF (t0,τ) .



BOUNDS OF CERTAIN DYNAMIC INEQUALITIES ON TIME SCALES 167

Theorem 2.3 Let a,b,c ∈ Crd(IT,R+) and

(2.26) u (t) ≤ c +
t∫

t0

a (τ)


u (τ) + τ∫

t0

b (s) u (s) ∆s +

α∫
t0

d (s)u (s) ∆s


 ∆τ,

(2.27) z =

α∫
t0

d (s)ea+b (t0,s) ∆s < 1,

then

(2.28) u(t) ≤
c

1 −z
ea+b (t0,τ) ,

for t ∈ IT.
Proof. Now define a function w(t) by right hand side of (2.26) then w(t0) =

c,u(t) ≤ w(t) and

w∆ (t) = a (t)


u (t) + t∫

t0

b (s) u (s) ∆s +

α∫
t0

d (s)u (s) ∆s




≤ a (t)


w (t) + t∫

t0

b (s) w (s) ∆s +

α∫
t0

d (s)w (s) ∆s


 ,(2.29)

for t ∈ IT . Now define a function v(t) by

(2.30) v(t) = w (t) +

t∫
t0

b (s) w (s) ∆s +

α∫
t0

d (s)w (s) ∆s,

then w(t) ≤ v(t), w∆ (t) ≤ a(t)v(t)

(2.31) v (t0) = c +

α∫
t0

d (s)w (s) ∆s,

and

v∆ (t) = w∆ (t) + b (t) w (t)

≤ a (t) v (t) + b (t) w (t)
≤ [a(t) + b(t)] w (t) .(2.32)

We get

(2.33) v (t) ≤ v (t0) ea+b ((t0, t)) ,

for t ∈ IT. Using (2.33) in w(t) ≤ v(t) we have

(2.34) w (t) ≤ v (t0) ea+b(t0, t).

Now from (2.33), (2.31) and from (2.27) we have

(2.35) v (t0) ≤
c

1 −z
,

Using (2.34) in (2.33) and we have u(t) ≤ w(t) we get (2.28).



168 PACHPATTE

3. Applications

Now we consider the following dynamic equation

y (t) = a (t) +

t∫
t0

G


t,τ,y (τ) , τ∫

t0

b (τ,s,y (τ)) ∆s


 ∆τ

+

α∫
t0

d (t,τ,x (τ))∆τ(3.1)

for t ∈ IT where y(t) is unknown function and α ∈ Crd(IT,R+), b,d ∈ Crd(IT ×
Rn,Rn), G ∈ Crd(IT ×Rn ×Rn,Rn).

Now we give the application of Theorem 2.2 for studying certain properties of
solution of equation (3.1).

Theorem 3.1 Suppose that the function a,b,d,G as in (3.1) satisfy the condi-
tions

(3.2) |a (t)| ≤ c,

(3.3) |b (t,τ,y)| ≤ h (t,τ) |y| ,

(3.4) |d (t,τ,y)| ≤ g (t,τ) |y| ,

(3.5) |G (t,τ,y,x)| ≤ f (t,τ) (|y| + |x|) ,
where f(t,τ),g(t,τ),h(t,τ) and c are as given in Theorem 2.2. Let k1(t) be as in
(2.14). If y(t), t ∈ IT is a solution of (3.1) then

(3.6) |y (t)| ≤
c

1 −k1 (t)
ef (τ,t0) ,

for t ∈ IT where F is defined by (2.15).
Proof. Since y(t) is solution of equation (3.1) and using (3.2) − (3.5) we get

|y (t)| ≤ c +
t∫

t0

f (t,τ)


|y (x)| + τ∫

t0

h (τ,s) |y (τ)|∆s


 ∆τ

+

α∫
t0

g (t,τ) |y (τ)|∆τ(3.7)

Now an application of Theorem 2.2 to (3.7) we get (3.6).

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Department of Mathematics, Dr. B.A.M. University, Aurangabad, Maharashtra

431004, India