Journal of Mathematics

Volume 2013 (2013), Article ID 902087, 5 pages

http://dx.doi.org/10.1155/2013/902087

Research Article

Estimates of Certain Integral Inequalities on Time Scales

Deepak B. Pachpatte

Department of Mathematics, Dr. B.A.M. University, Aurangabad, Maharashtra 431004, India

Received 13 August 2012; Accepted 22 October 2012

Academic Editor: Abdul Hamid Kara

Copyright © 2013 Deepak B. Pachpatte. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main objective of this paper is to establish explicit bounds on certain integral inequalities on time scales, which can be used as tools in the study of certain classes of integral equations on time scales. Some applications of our results are also given.

1. Introduction

Recently, many authors have studied various aspects of dynamic inequalities on time scales using various techniques [1–4]. In this paper, we obtain explicit bounds on certain integral inequalities on time scales. In the present paper we offer some fundamental dynamic integral inequalities on time scales which can be used as tools for handling the qualitative behavior of solutions of certain dynamic equations on time scales. Excellent information about introduction to time scales can be found in [5, 6]. In what follows, denotes the set of real numbers, the set of integers, and denotes arbitrary time scales. Now following [3, 4] we give some basic definitions about calculus on time scales in two variables.

We say that is rd-continuous provided is continuous at each right-dense point of and has a finite left sided limit at each left-dense point of and will be denoted by . Let and be two time scales with at least two points and consider the time scales intervals and for and and . Let ,  ,   and ,  ,   denote the forward jump operators, backward jump operators and the delta differentiation operator, respectively, on and . Let are points in , are points in , is the half closed bounded interval in , and is the half closed bounded interval in .

We say that a real-valued function on at has a partial derivative with respect to if for each there exists a neighborhood of such that

for all . We say that on at has a partial derivative with respect to if for each there exists a neighborhood of such that

for all . The function is called rd-continuous in if for every , the function is rd-continuous on . The function is called rd-continuous in if for every the function is rd-continuous on .

We require that the following lemma proved in [1, 2] holds.

Lemma 1. Let , , , and nonnegative constants. If

for , then

for , where

for .

Lemma 2 (see [1, Lemma]). Let ,  ,   and is nondecreasing in each of its variables. If

for , then

for .

2. Main Results

Our main results are given in the following theorems.

Theorem 3. Let ,  ,    , ,   . If

for , then

for , where

for and defined by (5).

Proof. Define a function by

then from (8), , and using this in (11) we get

where is defined by (10). Clearly is nonnegative, rd-continuous and nondecreasing in , for nondecreasing. First we assume that for . From (12), we have

Now as an application of Lemma 1 we have

The desired inequality (9) follows from (14) and the fact that . If , we carry out the above procedure with instead of , where is an arbitrary small constants, and then subsequently pass to the limit as to obtain (9). The following theorem deals with two independent versions of inequalities established in Theorem 3.

Theorem 4. Let , , , ,  , and be a nonnegative constants. If

for , then

where

for .

Proof. Let and define a function by the right-hand side of (15); then , , and

Define a function by

then , , , , is nondecreasing for , and

where is defined by (17).

Using the fact that ,   ,   , and we have

Now keeping fixed and delta integrating with respect to from to , we obtain estimates

keeping fixed in (22) and delta integrating from to , we get

Then Lemma 2 implies

Using (24) in (18) and integrating the resulting inequality from to and then from to for , we get

Using (25) in , we get the required inequality in (16).

Theorem 5. Let ,   ,    and ,   ,   ,  . If

For , then

where

for and as defined in (17).

Proof. Define a function by the right-hand side of (26)

then from (26) we have , and using this in (28) we get

where is defined by (28). Here nondecreasing for , then from (30) we have

Now as an application of inequality in Theorem 3 to (31) we get

The required inequality in (27) follows from (32) and using the fact that .

3. Applications

In this section, we give application of inequality in Theorem 4 to study certain properties of solution of nonlinear partial integrodifferential equation on time scales,

with the initial boundary condition

where , , and .

The following theorem deals with the estimate on the solution of (33)-(34).

Theorem 6. Assume that

where , , and are as defined in (9). If , is any solution of (33)-(34), then

for , where is defined by (17).

Proof. The solution of (33)-(34) can be written as

Using (35) in (37), we have

Now an application of inequality in Theorem 4 yields the estimate in (36).

In the next result, we give uniqueness of solutions of (33)-(34).

Theorem 7. Suppose that the functions , in (33) satisfy the conditions

where and are as in Theorem 4. Then the problem (33)-(34) has at most one solution on .

Proof. Let and be two solutions of (33)-(34) on , then we have

From (39) and (40), we have

As an application of inequality in Theorem 4 with yields , therefore ; that is, there is at most one solution of (33)-(34) on .

References

R. A. C. Ferreira and D. F. M. Torres, “Some linear and nonlinear integral inequalities on time scales in two independent variables,” Nonlinear Dynamics and Systems Theory, vol. 9, no. 2, pp. 161–169, 2009.

D. B. Pachpatte, “On a nonlinear dynamic integrodifferential equation on time scales,” Journal of Applied Analysis, vol. 16, pp. 279–294, 2010.

W. Liu, Q. A. Ngo, and W. Chen, “Ostrowski type inequalities on time scales for double integrals,” Acta Applicandae Mathematicae, vol. 110, no. 1, pp. 477–497, 2010.

U. M. Özkan and H. Yildirim, “Ostrowski type inequality for double integrals on time scales,” Acta Applicandae Mathematicae, vol. 110, no. 1, pp. 283–288, 2010.

M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhauser, Berlin, Germany, 2001.

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Berlin, Germany, 2003.