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Journal of Mathematics
Volume 2013 (2013), Article ID 902087, 5 pages
Estimates of Certain Integral Inequalities on Time Scales
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.
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.
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 .
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 .
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 .
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 .
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