Abstract
The main objective of the paper is to study new integral inequality on time scales which is used for the study of some partial dynamic equations. Some applications of our results are also given.
1. Introduction
During past few decades many authors have established various dynamic inequalities useful in the development of differential and integral equations. Mathematical inequalities on time scales play an important role in the theory of dynamic equations. The study of time scale was initiated by Hilger [1] in 1990 in his Ph.D. thesis which unifies continuous and discrete calculus. Since then, many authors have studied various properties of dynamic equations on time scales [2–9].
In what follows, let denotes the set of real numbers and let denote the arbitrary time scales. Let , , and be subsets of and . Let denote the set of rd-continuous function. The partial delta derivative of for with respect to , , and is denoted by , , and . We assume here understanding of time scales calculus and notations. Further information about time scales calculus can be found in [1, 5, 10].
We require the following lemmas given in [5, 6].
Lemma 1 (see [5], Theorem 2.6). Let , , and for all ; then for all .
Lemma 2 (see [6], Lemma 2.1). Let and is nondecreasing in and for ; then where for .
2. Main Results
Now in this section we give our main results.
Theorem 3. Let , , , , and suppose that for , where is a constant. If where for , then for .
Proof. Define a function by
Then (6) is
It is easy to see that is nonnegative, rd-continuous, and nondecreasing function for . Treating fixed and using Lemma 1 we get
for , where is defined by (9). From (11), (12), and the fact that , we have
Define a function by right hand side of (14). Then , . One has
Define a function by
then , ,
By keeping fixed in (18), taking and delta integrating with second variable from to . Using the fact that and is nondecreasing in , we have
Let
then (20) gives
Now treating fixed in (21) and applying Lemma 1, we have
From (18), (22), and (7), it is easy to see that
Using (23) in (22) and the fact that and we get the inequality in (10).
This completes the proof.
3. Applications
Now we give some application of theorem to study properties of solutions of initial value problem: where , for , , , , is delta differentiable with respect to .
We observe that (24) is equivalent to where The following theorem deals with estimate on solution (24).
Theorem 4. Suppose where , , which are as in Theorem 3 and is rd-continuous function defined on such that . Let where for . If is any solution of (24), then where .
Proof. The solution of (24) satisfies (25). Using (27) in (25) we have
Now an application of Theorem 3 (with ) to (32) yields (30).
This completes the proof.
Now we establish the uniqueness of solutions of (24).
Theorem 5. Suppose that where , , and are as in Theorem 4. Let and be as in (28) and (30). Then (24) has at most one solution on .
Proof. Let and be two solutions of (24) on ; then we have From (34) and (33) we obtain Applying Theorem 3 (with , ) yields Therefore ; there is at most one solution of (24) in .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.