Nonlinear Integral Inequalities in Two Independent Variables on Time Scales
© Wei Nian Li. 2011
Received: 7 December 2010
Accepted: 18 February 2011
Published: 10 March 2011
We investigate some nonlinear integral inequalities in two independent variables on time scales. Our results unify and extend some integral inequalities and their corresponding discrete analogues which established by Pachpatte. The inequalities given here can be used as handy tools to study the properties of certain partial dynamic equations on time scales.
The theory of dynamic equations on time scales unifies existing results in differential and finite difference equations and provides powerful new tools for exploring connections between the traditionally separated fields. During the last few years, more and more scholars have studied this theory. For example, we refer the reader to [1, 2] and the references cited therein. At the same time, some integral inequalities used in dynamic equations on time scales have been extended by many authors [3–11].
On the other hand, a few authors have focused on the theory of partial dynamic equations on time scales [12–17]. However, only [10, 11] have studied integral inequalities useful in the theory of partial dynamic equations on time scales, as far as we know. In this paper, we investigate some nonlinear integral inequalities in two independent variables on time scales, which can be used as handy tools to study the properties of certain partial dynamic equations on time scales.
2. Main Results
In what follows, is an arbitrary time scale, denotes the set of rd-continuous functions, denotes the set of all regressive and rd-continuous functions, for all , denotes the set of real numbers, , and denotes the set of nonnegative integers. We use the usual conventions that empty sums and products are taken to be 0 and 1, respectively. Throughout this paper, we always assume that and are time scales, , , , , , and we write for the partial delta derivatives of with respect to , and for the partial delta derivatives of with respect to .
The following two lemmas are useful in our main results.
Lemma 2.1 (see ).
Lemma 2.2 (Comparison Theorem ).
Next, we establish our main results.
Letting and , respectively, we easily see that Theorem 2.3 reduces to Theorem 2.3.3 and Theorem 5.2.4 in .
By Theorem 2.3, from (2.21), we easily obtain the desired (2.19). This completes the proof of Theorem 2.5.
If in Theorem 2.5, then we easily obtain Theorem 2.3.3 in .
Letting and in Theorem 2.7, respectively, we can obtain Theorem 2.3.4 and Theorem 5.2.4 in .
where is defined by (2.25). Obviously, is nonnegative, right-dense continuous, and nondecreasing for . The remainder of the proof is similar to that of Theorem 2.3, and we omit it here. This completes the proof of Theorem 2.9.
We note that when , Theorem 2.9 reduces to Theorem 2.3.4 in .
3. Some Applications
In this section, we present two applications of our main results.
Using Theorem 2.3, from (3.8), we easily obtain (3.4).
Therefore, by Theorem 2.5, from (3.14), we immediately obtain (3.11).
This work is supported by the National Natural Science Foundation of China (10971018), the Natural Science Foundation of Shandong Province (ZR2009AM005), China Postdoctoral Science Foundation Funded Project (20080440633), Shanghai Postdoctoral Scientific Program (09R21415200), the Project of Science and Technology of the Education Department of Shandong Province (J08LI52), and the Doctoral Foundation of Binzhou University (2006Y01). The author thanks the referees very much for their careful comments and valuable suggestions on this paper.
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