Discrete Dynamics in Nature and SocietyVolume 2011 (2011), Article ID 732164, 21 pagesdoi:10.1155/2011/732164

Research Article

Bounds of Double Integral Dynamic Inequalities in Two Independent Variables on Time Scales

S. H. Saker

College of Science Research Centre, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia

Received 4 October 2011; Revised 29 October 2011; Accepted 31 October 2011

Academic Editor: Delfim F. M. Torres

Copyright © 2011 S. H. Saker. 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.

Our aim in this paper is to establish some explicit bounds of the unknown function in a certain class of nonlinear dynamic inequalities in two independent variables on time scales which are unbounded above. These on the one hand generalize and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of partial dynamic equations on time scales. Some examples are considered to demonstrate the applications of the results.

1. Introduction

During the past decade, a number of dynamic inequalities have been established by some authors which are motivated by some applications, for example, when studying the behavior of solutions of certain classes of dynamic equations, the bounds provided by earlier inequalities are inadequate in applications and some new and specific type of dynamic inequalities on time scales are required. The general idea is to prove a result for a dynamic inequality where the domain of the unknown function is a so-called time scale , which may be an arbitrary closed subset of the real numbers . In [1, Theorem 6.1], it is proved that if , , and and , then implies where and is the class of rd-continuous and regressive functions. A function is said to be right-dense continuous (rd-continuous) provided is continuous at right-dense points and at left-dense points in , left-hand limits exist and are finite. The set of all such rd-continuous functions is denoted by . The graininess function for a time scale is defined by , and, for any function , the notation denotes , where is the forward jump operator defined by . We say that a function is regressive provided . The set of all regressive functions on a time scale forms an Abelian group under the addition defined by . Throughout this paper, we will assume that and define the time scale interval by . The exponential function on time scales is defined by where is the cylinder transformation, which is given by Alternatively, for , one can define the exponential function to be the unique solution of the IVP , with . If , then is real-valued and nonzero on . If , then is always positive, , and . Note that The book on the subject of time scales by Bohner and Peterson [1] summarizes and organizes much of time scale calculus. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see [2]), that is, when , and , where .

In this paper, we will refer to the (delta) integral which we can define as follows: If , then the Cauchy (delta) integral of is defined by . It can be shown (see [1]) that if , then the Cauchy integral exists, , and satisfies , . There are applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, and combinatorics. A recent cover story article in New Scientist [3] discusses several possible applications. Also, in [1, Theorem 6.4], it is proved that if , and and , then implies that Since (1.7) provides an explicit bound to the unknown function and a tool to the study of many qualitative as well as quantitative properties of solutions of dynamic equations, it has become one of the very few classic and most influential results in the theory and applications of dynamic inequalities. Because of its fundamental importance, over the years, many generalizations and analogous results of (1.7) have been established. Since the discovery of the inequalities (1.1)–(1.7), much work has been done, and many papers which deal with various generalizations and extensions have appeared in the literature, we refer the reader to [4–9] and the references cited therein. On the other hand, a few authors have focused on the theory of partial dynamic equations on time scales [10–15]. However, to the best of author’s knowledge, only [16–19] have studied integral inequalities useful in the theory of partial dynamic equations on time scales. Before, we give a brief summary of some of the results of dynamic inequalities in two independent variables, we present some basic definitions about calculus in two variables on time scales (for more details, we refer to [12]).

Let and be two time scales with at least two points, and consider the time scale intervals and for and . Let and denote the forward jump operators, backward jump operators, and the delta differentiation operator, respectively, on and . 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 In this case, we say is the (partial delta) derivative of at . 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 In this case, we say is the (partial delta) derivative of at . 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 . Now, we are ready to present some results for dynamic inequalities in two independent variables on times scales which are related to the main results in our paper. In [18], the authors proved that if , and are positive rd-continuous functions and is nonnegative and nondecreasing in each of its variables, then for all , implies In [19], the author proved that if , and are positive continuous real functions defined on and is a real constant, then implies for all , where In this paper, we are concerned with bounds of the double integral nonlinear dynamic inequality in two independent variables for all . For (1.15), we will assume the following hypotheses:

The main aim in this paper is to establish some explicit bounds of the unknown function of the inequality (1.15). Our results not only complement the results in [18, 19] but also improve the results in [19], in the sense that the results can be applied in the cases when . The main results will be proved by employing the Bernoulli inequality [20, Bernoulli’s inequality]: the Young inequality [20]: and the algebraic inequalities [20]: Some examples are considered to illustrate the main results.

2. Main Results

Before, we stated and proved the main results and we proved some Lemmas which play important roles in the proofs of the main results. We will assume that the equations or the inequalities possess such nontrivial solutions.

Lemma 2.1. Let be an unbounded time scale with and . Let for be functions with for , where for , whenever . Let , be differentiable with for all . Then, implies for all .

Proof. The proof is by induction and similar to the proof of Theorem 6.9 in [1] and hence is omitted.

Lemma 2.2. Let be an unbounded time scale with and . Suppose that is nondecreasing for and is such that is rd-continuous. Let be rd-continuous and positive for and differentiable. Then, implies for all , where solves the initial value problem

Proof. Let for all . Then, for all and so that for all . Since , the comparison Lemma 2.1 yields for all , where solves the initial value problem (2.3). Hence, since , we obtain . The proof is complete.

Now, we are ready to state and prove the main results in this paper. First, we consider the case when and . For simplicity, we introduce the following notations:

Theorem 2.3. Let be an unbounded time scale with . Assume that holds,   and . Then, for ,  implies that where solves the initial value problem

Proof. Define a function by This reduces (2.8) to This implies that Applying the inequality (1.17) (noting that , we see that From (2.13), we obtain Applying inequality (1.17) on (2.15) (where , we obtain for that Also, from (2.13), we obtain Applying inequality (1.17) on (2.17) (where , we have for that Combining (2.11), (2.16), and (2.18) and applying the inequality (1.19) (noting that , we have This implies that Now an application of Lemma 2.2 (with and gives that where solves the initial value problem (2.10). Substituting (2.21) into (2.14), we obtain the desired inequality (2.9). The proof is complete.

Theorem 2.4. Let be an unbounded time scale with . Assume that holds, and . Then, (2.8) implies where solves the initial value problem where is defined as in (2.7) and

Proof. Define a function by (2.11) and proceed as in the proof of Theorem 2.3 to obtain Applying the inequality (1.20), we see that From (2.25), we obtain Applying inequality (1.20) on (2.27) (where , we obtain for that Also, from (2.25), we have by (1.20) that Combining (2.11), (2.28), and (2.29) and applying the inequality (1.19) (noting that , we have This implies that for . Now, an application of Lemma 2.2 (with , and gives that where solves the initial value problem (2.23). Substituting (2.32) into (2.26), we obtain the desired inequality (2.22). The proof is complete.

As in the proof of Theorem 2.3 by employing the inequality (1.20) instead of the inequality (1.19), we can obtain an explicit bound for when . This will be presented below in Theorem 2.5 without proof since the proof is similar to the proof of Theorem 2.3. For simplicity, we introduce the following notations:

Theorem 2.5. Let be an unbounded time scale with . Assume that holds, , and . Then, (2.8) implies that where solves the initial value problem

In the following, we apply the Young inequality (1.18) to find a new explicit upper bound for of (2.8) when and . First, we consider the case when and assume that and .

Theorem 2.6. Let be an unbounded time scale with . Assume that holds, and such that and . Then, (2.8) implies that where and , and are defined as in (2.7) and (2.24).

Proof. Define a function by (2.11) and proceed as in the proof of Theorem 2.4 to obtain where , and are defined as in (2.7) and (2.24). Applying the Young inequality (1.18) on the term with and , we see that Again applying the Young inequality (1.18) on the term with and , we see that Substituting (2.40) and (2.41) into (2.39), we have From the definitions of and , we get that Applying the inequality (1.10) with , we have Substituting (2.44) into (2.38), we get the desired inequality (2.36). The proof is complete.

Theorem 2.7. Let be an unbounded time scale with . Assume that holds, , and . Then, (2.8) implies that where and and are defined as in (2.33).

Proof. Define a function by (2.11) and proceed as in the proof of Theorem 2.3 to obtain where and are defined in (2.33). Applying the Young inequality (1.18) on the term with and , we see that This and (2.48) imply that Using the definition of , we get that Applying the inequality (1.10) with , we have Substituting (2.52) into (2.47), we get the desired inequality (2.45). The proof is complete.

Next in the following, we consider the case when and establish some new explicit bounds of the unknown function of (2.8).

Theorem 2.8. Let be an unbounded time scale with . Assume that holds, , and . Then, (2.8) implies that where and

Proof. Define a function by This reduces (2.8) to This implies that Applying the inequality (1.19) (noting that , we see that From (2.58), we obtain Also, from (2.58), we obtain Combining (2.55), (2.59), and (2.60) and applying the inequality (1.19) (noting that , we have This implies (noting that and that Applying the Young inequality (1.18) on the term with and , we see that where . Again applying the Young inequality (1.18) on the term with and , we see that where . Combining (2.62)–(2.64), we have Applying the inequality (1.10) on (2.65) with , we have Substituting (2.66) into (2.56), we get the desired inequality (2.45). The proof is complete.

Remark 2.9. The above results can be extended to the general inequality when In fact, by using the new substitution the inequality (2.67) can be written as and then, since , we have where and . This implies that Using (2.68) in the last inequality, we get an inequality similar to the inequality (2.8) and then follow the proof of the above results to find some explicit bounds of (2.8). The details are left to the reader.

3. Applications

In this section, we present some applications of our main results.

Example 3.1. Consider the partial dynamic equation on time scales with initial boundary conditions where is a constant and and are rd-continuous functions on and are rd-continuous functions. Assume that where and are nonnegative rd-continuous functions for and . If is a solution of (3.1)-(3.2), then satisfies where In fact, the solution of (3.1)-(3.2) satisfies for Therefore, It follows from (3.3) and (3.7) that for . Applying Theorem 2.6 on (3.8) with and , we obtain (3.4).

Example 3.2. Consider the equation where is a constant and and are rd-continuous on and are rd-continuous functions. Assume that where and are nonnegative rd-continuous functions for and . If is a solution of (3.1)-(3.2), then satisfies where In fact, the solution of (3.9) satisfies It follows from (3.10) and (3.13) that for . Applying Theorem 2.7 on (3.14) with , we obtain (3.11).

Example 3.3. Assume that and consider the partial differential equation where , with initial boundary conditions Assume that is a constant and and , and are continuous functions. Also, we assume that where and are nonnegative continuous functions for and . If is a solution of (3.1)-(3.2), then satisfies where In fact, the solution formula of (3.15)-(3.16), after integration twice, is given by Therefore, It follows from (3.17) and (3.21) that for . Applying Theorem 2.6 on (3.22) with and , we obtain (3.18).

Acknowledgments

The author is very grateful to the referees for their valuable remarks and comments which significantly contributed to the quality of the paper. This project was supported by King Saud University, Dean-ship of Scientific Research, College of Science Research Centre.

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