About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 362526, 23 pages
http://dx.doi.org/10.1155/2012/362526
Research Article

New Inequalities of Opial's Type on Time Scales and Some of Their Applications

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

Received 31 December 2011; Accepted 14 March 2012

Academic Editor: Hassan A. El-Morshedy

Copyright © 2012 Samir 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

We will prove some new dynamic inequalities of Opial's type on time scales. The results not only extend some results in the literature but also improve some of them. Some continuous and discrete inequalities are derived from the main results as special cases. The results will be applied on second-order half-linear dynamic equations on time scales to prove several results related to the spacing between consecutive zeros of solutions and the spacing between zeros of a solution and/or its derivative. The results also yield conditions for disfocality of these equations.

1. Introduction

In 1960 Opial [1] proved that if is an absolutely continuous function on with then In further simplifying the proof of the Opial inequality which had already been simplified by Olech [2], Beesack [3], Levinson [4], Mallows [5], and Pederson [6], it is proved that if is real absolutely continuous on and with , then Since the discovery of Opial’s inequality much work has been done, and many papers which deal with new proofs, various generalizations, extensions, and their discrete analogues have appeared in the literature. The discrete analogy of the (1.1) has been proved in [7] and the discrete analogy of (1.2) has been proved in [8, Theorem  5.2.2]. It is worth to mention here that many results concerning differential inequalities carry over quite easily to corresponding results for difference inequalities, while other results seem to be completely different from their continuous counterparts.

In recent years, there has been much research activity concerning the qualitative theory of dynamic equations on time scales. It has been created in [9] in order to unify the study of differential and difference equations, and it also extends these classical cases to cases “in between,” for example, to the so-called -difference equations. The general idea is to prove a result for a dynamic equation or 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 . A cover story article in New Scientist [10] discusses several possible applications of time scales. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see Kac and Cheung [11]), that is, when and where .

One of the main subjects of the qualitative analysis on time scales is to prove some new dynamic inequalities. 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 dynamic equations on time scales. In the following, we recall some results obtained for dynamic inequalities on time scales that serve and motivate the contents of this paper.

Bohner and Kaymakçalan in [12] established the time scale analogy of (1.2) and proved that if is delta differentiable with , then Also in [12] the authors proved that if and are positive rd-continuous functions on nonincreasing and is delta differentiable with , then Since the discovery of the inequalities (1.3) and (1.4) some papers which deal with new proofs, various generalizations, and extensions of (1.3) and (1.4) have appeared in the literature, we refer to the results in [1315] and the references cited therein. Karpuz et al. [13] proved an inequality similar to the inequality (1.4) of the form where is a positive rd-continuous function on , and is delta differentiable with , and Wong et al. [14] and Sirvastava et al. [15] proved that if is a positive rd-continuous function on , we have where is delta differentiable with .

In [16] the author proved that if is delta differentiable with , then where are positive real numbers such that , and let be nonnegative rd-continuous functions on such that and If is replaced by , then where For contributions of different types of inequalities on time scales, we also refer the reader to the papers [1624] and the references cited therein.

The paper is organized as follows: in Section 2, we will prove some new dynamic inequalities of Opial’s type on time scales of the form where are positive real numbers such that and is the coefficient of the inequality. As special cases, we derive some differential and discrete inequalities on continuous and discrete time scales. In Section 3, we will apply the obtained inequalities in Section 2 on the second-order half-linear dynamic equation where is an arbitrary time scale, is a quotient of odd positive integers, and are real rd-continuous functions defined on with . In particular, we will prove several results related to the problems:(i)obtain lower bounds for the spacing where is a solution of (1.13) and satisfies , or ,(ii)obtain lower bounds for the spacing between consecutive zeros of solutions of (1.13).

Our motivation comes from that fact that the inequalities obtained in the literature cannot be applied on the half-linear dynamic equation (1.13) to prove results related to the problems (i)-(ii).

2. Main Results

The main inequalities will be proved in this section by making use of the Hölder inequality (see [25, Theorem 6.13]) where and and , and the inequality (see [8, page 51]) For completeness, we recall the following concepts related to the notion of time scales. A time scale is an arbitrary nonempty closed subset of the real numbers . We assume throughout that has the topology that it inherits from the standard topology on the real numbers . The forward jump operator and the backward jump operator are defined by where . A point is said to be left dense if and , is right dense if , is left scattered if and right scattered if .

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 . We will assume that and define the time scale interval by .

Fix and let . Define to be the number (if it exists) with the property that given any there is a neighborhood of with In this case, we say is the (delta) derivative of at and that is (delta) differentiable at .

We will frequently use the results in the following theorem which is due to Hilger [9].

Assume that and let .(i)If is differentiable at , then is continuous at .(ii)If is continuous at and is right scattered, then is differentiable at with (iii)If is differentiable and is right-dense, then (iv)If is differentiable at , then .

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 We will make use of the following product and quotient rules for the derivative of the product and the quotient (where , here ) of two differentiable functions and We say that a function is regressive provided . The integration by parts formula is given by It can be shown (see [25]) that if , then the Cauchy integral exists, , and satisfies . The integration on discrete time scales is defined by To prove the main results, we need the formula which is a simple consequence of Keller’s chain rule [25, Theorem 1.90]. The books on the subject of time scales by Bohner and Peterson [25, 26] summarize and organize much of time scale calculus and contain some results for dynamic equations on time scales. To admit functions such that and may change sign on , we note that if with equality if and only if does not change sign. (The same result holds if and .) Now, we are ready to state and prove the main results.

Theorem 2.1. Let be a time scale with and be positive real numbers such that and let be nonnegative rd-continuous functions on such that . If is delta differentiable with , then one has where

Proof. Since and is nonnegative on , then it follows from the Hölder inequality (2.1) with that Then, for , we get (noting that that Since , we have Applying the inequality (2.2), we get (where ) that Setting we see that , and From this, we get that Thus, since is nonnegative on , we have from (2.20) and (2.23) that This implies that Supposing that the integrals in (2.25) exist and again applying the Hölder inequality (2.1) with indices and on the first integral on the right-hand side of (2.25), we have From (2.22), and the chain rule (2.11), we obtain Substituting (2.27) into (2.26) and using the fact that , we have that Using (2.21), we have from the last inequality that which is the desired inequality (2.13). The proof is complete.

Here, we only state the following theorem, since its proof is the same as that of Theorem 2.1, with replaced by and .

Theorem 2.2. Let be a time scale with and let be positive real numbers such that and let be nonnegative rd-continuous functions on such that . If is delta differentiable with , then one has where

Note that when , we have and . Then from Theorems 2.1 and 2.2 we have the following differential inequalities.

Corollary 2.3. Assume that are positive real numbers such that and let be nonnegative continuous functions on   such that . If is differentiable with , then one has where

Corollary 2.4. Assume that are positive real numbers such that and let be nonnegative continuous functions on such that . If is delta differentiable with , then one has where

In the following, we assume that there exists which is the unique solution of the equation where and are defined as in Theorems 2.1 and 2.2.

Theorem 2.5. Let be a time scale with and let be positive real numbers such that and let be nonnegative rd-continuous functions on such that . If is delta differentiable with , then one has

Proof. Since then the rest of the proof will be a combination of Theorems 2.1 and 2.2 and hence is omitted. The proof is complete.

As a special case if in Theorem 2.1, then we obtain the following result.

Corollary 2.6. Let be a time scale with and let be positive real numbers such that and let be a nonnegative rd-continuous function on such that . If is delta differentiable with , then one has where

From Theorems 2.2 and 2.5 one can derive similar results by setting . The details are left to the reader.

On a time scale , we note from the chain rule (2.11) that This implies that From this and (2.40) (by putting ), we get that So setting in (2.39) and using (2.43), we have the following result.

Corollary 2.7. Let be a time scale with and let be positive real numbers such that and . If is delta differentiable with , then one has where

Remark 2.8. Note that when , we have , and then the inequality (2.44) becomes Note also that when and , then the inequality (2.46) becomes which is the Opial inequality (1.2).

When , we have form (2.44) the following discrete Opial’s type inequality.

Corollary 2.9. Assume that are positive real numbers such that and are a nonnegative real sequence. If is a sequence of positive real numbers with , then

The inequality (2.44) has an immediate application to the case where . Choose and apply (2.40) to and and adding we obtain the following inequality.

Corollary 2.10. Let be a time scale with and let be positive real numbers such that and . If is delta differentiable with , then one has where

From this inequality, we have the following discrete Opial type inequality.

Corollary 2.11. Assume that are positive real numbers such that and . If is a sequence of real numbers with , then

By setting in (2.49) we have the following Opial type inequality on a time scale.

Corollary 2.12. Let be a time scale with . If is delta differentiable with , then one has

As special cases from (2.52) on the continuous and discrete spaces, that is, when and , one has the following inequalities.

Corollary 2.13. If is differentiable with , then one has the Opial inequality

Corollary 2.14. If is a sequence of real numbers with , then

3. Applications

Our aim in this section, is to apply the dynamic inequalities of Opial’s type proved in Section 2 to prove several results related to the problems (i)-(ii) for the second-order half-linear dynamic equation on an arbitrary time scale , where is a quotient of odd positive integers, and are real rd-continuous functions defined on with . The terminology half-linear arises because of the fact that the space of all solutions of (3.1) is homogeneous, but not generally additive. Thus, it has just “half” of the properties of a linear space. It is easily seen that if is a solution of (3.1), then so also is .

By a solution of (3.1) on an interval , we mean a nontrivial real-valued function , which has the property that and satisfies (3.1) on . We say that a solution of (3.1) has a generalized zero at if and has a generalized zero in in case and . Equation (3.1) is disconjugate on the interval , if there is no nontrivial solution of (3.1) with two (or more) generalized zeros in .

Equation (3.1) is said to be nonoscillatory on if there exists such that this equation is disconjugate on for every . In the opposite case (3.1) is said to be oscillatory on . The oscillation of solutions of (3.1) may equivalently be defined as follows: a nontrivial solution of (3.1) is called oscillatory if it has infinitely many (isolated) generalized zeros in ; otherwise it is called nonoscillatory. So that the solution of (3.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory. This means that the property of oscillation or nonoscillation is the behavior in the neighborhood of the infinite points.

We say that (3.1) is right disfocal (left disfocal) on if the solutions of (3.1) such that have no generalized zeros in .

Note that (3.1) in its general form involves some different types of differential and difference equations depending on the choice of the time scale . For example when , (3.1) becomes a second-order half-linear differential equation and . When , (3.1) becomes a second-order half-linear difference equation and . When , (3.1) becomes a generalized difference equation and . When , (3.1) becomes a quantum difference equation (see [11]) and . Note also that the results in this paper can be applied on the time scales , and when where is the set of harmonic numbers. In these cases we see that when , we have and when where is the harmonic numbers that are defined by and , we have . When , we have , and when , we have .

Perhaps the best known existence results of types (i)-(ii) for a special case of (3.1) (when and ) is due to Bohner et al. [27], where they extended the Lyapunov inequality obtained for differential equations in [28]. In particular the authors in [27] considered the dynamic equation where is a positive rd-continuous function defined on and proved that if is a solution of (3.2) with, then Karpuz et al. [13] proved that if and is a solution of (3.2) with , then Saker [22] considered the second-order half-linear dynamic equation on an arbitrary time scale , where is a positive constant, and are real rd-continuous positive functions defined on and proved that if is a positive solution of (3.1) which satisfies for and has a maximum at a point , then Of particular interest in this paper is when is oscillatory which is different from the conditions imposed on in [12, 22, 27]. The results also yield conditions for disfocality for (3.1) on time scales. As special cases, the results include some results obtained for differential equations and give new results for difference equations on discrete time scales.

Now, we are ready to state and prove the main results in this section. To simplify the presentation of the results, we define Note that when , we have , and when , we have

Theorem 3.1. Suppose that is a nontrivial solution of (3.1). If , then where . If , then where.

Proof. We prove (3.9). Without loss of generality we may assume that in . Multiplying (3.1) by and integrating by parts, we have Using the assumptions that and , we have Integrating by parts the right-hand side (see (2.9)), we see that Again using the facts that , we obtain Applying the chain rule formula (2.11) and the inequality (2.2), we see that This and (3.14) imply that Applying the inequality (2.13) with and , we have where Then, we have from (3.17) after cancelling the term , that which is the desired inequality (3.9). The proof of (3.10) is similar to (3.9) by using the integration by parts and (2.30) of Theorem 2.2 and (2.31) instead of (2.14). The proof is complete.

As a special case of Theorem 3.1, when , we have the following result.

Corollary 3.2. Suppose that is a nontrivial solution of If , then where . If , then where .

As a special case of Theorem 3.1, when , we have the following result.

Corollary 3.3. Suppose that is a nontrivial solution of If , then where. If , then where.

As a special case when , we have and then the results in Corollary 3.3 reduce to the following results obtained by Brown and Hinton [29] for the second-order differential equation

Corollary 3.4 (see [29]). If is a solution of (3.26) such that , then where . If instead , then where .

Remark 3.5. Note that if , then and (3.1) (when ) becomes and as a special case of Corollary 3.3, we have the following result.

Corollary 3.6. If is a solution of (3.29) such that , then where . If instead , then where .

Remark 3.7. By using the maximum of on and in (3.21) and (3.22) with and , we have the following results.

Corollary 3.8. Suppose that is a nontrivial solution of (3.20), where is a quotient of odd positive integers. If , then and if , then

As a special when , we have