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 can be applied on the study of distribution of generalized zeros of half-linear dynamic equations on time scales.

1. Introduction

In 1960 Opial [1] proved that if is absolutely continuous on with then Since the discovery of Opial’s inequality much work has been done and many papers which deal with new proofs, various generalizations, and extensions have appeared in the literature. 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 These inequalities and their extensions and generalizations are the most important and fundamental inequalities in the analysis of qualitative properties of solutions of different types of differential equations.

In recent decades the asymptotic behavior of difference equations and inequalities and their applications have been and still are receiving intensive attention. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. So it is expected to see the discrete versions of the above inequalities. In fact, the discrete version of (1.1) which has been proved by Lasota [7] is given by where is a sequence of real numbers with and is the greatest integer function. The discrete version of (1.2) is proved in [8, Theorem 5.2.2] and states that for a real sequence with , we have These difference inequalities and their generalizations are also important and fundamental in the analysis of qualitative properties of solutions of difference equations.

Since the continuous and discrete inequalities are important in the analysis of qualitative properties of solutions of differential and difference equations, we also believe that the unification of these inequalities on time scales, which leads to dynamic inequalities on time scales, will play the same effective act in the analysis of qualitative properties of solutions of dynamic equations. The study of dynamic inequalities on time scales helps avoid proving results twice—once for differential inequality and once again for difference inequality. 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 . The cases when the time scale is equal to the reals or to the integers represent the classical theories of integral and of discrete inequalities. A cover story article in New Scientist [9] discusses several possible applications.

The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see Kac and Cheung [10]), that is, when , , and , where . For more details of time scale analysis we refer the reader to the two books by Bohner and Peterson [11, 12] which summarize and organize much of the time scale calculus.

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 is 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 .

Definition 1.1. 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 [13].

Theorem 1.2. 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.

Definition 1.3. If , then the Cauchy (delta) integral of is defined by

It can be shown (see [11]) that if , then the Cauchy integral exists, , and satisfies , . An infinite integral is defined as and the integration on discrete time scales is defined by However, the study of dynamic inequalities of the Opial types on time scales has been started by Bohner and Kaymakçalan [14] in 2001, only recently received a lot of attention and few papers have been written, see [14–17] and the references cited therein. For contributions of different types of inequalities on time scales, we refer also the reader to the papers [18–22] and the references cited therein. In the following, we recall some of the related results that have been established for dynamic inequalities on time scales that serve and motivate the contents of this paper.

In [14] the authors extended the inequality (1.1) on time scales and proved that if is delta differentiable with , then Also in [14] the authors proved that if and are positive rd-continuous functions on ,   nonincreasing, and is delta differentiable with , then Karpuz et al. [15] proved an inequality similar to inequality (1.13) replaced by of the form where is a positive rd-continuous function on , and is delta differentiable with and Wong et al. [16] and Sirvastava et al. [17] proved that if is a positive rd-continuous function on , we have where is delta differentiable with .

Following this trend, to develop the qualitative theory of dynamic inequalities on time scales, we will prove some new inequalities of Opial^’s type. Some special cases on continuous and discrete spaces are derived and compared by previous results. The main results in this paper can be considered as the continuation of the paper [23] that has been published by the author and can be applied on the study of distribution of the generalized zeros of the half-linear dynamic equation: and according to the limited space the applications of these inequalities will be discussed in a different paper.

2. Main Results

In this section, we will prove the main results and this will be done by making use of the Hölder inequality (see [11, Theorem 6.13]): where , and ,   and , and inequality (see [24, page 500]) where , are positive real numbers. We also need the formula which is a simple consequence of Keller’s chain rule [11, Theorem 1.90]. 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 , (and   does not change sign in , then one has where

Proof. Since does not change sign in , we have This implies that Now, since is nonnegative on , then it follows from the Hölder inequality (2.1) with that Then, for , we get (note that that Since , we have Applying inequality (2.2), we get (where ) that Setting we see that , and From this, we get that Also since is nonnegative on , we have from (2.12) and (2.15) that This implies that Supposing that the integrals in (2.17) exist and again applying the Hölder inequality (2.1) with indices and on the first integral on the right hand side, we have From (2.14), and the chain rule (2.3), we obtain Substituting (2.19) into (2.18) and using the fact that , we have that Using (2.13), we have from the last inequality that which is the desired inequality (2.4). 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 be positive real numbers such that , and let be nonnegative rd-continuous functions on   such that . If is delta differentiable with , (and does not change sign in ), then one has where

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

Corollary 2.3. Assume that be positive real numbers such that , and let be nonnegative continuous functions on  such that .   If is differentiable with , (and does not change sign in ),  then one has where

Corollary 2.4. Assume that be positive real numbers such that , and let be nonnegative continuous functions on such that . If is delta differentiable with , (and does not change sign in ), 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. Note that since then the proof will be a combination of Theorems 2.1 and 2.2.

Theorem 2.5. 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 , (and does not change sign in ), then one has

For in Theorem 2.1, we obtain the following result.

Corollary 2.6. Let be a time scale with and be positive real numbers such that , and let   be a nonnegative rd-continuous function on    such that . If   is delta differentiable with , (and   does not change sign in )  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 (2.3) that This implies that From this and (2.32) (by putting , we get that that So setting in (2.31) and using (2.35), we have the following result.

Corollary 2.7. Let be a time scale with and be positive real numbers such that . If is delta differentiable with , (and does not change sign in ), then one has where

Remark 2.8. Note that when , we have , and then the inequality (2.36) becomes Note also that when and , then the inequality (2.38) becomes which is the Opial inequality (1.2).
When , we have form (2.36) the following discrete Opial’s type inequality.

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