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Abstract and Applied Analysis
Volume 2011, Article ID 265316, 19 pages
http://dx.doi.org/10.1155/2011/265316
Research Article

Some Opial-Type Inequalities on Time Scales

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

Received 30 January 2011; Accepted 31 March 2011

Academic Editor: W. A. Kirk

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

We will prove some dynamic inequalities of Opial type on time scales which not only extend some results in the literature but also improve some of them. Some discrete inequalities are derived from the main results as special cases.

1. Introduction

In 1960, Opial [1] proved the following inequality:𝑏𝑎||||||𝑥𝑥(𝑡)||(𝑡)𝑑𝑡(𝑏𝑎)4𝑏𝑎||𝑥||(𝑡)2𝑑𝑡,(1.1) where 𝑥 is absolutely continuous on [𝑎,𝑏] and 𝑥(𝑎)=𝑥(𝑏)=0 with a best constant 1/4. Since the discovery of Opial 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], Beescak [3], Levinson [4], Mallows [5], and Pederson [6], it is proved that if 𝑥 is real absolutely continuous on (0,𝑏) and with 𝑥(0)=0, then𝑏0||||||𝑥𝑥(𝑡)||𝑏(𝑡)𝑑𝑡2𝑏0||𝑥||(𝑡)2𝑑𝑡.(1.2) 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 analogy of (1.1) which has been proved by Lasota [7] is given by 1𝑖=1||𝑥𝑖Δ𝑥𝑖||12+121𝑖=0||Δ𝑥𝑖||2,(1.3) where {𝑥𝑖}0𝑖 is a sequence of real numbers with 𝑥0=𝑥=0 and [] is the greatest integer function. The discrete analogy of (1.2) is proved in [8, Theorem  5.2.2] and given by1𝑖=1||𝑥𝑖Δ𝑥𝑖||121𝑖=0||Δ𝑥𝑖||2,(1.4) where {𝑥𝑖}0𝑖 is a sequence of real numbers with 𝑥0=0. These difference inequalities and their generalizations are also important and fundamental in the analysis of the 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 dyanmic 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 the so-called time scale 𝕋, which may be an arbitrary closed subset of the real numbers .

In this paper, we are concerned with a certain class of Opial-type dynamic inequalities on time scales and their extensions. If the time scale equals the reals (or to the integers), the results represent the classical results for differential (or difference) 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 𝕋=,  𝕋=, and 𝕋=𝑞0={𝑞𝑡𝑡0}, where 𝑞>1. 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.

In the following, we recall some of the related results that have been established for differential inequalities and dynamic inequalities on time scales that serve and motivate the contents of this paper. For a generalization of (1.1), Beescak [3] proved that if 𝑥 is an absolutely continuous function on [𝑎,𝑋] with 𝑥(𝑎)=0, then𝑋𝑎||||||𝑥𝑥(𝑡)||1(𝑡)𝑑𝑡2𝑋𝑎1𝑟(𝑡)𝑑𝑡𝑋𝑎||𝑥𝑟(𝑡)||(𝑡)2𝑑𝑡,(1.5) where 𝑟(𝑡) is positive and continuous function with 𝑋𝑎𝑑𝑡/𝑟(𝑡)<, and if 𝑥(𝑏)=0, then𝑏𝑋||||||𝑥𝑥(𝑡)||1(𝑡)𝑑𝑡2𝑏𝑋1𝑟(𝑡)𝑑𝑡𝑏𝑋||𝑥𝑟(𝑡)||(𝑡)2𝑑𝑡.(1.6) Yang [13] simplified the Beesack proof and extended the inequality (1.5) and proved that if 𝑥 is an absolutely continuous function on (𝑎,𝑏) with 𝑥(𝑎)=0, then𝑏𝑎||||||𝑥𝑞(𝑡)𝑥(𝑡)||1(𝑡)𝑑𝑡2𝑏𝑎1𝑟(𝑡)𝑑𝑡𝑏𝑎||𝑥𝑟(𝑡)𝑞(𝑡)||(𝑡)2𝑑𝑡,(1.7) where 𝑟(𝑡) is a positive and continuous function with 𝑋𝑎𝑑𝑡/𝑟(𝑡)< and 𝑞(𝑡) is a positive, bounded, and nonincreasing function on [𝑎,𝑏].

Hua [14] extended the inequality (1.2) and proved that if 𝑥 is an absolutely continuous function with 𝑥(𝑎)=0, then𝑏𝑎||||𝑥(𝑡)𝑝||𝑥||(𝑡)𝑑𝑡(𝑏𝑎)𝑝𝑝+1𝑏𝑎||𝑥||(𝑡)𝑝+1𝑑𝑡,(1.8) where 𝑝 is a positive integer. We mentioned here that the results in [14] fail to apply for general values of 𝑝.

Maroni [15] generalized (1.5) and proved that if 𝑥 is an absolutely continuous function on [𝑎,𝑏] with 𝑥(𝑎)=0=𝑥(𝑏), then𝑏𝑎||𝑥||||𝑥(𝑡)||1(𝑡)𝑑𝑡2𝑏𝑎1𝑟(𝑡)1𝑑𝑡2/𝑋𝑎𝑟||𝑥(𝑡)||(𝑡)𝜈𝑑𝑡2/𝜈,(1.9) where 𝑏𝑎(1/𝑟(𝑡))𝛼1𝑑𝑡<, 𝛼1 and 1/𝛼+1/𝜈=1.

Boyd and Wong [16] extended the inequality (1.8) for general values of 𝑝>0 and proved that if 𝑥 is an absolutely continuous function on [𝑎,𝑏] with 𝑥(0)=0, then𝑎0𝑠||𝑥||(𝑡)(𝑡)𝑝||𝑥||1(𝑡)𝑑𝑡𝜆0(𝑝+1)𝑎0𝑟||𝑥(𝑡)||(𝑡)𝑝+1𝑑𝑡,(1.10) where 𝑟 and 𝑠 are nonnegative functions in 𝐶1[0,𝑎] and 𝜆0 is the smallest eigenvalue of the boundary value problem𝑢𝑟(𝑡)(𝑡)𝑝=𝜆𝑠(𝑡)𝑢𝑝(𝑡),(1.11) with 𝑢(0)=0 and 𝑟(𝑎)(𝑢(𝑎))𝑝=𝜆𝑠(𝑎)𝑢𝑝(𝑎) for which 𝑢>0 in [0,𝑎].

Yang [13] extended the inequality (1.8) and proved that if 𝑥 is an absolutely continuous function on [𝑎,𝑏] with 𝑥(𝑎)=0, 𝑝0 and 𝑞1, then 𝑏𝑎||||𝑥(𝑡)𝑝||𝑥||(𝑡)𝑞𝑞𝑑𝑡𝑝+𝑞(𝑏𝑎)𝑝𝑏𝑎||𝑥||(𝑡)𝑝+𝑞𝑑𝑡.(1.12) Yang [17] extended the inequality (1.12) and proved that if 𝑟(𝑡) is a positive, bounded function and 𝑥 is an absolutely continuous on [𝑎,𝑏] with 𝑥(𝑎)=0, 𝑝0,𝑞1, then𝑏𝑎||||𝑟(𝑡)𝑥(𝑡)𝑝||𝑥||(𝑡)𝑞𝑞𝑑𝑡𝑝+𝑞(𝑏𝑎)𝑝𝑏𝑎||𝑥𝑟(𝑡)||(𝑡)𝑝+𝑞𝑑𝑡.(1.13) However, as mentioned by Beesack and Das [18], the inequalities (1.12) and (1.13) are sharp when 𝑞=1 but are not sharp for 𝑞>1. Considering this problem, Beesack and Das [18] extended and improved the inequalities (1.12) and (1.13) when 𝑥(𝑎)=0 or 𝑥(𝑏)=0 or both. In fact, the extensions dealt with integral inequalities of the form𝑏𝑎||||𝑠(𝑡)𝑦(𝑡)𝑝||𝑦||(𝑡)𝑞𝑑𝑡𝐶(𝑝,𝑞)𝑏𝑎||𝑦𝑟(𝑡)||(𝑡)𝑝+𝑞𝑑𝑡,(1.14) where the functions 𝑟,𝑠 are nonnegative measurable functions on 𝐼=[𝑎,𝑏], 𝑦 is absolutely continuous on 𝐼,  𝑝𝑞>0, and either 𝑝+𝑞1 or 𝑝+𝑞<0, with a sharp constant 𝐶(𝑝,𝑞) depends on 𝑟,𝑠,𝑝, and 𝑞. For applications of the inequality (1.14) on zeros of differential equations, we refer the reader to the paper [19].

However, the study of dynamic inequalities of Opial types on time scales is initiated by Bohner and Kaymakçalan [20], and only recently received a lot of attention and few papers have been written, see [2024] and the references cited therein. For contribution of different types of inequalities on time scales, we refer the reader to the papers [2528] and the references cited therein.

Throughout the paper, we denote 𝑓𝜎=𝑓𝜎, where the forward jump operator 𝜎 is defined by 𝜎(𝑡)=inf{𝑠𝕋𝑠>𝑡}. By 𝑥𝕋 is rd-continuous, we mean 𝑥 is continuous at all right-dense points 𝑡𝕋 and at all left-dense points 𝑡𝕋 left hand limits exist (finite). The graininess function 𝜇𝕋+ is defined by 𝜇(𝑡)=𝜎(𝑡)𝑡. Also 𝕋𝜅=𝕋{𝑚} if 𝕋 has a left-scattered maximum 𝑚, otherwise 𝕋𝜅=𝕋. We will assume that sup𝕋=, and define the time scale interval [𝑎,𝑏]𝕋 by [𝑎,𝑏]𝕋=[𝑎,𝑏]𝕋.

In [20], the authors extended the inequality (1.1) on time scales and proved that if 𝑥[0,𝑏]𝕋 is delta differentiable with 𝑥(0)=0, then0||𝑥(𝑡)+𝑥𝜎||||𝑥(𝑡)Δ||(𝑡)Δ𝑡0||𝑥Δ||(𝑡)2Δ𝑡.(1.15) Also in [20] the authors extended the inequality (1.7) of Yang and proved that if 𝑟 and 𝑞 are positive rd-continuous functions on [0,𝑏],   𝑏0(Δ𝑡/𝑟(𝑡))<,𝑞 nonincreasing and 𝑥[0,𝑏]𝕋 is delta differentiable with 𝑥(0)=0, then𝑏0𝑞𝜎||(𝑡)(𝑥(𝑡)+𝑥𝜎(𝑡))𝑥Δ||(𝑡)Δ𝑡𝑏0Δ𝑡𝑟(𝑡)𝑏0||𝑥𝑟(𝑡)𝑞(𝑡)Δ||(𝑡)2Δ𝑡.(1.16)Karpuz et al. [21] proved an inequality similar to the inequality (1.16) replaced 𝑞𝜎(𝑡) by 𝑞(𝑡) of the form𝑏𝑎||𝑞(𝑡)(𝑥(𝑡)+𝑥𝜎(𝑡))𝑥Δ||(𝑡)Δ𝑡𝐾𝑞(𝑎,𝑏)𝑏𝑎||𝑥Δ||(𝑡)2Δ𝑡,(1.17) where 𝑞 is a positive rd-continuous function on, 𝑥[0,𝑏]𝕋 is delta differentiable with 𝑥(𝑎)=0, and𝐾𝑞2(𝑎,𝑏)=𝑏𝑎𝑞2(𝑢)(𝜎(𝑢)𝑎)Δ𝑢1/2.(1.18) We note that when 𝕋=, we have 𝜎(𝑡)=𝑡,𝑥Δ(𝑡)=𝑥(𝑡), and then the inequality (1.15) becomes the opial inequality (1.2). When 𝕋=, we have 𝜎(𝑡)=𝑡+1,   𝑥Δ(𝑡)=Δ𝑥(𝑡) and the inequality (1.15) reduces to the inequality1𝑖=1||𝑥𝑖+𝑥𝑖+1Δ𝑥𝑖||(1)1𝑖=0||Δ𝑥𝑖||2,(1.19) which is different from the inequality (1.4). This means that the extensions obtained by Bohner and Kaymakçalan [20] and Karpuz et al. [21] do not give a unification of differential and difference inequalities. So, the natural question now is: if it possible to find new Opial dynamic inequalities which contains (1.2) and (1.4) as special cases? One of our aims in this paper is to give an affirmative answer to this question.

Srivastava et al. [22] extended the Maroni inequality on a time scale and proved that if 𝑥[0,𝑏]𝕋 is delta differentiable with 𝑥(𝑎)=0, then𝑏𝑎||||𝑠(𝑡)𝑥(𝑡)𝑝||𝑥Δ||1(𝑡)Δ𝑡𝑟+1𝑏𝑎1𝑟𝛼1(𝑡)Δ𝑡(1+𝑝)/𝛼×𝑋𝑎𝑟(𝑡)𝑠𝜈/1+𝑝||𝑥(𝑡)Δ||(𝑡)𝜈Δ𝑡(1+𝑝)/𝜈,(1.20) where 𝑞 and 𝑟 are positive rd-continuous functions on [0,𝑏], such that 𝑠(𝑡) is bounded and decreasing, 𝑏𝑎(1/𝑟(𝑡))𝛼1Δ𝑡<,   𝛼1 and 1/𝛼+1/𝜈=1. For the interested reader, it would be interesting to extend the inequality (1.20) with the term 𝑏𝑎𝑠(𝑡)|𝑥(𝑡)|𝑝|𝑥Δ(𝑡)|Δ𝑡 replaced by 𝑏𝑎𝑠(𝑡)|𝑥(𝑡)|𝑝|𝑥Δ(𝑡)|𝑞Δ𝑡.

Wong et al. [23] and Srivastava et al. [24] extended the Yang inequality on a time scale and proved that if 𝑟 is a positive rd-continuous function on [0,𝑏], we have𝑏𝑎||||𝑟(𝑡)𝑥(𝑡)𝑝||𝑥Δ||(𝑡)𝑞𝑞Δ𝑡𝑝+𝑞(𝑏𝑎)𝑝𝑏𝑎||𝑥𝑟(𝑡)Δ||(𝑡)𝑝+1Δ𝑡,(1.21) where 𝑥[0,𝑏]𝕋 is delta differentiable with 𝑥(𝑎)=0. But according to Beesack and Das [18], the inequality (1.21) is only sharp when 𝑞=1. So, the natural question now is: if it is possible to prove new inequality of type (1.21) with two different functions 𝑟 and 𝑠, instead of 𝑟 with a best constant? One of our aims in this paper is to give an affirmative answer to this question. Also one of our motivations comes from the fact that the inequality of type (1.21) cannot be applied on the study of the distribution of the generalized zeros of the general dynamic equation 𝑟(𝑡)𝑥Δ(𝑡)Δ[]+𝑞(𝑡)𝑥(𝑡)=0,𝑡𝛼,𝛽𝕋,(1.22)

since this study needs an inequality with two different functions instead of a single function.

The paper is organized as follows. First, we will extend the inequality (1.10) on a time scale which gives a connection between a dynamic Opial-type inequality and a boundary value problem on time scales. Second, we prove some new Opial-type inequalities on time scales with best constants of type (1.14). The main results give the affirmative answers of the above posed questions. Throughout the paper, some special cases on continuous and discrete spaces are derived and compared by previous results.

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])𝑎||𝑓||(𝑡)𝑔(𝑡)Δ𝑡𝑎||||𝑓(𝑡)𝛾Δ𝑡1/𝛾𝑎||||𝑔(𝑡)𝜈Δ𝑡1/𝜈,(2.1) where 𝑎,𝕋 and 𝑓;𝑔𝐶rd(𝕀,),𝛾>1  and 1/𝜈+1/𝛾=1, and the inequality (see [29, page 39])𝐴𝑝+1+(𝑝+1)𝐵𝑝+1(𝑝+1)𝐴𝐵𝑝0,𝐴𝐵>0,𝑝>0.(2.2) We also need the product and quotient rules for the derivative of the product 𝑓𝑔 and the quotient 𝑓/𝑔 (where 𝑔𝑔𝜎0) of two differentiable functions 𝑓 and 𝑔(𝑓𝑔)Δ=𝑓Δ𝑔+𝑓𝜎𝑔Δ=𝑓𝑔Δ+𝑓Δ𝑔𝜎,𝑓𝑔Δ=𝑓Δ𝑔𝑓𝑔Δ𝑔𝑔𝜎,(2.3) and the formula(𝑥𝛾(𝑡))Δ=𝛾10[𝑥𝜎]+(1)𝑥𝛾1𝑑𝑥Δ(𝑡),(2.4) which is a simple consequence of Keller's chain rule [11, Theorem  1.90]. We will assume that the boundary value problem𝑢𝑟(𝑡)Δ(𝑡)𝑝Δ=𝛽𝑠Δ(𝑡)𝑢𝑝𝑢(𝑡),𝑢(0)=0,𝑟(𝑏)Δ(𝑏)𝑝=𝛽𝑠(𝑏)𝑢𝑝(𝑏),(2.5) has a solution 𝑢(𝑡) such that 𝑢Δ(𝑡)0 on the interval [0,𝑏]𝕋, where 𝑟,𝑠 are nonnegative rd-continuous functions on (0,𝑏)𝕋.

Now, we are ready to state and prove one of our main results in this section. We begin with an inequality of Opial type which gives a connection with boundary value problems (2.5) on time scales.

Theorem 2.1. Let 𝕋 be a time scale with 0,b𝕋 and let r,s be nonnegative rd-continuous functions on (0,b)𝕋 such that (2.5) has a solution for some 𝛽>0. If x[0,b]𝕋 is delta differentiable with x(0)=0, then for p>0, 𝑏0||||𝑠(𝑡)𝑥(𝑡)𝑝||𝑥Δ||1(𝑡)Δ𝑡(𝑝+1)𝛽0𝑏0||𝑥𝑟(𝑡)Δ||(𝑡)𝑝+1+𝑝Δ𝑡(𝑝+1)𝛽0𝑏0𝑟(𝑡)𝑤(𝑝+1)/p(𝑡)𝑟𝜎(𝑡)𝑤𝜎(𝑡)𝑤1/p||𝑥(𝑡)𝜎||(𝑡)𝑝+1Δ𝑡.(2.6)

Proof. Let 𝑢(𝑡) be a solution of the boundary value problem (2.5) and denote ||𝑥𝑓(𝑡)=Δ||(𝑡),𝐹(𝑡)=𝑡0𝑢𝑓(𝑡)Δ𝑡,𝑤(𝑡)=Δ(𝑡)𝑢(𝑡)𝑝.(2.7) Using the inequality (2.2) and substituting 𝑓 for 𝐴 and 𝑤1/𝑝𝐹𝜎 for 𝐵, we obtain 𝑓𝑝+1+𝑝𝑤𝜆(𝐹𝜎)𝑝+1(𝑝+1)𝑓𝑤(𝐹𝜎)𝑝0,where𝜆=(𝑝+1)𝑝.(2.8) Multiplying this inequality by 𝑟(𝑡), integrating from 0 to 𝑏, and using the fact that 𝐹Δ(𝑡)=𝑓(𝑡)>0, we have 𝑏0𝑟(𝑡)𝑓𝑝+1(𝑡)Δ𝑡+𝑝𝑏0𝑟(𝑡)𝑤𝜆(𝑡)(𝐹𝜎(𝑡))𝑝+1Δ𝑡(𝑝+1)𝑏0𝑟(𝑡)𝑤(𝑡)𝑓(𝑡)(𝐹𝜎(𝑡))𝑝Δ𝑡=(𝑝+1)𝑏0𝑟(𝑡)𝑤(𝑡)(𝐹𝜎(𝑡))𝑝𝐹Δ(𝑡)Δ𝑡.(2.9) By the chain rule (2.4) and the fact that 𝐹Δ(𝑡)>0, we obtain 𝐹𝑝+1(𝑡)Δ=(𝑝+1)10[(1)𝐹(𝑡)+𝐹𝜎](𝑡)𝑝𝑑𝐹Δ(𝑡).(2.10) Noting that if 𝑓 is rd-continuous and 𝐹Δ=𝑓, we see that 𝑡𝜎(𝑡)𝑓(𝑠)Δ𝑠=𝐹(𝜎(𝑡))𝐹(𝑡)=𝜇(𝑡)𝐹Δ(𝑡)=𝜇(𝑡)𝑓(𝑡)>0.(2.11) From the definition of 𝐹(𝑡), we see that 𝐹𝜎(𝑡)=0𝜎(𝑡)𝑓(𝑡)Δ𝑡=𝑡0𝑓(𝑡)Δ𝑡+𝑡𝜎(𝑡)𝑓(𝑡)Δ𝑡=𝐹(𝑡)+𝜇(𝑡)𝑓(𝑡)𝐹(𝑡).(2.12) Substituting into (2.10), we see that [](𝑝+1)𝐹(𝑡)𝑝𝐹Δ𝐹(𝑡)𝑝+1(𝑡)Δ[𝐹(𝑝+1)𝜎](𝑡)𝑝𝐹Δ(𝑡).(2.13) Substituting (2.13) into (2.9), we have 𝑏0𝑟(𝑡)𝑓𝑝+1(𝑡)Δ𝑡+𝑝𝑏0𝑟(𝑡)𝑤𝜆(𝑡)(𝐹𝜎(𝑡))𝑝+1Δ𝑡𝑏0𝐹𝑟(𝑡)𝑤(𝑡)𝑝+1(𝑡)ΔΔ𝑡.(2.14) Integrating by parts and using the assumption 𝐹(0)=0, we see that 𝑏0𝐹𝑟(𝑡)𝑤(𝑡)𝑝+1(𝑡)ΔΔ𝑡=𝑟(𝑡)𝑤(𝑡)𝐹𝑝+1||(𝑡)𝑏0𝑏0(𝑟(𝑡)𝑤(𝑡))Δ(𝐹𝜎(𝑡))𝑝+1Δ𝑡=𝑟(𝑏)𝑤(𝑏)𝐹𝑝+1(𝑏)𝑏0(𝑟(𝑡)𝑤(𝑡))Δ(𝐹𝜎(𝑡))𝑝+1Δ𝑡.(2.15) From (2.14) and (2.15), we see that 𝑏0𝑟(𝑡)𝑓𝑝+1(𝑡)Δ𝑡+𝑝𝑏0𝑟(𝑡)𝑤𝜆(𝑡)(𝐹𝜎(𝑡))𝑝+1Δ𝑡𝑟(𝑏)𝑤(𝑏)𝐹𝑝+1(𝑏)𝑏0(𝑟(𝑡)𝑤(𝑡))Δ(𝐹𝜎(𝑡))𝑝+1Δ𝑡.(2.16) From the definition of the function 𝑤(𝑡), we see that 𝑢𝑟(𝑡)𝑤(𝑡)=𝑟(𝑡)Δ(𝑡)𝑝𝑢𝑝.(𝑡)(2.17) From this and (2.3), we obtain (𝑟(𝑡)𝑤(𝑡))Δ=1𝑢𝑝𝑢(𝑡)𝑟(𝑡)Δ(𝑡)𝑝Δ+𝑟𝑢Δ𝑝𝜎(𝑢𝑝(𝑡))Δ𝑢𝑝(𝑡)𝑢𝑝.(𝜎(𝑡))(2.18) In view of (2.5) and (2.18), we get that (𝑟(𝑡)𝑤(𝑡))Δ=𝛽𝑠Δ𝑟𝑢(𝑡)Δ𝑝𝜎(𝑢𝑝(𝑡))Δ𝑢𝑝(𝑡)𝑢𝑝.(𝜎(𝑡))(2.19) Using the fact that 𝑢Δ(𝑡)0 and the chain rule (2.4), we see that (𝑢𝑝(𝑡))Δ=𝑝10[𝑢𝜎]+(1)𝑢𝑝1𝑢Δ(𝑡)𝑑𝑝10[]𝑢+(1)𝑢𝑝1𝑢Δ(𝑡)𝑑=𝑝(𝑢(𝑡))𝑝1𝑢Δ(𝑡).(2.20) It follows from (2.19) and (2.20) that (𝑟(𝑡)𝑤(𝑡))Δ𝛽𝑠Δ𝑟𝑢(𝑡)Δ𝑝𝜎𝑝(𝑢(𝑡))𝑝1𝑢Δ(𝑡)𝑢𝑝(𝑡)𝑢𝑝(𝜎(𝑡))=𝛽𝑠Δ𝑝𝑟𝑢(𝑡)Δ𝑝𝜎𝑢Δ(𝑡)𝑢(𝑡)𝑢𝑝(𝜎(𝑡))=𝛽𝑠Δ(𝑡)𝑝𝑟𝜎𝑢(𝑡)Δ𝑝𝜎𝑢Δ(𝑡)𝑢(𝑡)𝑢𝑝(𝜎(𝑡))=𝛽𝑠Δ(𝑡)𝑝𝑟𝜎(𝑡)𝑤𝜎(𝑡)𝑤1/𝑝(𝑡).(2.21) From (2.21) and (2.16), we have 𝑏0𝑟(𝑡)𝑓𝑝+1(𝑡)Δ𝑡+𝑝𝑏0𝑟(𝑡)𝑤𝜆(𝑡)(𝐹𝜎(𝑡))𝑝+1Δ𝑡𝑟(𝑏)𝑤(𝑏)𝐹𝑝+1(𝑏)𝑏0𝛽𝑠Δ(𝑡)(𝐹𝜎(𝑡))𝑝+1Δ𝑡+𝑝𝑏0𝑟𝜎(𝑡)𝑤𝜎(𝑡)𝑤1/𝑝(𝑡)(𝐹𝜎(𝑡))𝑝+1Δ𝑡.(2.22) This implies that 𝑏0𝑟(𝑡)𝑓𝑝+1(𝑡)Δ𝑡+𝑝𝑏0𝑟(𝑡)𝑤𝜆(𝑡)𝑟𝜎(𝑡)𝑤𝜎(𝑡)𝑤1/𝑝(𝑡)(𝐹𝜎(𝑡))𝑝+1Δ𝑡𝑟(𝑏)𝑤(𝑏)𝐹𝑝+1(𝑏)𝑏0𝛽𝑠Δ(𝑡)(𝐹𝜎(𝑡))𝑝+1Δ𝑡.(2.23) Using the integration by parts again and using (2.13), we see that 𝛽𝑏0𝑠Δ(𝑡)(𝐹𝜎(𝑡))𝑝+1Δ𝑡=𝛽𝑠(𝑡)(𝐹(𝑡))𝑝+1||𝑏0+𝑏0𝐹𝑠(𝑡)𝑝+1(𝑡)ΔΔ𝑡=𝛽𝑠(𝑏)(𝐹(𝑏))𝑝+1+𝑏0𝐹𝑠(𝑡)𝑝+1(𝑡)ΔΔ𝑡𝛽𝑠(𝑏)(𝐹(𝑏))𝑝+1+𝛽(𝑝+1)𝑏0[]𝑠(𝑡)𝐹(𝑡)𝑝𝐹Δ(𝑡)Δ𝑡.(2.24) Substituting (2.24) into (2.23), we have 𝑏0𝑟(𝑡)𝑓𝑝+1(𝑡)Δ𝑡+𝑝𝑏0𝑟(𝑡)𝑤𝜆(𝑡)𝑟𝜎(𝑡)𝑤𝜎(𝑡)𝑤1/𝑝(𝑡)(𝐹𝜎(𝑡))𝑝+1[]𝐹Δ𝑡𝑟(𝑏)𝑤(𝑏𝛽𝑠(𝑏)𝑝+1(𝑏)+(𝑝+1)𝛽𝑏0[]𝑠(𝑡)𝐹(𝑡)𝑝𝐹Δ(𝑡)Δ𝑡.(2.25) From this, we obtain 𝑏0𝑟(𝑡)𝑓𝑝+1(𝑡)Δ𝑡+𝑝𝑏0𝑟(𝑡)𝑤𝜆(𝑡)𝑟𝜎(𝑡)𝑤𝜎(𝑡)𝑤1/𝑝(𝑡)(𝐹𝜎(𝑡))𝑝+1Δ𝑡(𝑝+1)𝛽𝑏0𝑠(𝑡)𝐹𝑝(𝑡)𝐹Δ(𝑡)Δ𝑡.(2.26) This implies that 𝑏0𝑟(𝑡)𝑓𝑝+1(𝑡)Δ𝑡+𝑝𝑏0𝑟(𝑡)𝑤𝜆(𝑡)𝑟𝜎(𝑡)𝑤𝜎(𝑡)𝑤1/𝑝(𝑡)(𝐹𝜎(𝑡))𝑝+1Δ𝑡(𝑝+1)𝛽0𝑏0𝑠(𝑡)𝐹𝑝(𝑡)𝐹Δ(𝑡)Δ𝑡,(2.27) which is the desired inequality (2.6) after replacing 𝑓 by 𝑥Δ(𝑡) and 𝐹 by 𝑥(𝑡). The proof is complete.

Remark 2.2. Note that when 𝕋=, we have 𝑟(𝑡)=𝑟𝜎(𝑡) and𝑤(𝑡)=𝑤𝜎(𝑡), so that 𝑟(𝑡)𝑤𝜆(𝑡)𝑟𝜎(𝑡)𝑤𝜎(𝑡)𝑤1/𝑝(𝑡)=0 and then (2.6) becomes the inequality (1.10) that has been proved by Boyed and Wong [16]. Note also that the inequality (2.6) can be applied for different values of r and s, and this will left to the interested reader.

When 𝕋=, then (2.6) reduces to the following discrete inequality.

Corollary 2.3. Assume that {ri}0iN and {si}0iN be nonnegative sequences such that the boundary value problem 𝑟(𝑛)Δ𝑢Δ(𝑛)𝑝=𝛽Δ𝑠(𝑛)𝑢𝑝(𝑛),𝑢(0)=0,𝑟(𝑏)(Δ𝑢(𝑏))𝑝=𝛽𝑠(𝑏)𝑢𝑝(𝑏),(2.28) has a solution u(n) such that Δu(n)0 on the interval [0,N] for some 𝛽>0. If {xi}0iN is a sequence of real numbers with x(0)=0, then for p>0, N1n=1||||s(n)x(n)p||||1Δx(n)(p+1)𝛽0N1n=1||||r(n)Δx(n)p+1+p(p+1)𝛽0N1n=0r(n)wp+1/p(n)r(n+1)w(n+1)w1/p||||(n)x(n+1)p+1.(2.29)

Next, in the following we will prove some Opial-type inequalities on time scales which can be considered as the extension of the inequality (1.14) obtained by Beesack and Das [18].

Theorem 2.4. Let 𝕋 be a time scale with 𝑎,𝑏𝕋 and 𝑝,𝑞 positive real numbers such that 𝑝+𝑞>1, and let 𝑟,𝑠 be nonnegative rd-continuous functions on (𝑎,𝑋)𝕋 such that Xar1/(p+q1)(t)Δt<. If y[a,X]𝕋 is delta differentiable with y(a)=0 (and yΔ does not change sign in (a,X)), then 𝑋𝑎||||𝑠(𝑥)𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞Δ𝑥𝐾1(𝑎,𝑋,𝑝,𝑞)𝑋𝑎||𝑦𝑟(𝑥)Δ||(𝑥)𝑝+𝑞Δ𝑥,(2.30) where 𝐾1𝑞(𝑎,𝑋,𝑝,𝑞)=𝑝+𝑞q/(𝑝+𝑞)×𝑋𝑎(𝑠(𝑥))(𝑝+𝑞)/p(𝑟(𝑥))(q/p)𝑥𝑎𝑟1/(𝑝+𝑞1)(𝑡)Δ𝑡(𝑝+𝑞1)Δ𝑥q/(𝑝+𝑞).(2.31)

Proof. Let ||𝑦||=(𝑥)𝑥𝑎||𝑦Δ||(𝑡)Δ𝑡=𝑥𝑎1(𝑟(𝑡))1/(𝑝+𝑞)(𝑟(𝑡))1/(𝑝+𝑞)||𝑦Δ||(𝑡)Δ𝑡.(2.32) Now, since 𝑟 is nonnegative on (𝑎,𝑋), it follows from the Hölder inequality (2.1) (assuming that the integrals exist) with 1𝑓(𝑡)=(𝑟(𝑡))1/(𝑝+𝑞),𝑔(𝑡)=(𝑟(𝑡))1/(𝑝+𝑞)||𝑦Δ||(𝑡),𝛾=𝑝+𝑞𝑝+𝑞1,𝜈=𝑝+𝑞,(2.33) that 𝑥𝑎||𝑦Δ||(𝑡)Δ𝑡𝑥𝑎1(𝑟(𝑡))1/(𝑝+𝑞1)Δ𝑡(𝑝+𝑞1)/(𝑝+𝑞)𝑥𝑎||𝑦𝑟(𝑡)Δ||(𝑡)𝑝+𝑞Δ𝑡1/(𝑝+𝑞).(2.34) Then, for 𝑎𝑥𝑋, we get that ||||𝑦(𝑥)𝑝𝑥𝑎1(𝑟(𝑡))1/(𝑝+𝑞1)Δ𝑡𝑝(𝑝+𝑞1/𝑝+𝑞)𝑥𝑎||𝑦𝑟(𝑡)Δ||(𝑡)𝑝+𝑞Δ𝑡𝑝/(𝑝+𝑞).(2.35) Setting 𝑧(𝑥)=𝑥𝑎𝑟||𝑦(𝑡)Δ||(𝑡)𝑝+𝑞Δ𝑡.(2.36) We see that 𝑧(𝑎)=0, and 𝑧Δ(||𝑦𝑥)=𝑟(𝑥)Δ(||𝑥)𝑝+𝑞>0.(2.37) This gives us ||𝑦Δ||(𝑥)𝑞=𝑧Δ(𝑥)𝑟(𝑥)𝑞/(𝑝+𝑞).(2.38) Thus, if 𝑠 is a nonnegative on (𝑎,𝑋), we have from (2.35) and (2.38) that ||||𝑠(𝑥)𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞1𝑠(𝑥)𝑟(𝑥)𝑞/(𝑝+𝑞)×𝑥𝑎1𝑟1/(𝑝+𝑞1)(𝑡)Δ𝑡𝑝(𝑝+𝑞1/𝑝+𝑞)(𝑧(𝑥))𝑝/(𝑝+𝑞)𝑧Δ(𝑥)𝑞/(𝑝+𝑞).(2.39) This implies that 𝑋𝑎||||𝑠(𝑥)𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞Δ𝑥𝑋𝑎1𝑠(𝑥)𝑟(𝑥)𝑞/(𝑝+𝑞)𝑥𝑎1𝑟1/(𝑝+𝑞1)(𝑡)Δ𝑡𝑝(𝑝+𝑞1/𝑝+𝑞)(𝑧(𝑥))𝑝/(𝑝+𝑞)𝑧Δ(𝑥)𝑞/(𝑝+𝑞)Δ𝑥.(2.40) Supposing that the integrals in (2.40) exist and again applying the Hölder inequality (2.1) with indices 𝑝+𝑞/𝑝 and 𝑝+𝑞/𝑞, we have 𝑋𝑎||||𝑠(𝑥)𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞Δ𝑥𝑋𝑎𝑠(𝑝+𝑞)/𝑝1(𝑥)𝑟(𝑥)𝑞/𝑝𝑥𝑎1𝑟1/(𝑝+𝑞1)(𝑡)Δ𝑡(𝑝+𝑞1)Δ𝑥𝑝/(𝑝+𝑞)×𝑋𝑎𝑧𝑝/𝑞(𝑥)𝑧Δ(𝑥)Δ𝑥𝑞/(𝑝+𝑞).(2.41) From (2.37), the chain rule (2.4), and the fact that 𝑧Δ(𝑡)>0, we obtain 𝑧𝑝/𝑞(𝑥)𝑧Δ𝑞(𝑥)𝑧𝑝+𝑞(𝑝+𝑞)/𝑞(𝑥)Δ.(2.42) Substituting (2.42) into (2.41) and using the fact that 𝑧(𝑎)=0, we have 𝑋𝑎||||𝑠(𝑥)𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞Δ𝑥𝑋𝑎𝑠(𝑝+𝑞)/𝑝1(𝑥)𝑟(𝑥)𝑞/𝑝𝑥𝑎1𝑟1/𝑝+𝑞1(𝑡)Δ𝑡(𝑝+𝑞1)𝑑𝑥𝑝/(𝑝+𝑞)×𝑝𝑝+𝑞𝑞/(𝑝+𝑞)𝑋𝑎𝑧(𝑝+𝑞)/𝑞(𝑡)ΔΔ𝑡𝑞/(𝑝+𝑞)=𝑋𝑎𝑠(𝑝+𝑞)/𝑝1(𝑥)𝑟(𝑥)𝑞/𝑝𝑥𝑎1𝑟1/(𝑝+𝑞1)(𝑡)Δ𝑡(𝑝+𝑞1)Δ𝑥𝑝/(𝑝+𝑞)×𝑞𝑝+𝑞𝑞/(𝑝+𝑞)𝑧(𝑋).(2.43) Using (2.36), we have from the last inequality that 𝑋𝑎||||𝑠(𝑥)𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞Δ𝑥𝐾1(𝑎,𝑏,𝑝,𝑞)𝑋𝑎||𝑦𝑟(𝑥)Δ||(𝑥)𝑝+𝑞Δ𝑥,(2.44) which is the desired inequality (2.30). The proof is complete.

Here, we only state the following theorem, since its proof is the same as that of Theorem 2.4, with [𝑎,𝑋] replaced by [𝑏,𝑋].

Theorem 2.5. Let 𝕋 be a time scale with 𝑎,𝑏𝕋 and 𝑝,𝑞 positive real numbers such that 𝑝+𝑞>1, and let 𝑟,𝑠 be nonnegative rd-continuous functions on (𝑏,𝑋)𝕋 such that 𝑏𝑋𝑟1/(𝑝+𝑞1)(𝑡)Δ𝑡<. If y[X,b]𝕋 is delta differentiable with y(b)=0, (and yΔ does not change sign in (X,b)), then one has 𝑏𝑋||||𝑠(𝑥)𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞Δ𝑥𝐾2(𝑋,𝑏,𝑝,𝑞)𝑏𝑋||𝑦𝑟(𝑥)Δ||(𝑥)𝑝+𝑞Δ𝑥,(2.45) where 𝐾2=𝑞(𝑋,𝑏,𝑝,𝑞)𝑝+𝑞𝑞/(𝑝+𝑞)𝑏𝑋(𝑠(𝑥))(𝑝+𝑞)/𝑝(𝑟(𝑥))(𝑞/𝑝)𝑏𝑥𝑟1/(𝑝+𝑞1)(𝑡)Δ𝑡(𝑝+𝑞1)Δ𝑥𝑝/(𝑝+𝑞).(2.46)

In the following, we assume that there exists (𝑎,𝑏) which is the unique solution of the equation𝐾(𝑝,𝑞)=𝐾1(𝑎,,𝑝,𝑞)=𝐾2(,𝑏,𝑝,𝑞)<,(2.47)

where 𝐾1(𝑎,,𝑝,𝑞) and 𝐾2(,𝑏,𝑝,𝑞) are defined as in Theorems 2.4 and 2.5. Note that since𝑏𝑎||||𝑠(𝑥)𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞Δ𝑥=𝑋𝑎||||𝑠(𝑥)𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞+Δ𝑥𝑏𝑋||||𝑠(𝑥)𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞Δ𝑥,(2.48)

then the proof of the following theorem will be a combination of Theorems 2.4 and 2.5 and due to the limited space we omit the details.

Theorem 2.6. Let 𝕋 be a time scale with 𝑎,𝑏𝕋 and p,q positive real numbers such that 𝑝𝑞>0 and 𝑝+𝑞>1, and let 𝑟,𝑠 be nonnegative rd-continuous functions on (𝑎,𝑏)𝕋 such that 𝑏𝑎𝑟1/(𝑝+𝑞1)(𝑡)Δ𝑡<. If y[𝑎,𝑏]𝕋 is delta differentiable with y(a)=0=y(b), (and yΔ does not change sign in (𝑎,𝑏), then one has 𝑏𝑎||||𝑠(𝑥)𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞Δ𝑥𝐾(𝑝,𝑞)𝑏𝑎||𝑦𝑟(𝑥)Δ||(𝑥)𝑝+𝑞Δ𝑥.(2.49)

For 𝑟=𝑠 in (2.30), we obtain the following special case from Theorem 2.4, which improves the inequality (1.21) obtained by Wong et al. [23].

Corollary 2.7. Let 𝕋 be a time scale with 𝑎,𝑏𝕋 and p,q positive real numbers such that 𝑝+𝑞>1, and let r be a nonnegative rd-continuous function on (𝑎,𝑋)𝕋 such that 𝑋𝑎𝑟1/(𝑝+𝑞1)(𝑡)Δ𝑡<. If y[𝑎,𝑋]𝕋 is delta differentiable with 𝑦(𝑎)=0, (and yΔ does not change sign in (a,X)), then one has 𝑋𝑎||||𝑟(𝑥)𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞Δ𝑥𝐾1(𝑎,𝑋,𝑝,𝑞)𝑋𝑎||𝑦𝑟(𝑥)Δ||(𝑥)𝑝+𝑞Δ𝑥,(2.50) where 𝐾1𝑞(𝑎,𝑋,𝑝,𝑞)=𝑝+𝑞𝑞/(𝑝+𝑞)×𝑋𝑎𝑟(𝑥)𝑥𝑎𝑟1/(𝑝+𝑞1)(𝑡)Δ𝑡(𝑝+𝑞1)Δ𝑥𝑝/(𝑝+𝑞).(2.51)

From Theorems 2.5 and 2.6 one can derive inequalities similar to the inequality in (2.50) by setting 𝑟=𝑠. The details are left to the reader.

On a time scale 𝕋, we note as a consequence from the chain rule (2.4) that(𝑡𝑎)𝑝+𝑞Δ=(𝑝+𝑞)10[](𝜎(𝑡)𝑎)+(1)(𝑡𝑎)𝑝+𝑞1𝑑(𝑝+𝑞)10[](𝑡𝑎)+(1)(𝑡𝑎)𝑝+𝑞1𝑑=(𝑝+𝑞)(𝑡𝑎)𝑝+𝑞1.(2.52)

This implies that 𝑋𝑎(𝑥𝑎)(𝑝+𝑞1)Δ𝑥𝑋𝑎1(𝑝+𝑞)(𝑥𝑎)𝑝+𝑞ΔΔ𝑥=(𝑋𝑎)𝑝+𝑞.(𝑝+𝑞)(2.53)

From this and (2.51) with 𝑟(𝑡)=1, one gets that𝐾1𝑞(𝑎,𝑋,𝑝,𝑞)=𝑝+𝑞𝑞/(𝑝+𝑞)×𝑋𝑎(𝑥𝑎)(𝑝+𝑞1)Δ𝑥𝑝/(𝑝+𝑞)𝑞𝑝+𝑞𝑞/(𝑝+𝑞)(𝑋𝑎)𝑝+𝑞(𝑝+𝑞)𝑝/(𝑝+𝑞)=𝑞𝑞/(𝑝+𝑞)𝑝+𝑞(𝑋𝑎)𝑝.(2.54) So setting 𝑟=1 in (2.50) and using (2.54), one has the following inequality.

Corollary 2.8. Let 𝕋 be a time scale with 𝑎,𝑏𝕋 and 𝑝,𝑞 be positive real numbers such that p+q>1. If y[a,X]𝕋 is delta differentiable with y(a)=0, (and yΔ does not change sign in (a,X)), then, one has 𝑋𝑎||||𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞𝑞Δ𝑥q/(p+q)𝑝+𝑞×(𝑋𝑎)𝑝𝑋𝑎||𝑦Δ||(𝑥)𝑝+𝑞Δ𝑥.(2.55)

Remark 2.9. Note that when 𝕋=, the inequality (2.55) becomes 𝑋𝑎||||𝑦(𝑥)𝑝||𝑦||(𝑥)𝑞𝑞𝑑𝑥𝑞/(𝑝+𝑞)𝑝+𝑞×(𝑋𝑎)𝑝𝑋𝑎||𝑦||(𝑥)𝑝+𝑞𝑑𝑥,(2.56) which gives an improvement of the inequality (1.13).

When 𝕋=, we have form (2.55) the following discrete Opial-type inequality.

Corollary 2.10. Assume that 𝑝,𝑞 be positive real numbers such that 𝑝+𝑞>1 and {ri}0iN  a nonnegative real sequence. If {yi}0iN is a sequence of real numbers with y(0)=0, then 𝑁1𝑛=1||||𝑟(𝑛)𝑦(𝑛)𝑝||||Δ𝑦(𝑛)𝑞𝑞𝑞/(𝑝+𝑞)𝑝+𝑞×(𝑁𝑎)𝑝𝑁1𝑛=0||||𝑟(𝑛)Δ𝑦(𝑛)𝑝+𝑞.(2.57)

The inequality (2.55) has immediate application to the case where 𝑦(𝑎)=𝑦(𝑏)=0. Choose 𝑋=(𝑎+𝑏)/2 and apply (2.51) to [𝑎,𝑐] and [𝑐,𝑏] and then add we obtain the following inequality.

Corollary 2.11. Let 𝕋 be a time scale with a,b𝕋 and p,q positive real numbers such that p+q>1. If y[a,X]𝕋 is delta differentiable with y(a)=0=y(b), then one has 𝑏𝑎||||𝑦(𝑥)𝑝||𝑦Δ||(𝑥)𝑞𝑞Δ𝑥q/(p+q)×𝑝+𝑞𝑏𝑎2𝑝𝑏𝑎||𝑦Δ||(𝑥)𝑝+𝑞Δ𝑥.(2.58)

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

Corollary 2.12. Assume that p,q be positive real numbers such that p+q>1. If {yi}0iN is a sequence of real numbers with y(0)=0=y(N), then 𝑁1𝑛=1||||𝑟(𝑛)𝑦(𝑛)𝑝||||Δ𝑦(𝑛)𝑞𝑞𝑞/(𝑝+𝑞)×𝑝+𝑞𝑁𝑎2𝑝𝑁1𝑛=0||||𝑟(𝑛)Δ𝑦(𝑛)𝑝+𝑞.(2.59)

By setting 𝑝=𝑞=1 in (2.58), we have the following Opial type inequality on a time scale.

Corollary 2.13. Let 𝕋 be a time scale with a,b𝕋. If y[a,X]𝕋 is delta differentiable with y(a)=0=y(b), then one has ba||||||yy(x)Δ||(x)Δx(ba)4ba||yΔ||(x)2Δx.(2.60)

When 𝕋= and 𝕋=, we have from (2.60) the following inequalities.

Corollary 2.14. If y[a,X] is differentiable with y(a)=0=y(b), then one has ba||||||yy(x)||(x)dx(ba)4ba||y||(x)2dx.(2.61)

Corollary 2.15. If {yi}0iN is a sequence of real numbers with y(0)=0=y(N), then N1n=1||||||||Ny(n)Δy(n)4N1n=0||||Δy(n)2.(2.62)

Acknowledgments

This project was supported by King Saud University, Dean-ship of Scientific Research, College of Science Research Centre.

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