Some Opial Dynamic Inequalities Involving Higher Order Derivatives on Time Scales

We will prove some new Opial dynamic inequalities involving higher order derivatives on time scales. The results will be proved by making use of Holder's inequality, a simple consequence of Keller's chain rule and Taylor monomials on time scales. Some continuous and discrete inequalities will be derived from our results as special cases.


Introduction
In the past decade a number of Opial dynamic inequalities have been established by some authors which are motivated by some applications; we refer to the papers 1-3 .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 T, which may be an arbitrary closed subset of the real numbers R, to avoid proving results twice, once on a continuous time scale which leads to a differential inequality and once again on a discrete time scale which leads to a difference inequality.The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus see 4 , that is, when T R, T N and T q N 0 {q t : t ∈ N 0 } where q > 1.A cover story article in New Scientist 5 discusses several possible applications of time scales.In this paper, we will assume that sup T ∞ and define the time scale interval a, b T by a, b T : a, b ∩ T. Since the continuous and discrete inequalities involving higher order derivatives are important in the analysis of qualitative properties of solutions of differential and difference equations 6-8 , we also believe that the dynamic inequalities involving higher order derivatives on time scales will play the same effective act in the analysis of qualitative properties of solutions of dynamic equations 2, 3, 9 .To the best of the author's knowledge there are few inequalities involving higher order derivatives established in the literature 10-13 .In the following, we recall some of these results that serve and motivate the contents of this paper.
In 13 the authors proved that if y : a, b T → R is delta differentiable n times with y Δ i a 0, for i 0, 1, . . ., n − 1, and h is a positive rd-continuous function on a, b T , then b a h t y t p y Δ n t q Δt ≤ q p q b − a np b a h t y Δ n t p q Δt. 1.1 In 10 it is proved that if y : a, b T → R is delta differentiable n times n odd with y Δ i a 0, for i 0, 1, . . ., n − 1, then where p, q > 1 and satisfy 1/p 1/q 1.Also in 10 it is proved that if y : a, b T → R is delta differentiable n times with y Δ i a 0, for i 0, 1, . . ., n − 1, and |y where p, q > 1 and satisfy 1/p 1/q 1.As a generalization of 1.3 it is proved in 10 that if y : a, b T → R is delta differentiable n times with y Δ m i a 0, for i 0, 1, . . ., n − m − 1, and where p, q > 1 and satisfy 1/p 1/q 1.In 12 the authors proved that if r and s are positive rd-continuous functions on a, b T such that s is nonincreasing, and y : where p > 0 and 1/γ 1/ν 1.For contributions of different types of dynamic inequalities on time scales, we refer the reader to the papers 1, 2, 14-17 and the references cited therein.
Following this trend, to develop the qualitative theory of dynamic inequalities on time scales, we will prove some new inequalities of Opial's type involving higher order derivatives by making use of the H ölder inequality see, 18, Theorem 6.13 : where a, h ∈ T and f; g ∈ C rd I, R , γ > 1 and 1/ν 1/γ 1, the formula which is a simple consequence of Keller's chain rule 18, Theorem 1.90 , and the Taylor monomials on time scales.The results in this paper extend and improve the pervious results in the sense that our results contain two different weighted functions and do not require the monotonicity condition on |y Δ n t | the results in 10 required that |y Δ n t | should be increasing .Some results on continuous and discrete spaces, which lead to differential and difference inequalities, will be derived from our results as special cases.This paper is a continuation of the papers 3, 10-13, 16 .

Main Results
In this section, we will prove the main results.For completeness, we recall the following concepts related to the notion of time scales.A time scale T is an arbitrary nonempty closed subset of the real numbers R. We assume throughout that T has the topology that it inherits from the standard topology on the real numbers R. is, when T R, T N and T q N 0 {q t : t ∈ N 0 } where q > 1.For more details of time scale analysis we refer the reader to the two books by Bohner and Peterson 18, 19 which summarize and organize much of the time scale calculus.
A function g : T → R is said to be right-dense continuous rd-continuous provided g is continuous at right-dense points and at left-dense points in T; left-hand limits exist and are finite.The set of all such rd-continuous functions is denoted by C rd T .
The graininess function μ for a time scale T is defined by μ t : σ t − t, and for any function f : T → R the notation f σ t denotes f σ t .We will assume that sup T ∞ and define the time scale interval a, b T by a, b T : a, b ∩ T. Fix t ∈ T and let y : T → R.
Define y Δ t to be the number if it exists with the property that given any > 0 there is a neighborhood U of t with In this case, we say y Δ t is the delta derivative of y at t and that y is delta differentiable at t.We will frequently use the following results which are due to Hilger 20  We will make use of the following product and quotient rules for the derivative of the product fg and the quotient f/g where gg σ / 0, here g σ g • σ of two differentiable functions f and g: In this paper, we will refer to the delta integral which we can define as follows: if G Δ t g t , then the Cauchy delta integral of g is defined by are defined recursively as follows.The function h 0 is defined by and given h k for k ∈ N 0 , the function h k 1 is defined by

2.10
If we let h Δ k t, s denote for each fixed s ∈ T, the derivative of h t, s with respect to t, then for each fixed s ∈ T. The above definition obviously implies

2.12
In the following, we give some formulas of h k t, s as determined in 18 .In the case when T R, then σ t t, μ t 0, y Δ t y t , and

2.13
In the case when T N, we see that σ t t 1, μ t 1, y Δ t Δy t y t 1 − y t , and where When T {t : t q n , n ∈ N, q > 1}, we have σ t qt, μ t q − 1 t, y Δ t Δ q y t y q t − y t / q − 1 t, and

2.16
In general for t ≥ s, we have that h k t, s ≥ 0, and We also consider the Taylor monomials g k : T × T → R, k ∈ N 0 N ∪ {0}, which are defined recursively as follows.The function g 0 is defined by and given g k for k ∈ N 0 , the function g k 1 is defined by

2.19
If we let g Δ k t, s denote for each fixed s ∈ T, the derivative of g t, s with respect to t, then for each fixed s ∈ T. By Theorem 1.112 in 18 , we see that

2.21
We denote by C n rd T the space of all functions f ∈ C rd T such that f Δ i ∈ C rd T for i 0, 1, 2, . . ., n for n ∈ N.For the function f : T → R, we consider the second derivative f Δ 2 provided f Δ is delta differentiable on T with derivative f Δ 2 f Δ Δ .Similarly, we define the nth order derivative f Δ n f Δ n−1 Δ .Now, we are ready to state the Taylor formula that we will need to prove the main results in this paper.This formula as proved in 23 states the following.Assuming that f ∈ C n rd T and s ∈ T, then

2.22
As a special case if m < n, then Now, we are ready to state and prove our main results in this paper.Throughout the rest of the paper, we will assume that all the integrals that will appear in the inequalities exist and are finite. Δt.

2.30
From 1.7 , we have note that z t > 0 and z Δ t > 0 that Δt.

2.34
Thus for x y Δ k , where y ∈ C n−k rd a, b , y Δ i 0, k ≤ i ≤ n − 1; then we have the following result.

Corollary 2.3. Let T be a time scale with a,b ∈ T and y
Δt.

2.35
Note that Theorem 2.1 can be extended to a general inequality with two different constants p and q that satisfy 1/p 1/q 1, to obtain the following result.

2.39
Applying the H ölder inequality with γ l m and ν l m / l m − 1 , we have

2.42
Define z t : Δs.This implies that z a 0, and From this and 2.42 , we have

2.45
From 1.7 , we have note that z t and z Δ t > 0; see also, page 116 17 that Δt,

2.49
where H t, s :

2.50
Note that Theorem 2.5 cannot be applied when l m 1.In the following theorem we prove a new inequality which can be applied in this case.

2.59
In the following, we will prove some inequalities with two different weighted functions.
Theorem 2.9.Let T be a time scale with a, b ∈ T and let l, m, r be positive real numbers such that l m > 1 and r > 1.Further, let p t and q t be positive rd-continuous functions defined on a, b ∩ T and where p −1/ r−1 s h n−1 t, σ s r/ r−1 Δs.

2.61
Proof.From the Taylor formula 2.22 , we see that y t ≤ t a p −1/r s h n−1 t, σ s p l/r s y Δ n s Δs.

2.62
Applying the H ölder inequality on the right hand side with r and r/r − 1, we have

2.63
This implies that q t y t l y Δ n t m ≤ q t P l r− z t l/r Δt.

2.67
Applying the H ölder inequality 1.6 with indices r/m and r/r − m, we obtain

2.68
Substituting into 2.67 , we have

2.69
As in the proof of Theorem 2.5, we have

2.70
This implies that b a q t y t l y which is the desired inequality 2.60 where Λ 1 l, m, p, q, r is defined as in 2.61 .The proof is complete.
Following Remark 2.2, we can obtain the following result.
Theorem 2.10.Let T be a time scale with a, b ∈ T and let l, m be positive real numbers such that l m > 1 and r > 1.Further, let p t and q t be positive rd-continuous functions defined on a, b ∩ T and where

2.73
Note that Theorem 2.10 cannot be applied when r 1 and r < m.In the following theorem we prove an inequality which can be applied in this case.Theorem 2.11.Let T be a time scale with a, b ∈ T and let α, β be positive real numbers such that α β > 1, and let p, q be nonnegative rd-continuous functions on a, b T and y where

2.75
Proof.From the Taylor formula, we see that

2.81
Since q is nonnegative on a, b T , we have from 2.79 and 2.81 that

2.82
This implies that b a q t y t α y Δt.

2.83
Applying the H ölder inequality 1.6 with indices α β /α and α β /β, we have b a q t y t α y

2.84
From 2.80 , the chain rule 1.7 and the fact that z Δ s > 0, we obtain

2.85
Substituting 2.85 into 2.84 and using the fact that z a 0, we have b a q t y t α y Following Remark 2.2, we can obtain the following result.
Theorem 2.12.Let T be a time scale with a, b ∈ T and let α, β be positive real numbers such that α β > 1, and let p, q be nonnegative rd-continuous functions on a, b T and y where

2.89
Instead of 2.25 , we can use the relation between g n and h n and define y t : −1 n b t g n−1 σ s , t y Δ n s Δs.

2.90
Proceeding as pervious by using the same arguments and using 2.90 one can obtain some results when y Δ i b 0, for i 0, 1, . . ., n − 1.For example one can get the following results.Δt.

2.92
Remark 2.15.Similar results as in Theorems 2.13 and 2.14 can be obtained from the results in the rest of the paper, but in this case one will use y Δ i b 0, for i 0, 1, . . ., n − 1, instead of If y Δ i a 0, for i 0, 1, . . ., n − 1, and g n s, t instead of h n t, s .
Remark 2.16.It is worth mentioning here that the results in this paper can be used to derive some inequalities on different time scales based on the definition of the corresponding function h k t, s .
For example if T R, then from Corollary 2.8, Theorems 2.11 and 2.12 and using 2.13 , we get the following inequalities of Opial's type in R.

2.97
When T Z, we have y Δ t Δy t y t 1 −y t and Δ i Δ Δ i−1 .Using the fact that h k t, s ≤ t − s k /k!, we get from Corollary 2.8 and Theorems 2.11 and 2.12 the following discrete inequalities.

2.102
Similar results when T hZ and T q N 0 {q t : t ∈ N 0 } where q > 1 and different time scales can be obtained as in Theorems 2.17 and 2.22.The details are left to the reader.Problem 1.It will be interesting to extend the pervious results and prove some inequalities of the form where Λ is the constant of the inequality that needs to be determined. b

Theorem 2 . 4 .
Letting T be a time scale with a, b ∈ T and y ∈ C n rd a, b ∩ T .If y Δ i a 0, for i 0, 1, . . ., n − 1, then b a y t y Δ n t Δt ≤ 1 n−1 t, σ s | p Δs Δt

Corollary 2 . 6 .
is the desired inequality 2.37 .The proof is complete.Following Remark 2.2, we can obtain the following result.Let T be a time scale with a, b ∈ T and let l,m be positive real numbers such that l m > 1 and y

Theorem 2 . 7 .
Let T be a time scale with a, b ∈ T and let l, m be positive real numbers such that l m 1 and y ∈ C n rd a, b ∩ T .If y Δ i a 0, for i 0, 1, . . ., n − 1, then |y Δ n s | α β Δs, we have from the last inequality that desired inequality 2.74 .The proof is complete.

Theorem 2 . 13 .Theorem 2 . 14 .
Letting T be a time scale with a, b ∈ T and y ∈ C n rd X, b ∩ T .If y Δ i b 0, for i 0, 1, . . ., n − 1, then b a y t y Δ n t Δt ≤ Let T be a time scale with a, b ∈ T and let l, m be positive real numbers such that l m > 1.Let y ∈ C n rd a, b ∩ T .If y Δ i b 0, for i 0, 1, . . ., n − 1, then

Theorem 2 . 17 . 93 Theorem 2 . 18 .Theorem 2 . 19 .
Let a, b ∈ R and let l, m be positive real numbers such that l m 1, and y∈ C n a, b ∩ R .If y i a 0, k ≤ i ≤ n − 1, then m m b − a l n−k n − k !l b a y n t dt .2.Let a, b ∈ R and let α, β be positive real numbers such that α β > 1, and let p, q be nonnegative continuous functions on a, b and y ∈C n a, b ∩ R .If y i a 0, for i 0, 1, . . ., n − 1, then s n−1 α β / α β−1 n − 1 !p 1/ α β−1 Let a, b ∈ Rand let α, β be positive real numbers such that α β > 1, and let p, q be nonnegative continuous functions on a, b and y ∈ C n a, b ∩ R .If y i a 0, k ≤ i ≤ n − 1, then

Theorem 2 . 20 . 98 Theorem 2 . 21 .Theorem 2 . 22 .
Let a, b ∈ N and let l, m be positive real numbers such that l m 1.If Δ i y a 0, k ≤ i ≤ n − 1, then b−1 t a Δ k y t l Δ n y t m ≤ m m b − a l n−k n − k !l b−1 t aΔ n y t .2.Let a, b ∈ N and let α, β be positive real numbers such that α β > 1, and let p, q be nonnegative sequences.If Δ i y a 0, for i 0, 1, . . ., n − 1, then Let a, b ∈ N and let α, β be positive real numbers such that α β > 1, and let p, q be nonnegative sequences.If Δ i y a 0, 0 ≤ k ≤ i ≤ n − 1, then Now, we define the Taylor monomials or generalized polynomials as defined originally by Agarwal and Bohner 21 .These types of monomials are important because they are intimately related to Cauchy functions for certain dynamic equations which are important in variations of constants formulas.The Taylor monomials h k rd T , then the Cauchy integral G t : t t 0 g s Δs exists, t 0 ∈ T, and satisfies G Δ t g t , t ∈ T.An infinite integral is defined as Theorem 2.1.Letting T be a time scale with a, b ∈ T and y ∈ C n t y t ≤ y Δ n t t a |h n−1 t, σ s | y Δ n s Δs. 2.26 Applying the H ölder inequality 1.6 with γ ν 2, we have y Δ n t y t ≤ y Δ n t t a |h n−1 t, σ s | 2 Δs t a |y Δ n s | 2 Δs.This implies that z a 0 and |y Δ n t | 2 z Δ t .From this and 2.28 , we have From the Taylor formula 2.22 and since y Δ q Δt. 2.36 Theorem 2.5.Let T be a time scale with a, b ∈ T and let l,m be positive real numbers such that l m > 1, and y ∈ C n rd a, b ∩ T .If y Δ i a 0, for i 0, 1, . . ., n − 1; then i a 0, for i 0, 1, . . ., n − 1, we have y t ≤ t a h n−1 t, σ s y Δ n s Δs.
Using the fact that |h n t, s | is increasing with respect to its first component for t ≥ σ s > a, we have from the Taylor formula 2.22 and y Δ i a 0, for i 0, 1, . . ., n − 1, that Δ n t Δt. 2.51 Proof.t a |y Δ n s |Δs.Then z a 0 and z Δ t |y Δ n t | so that which is the desired inequality 2.51 .The proof is complete.Following Remark 2.2, we can obtain the following result.Corollary 2.8.Let T be a time scale with a, b ∈ T and let l, m be positive real numbers such that l m 1, and y