# Some Opial Dynamic Inequalities Involving Higher Order Derivatives on Time Scales

Authors: Samir H. Saker
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

We will prove some new Opial dynamic inequalities involving higher order derivatives on time scales. The results will be proved by making use of Hölder'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.


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## Body

## 1. 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=qℕ0={qt:t∈ℕ0} 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 sup𝕋=∞ 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 yΔi(a)=0, for i=0,1,…,n-1, and h is a positive rd-continuous function on [a,b]𝕋, then
(1.1)∫ab‍h(t)|y(t)|p|yΔn(t)|qΔt≤(qp+q)(b-a)np∫ab‍h(t)|yΔn(t)|p+qΔt.
In [10] it is proved that if y:[a,b]𝕋→ℝ is delta differentiable n times (n odd) with yΔi(a)=0, for i=0,1,…,n-1, then
(1.2)∫ab‍|y(t)|Δt≤(∫ab‍(∫at‍hn-1p(t,σ(s))Δs)q/pΔt)∫ab‍|yΔn(t)|qΔt,
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 yΔi(a)=0, for i=0,1,…,n-1, and |yΔn(t)| is increasing, then
(1.3)∫ab‍|y(t)||yΔn(t)|Δt≤(b-a)1/q(∫ab‍(∫at‍hn-1p(t,σ(s))Δs)1/pΔt)(∫ab‍|yΔn(t)|2qΔt)1/q,
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 yΔm+i(a)=0, for i=0,1,…,n-m-1, and |yΔn(t)| is increasing, then
(1.4)∫ab‍|yΔm(t)||yΔn(t)|Δt≤(b-a)1/q(∫ab‍(∫at‍hn-m-1p(t,σ(s))Δs)1/pΔt)(∫ab‍|yΔn(t)|2qΔt)1/q,
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:[a,b]𝕋→ℝ is delta differentiable n times with yΔi(a)=0, for i=0,1,…,n-1, then
(1.5)∫ab‍s(x)|y(t)|p|yΔn(t)|Δt≤1p+1(b-a)n-1(∫ab‍r1-γ(t)Δt)(1+p)/γ(∫ab‍r(t)(s(t))p/(p+1)|yΔn(t)|νΔt)(1+p)/ν,
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 Hölder inequality (see, [18, Theorem 6.13]):
(1.6)∫ah‍|f(t)g(t)|Δt≤[∫ah‍|f(t)|γΔt]1/γ[∫ah‍|g(t)|νΔt]1/ν,
where a,h∈𝕋 and f;g∈Crd(𝕀,ℝ),γ>1 and 1/ν+1/γ=1, the formula
(1.7)(xγ(t))Δ=γ∫01‍[hxσ+(1-h)x]γ-1dhxΔ(t),
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 |yΔn(t)| (the results in [10] required that |yΔn(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].


## 2. 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 scale𝕋 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. The forward jump operator and the backward jump operator are defined by
(2.1)σ(t):=inf{s∈𝕋:s>t},ρ(t):=sup{s∈𝕋:s<t},
where sup∅=inf𝕋. A point t∈𝕋 is said to be left-dense if ρ(t)=t and t>inf𝕋, is right-dense if σ(t)=t, is left-scattered if ρ(t)<t and right-scattered if σ(t)>t. 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=qℕ0={qt:t∈ℕ0} 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 functiong: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 Crd(𝕋).The graininess functionμ for a time scale 𝕋 is defined by μ(t):=σ(t)-t, and for any function f:𝕋→ℝ the notation fσ(t) denotes f(σ(t)). We will assume that sup𝕋=∞ 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 yΔ(t) to be the number (if it exists) with the property that given any ϵ>0 there is a neighborhood U of t with
(2.2)|[y(σ(t))-y(s)]-yΔ(t)[σ(t)-s]|≤ϵ|σ(t)-s|,∀s∈U.
In this case, we say yΔ(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]. Assume that y:𝕋→ℝ and let t∈𝕋.(i)
Ify is differentiable at t, then y is continuous at t.(ii)
Ify is continuous at t and t is right-scattered, then y is differentiable at t with
(2.3)yΔ(t)=y(σ(t))-y(t)μ(t).(iii)
Ify is differentiable and t is right-dense, then
(2.4)yΔ(t)=lims→ty(t)-y(s)t-s.(iv)
Ify is differentiable at t, then y(σ(t))=y(t)+μ(t)yΔ(t).We will make use of the following product and quotient rules for the derivative of the productfg and the quotient f/g (where ggσ≠0, here gσ=g∘σ) of two differentiable functions f and g:
(2.5)(fg)Δ=fΔg+fσgΔ=fgΔ+fΔgσ,(fg)Δ=fΔg-fgΔggσ.
In this paper, we will refer to the (delta) integral which we can define as follows: if GΔ(t)=g(t), then the Cauchy (delta) integral of g is defined by
(2.6)∫at‍g(s)Δs:=G(t)-G(a).
It can be shown (see [18]) that if g∈Crd(𝕋), then the Cauchy integral G(t):=∫t0t‍g(s)Δs exists, t0∈𝕋, and satisfies GΔ(t)=g(t), t∈𝕋. An infinite integral is defined as
(2.7)∫a∞‍f(t)Δt=limb→∞∫ab‍f(t)Δt,
and the integration on discrete time scales is defined by
(2.8)∫ab‍f(t)Δt=∑t∈[a,b)‍μ(t)f(t).
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 hk:𝕋×𝕋→ℝ, k∈ℕ0=ℕ∪{0} are defined recursively as follows. The function h0 is defined by
(2.9)h0(t,s)=1,∀s,t∈𝕋,
and given hk for k∈ℕ0, the function hk+1 is defined by
(2.10)hk+1(t,s)=∫st‍hk(τ,s)Δτ,∀s,t∈𝕋.
If we let hkΔ(t,s) denote for each fixed s∈𝕋, the derivative of h(t,s) with respect to t, then
(2.11)hkΔ(t,s)=hk-1(t,s),k∈ℕ,t∈𝕋,
for each fixed s∈𝕋. The above definition obviously implies
(2.12)h1(t,s)=t-s,∀s,t∈𝕋.
In the following, we give some formulas of hk(t,s) as determined in [18]. In the case when 𝕋=ℝ, then σ(t)=t, μ(t)=0, yΔ(t)=y′(t), and
(2.13)hk(t,s)=(t-s)kk!,∀s,t∈ℝ.
In the case when 𝕋=ℕ, we see that σ(t)=t+1, μ(t)=1, yΔ(t)=Δy(t)=y(t+1)-y(t), and
(2.14)hk(n,s):=(n-s)(k)k!,k=0,1,2,…,t>s,
where t(k)=t(t-1)⋯(t-k+1) is the so-called falling function (cf. [22]). When 𝕋={t:t=qn,n∈ℕ,q>1}, we have σ(t)=qt, μ(t)=(q-1)t, yΔ(t)=Δqy(t)=(y(qt)-y(t))/(q-1)t, and
(2.15)hk(t,s)=∏m=0k-1‍t-qms∑j=0m‍qj,∀s,t∈𝕋.
If 𝕋=hℕ,h>0,we see that σ(t)=t+h, μ(t)=h, yΔ(t)=Δhy(t)=(y(t+h)-y(t))/h, and
(2.16)hk(t,s)=∏i=0k-1‍(t-ih-s)k!,∀s,t∈𝕋,t>s.
In general for t≥s, we have that hk(t,s)≥0, and
(2.17)hk(t,s)≤(t-s)kk!,∀t>s,k∈ℕ0.
We also consider the Taylor monomials gk:𝕋×𝕋→ℝ, k∈ℕ0=ℕ∪{0}, which are defined recursively as follows. The function g0 is defined by
(2.18)g0(t,s)=1,∀s,t∈𝕋,
and given gk for k∈ℕ0, the function gk+1 is defined by
(2.19)gk+1(t,s)=∫st‍gk(σ(τ),s)Δτ,∀s,t∈𝕋.
If we let gkΔ(t,s) denote for each fixed s∈𝕋, the derivative of g(t,s) with respect to t, then
(2.20)gkΔ(t,s)=gk-1(σ(t),s),k∈ℕ,t∈𝕋,
for each fixed s∈𝕋. By Theorem  1.112 in [18], we see that
(2.21)hk(t,s)=(-1)kgk(s,t).
We denote by Crd(n)(𝕋) the space of all functions f∈Crd(𝕋) such that fΔi∈Crd(𝕋) for i=0,1,2,…,n for n∈ℕ. For the function f:𝕋→ℝ, we consider the second derivative fΔ2 provided fΔ is delta differentiable on 𝕋 with derivative fΔ2=(fΔ)Δ. Similarly, we define the nth order derivative fΔn=(fΔn-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∈Crd(n)(𝕋) and s∈𝕋, then(2.22)f(t)=∑k=0n-1‍fΔk(s)hk(t,s)+∫st‍hn-1(t,σ(τ))fΔn(τ)Δτ.
As a special case if m<n, then
(2.23)fΔm(t)=∑k=0n-m-1‍fΔk+m(s)hk(t,s)+∫st‍hn-m-1(t,σ(τ))fΔn(τ)Δτ.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.Theorem 2.1.
Letting𝕋 be a time scale with a,b∈𝕋 and y∈Crd(n)([a,b]∩𝕋). If yΔi(a)=0, for i=0,1,…,n-1, then
(2.24)∫ab‍|y(t)||yΔn(t)|Δt≤12(∫ab‍(∫at‍|hn-1(t,σ(s))|2Δs)Δt)1/2∫ab‍|yΔn(t)|2Δt.Proof.
From the Taylor formula (2.22), since yΔi(a)=0, for i=0,1,…,n-1, we have
(2.25)y(t):=∫at‍hn-1(t,σ(s))yΔn(s)Δs.
This implies that
(2.26)|yΔn(t)||y(t)|≤|yΔn(t)|∫at‍|hn-1(t,σ(s))||yΔn(s)|Δs.
Applying the Hölder inequality (1.6) with γ=ν=2, we have
(2.27)|yΔn(t)||y(t)|≤|yΔn(t)|(∫at‍|hn-1(t,σ(s))|2Δs)1/2(∫at‍|yΔn(s)|2Δs)1/2.
Then
(2.28)∫ab‍|yΔn(t)||y(t)|Δt≤∫ab‍(∫at‍|hn-1(t,σ(s))|2Δs)1/2|yΔn(t)|(∫at‍|yΔn(s)|2Δs)1/2Δt.
Define z(t):=∫at‍|yΔn(s)|2Δs. This implies that z(a)=0 and |yΔn(t)|2=zΔ(t). From this and (2.28), we have
(2.29)∫ab‍|yΔn(t)||y(t)|Δt≤∫ab‍(∫at‍|hn-1(t,σ(s))|2Δs)1/2(z(t)zΔ(t))1/2Δt.
Applying the Hölder inequality again γ=ν=2, we obtain
(2.30)∫ab‍|yΔn(t)||y(t)|Δt≤(∫ab‍(∫at‍|hn-1(t,σ(s))|2Δs)Δt)1/2(∫ab‍z(t)zΔ(t)Δt)1/2.
From (1.7), we have (note that z(t)>0 and zΔ(t)>0) that
(2.31)(z2(t))Δ=2zΔ(t)∫01‍[hzσ+(1-h)z]dh≥2zΔ(t)z(t).
Then
(2.32)∫ab‍z(t)zΔ(t)Δt≤12∫ab‍(z2(t))ΔΔt=12z2(b),
where z(a)=0. Substituting (2.32) into (2.30), we have
(2.33)∫ab‍|yΔn(t)||y(t)|Δt≤12(∫ab‍(∫at‍|hn-1(t,σ(s))|2Δs)Δt)1/2z(b)=12(∫ab‍(∫at‍|hn-1(t,σ(s))|2Δs)Δt)1/2∫ab‍|yΔn(t)|2Δt,
which is the desired inequality (2.24). The proof is complete.Remark 2.2.
Let0≤k<n, but fixed, and let x(t)∈Crd(n-k)([a,b]∩𝕋) be such that x(Δi)(a)=0, 0≤i≤n-k-1. Then from (2.24) it follows that
(2.34)∫ab‍|x(t)||xΔn-k(t)|Δt≤12(∫ab‍(∫at‍|hn-k-1(t,σ(s))|2Δs)Δt)1/2∫ab‍|xΔn-k(t)|2Δt.
Thus for x=yΔk, where y∈Crd(n-k)[a,b], y(Δi)=0, k≤i≤n-1; then we have the following result.Corollary 2.3.
Let𝕋 be a time scale with a,b∈𝕋 and y∈Crd(n)([a,b]∩𝕋). If y(Δi)(a)=0, k≤i≤n-1, then
(2.35)∫ab‍|yΔk(t)||yΔn(t)|Δt≤12(∫ab‍(∫at‍|hn-k-1(t,σ(s))|2Δs)Δt)1/2∫ab‍|yΔn(t)|2Δt.Note that Theorem2.1 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.Theorem 2.4.
Letting𝕋 be a time scale with a,b∈𝕋 and y∈Crd(n)([a,b]∩𝕋). If yΔi(a)=0, for i=0,1,…,n-1, then
(2.36)∫ab‍|y(t)||yΔn(t)|Δt≤(12)1/q(∫ab‍(∫at‍|hn-1(t,σ(s))|pΔs)Δt)1/p∫ab‍|yΔn(t)|qΔt.Theorem 2.5.
Let𝕋 be a time scale with a,b∈𝕋 and let l,m be positive real numbers such that l+m>1, and y∈Crd(n)([a,b]∩𝕋). If yΔi(a)=0, for i=0,1,…,n-1; then
(2.37)∫ab‍|y(t)|l|yΔn(t)|mΔt≤(ml+m)m/(l+m)(∫ab‍H(l+m-1)(t,s)Δt)l/(l+m)∫ab‍|yΔn(t)|l+mΔt,
where
(2.38)H(t,s):=∫at‍(hn-1(t,σ(s)))(l+m)/(l+m-1)Δs.Proof.
From the Taylor formula (2.22) and since yΔi(a)=0,for i=0,1,…,n-1, we have
(2.39)|y(t)|≤∫at‍hn-1(t,σ(s))|yΔn(s)|Δs.
Applying the Hölder inequality with γ=l+m and ν=(l+m)/(l+m-1), we have
(2.40)|y(t)|≤(∫at‍(hn-1(t,σ(s)))(l+m)/(l+m-1)Δs)(l+m-1)/(l+m)(∫at‍|yΔn(s)|l+mΔs)1/(l+m).
This implies that
(2.41)|y(t)|l|yΔn(t)|m≤|yΔn(t)|m(∫at‍(hn-1(t,σ(s)))(l+m)/(l+m-1)Δs)l((l+m-1)/(l+m))(∫at‍|yΔn(s)|l+mΔs)l/(l+m).
Then
(2.42)∫ab‍|yΔn(t)|m|y(t)|lΔt≤∫ab‍(∫at‍(hn-1(t,σ(s)))(l+m)/(l+m-1)Δs)l((l+m-1)/(l+m))|yΔn(t)|m(∫at‍|yΔn(s)|l+mΔs)l/(l+m)Δt.
Define z(t):=∫at‍|yΔn(s)|l+mΔs. This implies that z(a)=0, and
(2.43)|yΔn(t)|m=(zΔ(t))m/(l+m)>0.
From this and (2.42), we have
(2.44)∫ab‍|y(t)|l|yΔn(t)|mΔt≤∫ab‍(∫at‍(hn-1(t,σ(s)))(l+m)/(l+m-1)Δs)l((l+m-1)/(l+m))(zΔ(t))m/(l+m)(z(t))l/(l+m)Δt.
Applying the Hölder inequality again γ=(l+m)/l and ν=(l+m)/m, we have
(2.45)∫ab‍|y(t)|l|yΔn(t)|mΔt≤(∫ab‍(∫at‍(hn-1(t,σ(s)))(l+m)/(l+m-1)Δs)l((l+m-1)/(l+m))((l+m)/l)Δt)l/(l+m)×(∫ab‍zΔ(t)(z(t))(l/m)Δt)m/(m+l).
From (1.7), we have (note that z(t) and zΔ(t)>0; see also, page 116 [17]) that
(2.46)(z(l+m)/m(t))Δ≥l+mm∫01‍[hzσ+(1-h)z]((l+m)/m)-1zΔ(t)≥l+mm(z(t))(l/m)zΔ(t).
Then
(2.47)∫ab‍z(l/m)(t)zΔ(t)Δt≤ml+m∫ab‍(z(l+m)/m(t))ΔΔt=ml+mz(l+m)/m(b),
where z(a)=0. Substituting (2.47) into (2.45), we have
(2.48)∫ab‍|y(t)|l|yΔn(t)|mΔt≤(ml+m)m/(l+m)(∫ab‍(∫at‍(hn-1(t,σ(s)))(l+m)/(l+m-1)Δs)l((l+m-1)/(l+m))((l+m)/l)Δt)l/(l+m)z(b)=(ml+m)m/(l+m)(∫ab‍(∫at‍(hn-1(t,σ(s)))(l+m)/(l+m-1)Δs)(l+m-1)Δt)l/(l+m)×∫ab‍|yΔn(t)|l+mΔt,
which is the desired inequality (2.37). The proof is complete.Following Remark2.2, we can obtain the following result.Corollary 2.6.
Let𝕋 be a time scale with a,b∈𝕋 and let l,m be positive real numbers such that l+m>1 and y∈Crd(n)([a,b]∩𝕋). If y(Δi)(a)=0, k≤i≤n-1; then
(2.49)∫ab‍|yΔk(t)|l|yΔn(t)|mΔt≤(ml+m)m/(l+m)(∫ab‍H(l+m-1)(t,s)Δt)l/(l+m)∫ab‍|yΔn(t)|l+mΔt,
where
(2.50)H(t,s):=∫at‍(hn-k-1(t,σ(s)))(l+m)/(l+m-1)Δs.Note that Theorem2.5 cannot be applied when l+m=1. In the following theorem we prove a new inequality which can be applied in this case.Theorem 2.7.
Let𝕋 be a time scale with a,b∈𝕋 and let l, m be positive real numbers such that l+m=1 and y∈Crd(n)([a,b]∩𝕋). If yΔi(a)=0, for i=0,1,…,n-1, then
(2.51)∫ab‍|y(t)|l|yΔn(t)|mΔt≤mm(∫ab‍hn-1(t,a)Δt)l∫ab‍|yΔn(t)|Δt.Proof.
Using the fact that|hn(t,s)| is increasing with respect to its first component for t≥σ(s)>a, we have from the Taylor formula (2.22) and yΔi(a)=0, for i=0,1,…,n-1, that
(2.52)|y(t)|≤hn-1(t,a)∫at‍|yΔn(s)|Δs.
This implies that
(2.53)|y(t)|l|yΔn(t)|m≤(hn-1(t,a))l|yΔn(t)|m(∫at‍|yΔn(s)|Δs)l.
Now applying the Hölder inequality (1.6) with γ=1/l and ν=1/m, we obtain
(2.54)∫ab‍|y(t)|l|yΔn(t)|mΔt≤(∫ab‍hn-1(t,a)Δt)l(∫ab‍|yΔn(t)|(∫at‍|yΔn(s)|Δs)l/mΔt)m.
Let z(t)=∫at‍|yΔn(s)|Δs. Then z(a)=0 and zΔ(t)=|yΔn(t)| so that
(2.55)∫ab‍|yΔn(t)|(∫at‍|yΔn(s)|Δs)l/mΔt=∫ab‍zΔ(t)(z(t))l/mΔt,
and hence
(2.56)∫ab‍|y(t)|l|yΔn(t)|mΔt≤(∫ab‍hn-1(t,a)Δt)l(∫ab‍zΔ(t)(z(t))l/mΔt)m.
As in the proof of Theorem 2.5, we have that
(2.57)∫ab‍zΔ(t)(z(t))l/mΔt≤∫ab‍zl/m(t)zΔ(t)Δt≤m∫ab‍(z(l+m)/m(t))ΔΔt=mz1/m(b)=m(∫ab‍|yΔn(t)|Δt)1/m.
Substituting into (2.56), we have
(2.58)∫ab‍|y(t)|l|yΔn(t)|mΔt≤mm(∫ab‍hn-1(t,a)Δt)l(∫ab‍|yΔn(t)|Δt),
which is the desired inequality (2.51). The proof is complete.Following Remark2.2, we can obtain the following result.Corollary 2.8.
Let𝕋 be a time scale with a,b∈𝕋 and let l, m be positive real numbers such that l+m=1, and y∈Crd(n)([a,b]∩𝕋). If y(Δi)(a)=0, k≤i≤n-1, then
(2.59)∫ab‍|yΔk(t)|l|yΔn(t)|mΔt≤mm(∫ab‍|hn-k-1(t,a)|Δt)l(∫ab‍|yΔn(t)|Δt).In the following, we will prove some inequalities with two different weighted functions.Theorem 2.9.
Let𝕋 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 y∈Crd(n)([a,b]∩𝕋). If yΔi(a)=0, for i=0,1,…,n-1, then
(2.60)∫ab‍q(t)|y(t)|l|yΔn(t)|mΔt≤Λ1(l,m,r,p,q)(∫ab‍p(s)|yΔn(s)|rΔs)(l+m)/r,
where
(2.61)Λ1(l,m,p,q,r):=(ml+m)m/r(∫ab‍qr/(r-m)(t)p-m/(r-m)(t)(P(t))l((r-1)/(r-m))Δt)(r-m)/r,P(t):=∫at‍p-1/(r-1)(s)(hn-1(t,σ(s)))r/(r-1)Δs.Proof.
From the Taylor formula (2.22), we see that
(2.62)|y(t)|≤∫at‍p-1/r(s)(hn-1(t,σ(s)))pl/r(s)|yΔn(s)|Δs.
Applying the Hölder inequality on the right hand side with r and r/r-1, we have
(2.63)|y(t)|≤∫at‍p-1/r(s)hn-1(t,σ(s))pl/r(s)|yΔn(s)|Δs≤(∫at‍p-1/(r-1)(s)(hn-1(t,σ(s)))r/(r-1)Δs)(r-1)/r(∫at‍p(s)|yΔn(s)|rΔs)1/r.
This implies that
(2.64)q(t)|y(t)|l|yΔn(t)|m≤q(t)Pl((r-1)/r)(t)|yΔn(t)|m(∫at‍p(s)|yΔn(s)|rΔs)l/r.
Integrating from a to b, we have
(2.65)∫ab‍q(t)|y(t)|l|yΔn(t)|mΔt≤∫ab‍q(t)Pl((r-1)/r)(t)|yΔn(t)|m(∫at‍p(s)|yΔn(s)|rΔs)l/rΔt.
Let
(2.66)z(t):=∫at‍p(s)|yΔn(s)|rΔs.
Then z(a)=0, zΔ(t)=p(t)|yΔn(t)|r and |yΔn(t)|m=(zΔ(t))m/rp-m/r(t). This implies that
(2.67)∫ab‍q(t)|y(t)|l|yΔn(t)|mΔt≤∫ab‍q(t)Pl((r-1)/r)(t)p-m/r(t)(zΔ(t))m/r(z(t))l/rΔt.
Applying the Hölder inequality (1.6) with indices r/m and r/r-m, we obtain
(2.68)∫ab‍q(t)Pl((r-1)/r)(t)p-m/r(t)(zΔ(t))m/r(z(t))l/rΔt≤(∫ab‍(zΔ(t))(z(t))l/mΔt)m/r(∫ab‍qr/(r-m)(t)Pl((r-1)/(r-m))(t)p-m/(r-m)(t)Δt)(r-m)/r.
Substituting into (2.67), we have
(2.69)∫ab‍q(t)|y(t)|l|yΔn(t)|mΔt≤(∫ab‍qr/(r-m)(t)Pl((r-1)/(r-m))(t)p-m/(r-m)(t)Δt)(r-m)/r(∫ab‍(zΔ(t))(z(t))l/mΔt)m/r.
As in the proof of Theorem 2.5, we have
(2.70)(∫ab‍zΔ(t)(z(t))l/mΔt)m/r≤(ml+m∫ab‍(z(l+m)/m(t))ΔΔt)m/r=(ml+m)m/r(z(b))(l+m)/r=(ml+m)m/r(∫ab‍p(s)|yΔn(s)|rΔs)(l+m)/r.
This implies that
(2.71)∫ab‍q(t)|y(t)|l|yΔn(t)|mΔt≤Λ1(l,m,p,q,r)(∫ab‍p(s)|yΔn(s)|rΔs)(l+m)/r,
which is the desired inequality (2.60) where Λ1(l,m,p,q,r) is defined as in (2.61). The proof is complete.Following Remark2.2, we can obtain the following result.Theorem 2.10.
Let𝕋 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 y∈Crd(n)([a,b]∩𝕋). If y(Δi)(a)=0, k≤i≤n-1, then
(2.72)∫ab‍q(t)|yΔk(t)|l|yΔn(t)|mΔt≤Λ2(l,m,r,p,q)(∫ab‍p(s)|yΔn(s)|rΔs)(l+m)/r,
where
(2.73)Λ2(l,m,p,q,r):=(ml+m)m/r(∫ab‍qr/(r-m)(t)p-m/(r-m)(t)(P(t))l((r-1)/(r-m))Δt)(r-m)/r,P(t):=∫at‍p-1/(r-1)(s)hn-k-1r/(r-1)(t,σ(s))Δs.Note that Theorem2.10 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.
Let𝕋 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∈Crd(n)([a,b]∩𝕋). If yΔi(a)=0, for i=0,1,…,n-1, then
(2.74)∫ab‍q(t)|y(t)|α|yΔn(t)|βΔt≤Λ3(a,b,α,β)∫ab‍p(t)|yΔn(t)|α+βΔt,
where
(2.75)Λ3(a,b,α,β)=(βα+β)β/(α+β)(∫ab‍q(α+β)/α(t)pβ/α(t)(∫at‍(hn-1(t,σ(s)))(α+β)/(α+β-1)p1/(α+β-1)(s)Δs)(α+β-1)Δt)α/(α+β).Proof.
From the Taylor formula, we see that(2.76)|y(t)|≤∫at‍hn-1(t,σ(s))|yΔn(s)|Δs=∫at‍hn-1(t,σ(s))(p(s))1/(α+β)(p(s))1/(α+β)|yΔn(s)|Δs.
Now, since p is nonnegative on (a,b)𝕋, it follows from the Hölder inequality (1.6) with
(2.77)f(s)=hn-1(t,σ(s))(p(s))1/(α+β),g(s)=(p(s))1/(α+β)|yΔn(s)|,γ=α+βα+β-1,ν=α+β
that
(2.78)|y(t)|≤(∫at‍(hn-1(t,σ(s)))(α+β)/(α+β-1)(p(s))1/(α+β-1)Δs)(α+β-1)/(α+β)(∫at‍p(s)|yΔn(s)|α+βΔs)1/(α+β).
Then, we get that
(2.79)|y(t)|α≤(∫at‍(hn-1(t,σ(s)))(α+β)/(α+β-1)(p(s))1/(α+β-1)Δs)α((α+β-1)/(α+β))(∫at‍p(s)|yΔn(s)|α+βΔs)α/(α+β).
Setting z(t):=∫at‍p(s)|yΔn(s)|α+βΔs, we see that z(a)=0, and
(2.80)zΔ(t)=p(t)|yΔn(t)|α+β>0.
This gives us
(2.81)|yΔn(t)|β=(zΔ(t)p(t))β/(α+β).
Since q is nonnegative on (a,b)𝕋, we have from (2.79) and (2.81) that
(2.82)q(t)|y(t)|α|yΔn(t)|β≤q(t)(1p(t))β/(α+β)(∫at‍(hn-1(t,σ(s)))(α+β)/(α+β-1)p1/(α+β-1)(s)Δs)α((α+β-1)/(α+β))(z(t))α/(α+β)(zΔ(t))β/(α+β).
This implies that
(2.83)∫ab‍q(t)|y(t)|α|yΔn(t)|βΔt≤∫ab‍q(t)(1p(t))β/(α+β)×(∫at‍(hn-1(t,σ(s)))(α+β)/(α+β-1)p1/(α+β-1)(s)Δs)α((α+β-1)/(α+β))(z(t))α/(α+β)(zΔ(t))β/(α+β)Δt.
Applying the Hölder inequality (1.6) with indices (α+β)/α and (α+β)/β, we have
(2.84)∫ab‍q(t)|y(t)|α|yΔn(t)|βΔt≤(∫ab‍s(α+β)/α(t)pβ/α(t)(∫at‍(hn-1(t,σ(s)))(α+β)/(α+β-1)p1/(α+β-1)(s)Δs)(α+β-1)Δt)α/(α+β)×(∫ab‍zα/β(t)zΔ(t)Δt)β/(α+β).
From (2.80), the chain rule (1.7) and the fact that zΔ(s)>0, we obtain
(2.85)zα/β(t)zΔ(t)≤βα+β(z(α+β)/β(t))Δ.
Substituting (2.85) into (2.84) and using the fact that z(a)=0, we have
(2.86)∫ab‍q(t)|y(t)|α|yΔn(t)|βΔt≤(∫ab‍q(α+β)/α(t)pβ/α(t)(∫at‍(hn-1(t,σ(s)))(α+β)/(α+β-1)p1/(α+β-1)(s)Δs)(α+β-1)dt)α/(α+β)×(αα+β)β/(α+β)(∫ab‍(z(α+β)/β(s))ΔΔs)β/(α+β)=(∫ab‍q(α+β)/α(t)pβ/α(t)(∫at‍(hn-1(t,σ(s)))(α+β)/(α+β-1)p1/(α+β-1)(s)Δs)(α+β-1)Δt)α/(α+β)×(βα+β)β/(α+β)z(b).
Using z(b):=∫ab‍p(s)|yΔn(s)|α+βΔs, we have from the last inequality that
(2.87)∫ab‍q(t)|y(t)|α|yΔn(t)|βΔt≤Λ3(a,b,α,β)∫ab‍p(t)|yΔn(t)|α+βΔt,
which is the desired inequality (2.74). The proof is complete.Following Remark2.2, we can obtain the following result.Theorem 2.12.
Let𝕋 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∈Crd(n)([a,b]∩𝕋). If y(Δi)(a)=0, k≤i≤n-1, then
(2.88)∫ab‍q(t)|yΔk(t)|α|yΔn(t)|βΔt≤Λ4(a,b,α,β)∫ab‍p(t)|yΔn(t)|α+βΔt,
where
(2.89)Λ4(a,b,α,β)=(βα+β)β/(α+β)(∫ab‍q(α+β)/α(t)pβ/α(t)(∫at‍(hn-k-1(t,σ(s)))(α+β)/(α+β-1)p1/(α+β-1)(s)Δs)(α+β-1)Δt)α/(α+β).Instead of (2.25), we can use the relation between gn and hn and define
(2.90)y(t):=(-1)n∫tb‍gn-1(σ(s),t)yΔn(s)Δs.
Proceeding as pervious by using the same arguments and using (2.90) one can obtain some results when yΔi(b)=0, fori=0,1,…,n-1. For example one can get the following results.Theorem 2.13.
Letting𝕋 be a time scale with a,b∈𝕋 and y∈Crd(n)([X,b]∩𝕋). If yΔi(b)=0, for i=0,1,…,n-1, then
(2.91)∫ab‍|y(t)||yΔn(t)|Δt≤12(∫ab‍(∫tb‍gn-12(σ(s),t)Δs)Δt)1/2∫ab‍|yΔn(t)|2Δt.Theorem 2.14.
Let𝕋 be a time scale with a,b∈𝕋 and let l, m be positive real numbers such that l+m>1. Let y∈Crd(n)([a,b]∩𝕋). If yΔi(b)=0, for i=0,1,…,n-1, then
(2.92)∫ab‍|y(t)|l|yΔn(t)|mΔt≤(∫ab‍(∫tb‍gn-1(l+m)/(l+m-1)(σ(s),t)Δs)(l+m-1)Δt)l/(l+m)∫ab‍|yΔn(t)|l+mΔt.Remark 2.15.
Similar results as in Theorems2.13 and 2.14 can be obtained from the results in the rest of the paper, but in this case one will use yΔi(b)=0, fori=0,1,…,n-1, instead of IfyΔi(a)=0, fori=0,1,…,n-1, and gn(s,t) instead of hn(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 functionhk(t,s).For example if𝕋=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.Theorem 2.17.
Leta,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
(2.93)∫ab‍|y(k)(t)|l|y(n)(t)|mdt≤mm(b-a)l(n-k)((n-k)!)l(∫ab‍|y(n)(t)|dt).Theorem 2.18.
Leta,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
(2.94)∫ab‍q(t)|y(t)|α|yΔn(t)|βdt≤Λ5(a,b,α,β)∫ab‍p(t)|yΔn(t)|α+βdt,
where
(2.95)Λ5(a,b,α,β)=(βα+β)β/(α+β)(∫ab‍q(α+β)/α(t)pβ/α(t)(∫at‍(t-s)(n-1)(α+β)/(α+β-1)(n-1)!p1/(α+β-1)(s)ds)(α+β-1)dt)α/(α+β).Theorem 2.19.
Leta,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, k≤i≤n-1, then
(2.96)∫ab‍q(t)|y(k)(t)|α|y(n)(t)|βdt≤Λ6(a,b,α,β)∫ab‍p(t)|y(n)(t)|α+βdt,
where
(2.97)Λ6(a,b,α,β)=(β(α+β))β/(α+β)(1(n-k-1)!)β/(α+β)×(∫ab‍q(α+β)/α(t)pβ/α(t)(∫at‍(t-s)(n-k-1)(α+β)/(α+β-1)p1/(α+β-1)(s)ds)(α+β-1)dt)α/(α+β).When𝕋=ℤ, we have yΔ(t)=Δy(t)=y(t+1)-y(t) and Δ(i)=Δ(Δ(i-1)). Using the fact that hk(t,s)≤(t-s)k/k!, we get from Corollary 2.8 and Theorems 2.11 and 2.12 the following discrete inequalities.Theorem 2.20.
Leta,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
(2.98)∑t=ab-1‍|Δ(k)y(t)|l|Δ(n)y(t)|m≤mm(b-a)l(n-k)((n-k)!)l∑t=ab-1‍|Δ(n)y(t)|.Theorem 2.21.
Leta,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
(2.99)∑t=ab-1‍q(t)|y(t)|α|Δ(n)y(t)|β≤Λ7(a,b,α,β)∑t=ab-1‍p(t)|Δ(n)y(t)|α+β,
where
(2.100)Λ7(a,b,α,β)=(βα+β)β/(α+β)(∑t=ab-1‍q(α+β)/α(t)pβ/α(t)(∑s=at-1‍(t-s)(n-1)(α+β)/(α+β-1)(n-1)!p1/(α+β-1)(s))(α+β-1))α/(α+β).Theorem 2.22.
Leta,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
(2.101)∑t=ab-1‍q(t)|Δ(k)y(t)|α|Δ(n)y(t)|βdt≤Λ8(a,b,α,β)∑t=ab-1‍p(t)|Δ(n)y(t)|α+β,
where
(2.102)Λ8(a,b,α,β)=(β(α+β))β/(α+β)(1(n-k-1)!)β/(α+β)×(∑t=ab-1‍q(α+β)/α(t)pβ/α(t)(∑s=at-1‍(t-s)(n-k-1)(α+β)/(α+β-1)p1/(α+β-1)(s))(α+β-1))α/(α+β).Similar results when𝕋=hℤ and 𝕋=qℕ0={qt:t∈ℕ0} 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(2.103)∫aX‍q(t)|yΔk(t)|α|yΔk+1(t)|βΔt≤Λ(a,b,α,β)∫at‍p(t)|yΔn(t)|α+pΔt,
where Λ is the constant of the inequality that needs to be determined.



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