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New Principles of Non-Linear Integral Inequalities on Time Scales

Published Online: 18 Jan 2021
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Received: 10 Apr 2020
Accepted: 18 Oct 2020
Journal Details
License
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Abstract

The concept of inequalities in time scales has attracted the attention of mathematicians for a quarter century. And these studies have inspired the solution of many problems in the branches of physics, biology, mechanics and economics etc. In this article, new principles of non-linear integral inequalities are presented in time scales via diamond-α dynamic integral and the nabla integral.

Keywords

MSC 2010

Introduction

For a quarter century, the theory of time scales has played an important role in the representation of differential calculus and integral inequalities. The concept of time scales was introduced by Stefan Hilger in 1988 [1]. Later, this theory was studied by many authors. They have demonstrated various aspects of integral inequalities [2,3,4,5,WongF.-H.YehC.-C.YuS.-L.HongC.-H.Young's inequality and related results on time scalesAppl. Math. Lett.200518983988' href="#j_amns.2021.1.00001_ref_006_w2aab3b7b1b1b6b1ab2ab6Aa">6,WongF.-H.YehC.-C.LianW.-C.An extension of Jensen's inequality on time scalesAdv. Dynam. Syst. Appl.200611113120' href="#j_amns.2021.1.00001_ref_007_w2aab3b7b1b1b6b1ab2ab7Aa">7,8,9,10,11,12,YangW.-G.A functional generalization of diamond-a integral Hölder's inequality on time scalesAppl. Math. Lett.20102312081212' href="#j_amns.2021.1.00001_ref_013_w2aab3b7b1b1b6b1ab2ac13Aa">13]. Dynamic equations and inequalities have many applications to quantum mechanics, phsical problems, wave equations, heat transfer and economic problems [SpeddingV.Taming nature's numbersNew ScientistJuly1920032831' href="#j_amns.2021.1.00001_ref_026_w2aab3b7b1b1b6b1ab2ac26Aa">26, 27, 28, 29]. For example; Aly R. Seadawy et al. have done a lot of research on the applications of dynamic equations in physics. As a result of these studies, they achieved good results [30]. The most important examples of time scale studies are differential calculus and inequalities [12]. Wong et al. [WongF.-H.YehC.-C.YuS.-L.HongC.-H.Young's inequality and related results on time scalesAppl. Math. Lett.200518983988' href="#j_amns.2021.1.00001_ref_006_w2aab3b7b1b1b6b1ab2ab6Aa">6, WongF.-H.YehC.-C.LianW.-C.An extension of Jensen's inequality on time scalesAdv. Dynam. Syst. Appl.200611113120' href="#j_amns.2021.1.00001_ref_007_w2aab3b7b1b1b6b1ab2ab7Aa">7] expressed some time scale integral inequalities. Yang [YangW.-G.A functional generalization of diamond-a integral Hölder's inequality on time scalesAppl. Math. Lett.20102312081212' href="#j_amns.2021.1.00001_ref_013_w2aab3b7b1b1b6b1ab2ac13Aa">13] obtained a generalization of the ⋄α-integral Hölder's inequality in time scales. Recently, Li Yin and Feng Qi [24] have introduced some non-linear integral inequalities under certain conditions.

Our aim of this article is to demonstrate new principles of non-linear integral inequalities in time scales via the ∇-integral and the ⋄α-integral.

Auxiliary Statements and Definitions

Now, let us briefly give information about time scales and give the necessary definitions and notations for our article. For more details, we refer the reader to the articles [1,2,3,4,5,WongF.-H.YehC.-C.YuS.-L.HongC.-H.Young's inequality and related results on time scalesAppl. Math. Lett.200518983988' href="#j_amns.2021.1.00001_ref_006_w2aab3b7b1b1b6b1ab2ab6Aa">6,WongF.-H.YehC.-C.LianW.-C.An extension of Jensen's inequality on time scalesAdv. Dynam. Syst. Appl.200611113120' href="#j_amns.2021.1.00001_ref_007_w2aab3b7b1b1b6b1ab2ab7Aa">7,8,9,10,11,12,YangW.-G.A functional generalization of diamond-a integral Hölder's inequality on time scalesAppl. Math. Lett.20102312081212' href="#j_amns.2021.1.00001_ref_013_w2aab3b7b1b1b6b1ab2ac13Aa">13,14,15,16,17,18,19,20,21,22,10.1186/1687-1847-2014-324.ChenG.WeiC.A functional generalization of diamond-a integral Dresher's inequality on time scalesAdv.Differ. Equ.201432410.1186/1687-1847-2014-324' href="#j_amns.2021.1.00001_ref_023_w2aab3b7b1b1b6b1ab2ac23Aa">23,24,25].

Let ω be a weight function on R, i.e., ω is a non-negative, almost everywhere positive on R and ∫R ω(y)∇y < ∞.

Let σ(t) be the forward jump operator and let ρ(t) be the backward jump operator in T (T is time scale) for tT. Respectively, they are defined by σ(t)=inf{sT:s>t}\sigma (t) = \inf \{ s \in T:s > t\} and ρ(t)=sup{sT:s>t}.\rho (t) = \sup \{ s \in T:s > t\} .

If σ : TT, σ(t) > t, then t is right-scattered. If ρ : TT, ρ(t) < t, then t is left-scattered. And, if σ : TT,σ(t) = t, then t is called right-dense, and if ρ : TT, ρ(t) = t, then t is called left-dense. Let two mappings μ,υ : TR+ such that μ(t) = σ(t) − t,υ (t) = tρ(t) are called graininess mappings. If T has a left-scattered maximum uR, then Tk = T /u. If not Tk = T. Briefly

If supT < ∞, then Tk = [ρ supT,supT ] and if supT = ∞, then Tk = T. By the same way

If |infT | < ∞, then Tk = [infT,σ infT ] and if infT = −∞, then Tk = T. Let f : TR and f σ : TR by f σ (t) = f (σ(t)) for ∀tT, i.e., f σ = fσ. And let f : TR and f ρ : TR by f ρ(t) = f (ρ(t)) for ∀tT, i.e., f ρ = fρ.

Assume that h : TR,tTk(tminT).

Let h is Δ-differentiable at point t and h is continuous at point t.

Let h is left continuous at point t. t is right-scattered and h is Δ-differentiable at point t, hΔ(t)=(hσ(t)h(t))/μ(t){h^\Delta }(t) = ({h^\sigma }(t) - h(t))/\mu (t)

Let t is right-dense, h is Δ-differentiable at point t and limsth(t)h(s)ts\mathop {\lim }\limits_{s \to t} {{h(t) - h(s)} \over {t - s}} then hΔ(t)=limsth(t)h(s)ts{h^\Delta }(t) = \mathop {\lim }\limits_{s \to t} {{h(t) - h(s)} \over {t - s}}

Let h is Δ-differentiable at point t, then hσ(t)=h(t)+μ(t)hΔ(t).{h^\sigma }(t) = h(t) + \mu (t){h^\Delta }(t).

Definition 2.1

[12] H : TR is called a Δ-antiderivative of h : TR. HΔ = h(t) holds for ∀s,tT. We define the Δ-integral of h by sth(τ)Δτ=H(t)H(s)\int_s^t h(\tau )\Delta \tau = H(t) - H(s) for s,tT.

Definition 2.2

[14] Let h : TkR is called a ∇-differentiable at tTk,h(t), if ɛ > 0 then there exists a neighborhood V of t such that |h(ρ(t))h(s)h(t)(ρ(t)s)|ε|ρ(t)s||h(\rho (t)) - h(s) - {h^\nabla }(t)(\rho (t) - s)| \le \varepsilon |\rho (t) - s| for ∀sV.

Assume that h : TR,tTk (tmaxT).

Let h is ∇-differentiable at point t and h is continuous at point t.

Let h is right continuous at point t. t is leftt-scattered and h is ∇-differentiable at point t, h(t)=h(t)hρ(t)ν(t){h^\nabla }(t) = {{h(t) - {h^\rho }(t)} \over {\nu (t)}}

Let t is left-dense, h is ∇-differentiable at point t and limsth(t)h(s)ts\mathop {\lim }\limits_{s \to t} {{h(t) - h(s)} \over {t - s}} then h(t)=limsth(t)h(s)ts{h^\nabla }(t) = \mathop {\lim }\limits_{s \to t} {{h(t) - h(s)} \over {t - s}}

Let h is ∇-differentiable at point t, then hρ(t)=h(t)ϑ(t)h(t).{h^\rho }(t) = h(t) - \vartheta (t){h^\nabla }(t).

Definition 2.3

[14] H : TR is called a ∇-antiderivative of h : TR. H = h(t) holds for ∀s,tT. Then, we define the ∇-integral of h by sth(τ)τ=H(t)H(s)\int_s^t h(\tau )\nabla \tau = H(t) - H(s) for s,tT.

Let h(t) be differentiable on T. And let b,tT. Then, ha(t)=bhΔ(t)+(1b)h(t),0b1.{h^{{\diamondsuit _a}}}(t) = b{h^\Delta }(t) + (1 - b){h^\nabla }(t),\quad \quad 0 \le b \le 1.

Proposition 2.4

[15] If we get f, g : TR, ⋄a-differentiable at tT, then

(f + g)a (t) = fa (t) + ga (t)

If cR, then (c f )a (t) = c fa (t).

(f g)a (t) = fa (t)g(t) + b f σ (t)gΔ(t) + (1 − b) f ρ(t)g(t).

Definition 2.5

[15] If we get b,tT, f : TR, then btf(γ)aγ=bbtf(γ)Δγ+(1b)btf(γ)γ,0b1.\int_b^t f(\gamma ){\diamondsuit _a}\gamma = b\int_b^t f(\gamma )\Delta \gamma + (1 - b)\int_b^t f(\gamma )\nabla \gamma ,\quad \quad 0 \le b \le 1.

Proposition 2.6

[15] Let u,v,tT, cR and if f (γ),g(γ) are ⋄a-integrable functions on [u,v]T, then the following statements are valid.

ut[f(γ)+g(γ)]aγ=utf(γ)aγ+utg(γ)aγ,\int_u^t [f(\gamma ) + g(\gamma )]{\diamondsuit _a}\gamma = \int_u^t f(\gamma ){\diamondsuit _a}\gamma + \int_u^t g(\gamma ){\diamondsuit _a}\gamma ,

utcf(γ)aγ=cutf(γ)aγ,\int_u^t cf(\gamma ){\diamondsuit _a}\gamma = c\int_u^t f(\gamma ){\diamondsuit _a}\gamma ,

utf(γ)aγ=tuf(γ)aγ,\int_u^t f(\gamma ){\diamondsuit _a}\gamma = - \int_t^u f(\gamma ){\diamondsuit _a}\gamma ,

utf(γ)aγ=uvf(γ)aγ+vtf(γ)aγ,\int_u^t f(\gamma ){\diamondsuit _a}\gamma = \int_u^v f(\gamma ){\diamondsuit _a}\gamma + \int_v^t f(\gamma ){\diamondsuit _a}\gamma ,

uuf(γ)aγ=0.\int_u^u f(\gamma ){\diamondsuit _a}\gamma = 0.

Lemma 2.7

[15] Let u,v,tT with u < v. Suppose that h(γ) and g(γ) are ⋄a-integrable functions on [u,v]T, then the following statements are valid.

If h(γ) ≥ 0 for ∀γ ∈ [u,v]T, then uvh(γ)aγ0\int_u^v h(\gamma ){\diamondsuit _a}\gamma \ge 0 .

If h(γ) ≤ g(γ) for ∀γ ∈ [u,v]T, then uvh(γ)aγuvg(γ)aγ\int_u^v h(\gamma ){\diamondsuit _a}\gamma \le \int_u^v g(\gamma ){\diamondsuit _a}\gamma .

If h(γ) ≥ 0 for ∀γ ∈ [u,v]T, then h(γ) = 0 iff uvh(γ)aγ=0\int_u^v h(\gamma ){\diamondsuit _a}\gamma = 0 .

Lemma 2.8

(For details, Lemma 2.5 in [24]) Let p > 1 or q < 0, while 1/p + 1/q = 1, if g,hCrd(T,R), g(y) > 0,h(y) > 0, while u,vT, then uv[g(y)]p[h(y)]p/qΔy(uvg(y)Δy)p(uvh(y)Δy)p/q.\int_u^v {{{{[g(y)]}^p}} \over {{{[h(y)]}^{p/q}}}}\Delta y \ge {{{{(\int_u^v g(y)\Delta y)}^p}} \over {{{(\int_u^v h(y)\Delta y)}^{p/q}}}}.

Qi F. et al. [25] proved some inequalities under the condition of Δ-differentiable. In the next section, we will prove these inequalities under the conditions of the ∇-differentiable and the ⋄a-differentiable.

Main Result

In this section, we will prove non-linear ∇-differentiable weighted integral inequalities under certain conditions. Later, we will prove their ⋄a-differentiable extensions. We have listed these studies in the references of the article for the relevant readers.

Theorem 3.1

Let h,wCrd(T,R) and let w be a weight function and let w(y),h(y) > 0, uvh(y)w(y)y<\int_u^v h(y)w(y)\nabla y < \infty and p > 1 or q < 0, while 1/p + 1/q = 1. Ifuvw(y)h(y)(vu)p1\int_u^v w(y)h(y) \ge {(v - u)^{p - 1}} , while u,vT, thenuv[h(y)w(y)]py(uvh(y)w(y)y)p1.\int_u^v {[h(y)w(y)]^p}\nabla y \ge {\left( {\int_u^v h(y)w(y)\nabla y} \right)^{p - 1}}.

Proof

Using Lemma 2.8, we obtain uv[h(y)w(y)]py=uv[h(y)w(y)]p1p1y[uvh(y)w(y)y]p[uv1y]p1(uvh(y)w(y)y)p1.\int_u^v {[h(y)w(y)]^p}\nabla y = \int_u^v {{{{[h(y)w(y)]}^p}} \over {{1^{p - 1}}}}\nabla y \ge {{{{[\int_u^v h(y)w(y)\nabla y]}^p}} \over {{{[\int_u^v 1\nabla y]}^{p - 1}}}} \ge {\left( {\int_u^v h(y)w(y)\nabla y} \right)^{p - 1}}.

Theorem 3.2

Let w be a weight function, w(y),g(y) > 0, uvg(y)w(y)y<\int_u^v g(y)w(y)\nabla y < \infty for y ∈ (u,v]and g,wC([u,v],R),∇-differantiable in (u,v),Let ɛ,ϕ be positive real numbers such that 1 < ϕ < ɛ. If[(wg)(εφ)/(φ1)(y)](εφ)φ1/(φ1)ε1{\left[ {{{(wg)}^{(\varepsilon - \varphi )/(\varphi - 1)}}(y)} \right]}^\prime \ge {{(\varepsilon - \varphi ){\varphi ^{1/(\varphi - 1)}}} \over {\varepsilon - 1}}fory ∈ (u,v), thenuv[g(y)w(y)]εy[uvg(y)w(y)y]φ.\int_u^v {[g(y)w(y)]^\varepsilon }\nabla y \ge {\left[ {\int_u^v g(y)w(y)\nabla y} \right]^\varphi }.

Proof

If we use Cauchy's Mean Value Theorem consecutively for δ ∈ (u,v) and θ ∈ (u,δ), then we obtain [uvg(y)w(y)y]φuv[g(y)]εy=φ[uδg(y)w(y)y]φ1w(y)g(δ)[w(y)g(δ)]ε={φ1/(φ1)uδw(y)g(y)y[w(y)g(δ)](ε1)/(φ1)}φ1={φ1/(φ1)w(θ)g(θ)ε1φ1[w(θ)g(θ)]εφφ1(wg)(θ)}φ1={(εφ)φ1/(φ1)/(φ1)[(wg)(εφ)/(φ1)(θ)]}φ1.\matrix{ {{{{{\left[ {\int_u^v g(y)w(y)\nabla y} \right]}^\varphi }} \over {\int_u^v {{[g(y)]}^\varepsilon }\nabla y}} = {{\varphi {{\left[ {\int_u^\delta g(y)w(y)\nabla y} \right]}^{\varphi - 1}}w(y)g(\delta )} \over {{{[w(y)g(\delta )]}^\varepsilon }}} = {{\left\{ {{{{\varphi ^{1/(\varphi - 1)}}\int_u^\delta w(y)g(y)\nabla y} \over {{{[w(y)g(\delta )]}^{(\varepsilon - 1)/(\varphi - 1)}}}}} \right\}}^{\varphi - 1}}} \cr { = {{\left\{ {{{{\varphi ^{1/(\varphi - 1)}}w(\theta )g(\theta )} \over {{{\varepsilon - 1} \over {\varphi - 1}}{{[w(\theta )g(\theta )]}^{{{\varepsilon - \varphi } \over {\varphi - 1}}}}(wg)'(\theta )}}} \right\}}^{\varphi - 1}} = {{\left\{ {{{(\varepsilon - \varphi ){\varphi ^{1/(\varphi - 1)}}/(\varphi - 1)} \over {[(wg{)^{(\varepsilon - \varphi )/(\varphi - 1)}}(\theta )]}}} \right\}}^{\varphi - 1}}.} \cr } thus, (3) inequality holds.

Theorem 3.3

Let w be a weight function, w(y),g(y) > 0, uvg(y)w(y)y<\int_u^v g(y)w(y)\nabla y < \infty for y ∈ (u,v],g,wC([u,v],R), ɛR. If ϕ = 1 and [w(y)g(y)]1−ϕ ≤ 1 fory ∈ (u,v), then (3) holds.

Proof

For ϕ = 1, inequality (3) reduced to uv[w(y)g(y)]εyuvw(y)g(y)y.\int_u^v {[w(y)g(y)]^\varepsilon }\nabla y \ge \int_u^v w(y)g(y)\nabla y.

If we use Cauchy's Mean Value Theorem, we obtain the following equation uv[w(y)g(y)]εyuvw(y)g(y)y=[w(δ)g(δ)]εw(δ)g(δ)=[w(δ)g(δ)]ε1.{{\int_u^v {{[w(y)g(y)]}^\varepsilon }\nabla y} \over {\int_u^v w(y)g(y)\nabla y}} = {{{{[w(\delta )g(\delta )]}^\varepsilon }} \over {w(\delta )g(\delta )}} = [w(\delta )g(\delta {)]^{\varepsilon - 1}}.

Theorem 3.4

Let w be a weight function,uvg(y)w(y)y<\int_u^v g(y)w(y)\nabla y < \infty for y ∈ (u,v], mN and 1 ≤ ϕm + 1, there exist (wg)m(y) derivative of the m-th order on [u,v] and (wg)m(y) is increasing, then g(m)(y) ≥ 0,g(i)(u) = 0 for 0 ≤ jm − 1. Ifw(y)g(y)[(yε)φ1φφ2]1/(εφ)w(y)g(y) \ge {\left[ {{{{{(y - \varepsilon )}^\varphi } - 1} \over {{\varphi ^{\varphi - 2}}}}} \right]^{1/(\varepsilon - \varphi )}} , then (3) holds.

Proof

If we use Cauchy's Mean Value Theorem together with the condition given in the theorem, we get the following. uv[w(y)g(y)]εy[uvw(y)g(y)y]φ=[w(c1)g(c1)]ε1φ[uc1w(y)g(y)y]φ1u<c1<v[(c1u)w(c1g(c1))]φ1/φφ2φ[uc1w(y)g(y)y]φ1=[(c1ε)w(c1)g(c1)φuc1w(y)g(y)y]φ1.\matrix{ {{{\int_u^v {{[w(y)g(y)]}^\varepsilon }\nabla y} \over {{{[\int_u^v w(y)g(y)\nabla y]}^\varphi }}} = {{{{[w({c_1})g({c_1})]}^{\varepsilon - 1}}} \over {\varphi {{[\int_u^{{c_1}} w(y)g(y)\nabla y]}^{\varphi - 1}}}}\quad \quad \quad \quad u < {c_1} < v} \cr { \ge {{{{[({c_1} - u)w({c_1}g({c_1}))]}^{\varphi - 1}}/{\varphi ^{\varphi - 2}}} \over {\varphi {{[\int_u^{{c_1}} w(y)g(y)\nabla y]}^{\varphi - 1}}}} = {{\left[ {{{({c_1} - \varepsilon )w({c_1})g({c_1})} \over {\varphi \int_u^{{c_1}} w(y)g(y)\nabla y}}} \right]}^{\varphi - 1}}.} \cr }

If we use Cauchy's Mean Value Theorem consecutively in (7), we obtain (c1ε)w(c1)g(c1)φuc1w(y)g(y)y=1+(c2ε)(wg)(c2)w(c2)g(c2)u<c2<c1...=m+(cm+1ε)(wg)(m)(cm+1)(wg)(m1)(cm+1)u<cm+1<cm.\matrix{ {{{({c_1} - \varepsilon )w({c_1})g({c_1})} \over {\varphi \int_u^{{c_1}} w(y)g(y)\nabla y}} = 1 + {{({c_2} - \varepsilon )(wg)'({c_2})} \over {w({c_2})g({c_2})}}\quad \quad \quad \quad u < {c_2} < {c_1}} \cr {...} \cr { = m + {{({c_{m + 1}} - \varepsilon )(wg{)^{(m)}}({c_{m + 1}})} \over {{{(wg)}^{(m - 1)}}({c_{m + 1}})}}\quad \quad \quad \quad \quad \quad u < {c_{m + 1}} < {c_m}.} \cr }

But (wg)(m−1)(k) = (wg)(m−1)(k)−(wg)(m−1)(u) = (ku)(wg)m(k1) for k1 ∈ (u,k). If (wg)m(k1) ≤ gm(k), then (wg)(m)(y) is increasing.

Hence (wg)(m)(k)(ku)(wg)(m1)(k)>0.{(wg)^{(m)}}(k)(k - u) \ge {(wg)^{(m - 1)}}(k) > 0.

Applying (9) to (8) yields (c1ε)g(c1)uc1g(y)ym+1.{{({c_1} - \varepsilon )g({c_1})} \over {\int_u^{{c_1}} g(y)\nabla y}} \ge m + 1.

Hence uv[g(y)]εy[uvg(y)y]φ(m+1φ)φ1{{\int_u^v {{[g(y)]}^\varepsilon }\nabla y} \over {{{[\int_u^v g(y)\nabla y]}^\varphi }}} \ge {\left( {{{m + 1} \over \varphi }} \right)^{\varphi - 1}} for 1 ≤ ɛm + 1.

Theorem 3.5

Suppose that w be a weight functionuvw(y)g(y)y<\int_u^v w(y)g(y)\nabla y < \infty for y ∈ (u,v), mN, 1 < ϕm + 1, there exist (wg)(m)(y) derivative of the m-th order on [u,v] and (wg)(m)(y) is increasing, then (wg)(m)(y) ≥ 0 and g(j)(u) = 0 for m − 1 ≥ j ≥ 0.

Ifw(y)g(y)[φ(yε)(φ1)(φ1)(φ2)]1/(εφ)w(y)g(y) \ge {\left[ {{{\varphi {{(y - \varepsilon )}^{(\varphi - 1)}}} \over {{{(\varphi - 1)}^{(\varphi - 2)}}}}} \right]^{1/(\varepsilon - \varphi )}} .

Proof

If w(y)g(y)[φ(yε)(φ1)(φ1)(φ2)]1/(εφ)w(y)g(y) \ge {\left[ {{{\varphi {{(y - \varepsilon )}^{(\varphi - 1)}}} \over {{{(\varphi - 1)}^{(\varphi - 2)}}}}} \right]^{1/(\varepsilon - \varphi )}} , (6) becomes uv[w(y)g(y)]εy[uvw(y)g(y)y]φ[(c1ε)w(c1)g(c1)(φ1)uc1w(y)g(y)y]φ1.{{\int_u^v {{[w(y)g(y)]}^\varepsilon }\nabla y} \over {{{[\int_u^v w(y)g(y)\nabla y]}^\varphi }}} \ge {\left[ {{{({c_1} - \varepsilon )w({c_1})g({c_1})} \over {(\varphi - 1)\int_u^{{c_1}} w(y)g(y)\nabla y}}} \right]^{\varphi - 1}}.

If all terms of (8) are positive, then (c1ε)w(c1)g(c1)uc1w(y)g(y)ym{{({c_1} - \varepsilon )w({c_1})g({c_1})} \over {\int_u^{{c_1}} w(y)g(y)\nabla y}} \ge m .

Now let's consider the ⋄a-integral in time scales.

Theorem 3.6

Let w be a weight function, h(y),w(y) > 0, uvw(y)h(y)y\int_u^v w(y)h(y)\nabla yfor y ∈ (u,v), p > 1 or q < 0, while 1/p + 1/q = 1 and h,wCrd(T,R). Ifuvw(y)h(y)ay(vu)p1\int_u^v w(y)h(y){\diamondsuit _a}y \ge {(v - u)^{p - 1}}for u,vT, thenuv[w(y)h(y)]pay[uvw(y)h(y)ay]p1.\int_u^v {[w(y)h(y)]^p}{\diamondsuit _a}y \ge {\left[ {\int_u^v w(y)h(y){\diamondsuit _a}y} \right]^{p - 1}}.

Proof

See proof of Theorem 3.1. Moreover, when α = 0, (11) reduce to (1).

Theorem 3.7

Let g,wCrd(T,R),⋄a-differantiabla on (u,v), and let w be a weight function,uvw(y)h(y)ay\int_u^v w(y)h(y){\diamondsuit _a}yfor y ∈ (u,v],w(y)g(y) > 0, and let ɛ,ϕ be positive real numbers such that 1 < ϕ < ɛ. If[(wg)(εφ)/(φ1)(y)](εφ)φ1/(φ1)ε1.{\left[ {{{(wg)}^{(\varepsilon - \varphi )/(\varphi - 1)}}(y)} \right]}^\prime \ge {{(\varepsilon - \varphi ){\varphi ^{1/(\varphi - 1)}}} \over {\varepsilon - 1}}.fory ∈ (u,v), thenuv[w(y)g(y)]εay[uvw(y)g(y)ay]φ.\int_u^v {[w(y)g(y)]^\varepsilon }{\diamondsuit _a}y \ge {\left[ {\int_u^v w(y)g(y){\diamondsuit _a}y} \right]^\varphi }.

Proof

See proof of Theorem 3.2. Moreover, (13) inequality is an extension of (3) inequality. When α = 0, (13) reduce to (3).

Theorem 3.8

Let aR, w be a weight function,uvw(y)g(y)y<\int_u^v w(y)g(y){\diamondsuit _y} < \infty for y ∈ (u,v), g(y),w(y) > 0, g,wC([u,v],R) and [w(y),g(y)], ⋄a-differantiable on (u,v). If ϕ = 1 and [w(y)g(y)]1−ϕ ≤ 1 fory ∈ (u,v), thenuv[w(y)g(y)]εay[uvw(y)g(y)ay]φ.\int_u^v {[w(y)g(y)]^\varepsilon }{\diamondsuit _a}y \ge {\left[ {\int_u^v w(y)g(y){\diamondsuit _a}y} \right]^\varphi }.

Proof

See proof of Theorem 3.3. Moreover, (14) inequality is an extension of (3) inequality. When α = 0, (14) reduce to (3).

Theorem 3.9

Suppose that w be a weight function,uvw(y)g(y)y<\int_u^v w(y)g(y){\diamondsuit _y} < \infty for y ∈ (u,v). mN,1 ≤ ϕm+1, there exist (wg)(m)(y) derivative of the m-th order on [u,v], (wg)(m)(y) is increasing and [w(y),g(y)],⋄a-differantiable in (u,v), then (wg)(m)(y) ≥ 0 and gj(u) = 0 for 0 ≤ jm − 1. Ifw(y)g(y)[(yε)φ1φφ2]1/(εφ),w(y)g(y) \ge {\left[ {{{{{(y - \varepsilon )}^{\varphi - 1}}} \over {{\varphi ^{\varphi - 2}}}}} \right]^{1/(\varepsilon - \varphi )}},thenuv[w(y)g(y)]εay[uvw(y)g(y)ay]φ.\int_u^v {[w(y)g(y)]^\varepsilon }{\diamondsuit _a}y \ge {\left[ {\int_u^v w(y)g(y){\diamondsuit _a}y} \right]^\varphi }.inequality holds.

Proof

See proof of Theorem 3.4. Moreover, (15) inequality is an extension of (3) inequality. When α = 0, (15) reduce to (3).

Theorem 3.10

Suppose that w be a weight function,uvw(y)g(y)y<\int_u^v w(y)g(y){\diamondsuit _y} < \infty for y ∈ (u,v). mN,1 ≤ ϕm + 1, there exist (wg)(m)(y) derivative of the m-th order on [u,v], (wg)(m)(y) is increasing and [w(y),g(y)],⋄a-differantiable in (u,v), then (wg)(m)(y) ≥ 0 and gj(u) = 0 for 0 ≤ jm − 1. Ifw(y)g(y)[φ(yε)φ1(φ1)φ2]1/(εφ),w(y)g(y) \ge {\left[ {{{\varphi {{(y - \varepsilon )}^{\varphi - 1}}} \over {{{(\varphi - 1)}^{\varphi - 2}}}}} \right]^{1/(\varepsilon - \varphi )}},thenuv[w(y)g(y)]εay[uvw(y)g(y)ay]φ.\int_u^v {[w(y)g(y)]^\varepsilon }{\diamondsuit _a}y \ge {\left[ {\int_u^v w(y)g(y){\diamondsuit _a}y} \right]^\varphi }.inequality holds.

Proof

See proof of Theorem 3.5. Moreover, (16) inequality is an extension of (3) inequality. When α = 0, (16) reduce to (3).

Conclusion

Integral inequalities and dynamic equations are the cornerstones of both time scales and harmonic analysis. Mathematicians proved many integral inequalities on time scales [4,5,WongF.-H.YehC.-C.YuS.-L.HongC.-H.Young's inequality and related results on time scalesAppl. Math. Lett.200518983988' href="#j_amns.2021.1.00001_ref_006_w2aab3b7b1b1b6b1ab2ab6Aa">6,WongF.-H.YehC.-C.LianW.-C.An extension of Jensen's inequality on time scalesAdv. Dynam. Syst. Appl.200611113120' href="#j_amns.2021.1.00001_ref_007_w2aab3b7b1b1b6b1ab2ab7Aa">7,8,9]. And they also showed generalized forms of these inequalities [10, 11, YangW.-G.A functional generalization of diamond-a integral Hölder's inequality on time scalesAppl. Math. Lett.20102312081212' href="#j_amns.2021.1.00001_ref_013_w2aab3b7b1b1b6b1ab2ac13Aa">13, 25]. Time scales theory has also been of interest in different sciences. For example, quantum mechanics, wave equations, physical problems, heat transfer, electrical engineering and economics [SpeddingV.Taming nature's numbersNew ScientistJuly1920032831' href="#j_amns.2021.1.00001_ref_026_w2aab3b7b1b1b6b1ab2ac26Aa">26,27,28,29,30]. In this article, we proved non-linear integral inequalities in time scales via the ∇-integral and the ⋄a-integral. We think that the multidimensional and multivariate cases of the inequalities proved in this article are also worth examining.

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