A Variety of Weighted Opial-Type Inequalities with Applications for Dynamic Equations on Time Scales

Abstract

Using higher order delta derivatives on time scales, we demonstrated a few dynamic inequalities of the Opial type in this paper. Our findings expanded upon and generalised earlier findings in the literature. Furthermore, we give the discrete and continuous inequalities as special cases. At the end of this paper, we apply our results to study the behaviour of the solution of an initial value problem. In selecting the best ways to solve dynamic inequalities, symmetry is crucial.

1. Introduction

In 1962, Beesack [1] stated the result.

Theorem 1 (Continuous Opial Inequality).

Letting ϝ be an abs continuous function on $[0,\vartheta ]$ with $ϝ\left(0\right)=0$, obtains

$${\int }_0^{\vartheta }|ϝ\left(x\right){ϝ}^{\prime }\left(x\right)|dx\le \frac{\vartheta }{2}{\int }_0^{\vartheta }|{ϝ}^{\prime }\left(x\right){|}^{2}dx,$$

for $ϝ\left(x\right)=cx$ the equality obtained.

In 1968, Losata [2] investigated the result as follows:

Theorem 2 (Discrete Opial Inequality).

Letting ${\left\{{ϝ}_i\right\}}_{0⩽i⩽\vartheta }\subset \mathbb{R}$ and ${ϝ}_0=0$ gets

$$\sum _{x=0}^{\vartheta -1}|{ϝ}_x({ϝ}_{x+1}-{ϝ}_x)|⩽\frac{\vartheta -1}{2}\sum _{x=0}^{\vartheta -1}|{ϝ}_{x+1}-{ϝ}_x|.$$

Bohner, et al. [3] discussed the dynamic version of Theorems 1 and 2.

Theorem 3 (Dynamic Opial inequality).

Letting $\mathbb{T}$ be a time scale with 0, $\vartheta \in \mathbb{T}$. If $ϝ:{[0,\vartheta ]}_{\mathbb{T}}\to \mathbb{R}$ is a Δ-diff. fun. and $ϝ\left(0\right)=0$, gets

$${\int }_0^{\vartheta }|\left[ϝ\left(x\right)+{ϝ}^{\sigma }\left(x\right)\right]{ϝ}^{\Delta }\left(x\right)|\Delta x\le \vartheta {\int }_0^{\vartheta }|{ϝ}^{\Delta }\left(x\right){|}^2\Delta x.$$

(1)

In 1965, Hua [4] extended Opial’s Theorem 1 and proved that

$${\int }_0^h|{ϝ}^{\ell }\left(x\right)||{ϝ}^{\prime }\left(x\right)|dx⩽\frac{h^{\ell }}{\ell +1}{\int }_0^h{|{ϝ}^{\prime }\left(x\right)|}^{\ell +1}dx,$$

(2)

with $\ell \ge 0$. In 1966, Yang [5] established that if $ϝ\left(\iota \right)=0$, then

$${\int }_{\iota }^{\varrho }q\left(x\right)\left|ϝ\left(x\right)\right||{ϝ}^{\prime }\left(x\right)|dx⩽\frac{1}{2}\left({\int }_{\iota }^{\varrho }\frac{1}{p\left(x\right)}dx\right)\left({\int }_{\iota }^{\varrho }p\left(x\right)q\left(x\right){\left|{ϝ}^{\prime }\left(x\right)\right|}^{2}dx\right).$$

(3)

Further Willet in [6] stated the following result.

$${\int }_0^h\left|ϝ\left(x\right)\right||{ϝ}^{\left(n\right)}\left(x\right)|dx⩽\frac{h^n}{4}{\int }_0^h{\left|{ϝ}^{\left(n\right)}\left(x\right)\right|}^{2}dx.$$

(4)

In [3], the authors studied the following inequalities.

$${\int }_0^{\vartheta }|\left[ϝ\left(x\right)+{ϝ}^{\sigma }\left(x\right)\right]{ϝ}^{\Delta }\left(x\right)|\Delta x\le \alpha {\int }_0^{\vartheta }|{ϝ}^{\Delta }\left(x\right){|}^2\Delta x+2\beta {\int }_0^{\vartheta }|{ϝ}^{\Delta }\left(x\right)|\Delta x.$$

(5)

$${\int }_0^{\vartheta }|\left[ϝ\left(x\right)+{ϝ}^{\sigma }\left(x\right)\right]{ϝ}^{\Delta }\left(x\right)|\Delta x\le \alpha {\int }_0^{\vartheta }|{ϝ}^{\Delta }\left(x\right){|}^2\Delta x.$$

(6)

In [3], the authors generalized (5).

$${\int }_0^{\vartheta }\left|\left\{\sum _{k=0}^{\ell }{ϝ}^k\left(x\right){\left({ϝ}^{\sigma }\right)}^{\ell -k}\left(x\right)\right\}{ϝ}^{{\Delta }^j}\left(x\right)\right|\Delta x\le {\vartheta }^{nl}\left\{{\int }_0^{\vartheta }|{ϝ}^{{\Delta }^j}\left(x\right){|}^{\ell +1}\Delta x\right\}.$$

(7)

As special cases of (7), if we take $\ell =1$ and $j=1,$ we obtain the following two inequalities, respectively:

$${\int }_0^{\vartheta }\left|(ϝ+{ϝ}^{\sigma }){ϝ}^{{\Delta }^j}\right|\left(x\right)\Delta x\le {\vartheta }^j{\int }_0^{\vartheta }|{ϝ}^{{\Delta }^j}\left(x\right){|}^2\Delta x.$$

(8)

$${\int }_0^{\vartheta }\left|\sum _{k=0}^{\ell }\left\{{ϝ}^k\left({ϝ}^{\sigma }\right){)}^{\ell -k}\right\}{ϝ}^{\Delta }\right|\left(x\right)\Delta x\le {\vartheta }^{\ell }{\int }_0^{\vartheta }|{ϝ}^{\Delta }\left(x\right){|}^{\ell +1}\Delta x.$$

(9)

Numerous scholars have been interested in, and are currently interested in, the study of opioid-type inequalities. The Opial Inequalities have undergone various extensions in recent years; for further information, see articles [7,8,9,10,11,12,13,14,15,16,17,18,19,20]). For more details on Opial-type inequalities, see the monograph [21].

Lemma 1

(See [22]). Supposing $\chi :\mathbb{T}\to \mathbb{R}$ is diff. fun. Letting $\eta \in \Re$ and fix $x_0\in \mathbb{T}$, then the following initial value problem has a unique solution: $e_{\eta }(x,x_0)$ (the exponential function):

$$\left\{\begin{array}{cc}{\chi }^{\Delta }\left(x\right)=\hfill & \eta \left(x\right)\chi \left(x\right),\hfill \\ \chi \left(x_0\right)=\hfill & 1.\hfill \end{array}\right.$$

(10)

The dynamic Hölder’s inequality is what follows [14].

Lemma 2.

Suppose $\alpha ,$$\beta \in \mathbb{T}$ and φ, $\psi \in C_{rd}({[\alpha ,\beta ]}_{\mathbb{T}},[0,\infty ))$. Assume p, $q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, then

$${\int }_{\alpha }^{\beta }\phi \left(x\right)\psi \left(x\right)\Delta x\le {\left[{\int }_{\alpha }^{\beta }{\phi }^p\left(x\right)\Delta x\right]}^{\frac{1}{p}}{\left[{\int }_{\alpha }^{\beta }{\psi }^q\left(x\right)\Delta x\right]}^{\frac{1}{q}}.$$

(11)

Below, is the dynamic integration by parts rule.

$${\int }_{\alpha }^{\beta }{{\Theta }_2}^{\Delta }\left(x\right){\Theta }_1\left(x\right)\Delta x={\Theta }_2\left(\beta \right){\Theta }_1\left(\beta \right)-{\Theta }_2\left(\alpha \right){\Theta }_1\left(a\right)-{\int }_{\alpha }^{\beta }{{\Theta }_2}^{\sigma }\left(x\right){\Theta }_1\left(x\right)\Delta x.$$

(12)

Supposing the function $\psi :\mathbb{T}\to \mathbb{T}$ with $j\in \mathbb{N}$, we have

$${\left({\psi }^j\right)}^{\Delta }\left(x\right)=\left\{\sum _{k=0}^{j-1}{\psi }^k\left(x\right){\left({\psi }^{\sigma }\left(x\right)\right)}^{j-1-k}\right\}{\psi }^{\Delta }\left(x\right).$$

(13)

In this manuscript, we extend the above-mentioned-qualities by applying some novel tools. Some previously known disparities are extended by and given more widespread new forms by these inequalities. In selecting the best ways to solve dynamic inequalities, symmetry is crucial.

2. Main Results

Here, we outline and provide evidence for our main conclusions.

Theorem 4.

Supposing ζ, $\delta \ge 0$ are right-dense-continuous fun. over ${[0,\vartheta ]}_{\mathbb{T}}$, ${\int }_0^{\vartheta }\zeta \left(x\right)\Delta x<\infty ,$ and δ nondecreasing. For diff. fun. $ϝ:{[0,\vartheta ]}_{\mathbb{T}}⟶\mathbb{R}$ on $ϝ\left(0\right)=0$ we get

$${\int }_0^{\vartheta }\left[{\displaystyle \frac{1}{{\delta }^{\sigma }}}\left|(ϝ+{ϝ}^{\sigma }){ϝ}^{\Delta }\right|\right]\left(x\right)\Delta x\le \left\{{\int }_0^{\vartheta }\zeta \left(x\right)\Delta x\right\}\left\{{\int }_0^{\vartheta }{\displaystyle \frac{1}{\zeta \left(x\right)\delta \left(x\right)}}{\left|{ϝ}^{\Delta }\left(x\right)\right|}^2\Delta x\right\}.$$

Proof. 

We consider

$$\daleth \left(x\right)={\int }_0^x{\displaystyle \frac{1}{\sqrt{{\delta }^{\sigma }(\aleph )}}}\left|{ϝ}^{\Delta }(\aleph )\right|\Delta \aleph .$$

Then, ${\daleth }^{\Delta }={\displaystyle \frac{1}{\sqrt{{\delta }^{\sigma }}}}\left|{ϝ}^{\Delta }\right|$ and $\sigma (\aleph )\le x$ and hence ${\delta }^{\sigma }(\aleph )\le \delta \left(x\right).$

$$\begin{array}{ccc}\hfill \left|ϝ\right(x\left)\right|& =& |ϝ\left(x\right)-ϝ\left(0\right)|\le {\int }_0^x\left|{ϝ}^{\Delta }(\aleph )\right|\Delta \aleph \le {\int }_0^x\sqrt{{\displaystyle \frac{\delta \left(x\right)}{{\delta }^{\sigma }(\aleph )}}}\left|{ϝ}^{\Delta }(\aleph )\right|\Delta \aleph =\sqrt{\delta \left(x\right)}{\int }_0^x{\displaystyle \frac{|{ϝ}^{\Delta }(\aleph )|}{\sqrt{{\delta }^{\sigma }(\aleph )}}}\Delta \aleph \hfill \\ & =& \sqrt{\delta \left(x\right)}\daleth \left(x\right).\hfill \end{array}$$

Thus, (observe that $x\le \sigma \left(x\right)$ implies $\delta \left(x\right)\le \delta \left(\sigma \right(x\left)\right)$).

$$\begin{array}{ccc}\hfill {\int }_0^{\vartheta }\left[{\displaystyle \frac{1}{{\delta }^{\sigma }}}\left|(ϝ+{ϝ}^{\sigma }){ϝ}^{\Delta }\right|\right]\left(x\right)\Delta x& \le & {\int }_0^{\vartheta }{\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\left(\left|ϝ\left(x\right)\right|+|{ϝ}^{\sigma }\left(x\right)|\right)\left|{ϝ}^{\Delta }\left(x\right)\right|\Delta x\hfill \\ & \le & {\int }_0^{\vartheta }{\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\left[\sqrt{\delta \left(x\right)}\daleth \left(x\right)+\sqrt{\delta \left(\sigma \right(x\left)\right)}\daleth \left(\sigma \left(x\right)\right)\right]\sqrt{{\delta }^{\sigma }\left(x\right)}{\daleth }^{\Delta }\left(x\right)\Delta x\hfill \\ & \le & {\int }_0^{\vartheta }{\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\left[\sqrt{\delta \left(\sigma \right(x\left)\right)}\daleth \left(x\right)+\sqrt{\delta \left(\sigma \right(x\left)\right)}\daleth \left(\sigma \left(x\right)\right)\right]\sqrt{{\delta }^{\sigma }\left(x\right)}{\daleth }^{\Delta }\left(x\right)\Delta x\hfill \\ & =& {\int }_0^{\vartheta }(\daleth \left(x\right)+\daleth \left(\sigma \left(x\right)\right){\daleth }^{\Delta }\left(x\right))\Delta x\hfill \\ & =& {\int }_0^{\vartheta }{\left({\daleth }^2\right)}^{\Delta }\Delta x={\daleth }^2\left(\vartheta \right)\hfill \\ & =& {\left\{{\int }_0^{\vartheta }{\displaystyle \frac{|{ϝ}^{\Delta }(\aleph )|}{\sqrt{{\delta }^{\sigma }(\aleph )}}}\Delta \aleph \right\}}^2\hfill \\ & =& {\left\{{\int }_0^{\vartheta }\sqrt{\zeta (\aleph )}{\displaystyle \frac{1}{\sqrt{\zeta (\aleph ){\delta }^{\sigma }(\aleph )}}}\left|{ϝ}^{\Delta }(\aleph )\right|\Delta \aleph \right\}}^2\hfill \\ & \le & \left\{{\int }_0^{\vartheta }\zeta \left(x\right)\Delta x\right\}\left\{{\int }_0^{\vartheta }{\displaystyle \frac{1}{\zeta \left(x\right)\delta \left(x\right)}}{\left|{ϝ}^{\Delta }\left(x\right)\right|}^2\Delta x\right\}.\hfill \end{array}$$

□

Corollary 1.

Supposing $\mathbb{T}=\mathbb{R}$, obtains

$${\int }_0^{\vartheta }{\displaystyle \frac{1}{\delta \left(x\right)}}\left|ϝ\left(x\right){ϝ}^{\prime }\left(x\right)\right|dt\le \frac{\vartheta }{2}\left\{{\int }_0^{\vartheta }\zeta \left(x\right)dt\right\}\left\{{\int }_0^{\vartheta }{\displaystyle \frac{1}{\zeta \left(x\right)\delta \left(x\right)}}{\left|{ϝ}^{\prime }\left(x\right)\right|}^{2}dx\right\}.$$

Corollary 2.

Taking $\mathbb{T}=\mathbb{Z}$ we obtain

$$\sum _{x=0}^{\vartheta -1}{\displaystyle \frac{1}{\delta (x+1)}}|(ϝ\left(x\right)+ϝ(x+1))\Delta ϝ\left(x\right)|\le \left\{\sum _{x=0}^{\vartheta -1}\zeta \left(x\right)\right\}\left\{\sum _{x=0}^{\vartheta -1}{\displaystyle \frac{1}{\zeta \left(x\right)\delta \left(x\right)}}|\Delta ϝ\left(x\right){|}^2\right\}.$$

We generalise the aforementioned findings to higher order dynamic inequalities in the sections that follow.

Theorem 5.

Let ℓ, $j\mathit{and}m\in \mathbb{N}$ and δ, $w\ge$ and right dense-continuous functions over ${[0,\vartheta ]}_{\mathbb{T}}$ and δ nondec. with ${\displaystyle {\int }_0^{\vartheta }w\left(x\right)\Delta x<\infty }$. Let $ϝ:{[0,\vartheta ]}_{\mathbb{T}}\to \mathbb{R}$ be j times differentiable and $ϝ\left(0\right)={ϝ}^{\Delta }\left(0\right)=\cdots ={ϝ}^{{\Delta }^{j-1}}\left(0\right)=0$, then

$${\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\right)}^m\left|\left\{\sum _{k=0}^{\ell }{ϝ}^k\left(x\right){\left({ϝ}^{\sigma }\right)}^{\ell -k}\left(x\right)\right\}{ϝ}^{{\Delta }^j}\left(x\right)\right|\Delta x\le {\vartheta }^{(j-1)\ell }{\left\{{\int }_0^{\vartheta }w\left(x\right)\Delta x\right\}}^{\ell }\left\{{\int }_0^{\vartheta }{\displaystyle \frac{1}{w^{\ell }\left(x\right){\delta }^m\left(\sigma \left(x\right)\right)}}|{ϝ}^{{\Delta }^j}\left(x\right){|}^{\ell +1}\Delta x\right\}.$$

(14)

Proof. 

Define

$$\daleth \left(x\right)={\int }_0^x{\int }_0^{x_{j-1}}\cdots {\int }_0^{x_2}{\int }_0^{x_1}{\left({\displaystyle \frac{1}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{m}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-1}.$$

Then,

$$\begin{array}{ccc}\hfill {\daleth }^{\Delta }\left(x\right)& =& {\int }_0^x{\int }_0^{x_{j-2}}\cdots {\int }_0^{x_2}{\int }_0^{x_1}{\left({\displaystyle \frac{1}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{m}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-2},\hfill \\ \hfill {\daleth }^{\Delta \Delta }\left(x\right)& =& {\int }_0^x{\int }_0^{x_{j-3}}\cdots {\int }_0^{x_2}{\int }_0^{x_1}{\left({\displaystyle \frac{1}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{m}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-3},\hfill \\ & ⋮& \\ \hfill {\daleth }^{{\Delta }^{j-1}}\left(x\right)& =& {\int }_0^x{\left({\displaystyle \frac{1}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{m}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph ,\hfill \\ \hfill {\daleth }^{{\Delta }^j}\left(x\right)& =& {\left({\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\right)}^{\frac{m}{\ell +1}}|{ϝ}^{{\Delta }^j}\left(x\right)|,\hfill \end{array}$$

and for $0\le x\le \vartheta$ (observe that $0\le \aleph <x_1<x_2<\cdots <x_{j-1}<x$ implies $\sigma (\aleph )<x$, and hence ${\delta }^{\sigma }(\aleph )<\delta \left(x\right).$

Now, we have

$$\begin{array}{ccc}\hfill |ϝ(x)|& =& |ϝ\left(x\right)-ϝ\left(0\right)|=|{\int }_0^x{ϝ}^{\Delta }\left(x_1\right)\Delta x_1|\hfill \\ & \le & {\int }_0^x\left|{ϝ}^{\Delta }\left(x_1\right)\right|\Delta x_1\hfill \\ & \le & {\int }_0^x{\int }_0^{x_1}\left|{ϝ}^{\Delta \Delta }\left(x_2\right)\right|\Delta x_2\Delta x_1\le \hfill \\ & ⋮& \\ & \le & {\int }_0^x{\int }_0^{x_{j-1}}\cdots {\int }_0^{x_2}{\int }_0^{x_1}\left|{ϝ}^{{\Delta }^j}(\aleph )\right|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-1}\hfill \\ & \le & {\int }_0^x{\int }_0^{x_{j-1}}\cdots {\int }_0^{x_2}{\int }_0^{x_1}{\left(\frac{\delta \left(x\right)}{{\delta }^{\sigma }(\aleph )}\right)}^{\frac{m}{\ell +1}}\left|{ϝ}^{{\Delta }^j}(\aleph )\right|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-1}\hfill \\ & =& {\delta }^{\frac{m}{\ell +1}}\left(x\right)\daleth \left(x\right)={\delta }^{\frac{m}{\ell +1}}\left(x\right){\int }_0^x{\daleth }^{\Delta }(\aleph )\Delta \aleph \le {\delta }^{\frac{m}{\ell +1}}\left(x\right){\int }_0^x{\daleth }^{\Delta }\left(x\right)\Delta \aleph \hfill \\ & \le & {\delta }^{\frac{m}{\ell +1}}\left(x\right){\int }_0^{\vartheta }{\daleth }^{\Delta }\left(x\right)\Delta \aleph =h{q}^{\frac{m}{\ell +1}}\left(x\right){\daleth }^{\Delta }\left(x\right)\hfill \\ & \le & {\vartheta }^2{\delta }^{\frac{m}{\ell +1}}\left(x\right){\daleth }^{\Delta \Delta }\left(x\right)\le \cdots \le {\vartheta }^{j-1}{\delta }^{\frac{m}{\ell +1}}\left(x\right){\daleth }^{{\Delta }^{j-1}}\left(x\right).\hfill \end{array}$$

Hence

$$\begin{array}{ccc}& & {\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\right)}^m\left|\left\{\sum _{k=0}^{\ell }{ϝ}^k\left(x\right){\left({ϝ}^{\sigma }\right)}^{\ell -k}\left(x\right)\right\}{ϝ}^{{\Delta }^j}\left(x\right)\right|\Delta x\le {\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\right)}^m\left\{\sum _{k=0}^{\ell }{\left|ϝ\left(x\right)\right|}^k{\left|{ϝ}^{\sigma }\left(x\right)\right|}^{\ell -k}\right\}|{ϝ}^{{\Delta }^j}\left(x\right)|\Delta x\hfill \\ & & \le {\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\right)}^m\left\{\sum _{k=0}^{\ell }{\left({\vartheta }^{j-1}{\delta }^{\frac{m}{\ell +1}}\left(x\right){\daleth }^{{\Delta }^{j-1}}\left(x\right)\right)}^k{\left({\vartheta }^{j-1}{\left({\delta }^{\sigma }\right)}^{\frac{m}{\ell +1}}\left(x\right){\daleth }^{{\Delta }^{j-1}}\left(\sigma \left(x\right)\right)\right)}^{\ell -k}\right\}{\left({\delta }^{\sigma }\right)}^{\frac{m}{\ell +1}}\left(x\right){\daleth }^{{\Delta }^j}\left(x\right)\Delta x\hfill \\ & & \le {\vartheta }^{(j-1)\ell }{\int }_0^{\vartheta }\left\{\sum _{k=0}^{\ell }{\left({\daleth }^{{\Delta }^{j-1}}\left(x\right)\right)}^k{\left({\daleth }^{{\Delta }^{j-1}}\left(\sigma \left(x\right)\right)\right)}^{\ell -k}\right\}{\daleth }^{{\Delta }^j}\left(x\right)\Delta x\hfill \\ & & ={\vartheta }^{(j-1)\ell }{\int }_0^{\vartheta }{\left[{\left({\daleth }^{{\Delta }^{j-1}}\left(x\right)\right)}^{\ell +1}\right]}^{\Delta }\Delta x\hfill \\ & & ={\vartheta }^{(j-1)\ell }{\left({\daleth }^{{\Delta }^{j-1}}\right)}^{\ell +1}\left(\vartheta \right)\hfill \\ & & ={\vartheta }^{(j-1)\ell }{\left\{{\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\right)}^m\left|{ϝ}^{{\Delta }^j}\left(x\right)\right|\Delta x\right\}}^{\ell +1}\hfill \\ & & ={\vartheta }^{(j-1)\ell }{\left\{{\int }_0^{\vartheta }\frac{1}{w^{\frac{\ell }{\ell +1}}\left(x\right){\left({\delta }^{\sigma }\left(x\right)\right)}^m}{w}^{\frac{\ell }{\ell +1}}\left(x\right)\left|{ϝ}^{{\Delta }^j}\left(x\right)\right|\Delta x\right\}}^{\ell +1}\hfill \\ & & \le {\vartheta }^{(j-1)\ell }{\left\{{\int }_0^{\vartheta }w\left(x\right)\Delta x\right\}}^{\ell }\left\{{\int }_0^{\vartheta }\frac{1}{w^{\ell }\left(x\right){\left({\delta }^{\sigma }\left(x\right)\right)}^m}|{ϝ}^{{\Delta }^j}\left(x\right){|}^{\ell +1}\Delta x\right\},\hfill \end{array}$$

where (11) (13) were employed. This concludes the evidence. □

Theorem 6.

Using the same conditions of Theorem 5, gets

$${\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\right)}^m\left|(ϝ+{ϝ}^{\sigma }){ϝ}^{{\Delta }^j}\right|\Delta x\le {\vartheta }^{(j-1)}\left\{{\int }_0^{\vartheta }w\left(x\right)\Delta x\right\}\left\{{\int }_0^{\vartheta }{\displaystyle \frac{1}{w\left(x\right){\delta }^m\left(\sigma \left(x\right)\right)}}|{ϝ}^{{\Delta }^j}\left(x\right){|}^2\Delta x\right\}.$$

(15)

Proof. 

That is Theorem 5 with $\ell =1.$ □

Theorem 7.

Using the same conditions of Theorem 5, gets

$${\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\right)}^m\left|\left\{\sum _{k=0}^{\ell }{ϝ}^k\left(x\right){\left({ϝ}^{\sigma }\right)}^{\ell -k}\left(x\right)\right\}{ϝ}^{\Delta }\left(x\right)\right|\Delta x\le {\left\{{\int }_0^{\vartheta }w\left(x\right)\Delta x\right\}}^{\ell }\left\{{\int }_0^{\vartheta }{\displaystyle \frac{1}{w^{\ell }\left(x\right){\delta }^m\left(\sigma \left(x\right)\right)}}|{ϝ}^{\Delta }\left(x\right){|}^{\ell +1}\Delta x\right\}.$$

(16)

Proof. 

This is Theorem 5 with $j=1.$ □

Remark 1.

Taking $\ell =j=m=1$ in Theorem 5, got result 4.

Remark 2.

Taking $\delta =w=1$ in Theorem 5, got result (7).

Remark 3.

Taking $\delta =w=1$ in Theorem 6, got result (8).

Remark 4.

Taking $\delta =w=1$ in Theorem 7, got result (9).

Corollary 3.

Taking $\mathbb{T}=\mathbb{R}$ in Theorem 5, obtains

$$\begin{array}{c}{\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{\delta \left(x\right)}}\right)}^m\left|{ϝ}^{\ell }\left(x\right){ϝ}^{\left(j\right)}\left(x\right)\right|dt\le \frac{{\vartheta }^{(j-1)\ell }}{\ell +1}{\left\{{\int }_0^{\vartheta }w\left(x\right)dt\right\}}^{\ell }\left\{{\int }_0^{\vartheta }{\displaystyle \frac{1}{w^{\ell }\left(x\right)\delta \left(x\right)}}|{ϝ}^{\left(j\right)}\left(x\right){|}^{\ell +1}dt\right\}.\end{array}$$

Corollary 4.

If $\mathbb{T}=\mathbb{Z}$ in Theorem 5, then, we obtain

$$\sum _{ϵ=0}^{\vartheta -1}{\left({\displaystyle \frac{1}{\delta (ϵ+1)}}\right)}^m\left|\left\{\sum _{k=0}^{\ell }{ϝ}^k\left(ϵ\right){ϝ}^{\ell -k}(ϵ+1)\right\}{\Delta }^{j}x\left(ϵ\right)\right|\le {\vartheta }^{(j-1)\ell }{\left\{\sum _{ϵ=0}^{\vartheta -1}\Theta \left(ϵ\right)\right\}}^{\ell }\left\{\sum _{ϵ=0}^{\vartheta -1}{\displaystyle \frac{1}{{\Theta }^{\ell }\left(ϵ\right)\delta (ϵ+1)}}|{\Delta }^{j}x\left(ϵ\right){|}^{\ell +1}\right\}.$$

Theorem 8.

Using the same conditions of Theorem 5, gets

$${\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{g\left(x\right)}}\right)}^m\left|\left\{\sum _{k=0}^{\ell }{ϝ}^k\left(x\right){\left({ϝ}^{\sigma }\right)}^{\ell -k}\left(x\right)\right\}{ϝ}^{{\Delta }^j}\left(x\right)\right|\Delta x\le {\vartheta }^{(j-1)\ell }{\left\{{\int }_0^{\vartheta }w\left(x\right)\Delta x\right\}}^{\ell }\left\{{\int }_0^{\vartheta }{\displaystyle \frac{1}{w^{\ell }\left(x\right)g^m\left(x\right)}}|{ϝ}^{{\Delta }^j}\left(x\right){|}^{\ell +1}\Delta x\right\}.$$

(17)

Proof. 

Define

$$\daleth \left(x\right)={\int }_x^{\vartheta }{\int }_{x_{j-1}}^{\vartheta }\cdots {\int }_{x_2}^{\vartheta }{\int }_{x_1}^{\vartheta }{\left({\displaystyle \frac{1}{g(\aleph )}}\right)}^{\frac{m}{\ell +1}}\left|{ϝ}^{{\Delta }^j(\aleph )}\right|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-1}.$$

Then

$$\begin{array}{ccc}\hfill {\daleth }^{\Delta }\left(x\right)& =& -{\int }_x^{\vartheta }{\int }_{x_{j-2}}^{\vartheta }\cdots {\int }_{x_2}^{\vartheta }{\int }_{x_1}^{\vartheta }{\left({\displaystyle \frac{1}{g(\aleph )}}\right)}^{\frac{m}{\ell +1}}\left|{ϝ}^{{\Delta }^j(\aleph )}\right|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-2},\hfill \\ \hfill {\daleth }^{\Delta \Delta }\left(x\right)& =& {\int }_x^{\vartheta }{\int }_{x_{j-3}}^{\vartheta }\cdots {\int }_{x_2}^{\vartheta }{\int }_{x_1}^{\vartheta }{\left({\displaystyle \frac{1}{g(\aleph )}}\right)}^{\frac{m}{\ell +1}}\left|{ϝ}^{{\Delta }^j(\aleph )}\right|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-3},\hfill \\ & ⋮& \\ \hfill {\daleth }^{{\Delta }^{j-1}}\left(x\right)& =& {(-1)}^{j-1}{\int }_x^{\vartheta }{\left({\displaystyle \frac{1}{g(\aleph )}}\right)}^{\frac{m}{\ell +1}}\left|{ϝ}^{{\Delta }^j}(\aleph )\right|\Delta \aleph ,\hfill \\ \hfill {\daleth }^{{\Delta }^j}\left(x\right)& =& {(-1)}^j{\left({\displaystyle \frac{1}{g\left(x\right)}}\right)}^{\frac{m}{\ell +1}}\left|{ϝ}^{{\Delta }^j}\left(x\right)\right|,\hfill \end{array}$$

and for $0\le x\le \vartheta$ (observe that $\aleph >x_1>x_2>\cdots >x_{j-1}>x$ implies $g(\aleph )<g\left(x\right)$).

Now, we have

$$\begin{array}{ccc}\hfill |ϝ(x)|& =& |ϝ\left(x\right)-ϝ\left(\vartheta \right)|=|ϝ\left(\vartheta \right)-ϝ\left(x\right)|=|{\int }_x^{\vartheta }{ϝ}^{\Delta }\left(x_1\right)\Delta x_1|\hfill \\ & \le & {\int }_x^{\vartheta }\left|{ϝ}^{\Delta }\left(x_1\right)\right|\Delta x_1\hfill \\ & \le & {\int }_x^{\vartheta }{\int }_{x_1}^{\vartheta }\left|{ϝ}^{\Delta \Delta }\left(x_2\right)\right|\Delta x_2\Delta x_1\le \hfill \\ & ⋮& \\ & \le & {\int }_x^{\vartheta }{\int }_{x_{j-1}}^{\vartheta }\cdots {\int }_{x_2}^{\vartheta }{\int }_{x_1}^{\vartheta }\left|{ϝ}^{{\Delta }^j}(\aleph )\right|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-1}\hfill \\ & \le & {\int }_x^{\vartheta }{\int }_{x_{j-1}}^{\vartheta }\cdots {\int }_{x_2}^{\vartheta }{\int }_{x_1}^{\vartheta }{\left(\frac{g\left(x\right)}{g(\aleph )}\right)}^{\frac{m}{\ell +1}}\left|{ϝ}^{{\Delta }^j}(\aleph )\right|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-1}\hfill \\ & =& g^{\frac{m}{\ell +1}}\left(x\right)\daleth \left(x\right)=-g^{\frac{m}{\ell +1}}\left(x\right){\int }_x^{\vartheta }{\daleth }^{\Delta }(\aleph )\Delta \aleph \le -g^{\frac{m}{\ell +1}}\left(x\right){\int }_x^{\vartheta }{\daleth }^{\Delta }\left(x\right)\Delta \aleph \hfill \\ & \le & -g^{\frac{m}{\ell +1}}\left(x\right){\int }_0^{\vartheta }{\daleth }^{\Delta }\left(x\right)\Delta \aleph =-\vartheta g^{\frac{m}{\ell +1}}\left(x\right){\daleth }^{\Delta }\left(x\right)\hfill \\ & \le & {\vartheta }^2{g}^{\frac{m}{\ell +1}}\left(x\right){\daleth }^{\Delta \Delta }\left(x\right)\le \cdots \le {(-1)}^{j-1}{\vartheta }^{j-1}{g}^{\frac{m}{\ell +1}}\left(x\right){\daleth }^{{\Delta }^{j-1}}\left(x\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}& & {\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{g\left(x\right)}}\right)}^m\left|\left\{\sum _{k=0}^{\ell }{ϝ}^k\left(x\right){\left({ϝ}^{\sigma }\right)}^{\ell -k}\left(x\right)\right\}{ϝ}^{{\Delta }^j}\left(x\right)\right|\Delta x\le {\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{g\left(x\right)}}\right)}^m\left\{\sum _{k=0}^{\ell }{\left|ϝ\left(x\right)\right|}^k{\left|{ϝ}^{\sigma }\left(x\right)\right|}^{\ell -k}\right\}|{ϝ}^{{\Delta }^j}\left(x\right)|\Delta x\hfill \\ & & \le {\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{g\left(x\right)}}\right)}^m\left\{\sum _{k=0}^{\ell }{\left({(-1)}^{j-1}{\vartheta }^{j-1}{\daleth }^{{\Delta }^{j-1}}\left(x\right)g^{\frac{m}{\ell +1}}\left(x\right)\right)}^k{\left({(-1)}^{j-1}{\vartheta }^{j-1}{\daleth }^{{\Delta }^{j-1}}\left(\sigma \left(x\right)\right){\left(g^{\sigma }\right)}^{\frac{m}{\ell +1}}\left(x\right)\right)}^{\ell -k}\right\}\hfill \\ & & \times {(-1)}^n{g}^{\frac{m}{\ell +1}}\left(x\right){\daleth }^{{\Delta }^j}\left(x\right)\Delta x\hfill \\ & & \le {(-1)}^{j\ell -\ell -j}{\vartheta }^{(j-1)\ell }{\int }_0^{\vartheta }\left\{\sum _{k=0}^{\ell }{\left({\daleth }^{{\Delta }^{j-1}}\left(x\right)\right)}^k{\left({\daleth }^{{\Delta }^{j-1}}\left(\sigma \left(x\right)\right)\right)}^{\ell -k}\right\}{\daleth }^{{\Delta }^j}\left(x\right)\Delta x\hfill \\ & & ={(-1)}^{j\ell -\ell -j}{\vartheta }^{(j-1)\ell }{\int }_0^{\vartheta }{\left[{\left({\daleth }^{{\Delta }^{j-1}}\left(x\right)\right)}^{\ell +1}\right]}^{\Delta }\Delta x\hfill \\ & & ={(-1)}^{j\ell -\ell -j+1}{\vartheta }^{(j-1)\ell }{\left({\daleth }^{{\Delta }^{j-1}}\right)}^{\ell +1}\left(0\right)\hfill \\ & & ={(-1)}^{2(nl-\ell )}{\vartheta }^{(j-1)\ell }{\left\{{\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{g\left(x\right)}}\right)}^{\frac{m}{\ell +1}}|{ϝ}^{{\Delta }^j}\left(x\right)|\Delta x\right\}}^{\ell +1}\hfill \\ & & ={\vartheta }^{(j-1)\ell }{\left\{{\int }_0^{\vartheta }{w}^{\frac{\ell }{\ell +1}}\left(x\right){\displaystyle \frac{1}{w^{\frac{\ell }{\ell +1}}\left(x\right)g^{\frac{m}{\ell +1}}\left(x\right))}}|{ϝ}^{{\Delta }^j}\left(x\right)|\Delta x\right\}}^{\ell +1}\hfill \\ & & \le {\vartheta }^{(j-1)\ell }{\left\{{\int }_0^{\vartheta }w\left(x\right)\Delta x\right\}}^{\ell }\left\{{\int }_0^{\vartheta }{\displaystyle \frac{1}{w^{\ell }\left(x\right)g^m\left(x\right)}}|{ϝ}^{{\Delta }^j}\left(x\right){|}^{\ell +1}\Delta x\right\}\hfill \end{array}$$

where (13) and (11) were employed. This concludes the evidence. □

Corollary 5.

Taking $\mathbb{R}=\mathbb{T}$ in Theorem 8, obtains

$${\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{g\left(x\right)}}\right)}^m\left|{ϝ}^{\ell }\left(x\right){ϝ}^{\left(j\right)}\left(x\right)\right|dt\le \frac{{\vartheta }^{(j-1)\ell }}{\ell +1}{\left\{{\int }_0^{\vartheta }w\left(x\right)dt\right\}}^{\ell }\left\{{\int }_0^{\vartheta }{\displaystyle \frac{1}{w^{\ell }\left(x\right)g^m\left(x\right)}}|{ϝ}^{\left(j\right)}\left(x\right){|}^{\ell +1}dt\right\}.$$

Corollary 6.

If $\mathbb{T}=\mathbb{Z}$ in Theorem 8, then, we obtain

$$\sum _{x=0}^{\vartheta -1}{\left({\displaystyle \frac{1}{g\left(x\right)}}\right)}^m\left|\left\{\sum _{k=0}^{\ell }{ϝ}^k\left(x\right){ϝ}^{\ell -k}(x+1)\right\}{\Delta }^{j}x\left(x\right)\right|\le {\vartheta }^{(j-1)\ell }{\left\{\sum _{x=0}^{\vartheta -1}w\left(x\right)\right\}}^{\ell }\left\{\sum _{x=0}^{\vartheta -1}{\displaystyle \frac{1}{w^{\ell }\left(x\right)g^m\left(x\right)}}|{\Delta }^{j}x\left(x\right){|}^{\ell +1}\right\}.$$

Theorem 9.

Using the same conditions of Theorem 8, gets

$${\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{g\left(x\right)}}\right)}^m\left|(ϝ+{ϝ}^{\sigma }){ϝ}^{{\Delta }^j}\right|\Delta x\le {\vartheta }^{(j-1)}\left\{{\int }_0^{\vartheta }w\left(x\right)\Delta x\right\}\left\{{\int }_0^{\vartheta }{\displaystyle \frac{1}{w\left(x\right)g^m\left(x\right)}}|{ϝ}^{{\Delta }^j}\left(x\right){|}^2\Delta x\right\}$$

(18)

Proof. 

That is Theorem 8 with $\ell =1.$ □

Theorem 10.

Using the same conditions of Theorem 8, gets

$${\int }_0^{\vartheta }{\left({\displaystyle \frac{1}{g\left(x\right)}}\right)}^m\left|\left\{\sum _{k=0}^{\ell }{ϝ}^k\left(x\right){\left({ϝ}^{\sigma }\right)}^{\ell -k}\left(x\right)\right\}{ϝ}^{\Delta }\left(x\right)\right|\Delta x\le {\left\{{\int }_0^{\vartheta }w\left(x\right)\Delta x\right\}}^{\ell }\left\{{\int }_0^{\vartheta }{\displaystyle \frac{1}{w^{\ell }\left(x\right)g^m\left(x\right)}}|{ϝ}^{\Delta }\left(x\right){|}^{\ell +1}\Delta x\right\}.$$

(19)

Proof. 

This is Theorem 8 with $j=1.$ □

In the following, we offer a generalization of the above inequalities where $ϝ\left(0\right)$ and $ϝ\left(\vartheta \right)$ do not need to be 0.

Theorem 11.

Using the same conditions of Theorem 8, gets

$${\int }_0^{\vartheta }{\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\left\{\sum _{k=0}^{\ell }{\left(\left|ϝ\left(x\right)\right|-\tilde{\beta }\right)}^k{\left(|{ϝ}^{\sigma }\left(x\right)|-\tilde{\beta }\right)}^{\ell -k}\right\}\left|{ϝ}^{{\Delta }^j}\left(x\right)\right|\Delta x\le {\vartheta }^{(j-1)\ell }{\tilde{\alpha }}^{\ell }{\int }_0^{\vartheta }{\displaystyle \frac{1}{w^{\ell }\left(x\right){\delta }^{\sigma }\left(x\right)}}|{ϝ}^{{\Delta }^j}\left(x\right){|}^{\ell +1}\Delta x,$$

(20)

where

$$\tilde{\alpha }\in \mathbb{T}with|\frac{\vartheta }{2}-\tilde{\alpha }|=dist\left(\frac{\vartheta }{2},\mathbb{T}\right),and\tilde{\beta }=\max \left\{\left|ϝ\right(0\left)\right|,\left|ϝ\right(\vartheta \left)\right|\right\}.$$

Proof. 

Define

$$\daleth \left(x\right)={\int }_0^x{\int }_0^{x_{j-1}}\cdots {\int }_0^{x_2}{\int }_0^{x_1}{\left({\displaystyle \frac{1}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-1},$$

and

$$z\left(x\right)={\int }_x^{\vartheta }{\int }_{x_{j-1}}^{\vartheta }\cdots {\int }_{x_2}^{\vartheta }{\int }_{x_1}^{\vartheta }{\left({\displaystyle \frac{1}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-1}.$$

Then,

$$\begin{array}{ccc}\hfill {\daleth }^{\Delta }\left(x\right)& =& {\int }_0^x{\int }_0^{x_{j-2}}\cdots {\int }_0^{x_2}{\int }_0^{x_1}{\left({\displaystyle \frac{1}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-2},\hfill \\ \hfill {\daleth }^{\Delta \Delta }\left(x\right)& =& {\int }_0^x{\int }_0^{x_{j-3}}\cdots {\int }_0^{x_2}{\int }_0^{x_1}{\left({\displaystyle \frac{1}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-3},\hfill \\ & ⋮& \\ \hfill {\daleth }^{{\Delta }^{j-1}}\left(x\right)& =& {\int }_0^x{\left({\displaystyle \frac{1}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph ,\hfill \\ \hfill {\daleth }^{{\Delta }^j}\left(x\right)& =& {\left({\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}\left(x\right)|,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill z^{\Delta }\left(\epsilon \right)& =& -{\int }_{\epsilon }^{\vartheta }{\int }_{{\epsilon }_{j-2}}^{\vartheta }\cdots {\int }_{{\epsilon }_2}^{\vartheta }{\int }_{{\epsilon }_1}^{\vartheta }{\left({\displaystyle \frac{1}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta {\epsilon }_1\Delta {\epsilon }_2\dots \Delta {\epsilon }_{j-2},\hfill \\ \hfill z^{\Delta \Delta }\left(\epsilon \right)& =& {\int }_{\epsilon }^{\vartheta }{\int }_{{\epsilon }_{j-3}}^{\vartheta }\cdots {\int }_{{\epsilon }_2}^{\vartheta }{\int }_{{\epsilon }_1}^{\vartheta }{\left({\displaystyle \frac{1}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta {\epsilon }_1\Delta {\epsilon }_2\dots \Delta {\epsilon }_{j-3},\hfill \\ & ⋮& \\ \hfill z^{{\Delta }^{j-1}}\left(x\right)& =& {(-1)}^{j-1}{\int }_x^{\vartheta }{\left({\displaystyle \frac{1}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph ,\hfill \\ \hfill z^{{\Delta }^j}\left(x\right)& =& {(-1)}^j{\left({\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}\left(x\right)|.\hfill \end{array}$$

and for $0\le x\le \vartheta$ (observe that $0\le \aleph <x_1<x_2<\cdots <x_{j-1}<x$ implies $\sigma (\aleph )<x$, and hence ${\delta }^{\sigma }(\aleph )<\delta \left(x\right)$.

$$\begin{array}{ccc}\hfill |ϝ(x)|& =& |ϝ\left(x\right)-ϝ\left(0\right)+ϝ\left(0\right)|\le |ϝ\left(x\right)-ϝ\left(0\right)|+|ϝ\left(0\right)|=|{\int }_0^x{ϝ}^{\Delta }\left(x_1\right)\Delta x_1|+|ϝ\left(0\right)|\hfill \\ & \le & {\int }_0^x|{ϝ}^{\Delta }\left(x_1\right)|\Delta x_1+|ϝ\left(0\right)|\hfill \\ & \le & {\int }_0^x{\int }_0^{x_1}|{ϝ}^{\Delta \Delta }\left(x_2\right)|\Delta x_2\Delta x_1+|ϝ\left(0\right)|\le \hfill \\ & ⋮& \\ & \le & {\int }_0^x{\int }_0^{x_{j-1}}\cdots {\int }_0^{x_2}{\int }_0^{x_1}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-1}+|ϝ\left(0\right)|\hfill \\ & \le & {\int }_0^x{\int }_0^{x_{j-1}}\cdots {\int }_0^{x_2}{\int }_0^{x_1}{\left({\displaystyle \frac{\delta \left(x\right)}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-1}+|ϝ\left(0\right)|\hfill \\ & =& {\delta }^{\frac{1}{\ell +1}}\left(x\right)\daleth \left(x\right)+|ϝ\left(0\right)|={\delta }^{\frac{1}{\ell +1}}\left(x\right){\int }_0^x{\daleth }^{\Delta }(\aleph )\Delta \aleph +|ϝ\left(0\right)|\le {\delta }^{\frac{1}{\ell +1}}\left(x\right){\int }_0^x{\daleth }^{\Delta }\left(x\right)\Delta \aleph +|ϝ\left(0\right)|\hfill \\ & \le & {\delta }^{\frac{1}{\ell +1}}\left(x\right){\int }_0^{\vartheta }{\daleth }^{\Delta }\left(x\right)\Delta \aleph +|ϝ\left(0\right)|=\vartheta {\delta }^{\frac{1}{\ell +1}}\left(x\right){\daleth }^{\Delta }\left(x\right)+|ϝ\left(0\right)|\hfill \\ & \le & {\vartheta }^2{\delta }^{\frac{1}{\ell +1}}\left(x\right){\daleth }^{\Delta \Delta }\left(x\right)+|ϝ\left(0\right)|\le \cdots \le {\vartheta }^{j-1}{\delta }^{\frac{1}{\ell +1}}\left(x\right){\daleth }^{{\Delta }^{j-1}}\left(x\right)+|ϝ\left(0\right)|,\hfill \end{array}$$

and $\left|ϝ\left(x\right)\right|-\tilde{\beta }\le {\vartheta }^{j-1}{\delta }^{\frac{1}{\ell +1}}\left(x\right){\daleth }^{{\Delta }^{j-1}}\left(x\right).$

Similarly, we have

$$\begin{array}{ccc}\hfill \left|ϝ\right(x\left)\right|& =& |ϝ\left(x\right)-ϝ\left(\vartheta \right)+ϝ\left(\vartheta \right)|\le |ϝ\left(x\right)-ϝ\left(\vartheta \right)|+|ϝ\left(\vartheta \right)|=|{\int }_x^{\vartheta }{ϝ}^{\Delta }\left(x_1\right)\Delta x_1|+|ϝ\left(\vartheta \right)|\hfill \\ & \le & {\int }_x^{\vartheta }|{ϝ}^{\Delta }\left(x_1\right)|\Delta x_1+|ϝ\left(\vartheta \right)|\hfill \\ & \le & {\int }_x^{\vartheta }{\int }_{x_1}^{\vartheta }|{ϝ}^{\Delta \Delta }\left(x_2\right)|\Delta x_2\Delta x_1+|ϝ\left(\vartheta \right)|\le \hfill \\ & ⋮& \\ & \le & {\int }_x^{\vartheta }{\int }_{x_{j-1}}^{\vartheta }\cdots {\int }_{x_2}^{\vartheta }{\int }_{x_1}^{\vartheta }|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta x_1\Delta x_2\dots \Delta x_{j-1}+|ϝ\left(\vartheta \right)|\hfill \\ & \le & {\int }_x^{\vartheta }{\int }_{x_{j-1}}^{\vartheta }\cdots {\int }_{x_2}^{\vartheta }{\int }_{x_1}^{\vartheta }{\left({\displaystyle \frac{\delta \left(x\right)}{{\delta }^{\sigma }(\aleph )}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}(\aleph )|\Delta \aleph \Delta x_1\Delta x_2\cdots \Delta x_{j-1}+|ϝ\left(\vartheta \right)|\hfill \\ & =& {\delta }^{\frac{1}{\ell +1}}\left(x\right)z\left(x\right)+|ϝ\left(\vartheta \right)|=-{\delta }^{\frac{1}{\ell +1}}\left(x\right){\int }_x^{\vartheta }{z}^{\Delta }(\aleph )\Delta \aleph +|ϝ\left(\vartheta \right)|\le -{\delta }^{\frac{1}{\ell +1}}\left(x\right){\int }_x^{\vartheta }{z}^{\Delta }\left(x\right)\Delta \aleph +|ϝ\left(\vartheta \right)|\hfill \\ & \le & -{\delta }^{\frac{1}{\ell +1}}\left(x\right){\int }_0^{\vartheta }{z}^{\Delta }\left(x\right)\Delta \aleph +|ϝ\left(\vartheta \right)|=-\vartheta {\delta }^{\frac{1}{\ell +1}}\left(x\right)z^{\Delta }\left(x\right)+|ϝ\left(\vartheta \right)|\hfill \\ & \le & {\vartheta }^2{\delta }^{\frac{1}{\ell +1}}\left(x\right)z^{\Delta \Delta }\left(x\right)+|ϝ\left(\vartheta \right)|\le \cdots \le {(-1)}^{j-1}{\vartheta }^{j-1}{\delta }^{\frac{1}{\ell +1}}\left(x\right)z^{{\Delta }^{j-1}}\left(x\right)+|ϝ\left(\vartheta \right)|.\hfill \end{array}$$

and thus $\left|ϝ\left(x\right)\right|-\tilde{\beta }\le {(-1)}^{j-1}{\vartheta }^{j-1}{\delta }^{\frac{1}{\ell +1}}\left(x\right)z^{{\Delta }^{j-1}}\left(x\right).$

Let $u\in {[0,\vartheta ]}_{\mathbb{T}}$. Then

$$\begin{array}{ccc}\hfill & & {\int }_0^u{\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\left\{\sum _{k=0}^{\ell }{\left(\left|ϝ\left(x\right)\right|-\tilde{\beta }\right)}^k{\left(|{ϝ}^{\sigma }\left(x\right)|-\tilde{\beta }\right)}^{\ell -k}\right\}\left|{ϝ}^{{\Delta }^j}\left(x\right)\right|\Delta x\hfill \\ \hfill & & \le {\int }_0^u{\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\left\{\sum _{k=0}^{\ell }{\left({\vartheta }^{j-1}{\delta }^{\frac{1}{\ell +1}}\left(x\right){\daleth }^{{\Delta }^{j-1}}\left(x\right)\right)}^k{\left({\vartheta }^{j-1}{\delta }^{\frac{1}{\ell +1}}\left(\sigma \left(x\right)\right){\daleth }^{{\Delta }^{j-1}}\left(\sigma \left(x\right)\right)\right)}^{\ell -k}\right\}{\delta }^{\frac{1}{\ell +1}}\left(\sigma \left(x\right)\right){\daleth }^{{\Delta }^j}\left(x\right)\Delta x\hfill \\ \hfill & & ={\vartheta }^{(j-1)\ell }{\int }_0^u\left\{\sum _{k=0}^{\ell }{\left({\daleth }^{{\Delta }^{j-1}}\left(x\right)\right)}^k{\left({\daleth }^{{\Delta }^{j-1}}\left(\sigma \left(x\right)\right)\right)}^{\ell -k}\right\}{\daleth }^{{\Delta }^j}\left(x\right)\Delta x\hfill \\ \hfill & & ={\vartheta }^{(j-1)\ell }{\int }_0^u{\left[{\left({\daleth }^{{\Delta }^{j-1}}\left(x\right)\right)}^{\ell +1}\right]}^{\Delta }\Delta x\hfill \\ \hfill & & ={\vartheta }^{(j-1)\ell }{\left({\daleth }^{{\Delta }^{j-1}}\right)}^{\ell +1}\left(u\right)\hfill \\ \hfill & & ={\vartheta }^{(j-1)\ell }{\left\{{\int }_0^u{\left({\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}\left(x\right)|\Delta x\right\}}^{\ell +1}\hfill \\ \hfill & & ={\vartheta }^{(j-1)\ell }{\left\{{\int }_0^u{w}^{\frac{\ell }{\ell +1}}\left(x\right){\left({\displaystyle \frac{1}{w^{\ell }\left(x\right){\delta }^{\sigma }\left(x\right)}}\right)}^{\frac{1}{\ell +1}}\left|{ϝ}^{{\Delta }^j}\left(x\right)\right|\Delta x\right\}}^{\ell +1}\hfill \\ & & \le {\vartheta }^{(j-1)\ell }{\left\{{\int }_0^{\vartheta }w\left(x\right)\Delta x\right\}}^{\ell }{\int }_0^u{\displaystyle \frac{1}{w^{\ell }\left(x\right){\delta }^{\sigma }\left(x\right)}}|{ϝ}^{{\Delta }^j}\left(x\right){|}^{\ell +1}\Delta x,\hfill \end{array}$$

(21)

the dynamicHölder inequality (11) and (13) were employed.

Similarly, we get

$$\begin{array}{ccc}\hfill & & {\int }_u^{\vartheta }{\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\left\{\sum _{k=0}^{\ell }{\left(\left|ϝ\right(x|-\tilde{\beta }\right)}^k{\left(|{ϝ}^{\sigma }\left(x\right)|-\tilde{\beta }\right)}^{\ell -k}\left(x\right)\right\}\left|{ϝ}^{{\Delta }^j}\left(x\right)\right|\Delta x\hfill \\ \hfill & & \le {\int }_u^{\vartheta }{\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\left\{\sum _{k=0}^{\ell }{\left({(-1)}^{j-1}{\vartheta }^{j-1}{\delta }^{\frac{1}{\ell +1}}\left(x\right)z^{{\Delta }^{j-1}}\left(x\right)\right)}^k{\left({(-1)}^{j-1}{\vartheta }^{j-1}{\delta }^{\frac{1}{\ell +1}}\left(\sigma \left(x\right)\right)z^{{\Delta }^{j-1}}\left(\sigma \left(x\right)\right)\right)}^{\ell -k}\right\}\hfill \\ \hfill & & \times \frac{{\delta }^{\frac{1}{\ell +1}}\left(\sigma \left(x\right)\right)z^{{\Delta }^j}\left(x\right)}{{(-1)}^j}\Delta x\hfill \\ \hfill & & ={(-1)}^{j\ell -\ell -j}{\vartheta }^{(j-1)\ell }{\int }_u^{\vartheta }\left\{\sum _{k=0}^{\ell }{\left(z^{{\Delta }^{j-1}}\left(x\right)\right)}^k{\left(z^{{\Delta }^{j-1}}\left(\sigma \left(x\right)\right)\right)}^{\ell -k}\right\}{z}^{{\Delta }^j}\left(x\right)\Delta x\hfill \\ \hfill & & ={(-1)}^{j\ell -\ell -j}{\vartheta }^{(j-1)\ell }{\int }_u^{\vartheta }{\left[{\left(z^{{\Delta }^{j-1}}\left(x\right)\right)}^{\ell +1}\right]}^{\Delta }\Delta x\hfill \\ \hfill & & ={(-1)}^{j\ell -\ell -j+1}{\vartheta }^{(j-1)\ell }{\left(z^{{\Delta }^{j-1}}\right)}^{\ell +1}\left(u\right)\hfill \\ \hfill & & ={(-1)}^{2(nl-\ell )}{\vartheta }^{(j-1)\ell }{\left\{{\int }_u^{\vartheta }{\left({\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}\left(x\right)|\Delta x\right\}}^{\ell +1}\hfill \\ \hfill & & ={\vartheta }^{(j-1)\ell }{\left\{{\int }_u^{\vartheta }{w}^{\frac{\ell }{\ell +1}}\left(x\right){\left({\displaystyle \frac{1}{w^{\ell }\left(x\right){\delta }^{\sigma }\left(x\right)}}\right)}^{\frac{1}{\ell +1}}|{ϝ}^{{\Delta }^j}\left(x\right)|\Delta x\right\}}^{\ell +1}\hfill \\ & & \le {\vartheta }^{(j-1)\ell }{\left\{{\int }_u^{\vartheta }w\left(x\right)\Delta x\right\}}^{\ell }{\int }_u^{\vartheta }{\displaystyle \frac{1}{w^{\ell }\left(x\right){\delta }^{\sigma }\left(x\right)}}|{ϝ}^{{\Delta }^j}\left(x\right){|}^{\ell +1}\Delta x.\hfill \end{array}$$

(22)

Using $v\left(u\right)=\max \{{\int }_0^{u}w\left(x\right)\Delta x,{\int }_u^{\vartheta }w\left(x\right)\Delta x\}$. by adding (21) and (22), gets

$${\int }_0^{\vartheta }{\displaystyle \frac{1}{{\delta }^{\sigma }\left(x\right)}}\left\{\sum _{k=0}^{\ell }{\left(\left|ϝ\left(x\right)\right|-\tilde{\beta }\right)}^k{\left(|{ϝ}^{\sigma }\left(x\right)|-\tilde{\beta }\right)}^{\ell -k}\right\}\left|{ϝ}^{{\Delta }^j}\left(x\right)\right|\Delta x\le {\vartheta }^{(j-1)\ell }{v}^{\ell }\left(u\right){\int }_0^{\vartheta }{\displaystyle \frac{1}{w^{\ell }\left(x\right){\delta }^{\sigma }\left(x\right)}}|{ϝ}^{{\Delta }^j}\left(x\right){|}^{\ell +1}\Delta x,$$

This holds true for any $u\in {[0,\vartheta ]}_{\mathbb{T}}$, hence it holds true if we swap out $v\left(u\right)$ for min $\underset{u\in {[0,\vartheta ]}_{\mathbb{T}}}{min}v\left(u\right)$. It is simple to demonstrate that the final quantity equals $\tilde{\alpha }$. That is the required results. □

Remark 5.

Taking $j=\delta =\ell =w=1$ and $ϝ\left(0\right)=ϝ\left(\vartheta \right)=0$ in Theorem 11, then we recapture (6).

Remark 6.

If we take $\delta =\ell =j=w=1$ in Theorem 11, then we recapture (5).

Corollary 7.

If $\mathbb{T}=\mathbb{R}$ in Theorem 11, then, we obtain

$${\int }_0^{\vartheta }{\displaystyle \frac{1}{\delta \left(x\right)}}{\left(|ϝ\left(x\right|)-\tilde{\beta }\right)}^{\ell }\left|{ϝ}^{\left(j\right)}\left(x\right)\right|dt\le \frac{{\vartheta }^{j\ell }}{2^{\ell }(\ell +1)}{\int }_0^{\vartheta }{\displaystyle \frac{1}{w^{\ell }\left(x\right)\delta \left(x\right)}}|{ϝ}^{\left(j\right)}\left(x\right){|}^{\ell +1}dt,$$

thus

$$\tilde{\beta }=\max \left\{\left|ϝ\right(0\left)\right|,\left|ϝ\right(\vartheta \left)\right|\right\}.$$

Corollary 8.

Taking $\mathbb{T}=\mathbb{Z}$ in Theorem 11, obtains

$$\sum _{x=0}^{\vartheta -1}{\displaystyle \frac{1}{\delta (x+1)}}\left\{\sum _{k=0}^{\ell }{\left(\left|ϝ\left(x\right)\right|-\tilde{\beta }\right)}^k{\left(\left|ϝ(x+1)\right|-\tilde{\beta }\right)}^{\ell -k}\right\}\left|{\Delta }^{j}x\left(x\right)\right|\le {\vartheta }^{(j-1)\ell }{\eta }^{\ell }\sum _{x=0}^{\vartheta -1}{\displaystyle \frac{1}{w^{\ell }(x+1)\delta (x+1)}}|{\Delta }^{j}x\left(x\right){|}^{\ell +1},$$

thus

$$\eta =\min \{\lfloor \vartheta /2\rfloor ,\lceil \vartheta /2\rceil \},and\tilde{\beta }=\max \left\{\left|ϝ\right(0\left)\right|,\left|ϝ\right(\vartheta \left)\right|\right\}.$$

Now, we give an application of Theorem 10.

3. An Application

Theorem 12.

Let ϝ be a solution for

$${ϝ}^{\Delta }\left(x\right)=1-\vartheta +x-\frac{{ϝ}^{2j}\left(x\right)}{{(x-\vartheta )}^{2j-1}},0<x\le \vartheta ,ϝ\left(\vartheta \right)=0,$$

(23)

where $j\in \mathbb{N}$. Gets $ϝ\left(x\right)\le {ϝ}_1\left(x\right)$ on ${[0,\vartheta ]}_{\mathbb{T}}$, where ${ϝ}_1\left(x\right)=x-\vartheta$ solves (23).

Proof. 

Obviously ${ϝ}_1$ solves (23). Now suppose that $ϝ$ is a solution of (23) and for brevity define $U\left(x\right)=1-x+\vartheta +{\int }_{\vartheta }^x|{ϝ}^{\Delta }(\aleph ){|}^{2j}\Delta \aleph$.

Then, for any $x\in {(0,\vartheta ]}_{\mathbb{T}}$, we have

$$\begin{array}{ccc}\hfill \left|{ϝ}^{\Delta }\left(x\right)\right|& =& |1-\vartheta +x-\frac{{ϝ}^{2j}\left(x\right)}{{(x-\vartheta )}^{2j-1}}|\le |1-\vartheta +x|+\frac{1}{{|x-\vartheta |}^{2j-1}}|{ϝ}^{2j}\left(x\right)|\hfill \\ & =& |1-\vartheta +x|+\frac{1}{{|x-\vartheta |}^{2j-1}}|{ϝ}^{2j}\left(x\right)-{ϝ}^{2j}\left(\vartheta \right)|\hfill \\ & =& |1-\vartheta +x|+\frac{1}{{|x-\vartheta |}^{2j-1}}|{\int }_{\vartheta }^x{\left({ϝ}^{2j}(\aleph )\right)}^{\Delta }\Delta \aleph |\le |1-\vartheta +x|+\frac{1}{{|x-\vartheta |}^{2j-1}}{\int }_{\vartheta }^x\left|{\left({ϝ}^{2j}(\aleph )\right)}^{\Delta }\right|\Delta \aleph \hfill \\ & =& |1-\vartheta +x|+\frac{1}{{|x-\vartheta |}^{2j-1}}{\int }_{\vartheta }^x\left|\left\{\sum _{k=0}^{2j-1}{ϝ}^k(\aleph ){\left({ϝ}^{\sigma }\right)}^{2j-1-k}(\aleph )\right\}{ϝ}^{\Delta }(\aleph )\right|\Delta \aleph \hfill \\ \hfill z& \le & 1-x+\vartheta +\frac{1}{{|x-\vartheta |}^{2j-1}}{\int }_{\vartheta }^x\left|\left\{\sum _{k=0}^{2j-1}{ϝ}^k(\aleph ){\left({ϝ}^{\sigma }\right)}^{2j-1-k}(\aleph )\right\}{ϝ}^{\Delta }(\aleph )\right|\Delta \aleph \hfill \\ & \le & 1-x+\vartheta +{\int }_{\vartheta }^x|{ϝ}^{\Delta }(\aleph ){|}^{2j}\Delta \aleph =U\left(x\right),\hfill \end{array}$$

where we applied the inequality (14) has been employed by putting $w=g=1$.

Thus, $U^{\Delta }\left(x\right)=-1+|{ϝ}^{\Delta }\left(x\right){|}^{2j}\le U^{2j}\left(x\right)-1$. Additionally, it is evident that $U\left(\vartheta \right)=1$.

Consider that the initial value problem’s unique solution is z

$$z^{\Delta }\left(x\right)=\left(U^j\left(x\right)+1\right)\left\{\sum _{k=0}^{j-1}{U}^k\left(x\right){\left(U^{\sigma }\right)}^{j-1-k}\left(x\right)\right\}z\left(x\right),z\left(\vartheta \right)=1.$$

Next, as stated by Lemma 1, this z exists because $1+\mu \left(x\right)\left(U^j\left(x\right)+1\right)\left\{{\sum }_{k=0}^{j-1}{U}^k\left(x\right){\left(U^{\sigma }\right)}^{j-1-k}\left(x\right)\right\}>0$.

In fact, $z\left(x\right)>0$ on ${[0,\vartheta ]}_{\mathbb{T}}$. Since $U^{\Delta }\left(x\right)\le U^{2j}\left(x\right)-1$, we have

$$\begin{array}{ccc}& & {\left(\frac{U^j\left(x\right)-1}{z\left(x\right)}\right)}^{\Delta }=\frac{{\left(U^j\left(x\right)-1\right)}^{\Delta }z\left(x\right)-\left(U^j\left(x\right)-1\right)z^{\Delta }\left(x\right)}{z\left(x\right)z^{\sigma }\left(x\right)}\hfill \\ & & =\frac{\left\{{\sum }_{k=0}^{j-1}{U}^k\left(x\right){\left(U^{\sigma }\right)}^{j-1-k}\left(x\right)\right\}{U}^{\Delta }\left(x\right)z\left(x\right)-\left(U^j\left(x\right)-1\right)\left(U^j\left(x\right)+1\right)\left\{{\sum }_{k=0}^{j-1}{U}^k\left(x\right){\left(U^{\sigma }\right)}^{j-1-k}\left(x\right)\right\}z\left(x\right)}{z\left(x\right)z^{\sigma }\left(x\right)}\hfill \\ & & \le \frac{\left\{{\sum }_{k=0}^{j-1}{U}^k\left(x\right){\left(U^{\sigma }\right)}^{j-1-k}\left(x\right)\right\}\left(U^{2j}\left(x\right)-1\right)z\left(x\right)-\left(U^j\left(x\right)-1\right)\left(U^j\left(x\right)+1\right)\left\{{\sum }_{k=0}^{j-1}{U}^k\left(x\right){\left(U^{\sigma }\right)}^{j-1-k}\left(x\right)\right\}z\left(x\right)}{z\left(x\right)z^{\sigma }\left(x\right)}\hfill \\ & & =0,\hfill \end{array}$$

where we used formula (13).

Hence

$$\frac{U^j\left(x\right)-1}{z\left(x\right)}=\frac{U^j\left(\vartheta \right)-1}{z\left(\vartheta \right)}+{\int }_{\vartheta }^x{\left(\frac{U^j(\aleph )-1}{z(\aleph )}\right)}^{\Delta }\Delta \aleph \le 0,$$

and thus $U^j\left(x\right)\le 1$, i.e., $U\left(x\right)\le 1$. Therefore ${ϝ}^{\Delta }\left(x\right)\le \left|{ϝ}^{\Delta }\left(x\right)\right|\le U\left(x\right)\le 1$ and $ϝ\left(x\right)={\int }_{\vartheta }^x{ϝ}^{\Delta }(\aleph )\Delta \aleph \le {\int }_{\vartheta }^x\Delta \aleph =x-\vartheta ={ϝ}_1\left(x\right)$, which is the desired result. □

4. Discussion

Using the integration by parts, Holder’s inequality, and other crucial techniques, we addressed a number of novel, higher order, opioid-type inequalities on temporal scales in this study. Several previously acknowledged inequalities are extended and given more general new shapes by these disparities. In order to create some extra inequalities as special cases, we also extended our inequalities to discrete and continuous calculus.