Asymptotic stability for quaternion-valued BAM neural networks via a contradictory method and two Lyapunov functionals

Ailing Li; Mengting Lv; Yifang Yan; Ailing Li; Mengting Lv; Yifang Yan

doi:10.3934/math.2022457

AIMS Mathematics

2022, Volume 7, Issue 5: 8206-8223. doi: 10.3934/math.2022457

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Research article

Asymptotic stability for quaternion-valued BAM neural networks via a contradictory method and two Lyapunov functionals

1.
College of Science, Hebei North University, Zhangjiakou, 075000, China
2.
School of Mathematics, Hunan University, Changsha, 410082, China

Received: 10 December 2021 Revised: 11 February 2022 Accepted: 14 February 2022 Published: 25 February 2022
MSC : 34K24

We explore the existence and asymptotic stability of equilibrium point for a class of quaternion-valued BAM neural networks with time-varying delays. Firstly, by employing Homeomorphism theorem and a contradictory method with novel analysis skills, a criterion ensuring the existence of equilibrium point of the considered quaternion-valued BAM neural networks is acquired. Secondly, by constructing two Lyapunov functionals, a criterion assuring the global asymptotic stability of equilibrium point for above discussed quaternion-valued BAM is presented. Applying a contradictory method to study the equilibrium point and applying two Lyapunov functionals to study stability of equilibrium point are completely new methods.

Keywords:

Citation: Ailing Li, Mengting Lv, Yifang Yan. Asymptotic stability for quaternion-valued BAM neural networks via a contradictory method and two Lyapunov functionals[J]. AIMS Mathematics, 2022, 7(5): 8206-8223. doi: 10.3934/math.2022457

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Abstract

1. Introduction

Riemann-Liouville fractional integral given by

$\begin{equation*} I_{a+}^{\alpha}\xi(\wp) = \frac{1}{\Gamma(\alpha)}\int_{a}^{\chi}(\chi-\wp)^{\alpha-1}\xi(\wp)dt. \end{equation*}$

Many different concepts of fractional derivative maybe found in ^[9,10,11]. In ^[12] studied a conformable derivative:

$\begin{equation*} \wp_{\alpha}f(\wp) = \lim\limits_{\epsilon\rightarrow 0}\frac{f(\wp+\epsilon \wp^{1-\alpha})-f(\wp)}{\epsilon}. \end{equation*}$

The time scale conformable derivatives was introduced by Benkhettou et al. ^[17].

Further, in recent years, numerous mathematicians claimed that non-integer order derivatives and integrals are well suited to describing the properties of many actual materials, such as polymers. Fractional derivatives are a wonderful tool for describing memory and learning. a variety of materials and procedures inherited properties is one of the most significant benefits of fractional ownership. For more concepts and definition on time scales see ^{[13,14,15,16,17,18,19,33,34,35]}.

Continuous version of Steffensen's inequality ^[7] is written as: For $0\leq g(\wp)\leq1$ on $\in[a, b]$ . Then

$\begin{equation} \int_{b-\lambda}^{b}f(\wp)dt\leq\int_{a}^{b}f(\wp)g(\wp)dt\leq\int_{a}^{a+\lambda}f(\wp)dt, \end{equation}$

(1.1)

where $\lambda = \int_{a}^{b}g(\wp)dt.$

Supposing $f$ is nondecreasing gets the reverse of (1.1).

Also, the discrete inequality of Steffensen ^[6] is: For $\lambda_2\leq \sum_{\ell = 1}^{n}g(\ell)\leq \lambda_1$ . Then

$\begin{equation} \sum\limits_{\ell = n-\lambda_2+1}^{n} f(\ell) \leq \sum\limits_{\ell = 1}^{n}f(\ell)g(\ell) \leq \sum\limits_{\ell = 1}^{\lambda_1} f(\ell). \end{equation}$

(1.2)

Recently, a large number of dynamic inequalities on time scales have been studied by a small number of writers who were inspired by a few applications (see ^{[1,2,3,4,8,28,29,30,31,32,36,37,40,41,42,44,48,49,50,51,52,53]}).

In ^[5] Jakšetić et al. proved that, if $\hat{\mu}([c, d]) = \int_{[a, b]}g(\wp)d \hat{\mu}(\wp)$ , where $[c, d]\subseteq[a, b]$ . Then

$\begin{equation*} \int_{[a,b]}f(\wp)g(\wp)d\hat{\mu}(\wp)\leq \int_{[c,d]}f(\wp)g(\wp)d\hat{\mu}(\wp)+\int_{[a,c]}\big(f(\wp)-f(d)\big)g(\wp)d\hat{\mu}(\wp), \end{equation*}$

and

$\begin{equation*} \int_{[c,d]}f(\wp)d\hat{\mu}(\wp)-\int_{[d,b]}\big(f(c)-f(\wp)\big)g(\wp)d\hat{\mu}(\wp)\leq \int_{[a,b]}f(\wp)g(\wp)d\hat{\mu}(\wp). \end{equation*}$

Anderson, in ^[3], studied the inequality:

$\begin{equation} \int_{b-\lambda}^{b}\phi(\wp)\nabla \wp\leq \int_{a}^{b}\phi(\wp)\psi(\wp)\nabla \wp\leq\int_{a}^{a+\lambda}\phi(\wp)\nabla \wp, \end{equation}$

(1.3)

In ^[47] the authors have proved, for

$\begin{equation*} \int_{m}^{m+\lambda_1}\zeta(\wp)d\wp = \int_{m}^{k}\zeta(\wp)g(\wp)d\wp, \end{equation*}$

and

$\begin{equation*} \int_{n-\lambda_2}^{n}\zeta(\wp)d\wp = \int_{k}^{n}\zeta(\wp)g(\wp)d\wp. \end{equation*}$

If there exists a constant $A$ such that $r(\wp)/\zeta(\wp)-At$ is monotonic on the intervals $[m, k]$ , $[k, n]$ , and

$\begin{equation*} \int_{m}^{n}tq(\wp)g(\wp)d\wp = \int_{m}^{m+\lambda_1}tq(\wp)d\wp+\int_{n-\lambda_2}^{n}tq(\wp)d\wp, \end{equation*}$

then

$\begin{equation*} \int_{m}^{n}r(\wp)g(\wp)d\wp\leq\int_{m}^{m+\lambda_1}r(\wp)d\wp+\int_{n-\lambda_2}^{n}r(\wp)d\wp. \end{equation*}$

In particularly, Anderson ^[3] proved

$\begin{equation*} \int_{n-\lambda}^{n}r(\wp)\nabla \wp\leq\int_{m}^{n}r(\wp)g(\wp)\nabla \wp\leq \int_{m}^{m+\lambda}r(\wp)\nabla \wp. \end{equation*}$

where $m, n\in\mathbb{T}_\kappa$ with $m < n$ , $r$ , $g:[m, n]_{\mathbb{T}}\rightarrow \mathbb{R}$ are $\nabla$ -integrable functions such that $r$ is of one sign and nonincreasing and $0\leq g(\wp)\leq1$ on $[m, n]_{\mathbb{T}}$ and $\lambda = \int_{m}^{n}g(\wp)\nabla \wp$ , $n-\lambda, m+\lambda \in \mathbb{T}$ .

We prove the next two needed results:

Theorem 1.1. Assume $q > 0$ with $0\leq g(\wp)\leq \zeta(\wp)$ $\forall\wp\in[m, n]_{\mathbb{T}}$ and $\lambda$ is given from $\int_{m}^{n}g(\wp)\Delta_{\alpha} \wp = \int_{m}^{m+\lambda}\zeta(\wp)\Delta_{\alpha} \wp$ , then

$\begin{equation} \int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp\leq \int_{m}^{m+\lambda}r(\wp)\zeta(\wp)\Delta_{\alpha} \wp. \end{equation}$

(1.4)

Also, provided with $0\leq g(\wp)\leq \zeta(\wp)$ and $\int_{n-\lambda}^{n}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{n}g(\wp)\Delta_{\alpha} \wp$ , we have

$\begin{equation} \int_{n-\lambda}^{n}r(\wp)\zeta(\wp)\Delta_{\alpha} \wp\leq \int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp. \end{equation}$

(1.5)

We get the reverse inequalities of (1.4) and (1.5) when assuming $r/\zeta$ is nondecreasing.

Theorem 1.2. Assume $\psi$ is integrable on time scales interval $[m, n]$ , with $\zeta(\wp)-\psi(\wp)\geq g(\wp)\geq\psi(\wp)\geq 0 \; \forall\wp\in[m, n]_{\mathbb{T}}$ and $\int_{m}^{m+\lambda}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{n}g(\wp)\Delta_{\alpha} \wp = \int_{n-\lambda}^{n}\zeta(\wp)\Delta_{\alpha} \wp$ and $g$ , $r$ and $\zeta$ are $\Delta_{\alpha}$ -integrable functions, $\zeta(\wp)\geq g(\wp)\geq0$ , we have

$\begin{eqnarray} &&\int_{n-\lambda}^{n}r(\wp)\zeta(\wp)\Delta_{\alpha} \wp+ \int_{m}^{n}\Big|\big(r(\wp)-r(n-\lambda)\big)\psi(\wp)\Big|\Delta_{\alpha} \wp\\ &&\leq \int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\leq \int_{m}^{m+\lambda}r(\wp)\zeta(\wp)\Delta_{\alpha} \wp-\int_{m}^{n}\Big|\big(r(\wp)-r(m+\lambda)\big)\psi(\wp)\bigg|\Delta_{\alpha} \wp, \end{eqnarray}$

(1.6)

and

$\begin{eqnarray} \int_{n-\lambda}^{n}r(\wp)\zeta(\wp)\Delta_{\alpha} \wp&\leq& \int_{n-\lambda}^{n}\Big[r(\wp)\zeta(\wp)-\big(r(\wp)-r(n-\lambda)\big)\Big]\big[\zeta(\wp)-g(\wp)\big]\Delta_{\alpha} \wp\\ &\leq&\int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp\\ &\leq& \int_{m}^{m+\lambda}\Big[r(\wp)\zeta(\wp)-\big(r(\wp)-r(m+\lambda)\big)\Big]\big[\zeta(\wp)-g(\wp)\big]\Delta_{\alpha} \wp\\ &\leq&\int_{m}^{m+\lambda}r(\wp)\zeta(\wp)\Delta_{\alpha} \wp. \end{eqnarray}$

(1.7)

Proof. The proof techniques of Theorems 1.6 and 1.7 are like to that in ^[4] and is removed.

Several authors proved conformable Hardy's inequality ^[20,21], conformable Hermite-Hadamard's inequality ^[22,23,24], conformable inequality of Opial's ^[26,27] and conformable inequality of Steffensen's ^[25]. In ^[45] Anderson proved the followong results:

Theorem 1.3. ^[45] Suppose $\alpha\in(0, 1]$ and $r_1$ , $r_2\in \mathbb{R}$ such that $0\leq r_1\leq r_2$ . Suppose $\prod: [r_1, r_2]\rightarrow [0, \infty)$ and $\Gamma: [r_1, r_2]\rightarrow [0, 1]$ are $\alpha$ -fractional integrable functions on $[r_1, r_2]$ with $\Pi$ is decreasing, we get

$\begin{equation*} \int_{r_2-\aleph}^{r_2}\Pi(\zeta)d_{\alpha}\zeta\leq \int_{r_1}^{r_2}\Pi(\zeta)\Gamma(\zeta)d_{\alpha}\zeta\leq \int_{r_1}^{r_1+\aleph}\Pi(\zeta)d_{\alpha}\zeta, \end{equation*}$

where $\aleph = \frac{\alpha(r_2-r_1)}{r_2^{\alpha}-r_1^{\alpha}}\int_{r_1}^{r_2}\Gamma(\zeta)d_{\alpha}\zeta\in[0, r_2-r_1]$ .

In ^[46] the authors gave an extension for Theorem 1.8:

Theorem 1.4. Assume $\alpha\in(0, 1]$ and $r_1$ , $r_2\in \mathbb{R}$ such that $0\leq r_1\leq r_2$ . Suppose $\prod, \Gamma, \Sigma: [r_1, r_2]\rightarrow [0, \infty)$ are integrable on $[r_1, r_2]$ with the decreasing function $\Pi$ and $0\leq \Gamma\leq \Sigma$ , we get

$\begin{equation*} \int_{r_2-\aleph}^{r_2}\Sigma(\zeta)\Pi(\zeta)d_{\alpha}\zeta\leq \int_{r_1}^{r_2}\Pi(\zeta)\Gamma(\zeta)d_{\alpha}\zeta\leq \int_{r_1}^{r_1+\aleph}\Sigma(\zeta)\Pi(\zeta)d_{\alpha}\zeta, \end{equation*}$

where $\aleph = \frac{(r_2-r_1)}{\int_{r_1}^{r_2}\Sigma(\zeta)d_{\alpha}\zeta}\int_{r_1}^{r_2}\Gamma(\zeta) d_{\alpha}\zeta\in[0, r_2-r_1]$ .

In this paper, we prove and explore several novel speculations of the Steffensen inequality obtained in ^[47] through the conformable integral containing time scale concept. We furthermore recover certain known results as special cases of our results.

2. Main results

Lemma 2.1. Assume $\zeta > 0$ is $rd$ -continuous function on $[m, n]\cap\mathbb{T}$ , $g$ , $r$ be $rd$ -continuous on $[m, n]\cap\mathbb{T}$ such that $r/\zeta$ nonincreasing function and $0\leq g(\wp)\leq 1$ $\forall\wp\in[m, n]\cap\mathbb{T}$ . Then

$(\Lambda_{1})$

$\begin{equation} \qquad \int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp\leq \int_{m}^{m+\lambda}r(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.1)

where $\lambda$ is given by

$\begin{equation*} \int_{m}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{m}^{m+\lambda}\zeta(\wp)\Delta_{\alpha} \wp. \end{equation*}$

$(\Lambda_{2})$

$\begin{equation} \qquad\int_{n-\lambda}^{n}r(\wp)\Delta_{\alpha} \wp\leq \int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.2)

such that

$\begin{equation*} \int_{n-\lambda}^{n}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

(2.1) and (2.2) are reversed when $r/\zeta$ is nondecreasing.

Proof. Putting $g(\wp)\mapsto \zeta(\wp)g(\wp)$ and $r(\wp)\mapsto r(\wp)/\zeta(\wp)$ in (1.4), (1.5) to get $(\Lambda_{1})$ and $(\Lambda_{2})$ simultaneously.

Lemma 2.2. Under the same hypotheses of Lemma 2.1. with $\psi$ be integrable functions on $[m, n]\cap\mathbb{T}$ and $0\leq \psi(\wp)\leq g(\wp)\leq 1-\psi(\wp)$ for all $\wp\in[m, n]_{\mathbb{T}}$ . Then

$\begin{eqnarray*} &&\int_{n-\lambda}^{n}r(\wp)\Delta_{\alpha} \wp+ \int_{m}^{n}\bigg|\bigg(\frac{r(\wp)}{\zeta(\wp)}-\frac{r(n-\lambda)}{\zeta(n-\lambda)}\bigg)\zeta(\wp)\psi(\wp)\bigg|\Delta_{\alpha} \wp\\ &&\leq \int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\leq \int_{m}^{m+\lambda}r(\wp)\Delta_{\alpha} \wp-\int_{m}^{n}\bigg|\bigg(\frac{r(\wp)}{\zeta(\wp)}-\frac{r(m+\lambda)}{\zeta(m+\lambda)}\bigg)\zeta(\wp)\psi(\wp)\bigg|\Delta_{\alpha} \wp, \end{eqnarray*}$

where $\lambda$ is obtained from

$\begin{equation*} \int_{m}^{m+\lambda}h(\wp)\Delta_{\alpha} \wp = \int_{m}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{n-\lambda}^{n}\zeta(\wp)\Delta_{\alpha} \wp. \end{equation*}$

Proof. Putting $g(\wp)\mapsto \zeta(\wp)g(\wp)$ , $r(\wp)\mapsto r(\wp)/h(\wp)$ and $\psi(\wp)\mapsto \zeta(\wp)\psi(\wp)$ in (1.6).

Lemma 2.3. Under the same conditions of Lemma 2.1. Then

$\begin{eqnarray*} \int_{n-\lambda}^{n}r(\wp)\Delta_{\alpha} \wp&\leq& \int_{n-\lambda}^{n}\bigg(r(\wp)-\bigg[\frac{r(\wp)}{\zeta(\wp)}-\frac{r(n-\lambda)}{\zeta(n-\lambda)}\bigg]\zeta(\wp)\big[1-g(\wp)\big]\bigg)\Delta_{\alpha} \wp\\ &\leq&\int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp\\ &\leq& \int_{m}^{m+\lambda}\bigg(r(\wp)-\bigg[\frac{r(\wp)}{\zeta(\wp)}-\frac{r(a+\lambda)}{\zeta(m+\lambda)}\bigg]\zeta(\wp)\big[1-g(\wp)\big]\bigg)\Delta_{\alpha} \wp\\ &\leq&\int_{m}^{m+\lambda}r(\wp)\Delta_{\alpha} \wp, \end{eqnarray*}$

where $\lambda$ is obtained from

$\begin{equation*} \int_{m}^{m+\lambda}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{n}g(\wp)\Delta_{\alpha} \wp = \int_{n-\lambda}^{n}\zeta(\wp)\Delta_{\alpha} \wp. \end{equation*}$

Proof. Taking $g(\wp)\mapsto \zeta(\wp)g(\wp)$ and $r(\wp)\mapsto r(\wp)/\zeta(\wp)$ in (1.7).

Theorem 2.1. Under the same conditions of Lemma 2.3 such that $k\in(m, n)$ and $\lambda_1$ , $\lambda_2$ are given from

$(\Lambda_{3})$

$\begin{equation*} \int_{m}^{m+\lambda_1}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{k}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp, \end{equation*}$

$\begin{equation*} \int_{n-\lambda_2}^{n}\zeta(\wp)\Delta_{\alpha} \wp = \int_{k}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{equation} \int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{m}^{m+\lambda_1}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.3)

then

$\begin{equation} \int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)\Delta_{\alpha} \wp. \end{equation}$

(2.4)

(2.4) is reversed if $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.3).

$(\Lambda_{4})$

$\begin{equation*} \int_{k-\lambda_1}^{k}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{k}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp, \end{equation*}$

$\begin{equation*} \int_{k}^{k+\lambda_2}\zeta(\wp)\Delta_{\alpha} \wp = \int_{k}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{equation} \int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{k-\lambda_1}^{k+\lambda_2}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.5)

then

$\begin{equation} \int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)\Delta_{\alpha} \wp. \end{equation}$

(2.6)

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.5) satisfied, then we reverse (2.6).

$(\Lambda_{5})$ If $\lambda_1$ , $\lambda_2$ be the same as in $(\Lambda_{3})$ and $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ so that

$\begin{eqnarray} &&\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\\ && = \int_{m}^{m+\lambda_1}\bigg(\phi(\wp)\zeta(\wp)-[\phi(\wp)-m-\lambda_1]\zeta(\wp)[1-g(\wp)]\bigg)\Delta_{\alpha} \wp \\ &&\quad +\int_{n-\lambda_2}^{n}\bigg(\phi(\wp)\zeta(\wp)-[\phi(\wp)-n+\lambda_2]\zeta(\wp)[1-g(\wp)]\bigg)\Delta_{\alpha} \wp, \end{eqnarray}$

(2.7)

then

$\begin{eqnarray} &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\leq\int_{m}^{m+\lambda_1}\bigg(r^{\sigma}(\wp) -\Big|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(m+\lambda_1)}{\zeta(m+\lambda_1)}\Big|\zeta(\wp)[1-g(\wp)]\bigg)\Delta_{\alpha} \wp\\ &&\quad +\int_{n-\lambda_2}^{n}\bigg(r^{\sigma}(\wp)-\Big|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(n-\lambda_2)}{\zeta(n-\lambda_2)}\Big|\zeta(\wp)[1-g(\wp)]\bigg)\Delta_{\alpha} \wp. \end{eqnarray}$

(2.8)

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.7) satisfied, the inequality in (2.8) is reversed.

$(\Lambda_{6})$ If $\lambda_1$ , $\lambda_2$ be defined as in $(\Lambda_{4})$ and $r^{\sigma}/\zeta\in\mathbb{AH}_1^k [m, n]$ and

$\begin{eqnarray} &&\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\\ && = \int_{k-\lambda_1}^{k}\Big(\phi(\wp)\zeta(\wp)-[\phi(\wp)-k+\lambda_1]\zeta(\wp)[1-g(\wp)]\Big)\Delta_{\alpha} \wp\\ && = \int_{m}^{m+\lambda_1}\Big(\phi(\wp)\zeta(\wp)-[\phi(\wp)-k+\lambda_2]\zeta(\wp)[1-g(\wp)]\Big)\Delta_{\alpha} \wp, \end{eqnarray}$

(2.9)

then

$\begin{eqnarray} &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\geq\int_{k-\lambda_1}^{k}\bigg(r^{\sigma}(\wp)-\Big[\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k-\lambda_1)}{\zeta(k-\lambda_1)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)\Delta_{\alpha} \wp\\ &&\quad +\int_{k}^{k+\lambda_2}\bigg(r^{\sigma}(\wp)-\Big[\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k+\lambda_2)}{\zeta(k+\lambda_2)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)\Delta_{\alpha} \wp. \end{eqnarray}$

(2.10)

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.9) satisfied, we reverse (2.10).

Proof. $(\Lambda_{3})$ Consider $r^{\sigma}/\zeta\in\mathbb{AH}_1^k [m, n]$ , and $R_{1}(\ell) = r^{\sigma}(\ell)-A\phi(\ell)\zeta(\ell)$ , since $A$ is given in Definition 2.1. Since $R_{1}/\zeta:[m, k]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ , using Lemma 2.1 $(\Lambda_{1})$ , we deduce

$\begin{eqnarray} 0&\leq&\int_{m}^{m+\lambda_1}R_{1}(\wp)\Delta_{\alpha} \wp-\int_{m}^{k}R_{1}(\wp)g(\wp)\Delta_{\alpha} \wp\\ & = &\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)\Delta_{\alpha} \wp-\int_{m}^{k}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&-A\bigg(\int_{m}^{m+\lambda_1}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp-\int_{m}^{k}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\bigg). \end{eqnarray}$

(2.11)

As $R_{1}/\zeta:[k, n]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ is nondecreasing, using Lemma 2.1 $(\Lambda_{2})$ , we obtain

$\begin{eqnarray} 0&\geq&\int_{k}^{n}R_{1}(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{n-\lambda_2}^{n}R_{1}(\wp)\Delta_{\alpha} \wp\\ & = &\int_{k}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)\Delta_{\alpha} \wp\\ &&-A\bigg(\int_{k}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{n-\lambda_2}^{n}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp\bigg). \end{eqnarray}$

(2.12)

(2.11) and (2.12) imply that

$\begin{eqnarray*} &&\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)\Delta_{\alpha} \wp-\int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\geq A\bigg(\int_{m}^{m+\lambda_1}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp -\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\bigg) \end{eqnarray*}$

Hence, if (2.3) is hold, then (2.4) holds. For $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ , we get the some steps.

$(\Lambda_{4})$ Let $r^{\sigma}/\zeta\in\mathbb{AH}_1^k [m, n]$ , also $R_{1}(x) = r^{\sigma}(x)-A\phi(x)\zeta(x)$ , where $A$ as in Definition 2.1. $R_{1}/\zeta:[m, k]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ is nonincreasing, so from Lemma 2.1 $(\Lambda_{1})$ we obtain

$\begin{eqnarray} 0&\leq&\int_{m}^{k}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{k-\lambda_1}^{k}r^{\sigma}(\wp)\Delta_{\alpha} \wp\\ &&-A\bigg(\int_{m}^{k}\phi(\wp)h(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{c-\lambda_1}^{k}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp\bigg). \end{eqnarray}$

(2.13)

Using Lemma 2.1 $(\Lambda_{1})$ we have

$\begin{eqnarray} 0&\geq&\int_{k}^{k+\lambda_2}r^{\sigma}(\wp)\Delta_{\alpha} \wp-\int_{k}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&-A\bigg(\int_{k}^{k+\lambda_2}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp-\int_{k}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\bigg). \end{eqnarray}$

(2.14)

Thus, from (2.13), (2.14), we get

$\begin{eqnarray*} &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)\Delta_{\alpha} \wp\\ &&\geq A\bigg(\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{k-\lambda_1}^{k+\lambda_2}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp\bigg) \end{eqnarray*}$

Therefore, if $\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{k-\lambda_1}^{k+\lambda_2}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp$ is satisfied, then (2.8) holds. Follow the same steps for $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ .

Using Lemma 2.3 and repeat the steps of Theorem 2.1 $(\Lambda_{3})$ and Theorem 2.1 $(\Lambda_{4})$ in the proof of $(\Lambda_{5})$ and $(\Lambda_{6})$ respectively.

Corollary 2.1. The inequalities (2.4), (2.6), (2.8) and (2.10) of Theorem 2.1 letting $\mathbb{T} = \mathbb{R}$ takes

$\begin{equation} (i)\qquad \int_{m}^{n}f^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)d_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)d_{\alpha} \wp. \end{equation}$

(2.15)

$\begin{equation} (ii)\qquad \int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)d_{\alpha} \wp. \end{equation}$

(2.16)

$\begin{eqnarray} (iii)\qquad &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\\ &&\leq\int_{m}^{m+\lambda_1}\bigg(r^{\sigma}(\wp)-\Big[\frac{r^{\sigma}(\wp)}{\zeta(\wp)} -\frac{r^{\sigma}(m+\lambda_1)}{\zeta(m+\lambda_1)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)d_{\alpha} \wp\\ &&\quad +\int_{n-\lambda_2}^{n}\bigg(r^{\sigma}(\wp)-\Big[\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(n-\lambda_2)}{\zeta(n-\lambda_2)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)d_{\alpha} \wp. \end{eqnarray}$

(2.17)

$\begin{eqnarray} (iv)\qquad &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\\ &&\geq\int_{k-\lambda_1}^{k}\bigg(r^{\sigma}(\wp)-\Big[\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k-\lambda_1)}{\zeta(k-\lambda_1)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)d_{\alpha} \wp\\ &&\quad +\int_{k}^{k+\lambda_2}\bigg(r^{\sigma}(\wp)-\Big[\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k+\lambda_2)}{\zeta(k+\lambda_2)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)d_{\alpha} \wp. \end{eqnarray}$

(2.18)

Corollary 2.2. We get [,Theorems 8,10,21 and 22], if we put $\alpha = 1$ and $\phi(\wp) = \wp$ in Corollary 2.1 $[(i), (ii), (iii), (iv)]$ simultaneously.

Corollary 2.3. In Corollary 2.1 taking $\mathbb{T} = \mathbb{Z}$ , the results (2.15)–(2.18) will be equivalent to

$\begin{equation*} (i)\qquad \sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\leq\sum\limits_{\wp = m}^{m+\lambda_1-1}r(\wp+1)+\sum\limits_{\wp = n-\lambda_2}^{n-1}r(\wp+1)\wp^{\alpha-1}. \end{equation*}$

$\begin{equation*} (ii)\qquad \sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\geq\sum\limits_{\wp = k-\lambda_1}^{k+\lambda_2-1}r(\wp+1)\wp^{\alpha-1}. \end{equation*}$

$\begin{eqnarray*} (iii)\qquad \nonumber&&\sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\\ \nonumber&&\leq\sum\limits_{\wp = m}^{m+\lambda_1-1}\bigg(r(\wp+1)-\Big[\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(a+\lambda_1+1)}{\zeta(m+\lambda_1)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)\wp^{\alpha-1}\\ &&\quad +\sum\limits_{\wp = n-\lambda_2}^{n-1}\bigg(r(\wp+1)-\Big[\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(n-\lambda_2+1)}{\zeta(n-\lambda_2)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)\wp^{\alpha-1}. \end{eqnarray*}$

$\begin{eqnarray*} (iv)\qquad \nonumber&&\sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp))\wp^{\alpha-1}\\ \nonumber&&\geq\sum\limits_{\wp = k-\lambda_1}^{k-1}\bigg(r(\wp+1)-\Big[\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(k-\lambda_1+1)}{\zeta(k-\lambda_1)}\Big]\zeta(\wp)[1-g(\wp)]\bigg))\wp^{\alpha-1}\\ &&\quad +\sum\limits_{\wp = k}^{k+\lambda_2-1}\bigg(r(\wp+1)-\Big[\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(k+\lambda_2+1)}{\zeta(k+\lambda_2)}\Big]\zeta(\wp)[1-g(\wp)]\bigg))\wp^{\alpha-1}. \end{eqnarray*}$

Theorem 2.2. Under the assumptions in Lemma 2.1 with $0\leq g(\wp)\leq \zeta(\wp)$ and $\lambda_1$ , $\lambda_2$ be defined as

$(\Lambda_{7})$

$\begin{equation*} \int_{m}^{m+\lambda_1}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{k}g(\wp)\Delta_{\alpha} \wp, \end{equation*}$

$\begin{equation*} \int_{n-\lambda_2}^{n}\zeta(\wp)\Delta_{\alpha} \wp = \int_{k}^{n}g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{equation} \int_{m}^{n}\phi(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{m}^{m+\lambda_1}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.19)

then

$\begin{equation} \int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)\zeta(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)\zeta(\wp)\Delta_{\alpha} \wp. \end{equation}$

(2.20)

$(\Lambda_{8})$

$\begin{equation*} \int_{k-\lambda_1}^{k}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{k}g(\wp)\Delta_{\alpha} \wp, \end{equation*}$

$\begin{equation*} \int_{k}^{k+\lambda_2}\zeta(\wp)\Delta_{\alpha} \wp = \int_{k}^{n}g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{equation} \int_{m}^{n}\phi(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{k-\lambda_1}^{k+\lambda_2}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.21)

then

$\begin{equation} \int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)\zeta(\wp)\Delta_{\alpha} \wp. \end{equation}$

(2.22)

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.19), (2.21) satisfied, we get the reverse of (2.20) and (2.22).

Proof. By using Theorem 2.1 $[(\Lambda_{3}), (\Lambda_{4})]$ and by putting $g\mapsto g/h$ and $f\mapsto fh$ , we get the proof of $(\Lambda_{7})$ and $(\Lambda_{8})$ .

Corollary 2.4. In Theorem 2.2 $[(\Lambda_{7}), (\Lambda_{8})]$ , assuming $\mathbb{T} = \mathbb{R}$ , the following results obtains:

$\begin{equation} (i)\qquad \int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)\zeta(\wp)d_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)\zeta(\wp)d_{\alpha} \wp. \end{equation}$

(2.23)

$\begin{equation} (ii)\qquad \int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)\zeta(\wp)d_{\alpha} \wp. \end{equation}$

(2.24)

Corollary 2.5. In Corollary 2.4 $[(i), (ii)]$ , when we put $\alpha = 1$ and $\phi(\wp) = \wp$ then [47,Theorems 16 and 17] gotten.

Corollary 2.6. In (2.23) and (2.24) letting $\mathbb{T} = \mathbb{Z}$ , gets

$\begin{equation*} (i)\qquad \sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\leq\sum\limits_{\wp = m}^{m+\lambda_1-1}r(\wp+1)h(\wp)+\sum\limits_{\wp = n-\lambda_2}^{n-1}r(\wp+1)h(\wp)\wp^{\alpha-1}. \end{equation*}$

$\begin{equation*} (ii)\qquad \sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\geq\sum\limits_{\wp = k-\lambda_1}^{k+\lambda_2-1}r(\wp+1)\zeta(\wp)\wp^{\alpha-1}. \end{equation*}$

Theorem 2.3. Using the same conditions in Lemma 2.3. Letting $w:[m, n]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ be integrable with $0\leq g(\wp)\leq w(\wp)$ $\forall \wp\in[m, n]\cap{\mathbb{T}}$ and

$\begin{equation*} (\Lambda_{9})\qquad \int_{m}^{m+\lambda_1}w(\wp)\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{k}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp, \end{equation*}$

$\begin{equation*} \int_{n-\lambda_2}^{n}w(\wp)\zeta(\wp)\Delta_{\alpha} \wp = \int_{k}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{equation} \int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{m}^{m+\lambda_1}\phi(\wp)w(\wp)\zeta(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}\phi(\wp)w(\wp)\zeta(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.25)

then

$\begin{equation} \int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)w(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)w(\wp)\Delta_{\alpha} \wp. \end{equation}$

(2.26)

$\begin{equation*} (\Lambda_{10})\qquad \int_{k-\lambda_1}^{k}w(\wp)\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{k}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp, \end{equation*}$

$\begin{equation*} \int_{k}^{k+\lambda_2}w(\wp)\zeta(\wp)\Delta_{\alpha} \wp = \int_{k}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{equation} \int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{k-\lambda_1}^{k+\lambda_2}\phi(\wp)w(\wp)\zeta(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.27)

$\begin{equation} \int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)w(\wp)\Delta_{\alpha} \wp. \end{equation}$

(2.28)

The inequalities in (2.26) and (2.28) are reversible if $r^{\sigma}/\zeta\in\mathbb{AH}_2^c [a, b]$ and (2.25), (2.27) hold.

Proof. In Theorem 2.1 $[(\Lambda_{3}), (\Lambda_{4})]$ , $\zeta$ changes $wq$ , $g$ changes $g/w$ and $r$ changes $rw$ .

Corollary 2.7. In (2.26) and (2.28). Letting $\mathbb{T} = \mathbb{R}$ , we have

$\begin{equation} (i)\qquad \int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)w(\wp)d_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)w(\wp)d_{\alpha} \wp. \end{equation}$

(2.29)

$\begin{equation} (ii)\qquad \int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)w(\wp)d_{\alpha} \wp. \end{equation}$

(2.30)

Corollary 2.8. In Corollary 2.7 $[(i), (ii)]$ , letting $\alpha = 1$ and $\phi(\wp) = \wp$ we get [47,Theorems 18 and 19].

Corollary 2.9. In (2.29) and (2.30), crossing $\mathbb{T} = \mathbb{Z}$ , gets

$\begin{equation*} (i)\qquad \sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\leq\sum\limits_{\wp = m}^{m+\lambda_1-1}r(\wp+1)w(\wp)+\sum\limits_{\wp = n-\lambda_2}^{n-1}r(\wp+1)w(\wp)\wp^{\alpha-1}. \end{equation*}$

$\begin{equation*} (ii)\qquad \sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\geq\sum\limits_{\wp = k-\lambda_1}^{k+\lambda_2-1}r(\wp+1)w(\wp)\wp^{\alpha-1}. \end{equation*}$

Theorem 2.4. Using the same conditions in Lemma 2.1, and Theorem 2.1 $[(\Lambda_{3}), (\Lambda_{4})]$ with $\psi:[m, n]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ be a integrable: $0\leq \psi(\wp)\leq g(\wp)\leq 1-\psi(\wp)$ .

$(\Lambda_{11})$ If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{eqnarray} \qquad &&\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\\ && = \int_{m}^{m+\lambda_1}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp-\int_{m}^{k}\big|\phi(\wp)-m-\lambda_1\big|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp +\int_{n-\lambda_2}^{n}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp\\ &&\quad +\int_{k}^{n}\big|\phi(\wp)-n+\lambda_2\big|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp, \end{eqnarray}$

(2.31)

then

$\begin{eqnarray} &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)\Delta_{\alpha} \wp -\int_{m}^{k}\Big|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(m+\lambda_1)}{\zeta(m+\lambda_1)}\Big|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)\Delta_{\alpha} \wp\\ &&\quad +\int_{k}^{n}\Big|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(n-\lambda_2)}{\zeta(n-\lambda_2)}\Big|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp. \end{eqnarray}$

(2.32)

$(\Lambda_{12})$ If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{eqnarray} &&\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\\ && = \int_{k-\lambda_1}^{k}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp-\int_{m}^{k}\big|\phi(\wp)-k+\lambda_1\big|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp\\ &&+\int_{k}^{n}\big|\phi(\wp)-k-\lambda_1\big|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp, \end{eqnarray}$

(2.33)

then

$\begin{eqnarray} &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)\Delta_{\alpha} \wp+\int_{m}^{k}\bigg|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k-\lambda_1)}{\zeta(k-\lambda_1)}\bigg|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp\\ &&\quad-\int_{k}^{n}\bigg|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k+\lambda_2)}{\zeta(k+\lambda_2)}\bigg|\zeta(\wp)\psi(\wp) \Delta_{\alpha} \wp. \end{eqnarray}$

(2.34)

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.31) and (2.33) satisfied, we get the reverse of (2.32) and (2.34).

Proof. The same steps of Theorem 2.1 $[(\Lambda_{3}), (\Lambda_{4})]$ with Lemma 2.1, $R_{1}/\zeta:[m, k]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ nonincreasing, $R_{1}/\zeta:[k, n]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ nondecreasing.

Corollary 2.10. In Theorem 2.4 $[(\Lambda_{11}), (\Lambda_{12})]$ , letting $\mathbb{T} = \mathbb{R}$ we get:

$\begin{eqnarray} (i)\qquad &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\\ &&\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)d_{\alpha} \wp -\int_{m}^{k}\Big|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(m+\lambda_1)}{\zeta(m+\lambda_1)}\Big|\zeta(\wp)\psi(\wp)d_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)d_{\alpha} \wp\\ &&\quad +\int_{k}^{n}\Big|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(n-\lambda_2)}{\zeta(n-\lambda_2)}\Big|\zeta(\wp)\psi(\wp)d_{\alpha} \wp. \end{eqnarray}$

(2.35)

$\begin{eqnarray} (ii)\qquad &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\\ &&\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)d_{\alpha} \wp+\int_{m}^{k}\bigg|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k-\lambda_1)}{\zeta(k-\lambda_1)}\bigg|\zeta(\wp)\psi(\wp)d_{\alpha} \wp\\ &&\quad-\int_{k}^{n}\bigg|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k+\lambda_2)}{\zeta(k+\lambda_2)}\bigg|\zeta(\wp)\psi(\wp)d_{\alpha} \wp. \end{eqnarray}$

(2.36)

Corollary 2.11. In (2.35) and (2.36), we put $\alpha = 1$ , with $\phi(\wp) = \wp$ we get [47,Theorems 23 and 24].

Corollary 2.12. Our results (2.35) and (2.36), by using $\mathbb{T} = \mathbb{Z}$ gets

$\begin{eqnarray*} (i)\qquad \nonumber&&\sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\\ \nonumber&&\leq\sum\limits_{\wp = m}^{m+\lambda_1-1}r(\wp+1)\wp^{\alpha-1} -\sum\limits_{\wp = m}^{k-1}\Big|\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(m+\lambda_1+1)}{\zeta(m+\lambda_1)}\Big|\zeta(\wp)\psi(\wp)\hat{\nabla} \wp\\ &&\quad+\sum\limits_{\wp = n-\lambda_2}^{n-1}r(\wp+1)\wp^{\alpha-1}+\sum\limits_{\wp = k}^{n-1}\Big|\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(n-\lambda_2+1)}{\zeta(n-\lambda_2)}\Big|\zeta(\wp)\psi(\wp)\wp^{\alpha-1}. \end{eqnarray*}$

$\begin{eqnarray*} (ii)\qquad\nonumber&&\sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\\ \nonumber&&\geq\sum\limits_{\wp = k-\lambda_1}^{k+\lambda_2-1}r(\wp+1)\wp^{\alpha-1}+\sum\limits_{\wp = m}^{k-1}\bigg|\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(k-\lambda_1+1)}{\zeta(k-\lambda_1)} \bigg|\zeta(\wp)\psi(\wp)\wp^{\alpha-1}\\ &&\quad-\sum\limits_{\wp = k}^{n-1}\bigg|\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(k+\lambda_2+1)}{\zeta(k+\lambda_2)}\bigg|h(\wp)\psi(\wp)\wp^{\alpha-1}. \end{eqnarray*}$

3. Conclusions

In this work, we explore new generalizations of the integral Steffensen inequality given in ^[38,39,43] by the utilization of the $\alpha$ -conformable derivatives and integrals, A few of these results are generalised to time scales. We also obtained the discrete and continuous case of our main results, in order to gain some fresh inequalities as specific cases.

Acknowledgments

The authors extend their appreciation to the Research Supporting Project number (RSP-2022/167), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no conflict of interest.

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