Electrical Resistivity Tomography Investigation of Permafrost Conditions in a Thermokarst Site in Fairbanks, Alaska

Abdallah Basiru; Shishay T Kidanu; Sergei Rybakov; Nicholas Hasson; Moustapha Kebe; Emmanuel Osei Acheampong; Abdallah Basiru; Shishay T Kidanu; Sergei Rybakov; Nicholas Hasson; Moustapha Kebe; Emmanuel Osei Acheampong

doi:10.3934/geosci.2024001

AIMS Geosciences

2024, Volume 10, Issue 1: 1-27. doi: 10.3934/geosci.2024001

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Research article Topical Sections

Electrical Resistivity Tomography Investigation of Permafrost Conditions in a Thermokarst Site in Fairbanks, Alaska

1.
Department of Civil, Geological, and Environmental Engineering, University of Alaska Fairbanks, 1760 Tanana Loop Duckering 245, USA
2.
Geophysical Institute, University of Alaska-Fairbanks, 903 N Koyukuk Drive, Fairbanks, AK, 99709, USA
3.
Department of Mining and Minerals Engineering, University of Alaska Fairbanks, 1760 Tanana Loop Duckering 245, USA
4.
Department of Geological Sciences, Ohio University, Clippinger Laboratories 316 Ohio University Athens, OH 45701, USA

Received: 12 September 2023 Revised: 23 November 2023 Accepted: 01 December 2023 Published: 09 January 2024

The degradation of permafrost poses severe environmental threats to communities in cold regions. As near-surface permafrost warms, extensive topographic variability is prevalent in the Arctic and Sub-Arctic communities. Geologic hazards such as thermokarst are formed due to varying rates of permafrost degradation, resulting in ground subsidence. This gradual subsidence or abrupt collapse of the earth causes a danger to existing infrastructure and the economic activities of communities in cold regions. Understanding the causes of thermokarst development and its dynamics requires imaging its underground morpho-structures and characterizing the surface and subsurface controls. In this study, we conducted a two-dimensional (2D) electrical resistivity tomography (ERT) survey to characterize the permafrost conditions in a thermokarst prone site located in Fairbanks, Alaska. To increase the reliability in the interpretability of the ERT data, borehole data and the depth-of-investigation (DOI) methods were applied. By using the 2D and three-dimensional (3D) ERT methods, we gained valuable information on the spatial variability of transient processes, such as the movement of freezing and thawing fronts. Resistivity imaging across the site exhibited distinct variations in permafrost conditions, with both low and high resistive anomalies observed along the transects. These anomalies, representing taliks and ice wedges, were characterized by resistivity values ranging from 50 Ωm and above 700 Ωm, respectively. The results from this study showed the effectiveness of ERT to characterize permafrost conditions and thermokarst subsurface morpho-structures. The insights gained from this research contribute to a better understanding of the causes and dynamics of thermokarst, which can be instrumental for engineers in developing feasible remedial measures.

Keywords:

Citation: Abdallah Basiru, Shishay T Kidanu, Sergei Rybakov, Nicholas Hasson, Moustapha Kebe, Emmanuel Osei Acheampong. Electrical Resistivity Tomography Investigation of Permafrost Conditions in a Thermokarst Site in Fairbanks, Alaska[J]. AIMS Geosciences, 2024, 10(1): 1-27. doi: 10.3934/geosci.2024001

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Abstract

1. Introduction

First we give the definitions of generalized fractional integral operators which are special cases of the unified integral operators defined in (1.9), (1.10).

Definition 1.1. ^[1] Let $f:[a, b]\rightarrow\mathbb{R}$ be an integrable function. Also let $g$ be an increasing and positive function on $(a, b]$ , having a continuous derivative $g^{\prime}$ on $(a, b)$ . The left-sided and right-sided fractional integrals of a function $f$ with respect to another function $g$ on $[a, b]$ of order ${\mu}$ where $\Re(\mu) > 0$ are defined by:

$\begin{equation} _{g}^{\mu}I_{a^+}f(x) = \frac{1}{\Gamma({\mu})}\int_{a}^{x}(g(x)-g(t))^{{\mu}-1}g'(t)f(t)dt,\,\, x \gt a, \end{equation}$

(1.1)

$\begin{equation} _{g}^{\mu}I_{b^-}f(x) = \frac{1}{\Gamma({\mu})}\int_{x}^{b}(g(t)-g(x))^{{\mu}-1}g'(t)f(t)dt,\,\ x \lt b, \end{equation}$

(1.2)

where $\Gamma(.)$ is the gamma function.

Definition 1.2. ^[2] Let $f:[a, b]\rightarrow\mathbb{R}$ be an integrable function. Also let $g$ be an increasing and positive function on $(a, b]$ , having a continuous derivative $g^{\prime}$ on $(a, b)$ . The left-sided and right-sided fractional integrals of a function $f$ with respect to another function $g$ on $[a, b]$ of order ${\mu}$ where $\Re(\mu), k > 0$ are defined by:

$\begin{equation} ^{\mu}_{g}I^{k}_{a^+}f(x) = \frac{1}{k\Gamma_k({\mu})}\int_{a}^{x}(g(x)-g(t))^{\frac{\mu}{k}-1}g'(t)f(t)dt,\,\, x \gt a, \end{equation}$

(1.3)

$\begin{equation} ^{\mu}_{g}I^{k}_{b^-}f(x) = \frac{1}{k\Gamma_k({\mu})}\int_{x}^{b}(g(t)-g(x))^{\frac{\mu}{k}-1}g'(t)f(t)dt,\,\ x \lt b, \end{equation}$

(1.4)

where $\Gamma_{k}(.)$ is defined as follows ^[3]:

$\begin{equation} \Gamma_k(x) = \int_{0}^{\infty}t^{x-1}e^{-\frac{t^{k}}{k}} dt,\Re(x) \gt 0. \end{equation}$

(1.5)

A fractional integral operator containing an extended generalized Mittag-Leffler function in its kernel is defined as follows:

Definition 1.3. ^[4] Let $\omega, \mu, \alpha, l, \gamma, c\in \mathbb{C}$ , $\Re(\mu), \Re(\alpha), \Re(l) > 0$ , $\Re(c) > \Re(\gamma) > 0$ with $p\geq0$ , $\delta > 0$ and $0 < k\leq\delta+\Re(\mu)$ . Let $f\in L_{1}[a, b]$ and $x\in[a, b].$ Then the generalized fractional integral operators $\epsilon_{\mu, \alpha, l, \omega, a^{+}}^{\gamma, \delta, k, c}f$ and $\epsilon_{\mu, \alpha, l, \omega, b^{-}}^{\gamma, \delta, k, c}f$ are defined by:

$\begin{equation} \left( \epsilon_{\mu,\alpha,l,\omega,a^{+}}^{\gamma,\delta,k,c}f \right)(x;p) = \int_{a}^{x}(x-t)^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(\omega(x-t)^{\mu};p)f(t)dt, \end{equation}$

(1.6)

$\begin{equation} \left( \epsilon_{\mu,\alpha,l,\omega,b^{-}}^{\gamma,\delta,k,c}f \right)(x;p) = \int_{x}^{b}(t-x)^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(\omega(t-x)^{\mu};p)f(t)dt, \end{equation}$

(1.7)

where

$\begin{equation} E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(t;p) = \sum\limits_{n = 0}^{\infty}\frac{\beta_{p}(\gamma+nk,c-\gamma)}{\beta(\gamma,c-\gamma)} \frac{(c)_{nk}}{\Gamma(\mu n +\alpha)} \frac{t^{n}}{(l)_{n \delta}} \end{equation}$

(1.8)

is the extended generalized Mittag-Leffler function and $(c)_{nk}$ is the Pochhammer symbol defined by $(c)_{nk} = \frac{\Gamma(c+nk)}{\Gamma(c)}$ .

Recently, a unified integral operator is defined as follows:

Definition 1.4. ^[5] Let $f, g:[a, b]\longrightarrow \mathbb{R}$ , $0 < a < b$ , be the functions such that $f$ be positive and $f\in L_{1}[a, b]$ , and $g$ be differentiable and strictly increasing. Also let $\frac{\phi}{x}$ be an increasing function on $[a, \infty)$ and $\alpha, l, \gamma, c$ $\in$ $\mathbb{C}$ , $p, \mu, \delta$ $\geq$ 0 and $0 < k\leq \delta + \mu$ . Then for $x\in[a, b]$ the left and right integral operators are defined by

$\begin{equation} (_gF_{\mu, \alpha, l, {a^+}}^{\phi, \gamma, \delta, k, c}f)(x,\omega;p) = \int_{a}^{x}K_{x}^{y}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)f(y)d(g(y)), \end{equation}$

(1.9)

$\begin{equation} (_gF_{\mu, \beta, l, {b^-}}^{\phi, \gamma, \delta, k, c}f)(x,\omega;p) = \int_{x}^{b}K_{y}^{x}(E^{\gamma, \delta, k, c}_{\mu, \beta, l},g;\phi)f(y)d(g(y)), \end{equation}$

(1.10)

where the involved kernel is defined by

$\begin{equation} K_{x}^{y}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi) = \dfrac{\phi(g(x)-g(y))}{g(x)-g(y)}E^{\gamma, \delta, k, c}_{\mu, \alpha, l}(\omega(g(x)-g(y))^{\mu}; p). \end{equation}$

(1.11)

For suitable settings of functions $\phi$ , $g$ and certain values of parameters included in Mittag-Leffler function, several recently defined known fractional and conformable fractional integrals studied in ^{[6,7,8,9,10,1,11,12,13,14,15,16,17]} can be reproduced, see [18,Remarks 6&7].

The aim of this study is to derive the bounds of all aforementioned integral operators in a unified form for $(s, m)$ -convex functions. These bounds will hold particularly for $m$ -convex, $s$ -convex and convex functions and for almost all fractional and conformable integrals defined in ^{[6,7,8,9,10,1,11,12,13,14,15,16,17]}.

Definition 1.5. ^[19] A function $f:[0, b]\rightarrow \mathbb{R}, \, b > 0$ is said to be $(s, m)$ -convex, where $(s, m)\in[0, 1]^{2}$ if

$\begin{equation} f(tx+m(1-t)y)\leq t^{s}f(x)+m(1-t)^{s}f(y) \end{equation}$

(1.12)

holds for all $x, y\in[0, b]\, \, and\, \, t\in[0, 1].$

Remark 1. 1. If we take $(s, m)$ = $(1, m)$ , then (1.12) gives the definition of $m$ -convex function.

2. If we take $(s, m)$ = $(1, 1)$ , then (1.12) gives the definition of convex function.

3. If we take $(s, m)$ = $(1, 0)$ , then (1.12) gives the definition of star-shaped function.

2. Properties of the kernel $K_{x}^{y}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l}, g;\phi)$

P1: Let $g$ and $\frac{\phi}{x}$ be increasing functions. Then for $x < t < y$ , $x, y\in[a, b]$ the kernel $K_{x}^{y}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l}, g;\phi)$ satisfies the following inequality:

$\begin{equation} K_{t}^{x}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)g'(t)\leq K_{y}^{x}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)g'(t). \end{equation}$

(2.1)

This can be obtained from the following two straightforward inequalities:

$\begin{equation} \dfrac{\phi(g(t)-g(x))}{g(t)-g(x)}g'(t)\leq\dfrac{\phi(g(y)-g(x))}{g(y)-g(x)}g'(t), \end{equation}$

(2.2)

$\begin{equation} E^{\gamma, \delta, k, c}_{\mu, \alpha, l}(\omega(g(t)-g(x))^{\mu}; p)\leq E^{\gamma, \delta, k, c}_{\mu, \alpha, l}(\omega(g(y)-g(x))^{\mu}; p). \end{equation}$

(2.3)

The reverse of inequality (1.9) holds when $g$ and $\frac{\phi}{x}$ are decreasing.

P2: Let $g$ and $\frac{\phi}{x}$ be increasing functions. If $\phi(0) = \phi'(0) = 0$ , then for $x, y\in[a, b], \, x < y$ ,

$K_{y}^{x}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l}, g;\phi)\geq0$ .

P3: For $p, q \in \mathbb{R}$ ,

$K_{y}^{x}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l}, g;p\phi_1+q\phi_2) = p K_{y}^{x}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l}, g;\phi_1)+q K_{y}^{x}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l}, g;\phi_2).$

The upcoming section contains the results which deal with the bounds of several integral operators in a compact form by utilizing $(s, m)$ -convex functions. A version of the Hadamard inequality in a compact form is presented, also a modulus inequality is given for differentiable function $f$ such that $|f'|$ is $(s, m)$ -convex function.

3. Main results

In this section first we will state the main results. The following result provides upper bound of unified integral operators.

Theorem 3.1. Let $f:[a, b]\longrightarrow \mathbb{R}$ , $0\leq a < b$ be a positive integrable $(s, m)$ -convex function, $m\in(0, 1]$ . Let $g:[a, b]\longrightarrow \mathbb{R}$ be differentiable and strictly increasing function, also let $\frac{\phi}{x}$ be an increasing function on $[a, b]$ . If $\alpha, \beta, l, \gamma, c \in\mathbb{C}$ , $p, \mu \geq 0, \delta \geq 0$ and $0 < k\leq \delta + \mu$ , then for $x\in(a, b)$ the following inequality holds for unified integral operators:

$\begin{align} &\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {a^+}}f\right)(x,\omega; p)+\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {b^-}}f\right)(x,\omega; p)\\ &\leq K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)\left(mf\left(\frac{x}{m}\right)g(x)-f(a)g(a)-\dfrac{\Gamma(s+1)}{(x-a)^{s}}\left(mf\left(\frac{x}{m}\right)\, ^{s}I_{x^{-}}g(a)-f(a)\, ^{s}I_{a^{+}}g(x)\right)\right)\\ &+K_{b}^{x}(E^{\gamma, \delta, k, c}_{\mu, \beta, l},g;\phi)\left(f(b)g(b)-mf\left(\frac{x}{m}\right)g(x)-\dfrac{\Gamma(s+1)}{(b-x)^{s}}\left(f(b)\, ^{s}I_{b^{-}}g(x)-mf\left(\frac{x}{m}\right)\, ^{s}I_{x^{+}}g(b)\right)\right). \end{align}$

(3.1)

Lemma 3.2. ^[20] Let $f: [0, \infty] \longrightarrow \mathbb{R}$ , be an $(s, m)$ -convex function, $m\in(0, 1]$ . If $f(x) = f(\frac{a+b-x}{m})$ , then the following inequality holds:

$\begin{equation} f\left(\frac{a+b}{2}\right)\leq \frac{1}{2^{s}}(1+m)f(x)\,\,\,\,\,\,\,x\in[a,b]. \end{equation}$

(3.2)

The following result provides generalized Hadamard inequality for $(s, m)$ -convex functions.

Theorem 3.3. Under the assumptions of Theorem 3.1, in addition if $f(x) = f\left(\dfrac{a+b-x}{m}\right)$ , $m\in(0, 1]$ , then the following inequality holds:

$\begin{align} &\dfrac{2^{s}}{\left(1+m\right)}f\left(\dfrac{a+b}{2}\right)\left(\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {b^-}} 1\right)(a,\omega;p)+\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {a^+}} 1\right)(b,\omega;p)\right)\\ &\leq \left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {b^-}} f\right)(a,\omega;p)+\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {a^+}} f\right)(b,\omega;p)\\ & \leq \left( K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)+K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)\right)\left(f(b)g(b)-mf\left(\dfrac{a}{m}\right)g(a)\right.\\ &\left.-\dfrac{\Gamma(s+1)}{(b-a)^{s}}\left(f(b)\, ^{s}I_{b^{-}}g(a)-mf\left(\dfrac{a}{m}\right)\, ^{s}I_{a^{+}}g(b)\right)\right). \end{align}$

(3.3)

Theorem 3.4. Let $f:[a, b]\longrightarrow \mathbb{R}$ , $0\leq a < b$ be a differentiable function. If $|f'|$ is $(s, m)$ -convex, $m\in(0, 1]$ and $g:[a, b]\longrightarrow \mathbb{R}$ be differentiable and strictly increasing function, also let $\frac{\phi}{x}$ be an increasing function on $[a, b]$ . If $\alpha, \beta, l, \gamma, c\in\mathbb{C}$ , $p, \mu \geq 0$ , $\delta \geq 0$ and $0 < k\leq \delta + \mu$ , then for $x\in(a, b)$ we have

$\begin{align} &\left|\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {a^+}}f\ast g\right)(x,\omega; p)+\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {b^-}}f\ast g\right)(x,\omega; p)\right| \\&\leq K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)\left(m\left|f'\left(\frac{x}{m}\right)\right|g(x)-|f'(a)|g(a)-\dfrac{\Gamma(s+1)}{(x-a)^{s}}\left(m\left|f'\left(\frac{x}{m}\right)\right|\, ^{s}I_{x^{-}}g(a)-|f'(a)|\, ^{s}I_{a^{+}}g(x)\right)\right)\\ \nonumber&+K_{b}^{x}(E^{\gamma, \delta, k, c}_{\mu, \beta, l},g;\phi)\left(|f'(b)|g(b)-m\left|f'\left(\frac{x}{m}\right)\right|g(x)-\dfrac{\Gamma(s+1)}{(b-x)^{s}}\left(|f'(b)|\, ^{s}I_{b^{-}}g(x)-m\left|f'\left(\frac{x}{m}\right)\right|\, ^{s}I_{x^{+}}g(b)\right)\right),\nonumber \end{align}$

(3.4)

where

$\begin{equation*} \left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {a^+}}f\ast g\right)(x,\omega; p): = \int_{a}^{x}K_{x}^{t}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)f'(t)d(g(t)), \end{equation*}$

$\begin{equation*} \left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {b^-}}f\ast g\right)(x,\omega; p): = \int_{x}^{b}K_{t}^{x}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)f'(t)d(g(t)). \end{equation*}$

4. Proofs of main results

In this section we give the proves of the results stated in aforementioned section.

Proof of Theorem 3.1. By $(\bf{P_1})$ , the following inequalities hold:

$\begin{equation} K_{x}^{t}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)g'(t)\leq K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)g'(t),\,\ a \lt t \lt x, \end{equation}$

(4.1)

$\begin{equation} K_{t}^{x}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)g'(t)\leq K_{b}^{x}(E^{\gamma, \delta, k, c}_{\mu, \beta, l},g;\phi)g'(t),\,\ x \lt t \lt b. \end{equation}$

(4.2)

For $(s, m)$ -convex function the following inequalities hold:

$\begin{equation} f(t)\leq \bigg(\dfrac{x-t}{x-a}\bigg)^{s}f(a)+m\bigg(\dfrac{t-a}{x-a}\bigg)^{s}f\bigg(\frac{x}{m}\bigg),\,\ a \lt t \lt x, \end{equation}$

(4.3)

$\begin{equation} f(t)\leq \bigg(\dfrac{t-x}{b-x}\bigg)^{s}f(b)+m\bigg(\dfrac{b-t}{b-x}\bigg)^{s}f\bigg(\frac{x}{m}\bigg),\,\ x \lt t \lt b. \end{equation}$

(4.4)

From (4.1) and (4.3), the following integral inequality holds true:

$\begin{align} &\int_{a}^{x} K_{x}^{t}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi) f(t)d(g(t))\leq f(a)K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)\\ &\times\int_{a}^{x}\left(\dfrac{x-t}{x-a}\right)^{s} d(g(t))+mf\left(\dfrac{x}{m}\right)K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)\int_{a}^{x}\left(\dfrac{t-a}{x-a}\right)^{s} d(g(t)). \end{align}$

(4.5)

Further the aforementioned inequality takes the form which involves Riemann-Liouville fractional integrals in the right hand side, provides the upper bound of unified left sided integral operator (1.1) as follows:

$\begin{align} &\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {a^+}}f\right)(x,\omega; p) \leq K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)\left(mf\left(\frac{x}{m}\right)g(x)-f(a)g(a)\right.\\ &\left.-\dfrac{\Gamma(s+1)}{(x-a)^{s}}\left(mf\left(\frac{x}{m}\right)\, ^{s}I_{x^{-}}g(a)-f(a)\, ^{s}I_{a^{+}}g(x)\right)\right). \end{align}$

(4.6)

On the other hand from (4.2) and (4.4), the following integral inequality holds true:

$\begin{align} &\int_{x}^{b} K_{t}^{x}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi) f(t)d(g(t))\leq f(b)K_{b}^{x}(E^{\gamma, \delta, k, c}_{\mu, \beta, l},g;\phi)\\ &\times\int_{x}^{b}\left(\dfrac{t-x}{b-x}\right)^{s} d(g(t))+mf\left(\dfrac{x}{m}\right)K_{x}^{b}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)\int_{x}^{b}\left(\dfrac{b-t}{b-x}\right)^{s} d(g(t)). \end{align}$

(4.7)

Further the aforementioned inequality takes the form which involves Riemann-Liouville fractional integrals in the right hand side, provides the upper bound of unified right sided integral operator (1.2) as follows:

$\begin{align} &\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {b^-}}f\right)(x,\omega; p) \leq K_{b}^{x}(E^{\gamma, \delta, k, c}_{\mu, \beta, l},g;\phi)\left(f(b)g(b)-mf\left(\frac{x}{m}\right)g(x)\right.\\ &\left.-\dfrac{\Gamma(s+1)}{(b-x)^{s}}\left(f(b)\, ^{s}I_{b^{-}}g(x)-mf\left(\frac{x}{m}\right)\, ^{s}I_{x^{+}}g(b)\right)\right). \end{align}$

(4.8)

By adding (4.6) and (4.8), (3.1) can be obtained.

Remark 2. (ⅰ) If we consider $(s, m)$ = (1, 1) in (3.1), [18,Theorem 1] is obtained.

(ⅱ) If we consider $p = \omega = 0$ in (3.1), [20,Theorem 1] is obtained.

(ⅲ) If we consider $\phi(t) = \Gamma(\alpha)t^{\alpha}$ , $p = \omega = 0$ and $(s, m)$ = (1, 1) in (3.1), [21,Theorem 1] is obtained.

(ⅳ) If we consider $\alpha = \beta$ in the result of (ⅲ), then [21,Corollary 1] is obtained.

(ⅴ) If we consider $\phi(t) = t^{\alpha}$ , $g(x) = x$ and $m = 1$ in (3.1), then [22,Theorem 2.1] is obtained.

(ⅵ) If we consider $\alpha = \beta$ in the result of (v), then [22,Corollary 2.1] is obtained.

(ⅶ) If we consider $\phi(t) = \frac{\Gamma (\alpha) t^\frac{\alpha}{k}}{k\Gamma_k(\alpha)}$ , $(s, m)$ = (1, 1), $g(x) = x$ and $p = \omega = 0$ in (3.1), then [23,Theorem 1] can be obtained.

(ⅷ) If we consider $\alpha = \beta$ in the result of (ⅶ), then [23,Corollary 1] can be obtained.

(ⅸ) If we consider $\phi(t) = \Gamma(\alpha)t^{\alpha}$ , $g(x) = x$ and $p = \omega = 0$ and $(s, m)$ = (1, 1) in (3.1), then [24,Theorem 1] is obtained.

(ⅹ) If we consider $\alpha = \beta$ in the result of (ⅸ), then [24,Corollary 1] can be obtained.

(ⅹⅰ) If we consider $\alpha = \beta = 1$ and $x = a\, \ or \, \ x = b$ in the result of (x), then [24,Corollary 2] can be obtained.

(ⅹⅱ) If we consider $\alpha = \beta = 1$ and $x = \dfrac{a+b}{2}$ in the result of (ⅹ), then [24,Corollary 3] can be obtained.

Proof of Theorem 3.3. By $(\bf{P_1})$ , the following inequalities hold:

$\begin{equation} K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)g'(x)\leq K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)g'(x),\,\ a \lt x \lt b, \end{equation}$

(4.9)

$\begin{equation} K_{b}^{x}(E^{\gamma, \delta, k, c}_{\mu, \beta, l},g;\phi)g'(x)\leq K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)g'(x)\,\ a \lt x \lt b. \end{equation}$

(4.10)

For $(s, m)$ -convex function $f$ , the following inequality holds:

$\begin{equation} f(x)\leq \bigg(\dfrac{x-a}{b-a}\bigg)^{s}f(b)+m\bigg(\dfrac{b-x}{b-a}\bigg)^{s}f\bigg(\frac{a}{m}\bigg),\,\,\ a \lt x \lt b. \end{equation}$

(4.11)

From (4.9) and (4.11), the following integral inequality holds true:

$\begin{align*} &\int_{a}^{b} K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)f(x)d(g(x))\\ \nonumber &\leq mf\left(\dfrac{a}{m}\right)K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)\int_{a}^{b}\left(\dfrac{b-x}{b-a}\right)^{s}d(g(x))+f(b)K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)\int_{a}^{b}\left(\dfrac{x-a}{b-a}\right)^{s} d(g(x)). \end{align*}$

$\begin{align} &\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {b^-}}f\right)(a,\omega; p) \leq K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi) \bigg(f(b)g(b)-mf\bigg(\frac{a}{m}\bigg)g(a)\\ &-\dfrac{\Gamma(s+1)}{(b-a)^{s}}\bigg(f(b)\, ^{s}I_{b^{-}}g(a)-mf\bigg(\frac{a}{m}\bigg)\, ^{s}I_{a^{+}}g(b)\bigg)\bigg). \end{align}$

(4.12)

On the other hand from (4.9) and (4.11), the following inequality holds which involves Riemann-Liouville fractional integrals on the right hand side and estimates of the integral operator (1.2):

$\begin{align} &\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {a^+}}f\right)(b,\omega; p) \leq K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi) \bigg(f(b)g(b)-mf\bigg(\frac{a}{m}\bigg)g(a)\\ &-\dfrac{\Gamma(s+1)}{(b-a)^{s}}\bigg(f(b)\, ^{s}I_{b^{-}}g(a)-mf\bigg(\frac{a}{m}\bigg)\, ^{s}I_{a^{+}}g(b)\bigg)\bigg). \end{align}$

(4.13)

By adding (4.12) and (4.13), following inequality can be obtained:

$\begin{align} &\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {b^-}}f\right)(a,\omega; p)+\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {a^+}}f\right)(b,\omega; p)\\ & \leq \left(K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)+K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)\right) \bigg(f(b)g(b)-mf\bigg(\frac{a}{m}\bigg)g(a)\\ &-\dfrac{\Gamma(\alpha+1)}{(b-a)^{s}}\bigg(f(b)\, ^{s}I_{b^{-}}g(b)-mf\bigg(\frac{a}{m}\bigg)\, ^{s}I_{a^{+}}g(b)\bigg)\bigg). \end{align}$

(4.14)

Multiplying both sides of (3.2) by $K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l}, g;\phi)g'(x)$ , and integrating over $[a, b]$ we have

$\begin{equation*} f\left(\dfrac{a+b}{2}\right)\int_{a}^{b} K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)d(g(x))\leq \left(\dfrac{1}{2^{s}}\right)\left(1+m\right)\int_{a}^{b} K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)f(x)d(g(x)). \end{equation*}$

From Definition 1.4, the following inequality is obtained:

$\begin{equation} f\left(\dfrac{a+b}{2}\right)\dfrac{2^{s}}{(1+m)}\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {b^-}} 1\right)(a,\omega;p)\leq \left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {b^-}} f\right)(a,\omega;p). \end{equation}$

(4.15)

Similarly multiplying both sides of (3.2) by $K_{b}^{x}(E^{\gamma, \delta, k, c}_{\mu, \beta, l}, g;\phi)g'(x)$ , and integrating over $[a, b]$ we have

$\begin{equation} f\left(\dfrac{a+b}{2}\right)\dfrac{2^{s}}{(1+m)}\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {a^+}} 1\right)(b,\omega;p)\leq \left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {a^+}} f\right)(b,\omega;p). \end{equation}$

(4.16)

By adding (4.15) and (4.16) the following inequality is obtained:

$\begin{align} &f\left(\dfrac{a+b}{2}\right)\dfrac{2^{s}}{(1+m)}\left(\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {a^+}} 1\right)(b,\omega;p)+\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {b^-}} 1\right)(a,\omega;p)\right)\\ &\leq \left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {a^+}} f\right)(b,\omega;p)+\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {b^-}} f\right)(a,\omega;p). \end{align}$

(4.17)

Using (4.14) and (4.17), inequality (3.3) can be obtained, this completes the proof.

Remark 3. (ⅰ) If we consider $(s, m)$ = (1, 1) in (3.3), [18,Theorem 2] is obtained.

(ⅱ) If we consider $p = \omega = 0$ in (3.3), [20,Theorem 3] is obtained.

(ⅲ) If we consider $\phi(t) = \Gamma(\alpha)t^{\alpha+1}$ , $p = \omega = 0$ and $(s, m)$ = (1, 1) in (3.3), [21,Theorem 3] is obtained.

(ⅳ) If we consider $\alpha = \beta$ in the result of (iii), then [21,Corollary 3] is obtained.

(ⅴ) If we consider $\phi(t) = t^{\alpha+1}$ , $g(x) = x$ and $m = 1$ in (3.3), then [22,Theorem 2.4] is obtained.

(ⅵ) If we consider $\alpha = \beta$ in the result of (v), then [22,Corollary 2.6] is obtained.

(ⅶ) If we consider $\phi(t) = \Gamma (\alpha) t^{\frac{\alpha}{k}+1}$ , $(s, m)$ = (1, 1), $g(x) = x$ and $p = \omega = 0$ in (3.3), then [23,Theorem 3] can be obtained.

(ⅷ) If we consider $\alpha = \beta$ in the result of (ⅶ), then [23,Corollary 6] can be obtained.

(ⅸ) If we consider $\phi(t) = \Gamma(\alpha)t^{\alpha+1}$ , $p = \omega = 0$ , $(s, m) = 1$ and $g(x) = x$ in (3.3), [24,Theorem 3] can be obtained.

(ⅹ) If we consider $\alpha = \beta$ in the result of (ⅸ), [24,Corrolary 6] can be obtained.

Proof of Theorem 3.4. For $(s, m)$ -convex function the following inequalities hold:

$\begin{equation} |f'(t)|\leq\bigg(\dfrac{x-t}{x-a}\bigg)^{s}|f'(a)|+m\bigg(\dfrac{t-a}{x-a}\bigg)^{s}\left|f'\bigg(\frac{x}{m}\bigg)\right|,\,\ a \lt t \lt x, \end{equation}$

(4.18)

$\begin{equation} |f'(t)|\leq\bigg(\dfrac{t-x}{b-x}\bigg)^{s}|f'(b)|+m\bigg(\dfrac{b-t}{b-x}\bigg)^{s}\left|f'\bigg(\frac{x}{m}\bigg)\right|,\,\ x \lt t \lt b. \end{equation}$

(4.19)

From (4.1) and (4.18), the following inequality is obtained:

$\begin{align} &\left|\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {a^+}}(f\ast g)\right)(x,\omega; p)\right| \leq\dfrac{K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)}{(x-a)^{s}}\\ &\times\left((x-a)^{s}\left(m\left|f'\left(\frac{x}{m}\right)\right|g(x)-|f'(a)|g(a)\right)-\Gamma(s+1)\left(m\left|f'\left(\frac{x}{m}\right)\right|\, ^{s}I_{x^{-}}g(a)-|f'(a)|\, ^{s}I_{a^{+}}g(x)\right)\right). \end{align}$

(4.20)

Similarly, from (4.2) and (4.19), the following inequality is obtained:

$\begin{align} &\left|\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {b^-}}(f\ast g)\right)(x,\omega; p)\right| \leq\dfrac{K_{b}^{x}(E^{\gamma, \delta, k, c}_{\mu, \beta, l},g;\phi)}{(b-x)^{s}}\\ &\times\left((b-x)^{s}\left(|f'(b)|g(b)-mf'\left|\left(\frac{x}{m}\right)\right| g(x)\right)-\Gamma(s+1)\left(|f'(b)|\, ^{s}I_{b^{-}}g(x)-mf'\left|\left(\frac{x}{m}\right)\right|\, ^{s}I_{x^{+}}g(b)\right)\right). \end{align}$

(4.21)

By adding (4.20) and (4.21), inequality (3.4) can be achieved.

Remark 4. (ⅰ) If we consider $(s, m)$ = (1, 1) in (3.4), then [18,Theorem 3] is obtained.

(ⅱ) If we consider $p = \omega = 0$ in (3.4), then [20,Theorem 2] is obtained.

(ⅲ) If we consider $\phi(t) = \Gamma(\alpha)t^{\alpha+1}$ , $p = \omega = 0$ and $(s, m)$ = (1, 1) in (3.4), then [21,Theorem 2] is obtained.

(ⅳ) If we consider $\alpha = \beta$ in the result of (iii), then [21,Corollary 2] is obtained.

(ⅴ) If we consider $\phi(t) = t^{\alpha}$ , $g(x) = x$ and $m = 1$ in (3.4), then [22,Theorem 2.3] is obtained.

(ⅵ) If we consider $\alpha = \beta$ in the result of (v), then [22,Corollary 2.5] is obtained.

(ⅶ) If we consider $\phi(t) = \Gamma (\alpha) t^{\frac{\alpha}{k}+1}$ , $(s, m)$ = (1, 1), $g(x) = x$ and $p = \omega = 0$ in (3.4), then [23,Theorem 2] can be obtained.

(ⅷ) If we consider $\alpha = \beta$ in the result of (ⅶ), then [23,Corollary 4] can be obtained.

(ⅸ) If we consider $\alpha = \beta = k = 1$ and $x = \dfrac{a+b}{2}$ , in the result of (ⅷ), then [23,Corollary 5] can be obtained.

(ⅹ) If we consider $\phi(t) = \Gamma(\alpha)t^{\alpha+1}$ , $g(x) = x$ and $p = \omega = 0$ and $(s, m)$ = (1, 1) in (3.4), then [24,Theorem 2] is obtained.

(ⅹⅰ) If we consider $\alpha = \beta$ in the result of (x), then [24,Corollary 5] can be obtained.

5. Boundedness and continuity

In this section, we have established boundedness and continuity of unified integral operators for $m$ -convex and convex functions.

Theorem 5.1. Under the assumptions of Theorem 1, the following inequality holds for $m$ -convex functions:

(5.1)

Proof. If we put $s = 1$ in (4.5), we have

$\begin{align} &\int_{a}^{x} K_{x}^{t}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi) f(t)d(g(t))\leq f(a)K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)\\ &\times\int_{a}^{x}\left(\dfrac{x-t}{x-a}\right) d(g(t))+mf\left(\dfrac{x}{m}\right)K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)\int_{a}^{x}\left(\dfrac{t-a}{x-a}\right) d(g(t)). \end{align}$

(5.2)

Further from simplification of (5.2), the following inequality holds:

$\begin{align} \left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {a^+}}f\right)(x,\omega; p)\leq K_{x}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi)(g(x)-g(a))\left(mf\left(\dfrac{x}{m}\right)+f(a)\right). \end{align}$

(5.3)

Similarly from (4.8), the following inequality holds:

$\begin{equation} \left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {b^-}}f\right)(x,\omega; p)\leq K_{b}^{x}(E^{\gamma, \delta, k, c}_{\mu, \beta, l},g;\phi)(g(b)-g(x))\left(mf\left(\dfrac{x}{m}\right)+f(b)\right). \end{equation}$

(5.4)

From (5.3) and (5.4), (5.1) can be obtained.

Theorem 5.2. With assumptions of Theorem 4, if $f\in L_{\infty}[a, b]$ , then unified integral operators for $m$ -convex functions are bounded and continuous.

Proof. From (5.3) we have

$\begin{equation*} \label{2.13-} \left|\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {a^+}}f\right)(x,\omega; p)\right| \leq K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l},g;\phi) (g(b)-g(a))(m+1)\|f\|_{\infty}, \end{equation*}$

which further gives

$\begin{equation*} \label{2.15} \left|\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {a^+}}f\right)(x,\omega; p)\right| \leq K \|f\|_{\infty}, \end{equation*}$

where $K = (g(b)-g(a))(m+1)K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l}, g;\phi)$ .

Similarly, from (5.4) the following inequality holds:

$\begin{equation*} \label{2.16} \left|\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {b^-}}f\right)(x,\omega; p)\right| \leq K \|f\|_{\infty}. \end{equation*}$

Hence the boundedness is followed, further from linearity the continuity of (1.9) and (1.10) is obtained.

Corollary 1. If we take $m = 1$ in Theorem 5, then unified integral operators for convex functions are bounded and continuous and following inequalities hold:

$\begin{equation*} \label{2.115} \left|\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \alpha, l, {a^+}}f\right)(x,\omega; p)\right| \leq K \|f\|_{\infty}, \end{equation*}$

$\begin{equation*} \label{2.116} \left|\left(_gF^{\phi, \gamma, \delta, k, c}_{\mu, \beta, l, {b^-}}f\right)(x,\omega; p)\right| \leq K \|f\|_{\infty}, \end{equation*}$

where $K = 2(g(b)-g(a))K_{b}^{a}(E^{\gamma, \delta, k, c}_{\mu, \alpha, l}, g;\phi)$ .

6. Conclusions

This paper has explored bounds of a unified integral operator for $(s, m)$ -convex functions. These bounds are obtained in a compact form which have further interesting consequences with respect to fractional and conformable integrals for convex, $m$ -convex and $s$ -convex functions. Furthermore by applying Theorems 3.1, 3.3 and 3.4 several associated results can be derived for different kinds of fractional integral operators of convex, $m$ -convex and $s$ -convex functions.

Acknowledgments

This work was sponsored in part by Social Science Planning Fund of Liaoning Province of China(L15AJL001, L16BJY011, L18AJY001), Scientific Research Fund of The Educational Department of Liaoning Province(2017LNZD07, 2016FRZD03), Scientific Research Fund of University of science and technology Liaoning(2016RC01, 2016FR01)

Conflict of interest

The authors declare that no competing interests exist.

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