Some properties of the interval-valued $\overline{g}$-integrals and a standard interval-valued $\overline{g}$-convolution

Abstract

Pap and Stajner (Fuzzy Sets Syst. 102:393-415, 1999) investigated a generalized pseudo-convolution of functions based on pseudo-operations. Jang (Fuzzy Sets Syst. 222:45-57, 2013) studied the interval-valued generalized fuzzy integral by using an interval-representable pseudo-multiplication.

In this paper, by using the concepts of interval-representable pseudo-multiplication and g-integral, we define the interval-valued $\overline{g}$-integral represented by its interval-valued generator $\overline{g}$ and a standard interval-valued $\overline{g}$-convolution by means of the corresponding interval-valued $\overline{g}$-integral. We also investigate some characterizations of the interval-valued $\overline{g}$-integral and a standard interval-valued $\overline{g}$-convolution.

MSC:28E10, 28E20, 03E72, 26E50, 11B68.

1 Introduction

Benvenuti and Mesiar [1], Daraby [2], Deschrijver [3], Grbic et al. [4], Klement et al. [5], Mesiar et al. [6], Stajner-Papuga et al. [7], Sugeno [8], Sugeno and Murofushi [9], Wu et al. [10, 11] have been studying pseudo-multiplications and various pseudo-integrals of measurable functions. Markova and Stupnanova [12], Maslov and Samborskij [13], and Pap and Stajner [14] introduced a general notion of pseudo-convolution of functions based on pseudo-mathematical operations and investigated the idempotent with respect to a pseudo-convolution.

Many researchers [3, 4, 15–29] have studied the pseudo-integral of measurable multi-valued function, for example, the Aumann integral, the fuzzy integral, and the Choquet integral of measurable interval-valued functions, in many different mathematical theories and their applications.

Recently, Jang [26] defined the interval-valued generalized fuzzy integral by using an interval-representable pseudo-multiplication and investigated their characterizations. The purpose of this study is to define the interval-valued $\overline{g}$-integral represented by its interval-valued generator $\overline{g}$ and a standard interval-valued $\overline{g}$-convolution by means of the corresponding interval-valued $\overline{g}$-integral, and to investigate an interval-valued idempotent function with respect to a standard interval-valued $\overline{g}$-convolution.

This paper is organized in five sections. In Section 2, we list definitions and some properties of a pseudo-addition, a pseudo-multiplication, a g-integral, and a g-convolution of functions by means of the corresponding g-integral. In Section 3, we define an interval-representable pseudo-addition, an interval-representable pseudo-multiplication, the interval-valued $\overline{g}$-integral represented by its interval-valued generator $\overline{g}$, and investigate some characterizations of the interval-valued $\overline{g}$-integral. In Section 4, we define a standard interval-valued $\overline{g}$-convolution by means of the corresponding interval-valued $\overline{g}$-integral and investigate some basic characterizations of them. In Section 5, we give a brief summary of results and some conclusions.

2 Definitions and preliminaries

Let X be a set, $\mathcal{A}$ be a σ-algebra of X, and $\mathfrak{F}(X)$ be a set of all measurable functions $f:X⟶[0,\mathrm{\infty })$. We introduce a pseudo-addition and a pseudo-multiplication (see [1–4, 6, 7, 12, 14, 26, 30]).

Definition 2.1 ([12])

• (1) A binary operation $\oplus :{[0,\mathrm{\infty }]}^2⟶[0,\mathrm{\infty }]$ is called a pseudo-addition if it satisfies the following axioms:

• (i) $x\oplus y=y\oplus x$ for all $x,y\in [0,\mathrm{\infty }]$,

• (ii) $x\le y⟹x\oplus z\le y\oplus z$ for all $x,y,z\in [0,\mathrm{\infty }]$,

• (iii) $(x\oplus y)\oplus z=x\oplus (y\oplus z)$ for all $x,y,z\in [0,\mathrm{\infty }]$,

• (iv) $\mathrm{\exists }\mathbf{0}\in [0,\mathrm{\infty }]$ such that $x\oplus \mathbf{0}=x$ for all $x\in [0,\mathrm{\infty }]$,

• (v) $x_n⟶x$, $y_n⟶y⟹x_n\oplus y_n⟶x\oplus y$.

• (2) A binary operation $\odot :{[0,\mathrm{\infty }]}^2⟶[0,\mathrm{\infty }]$ is called a pseudo-multiplication with respect to ⊕ if it satisfies the following axioms:

• (i) $x\odot y=y\odot x$ for all $x,y\in [0,\mathrm{\infty }]$,

• (ii) $x\odot (y\odot z)=x\odot (y\odot z)$ for all $x,y,z\in [0,\mathrm{\infty }]$,

• (iii) $\mathrm{\exists }\mathbf{1}\in [0,\mathrm{\infty }]$ such that $x\odot \mathbf{1}=x$ for all $x\in [0,\mathrm{\infty }]$,

• (iv) $(x\odot y)\oplus z=(x\odot y)\oplus (x\odot z)$ for all $x,y,z\in [0,\mathrm{\infty }]$,

• (v) $x\odot \mathbf{0}=\mathbf{0}$ for all $x\in [0,\mathrm{\infty }]$,

• (vi) $x\le y⟹x\odot z\le y\odot z$ for all $x,y,z\in [0,\mathrm{\infty }]$.

Remark 2.1 ([6, 12, 14])

If $g:[0,\mathrm{\infty }]⟶[0,\mathrm{\infty }]$ is a generating function for a semigroup $([0,\mathrm{\infty }],\oplus ,\odot )$, then the pseudo-operations are of the following forms:

$$x\oplus y=g^{-1}(g(x)+g(y))$$

(1)

and

$$x\odot y=g^{-1}(g(x)g(y)).$$

(2)

Definition 2.2 ([6])

A set function $\mu :\mathcal{A}⟶[0,\mathrm{\infty }]$ is called a $\sigma -\oplus$-measure if it satisfies the following axioms:

• (i) $\mu (\mathrm{\varnothing })=0$,

• (ii) $\mu {({\bigcup }_{i=1}^{\mathrm{\infty }}A)}_i={⨁}_{i=1}^{\mathrm{\infty }}\mu (A_i)$ for any sequence $\{A_i\}$ of pairwise disjoint sets from $\mathcal{A}$, where ${⨁}_{i=1}^{\mathrm{\infty }}{x}_i={lim}_{n\to \mathrm{\infty }}{⨁}_{i=1}^n{x}_i$.

Let $\mathfrak{F}(X)$ be the set of all measurable functions $f:X⟶[0,\mathrm{\infty })$. We introduce the g-integral with respect to a fuzzy measure induced by a pseudo-addition ⊕ and a pseudo-multiplication ⊙ in Remark 2.1.

Definition 2.3 ([6])

• (1) Let $g:[0,\mathrm{\infty }]⟶[0,\mathrm{\infty }]$ be a continuous strictly monotone increasing surjection function such that $g(0)=0$ and $f\in \mathfrak{F}(X)$. The g-integral of f on A is defined by

  $${\int }_A^{\oplus }f\odot d\mu =g^{-1}{\int }_{A}g(f(x))dx,$$

  (3)

where dx is related to the Lebesgue measure and the integral on the right-hand side is the Lebesgue integral.

• (2) f is said to be integrable if ${\int }_A^{\oplus }f\odot d\mu \in [0,\mathrm{\infty })$.

Let ${\mathfrak{F}}^{\ast }(X)$ be the set of all integrable functions. Then we obtain some basic properties of the g-integral with respect to a fuzzy measure.

Theorem 2.2 (1) If $A\in \mathcal{A}$, $f,h\in {\mathfrak{F}}^{\ast }(X)$ and $f\le h$, then we have

$${\int }_A^{\oplus }f\odot d\mu \le {\int }_A^{\oplus }h\odot d\mu .$$

(4)

• (2) Let $g:[0,\mathrm{\infty }]⟶[0,\mathrm{\infty }]$ be a continuous strictly monotone increasing surjection function such that $g(0)=0$, ⊕, ⊙ are the same pseudo-operations as in Remark  2.1. If $A\in \mathcal{A}$, $f,h\in {\mathfrak{F}}^{\ast }(X)$, then we have

  $${\int }_A^{\oplus }(f\oplus h)\odot d\mu ={\int }_A^{\oplus }f\odot d\mu \oplus {\int }_A^{\oplus }h\odot d\mu .$$

  (5)

• (3) Let $g:[0,\mathrm{\infty }]⟶[0,\mathrm{\infty }]$ be a continuous strictly monotone increasing surjection function such that $g(0)=0$, ⊕, ⊙ are the same pseudo-operations as in Remark  2.1, and $u\otimes v=g^{-1}(g(u)g(v))$ for $u,v\in [0,\mathrm{\infty })$. If $A\in \mathcal{A}$, $c\in [0,\mathrm{\infty })$, $h\in {\mathfrak{F}}^{\ast }(X)$, then we have

  $${\int }_A^{\oplus }(c\otimes h)\odot d\mu =c\otimes {\int }_A^{\oplus }h\odot d\mu .$$

  (6)

Proof (1) Note that if $f,h\in {\mathfrak{F}}^{\ast }(X)$ and $f\le h$, then

$$g(f(x))\le g(h(x))\text{and}{g}^{-1}(f(x))\le g^{-1}(h(x)).$$

(7)

By Definition 2.3(1), (7), and the monotonicity of the Lebesgue integral,

$$\begin{array}{rcl}{\int }_A^{\oplus }f\odot d\mu & =& g^{-1}{\int }_{A}g(f(x))dx\\ \le & g^{-1}{\int }_{A}g(h(x))dx\\ =& {\int }_A^{\oplus }h\odot d\mu .\end{array}$$

(8)

• (2) By Definition 2.3(1) and the additivity of the Lebesgue integral,

  $$\begin{array}{rcl}{\int }_A^{\oplus }(f\oplus h)\odot d\mu & =& g^{-1}{\int }_{A}g\left(g^{-1}(g(f(x))+g(h(x)))\right)dx\\ =& g^{-1}{\int }_A(g(f(x))+g(h(x)))dx\\ =& g^{-1}[{\int }_{A}g(f(x))dx+{\int }_{A}g(h(x))dx]\\ =& g^{-1}g{g}^{-1}{\int }_{A}g(f(x))dx+g{g}^{-1}{\int }_{A}g(h(x))dx\\ =& g^{-1}(g{\int }_A^{\oplus }f\odot d\mu +g{\int }_A^{\oplus }h\odot d\mu )\\ =& {\int }_A^{\oplus }f\odot d\mu \oplus {\int }_A^{\oplus }h\odot d\mu .\end{array}$$

  (9)

• (3) By Definition 2.3(1) and the linearity of the Lebesgue integral,

  $$\begin{array}{rcl}{\int }_A^{\oplus }(c\otimes h)\odot d\mu & =& g^{-1}{\int }_{A}g\left(g^{-1}(g(c)g(h))\right)dx\\ =& g^{-1}({\int }_{A}g(c)g(h)dx)\\ =& g^{-1}g(c){\int }_{A}g(h)dx\\ =& g^{-1}g(c)g{g}^{-1}({\int }_{A}g(h)dx)\\ =& g^{-1}g(c)g({\int }_A^{\oplus }h\odot d\mu )\\ =& c\otimes {\int }_A^{\oplus }h\odot d\mu .\end{array}$$

  (10)

By using the g-integral, we define the g-convolution of functions by means of the corresponding g-integral (see [2, 12–14]). □

Definition 2.4 ([14])

Let g be the same function as in Theorem 2.2, let ⊕, ⊙ be the same pseudo-operations as in Remark 2.1, $u\otimes v=g^{-1}(g(u)g(v))$ for $u,v\in [0,\mathrm{\infty })$, and $f,h\in {\mathfrak{F}}^{\ast }(X)$. The g-convolution of f and h by means of the g-integral is defined by

$$(f\ast h)(t)={\int }_{[0,t]}^{\oplus }[f(t-u)\otimes h(u)]\odot d\mu (u)$$

(11)

for all $t\in [0,\mathrm{\infty })$.

Finally, we introduce the following basic characterizations of the g-convolution in [14].

Theorem 2.3 ([14])

If g is the same function as in Theorem  2.2, ⊕, ⊙ are the same pseudo-operations as in Remark  2.1, $u\otimes v=g^{-1}(g(u)g(v))$ for $u,v\in [0,\mathrm{\infty })$, and $f,h\in {\mathfrak{F}}^{\ast }(X)$, then we have

$$(f\ast h)(t)=g^{-1}{\int }_0^{t}g(f(t-u))g(h(u))du$$

(12)

for all $t\in [0,\mathrm{\infty })$.

Theorem 2.4 ([14])

If g is the same function as in Theorem  2.2, ⊕, ⊙ are the same pseudo-operations as in Remark  2.1, $u\otimes v=g^{-1}(g(u)g(v))$ for $u,v\in [0,\mathrm{\infty })$, and $f,h,k\in {\mathfrak{F}}^{\ast }(X)$, then we have

$$f\ast h=h\ast f$$

(13)

and

$$(f\ast h)\ast k=f\ast (h\ast k).$$

(14)

3 The interval-valued $\overline{g}$-integrals

In this section, we consider the intervals, a standard interval-valued pseudo-addition, and a standard interval-valued pseudo-multiplication. Let $I(Y)$ be the set of all closed intervals (for short, intervals) in Y as follows:

$$I(Y)=\{\overline{a}=[a_l,a_r]\mid a_l,a_r\in Y\text{ and }{a}_l\le a_r\},$$

(15)

where Y is $[0,\mathrm{\infty })$ or $[0,\mathrm{\infty }]$. For any $a\in Y$, we define $a=[a,a]$. Obviously, $a\in I(Y)$ (see [1, 21–29]).

Definition 3.1 ([26])

If $\overline{a}=[a_l,a_r],\overline{b}=[b_l,b_r],{\overline{a}}_n=[a_{nl},a_{nr}],{\overline{a}}_{\alpha }=[a_{\alpha l},a_{\alpha r}]\in I(Y)$ for all $n\in \mathbb{N}$ and $\alpha \in [0,\mathrm{\infty })$, and $k\in [0,\mathrm{\infty })$, then we define arithmetic, maximum, minimum, order, inclusion, superior, and inferior operations as follows:

• (1) $\overline{a}+\overline{b}=[a_l+b_l,a_r+b_r]$,

• (2) $k\overline{a}=[k{a}_l,k{a}_r]$,

• (3) $\overline{a}\overline{b}=[a_l{b}_l,a_r{b}_r]$,

• (4) $\overline{a}\vee \overline{b}=[a_l\vee b_l,a_r\vee b_r]$,

• (5) $\overline{a}\wedge \overline{b}=[a_l\wedge b_l,a_r\wedge b_r]$,

• (6) $\overline{a}\le \overline{b}$ if and only if $a_l\le b_l$ and $a_r\le b_r$,

• (7) $\overline{a}<\overline{b}$ if and only if $a_l\le b_l$ and $a_l\ne b_l$,

• (8) $\overline{a}\subset \overline{b}$ if and only if $b_l\le a_l$ and $a_r\le b_r$,

• (9) ${sup}_n{\overline{a}}_n=[{sup}_n{a}_{nl},{sup}_n{a}_{nr}]$,

• (10) ${inf}_n{\overline{a}}_n=[{inf}_n{a}_{nl},{inf}_n{a}_{nr}]$,

• (11) ${sup}_{\alpha }{\overline{a}}_{\alpha }=[{sup}_{\alpha }{a}_{\alpha l},{sup}_{\alpha }{a}_{\alpha r}]$, and

• (12) ${inf}_{\alpha }{\overline{a}}_{\alpha }=[{inf}_{\alpha }{a}_{\alpha l},{inf}_{\alpha }{a}_{\alpha r}]$.

Definition 3.2 ([26])

• (1) A binary operation $⨁:I{([0,\mathrm{\infty }])}^2⟶I([0,\mathrm{\infty }])$ is called a standard interval-valued pseudo-addition if there exist pseudo-additions ${\oplus }_l$ and ${\oplus }_r$ such that $x{\oplus }_{l}y\le x{\oplus }_{r}y$ for all $x,y\in [0,\mathrm{\infty }]$, and such that for all $\overline{a}=[a_l,a_r],\overline{b}=[b_l,b_r]\in I([0,\mathrm{\infty }])$,

  $$\overline{a}⨁\overline{b}=[a_l{\oplus }_l{b}_l,a_r{\oplus }_r{b}_r].$$

  (16)

Then ${\oplus }_l$ and ${\oplus }_r$ are called the representants of ⨁.

• (2) A binary operation $⨀:I{([0,\mathrm{\infty }])}^2⟶I([0,\mathrm{\infty }])$ is called a standard interval-valued pseudo-multiplication if there exist pseudo-multiplications ${\odot }_l$ and ${\odot }_r$ such that $x{\odot }_{l}y\le x{\odot }_{r}y$ for all $x,y\in [0,\mathrm{\infty }]$, and such that for all $\overline{a}=[a_l,a_r],\overline{b}=[b_l,b_r]\in I([0,\mathrm{\infty }])$,

  $$\overline{a}⨀\overline{b}=[a_l{\odot }_l{b}_l,a_r{\odot }_r{b}_r].$$

  (17)

Then ${\odot }_l$ and ${\odot }_r$ are called the representants of ⨀.

Theorem 3.1 If two pseudo-additions ${\oplus }_l$ and ${\oplus }_r$ are representants of a standard interval-valued pseudo-addition ⨁, two pseudo-multiplications ${\odot }_l$ and ${\odot }_r$ are representants of a standard interval-valued pseudo-multiplication ⨀, then we have

• (1) $\overline{x}⨁\overline{y}=\overline{y}⨁\overline{x}$ for all $\overline{x},\overline{y}\in I([0,\mathrm{\infty }])$,

• (2) $(\overline{x}⨁\overline{y})⨁\overline{z}=\overline{x}⨁(\overline{y}⨁\overline{z})$ for all $\overline{x},\overline{y},\overline{z}\in I([0,\mathrm{\infty }])$,

• (3) $\overline{x}⨀\overline{y}=\overline{y}⨀\overline{x}$ for all $\overline{x},\overline{y}\in I([0,\mathrm{\infty }])$,

• (4) $(\overline{x}⨀\overline{y})⨀\overline{z}=\overline{x}⨀(\overline{y}⨀\overline{z})$ for all $\overline{x},\overline{y},\overline{z}\in I([0,\mathrm{\infty }])$,

• (5) $\overline{x}⨀(\overline{y}⨁\overline{z})=(\overline{x}⨀\overline{y})⨁(\overline{x}⨀\overline{z})$ for all $\overline{x},\overline{y},\overline{z}\in I([0,\mathrm{\infty }])$.

Proof (1) By the commutativity of ${\oplus }_l$ and ${\oplus }_r$, for any $\overline{x},\overline{y}\in I([0,\mathrm{\infty }])$, we have

$$\begin{array}{rcl}\overline{x}⨁\overline{y}& =& [x_l{\oplus }_l{y}_l,x_r{\oplus }_r{y}_r]\\ =& [y_l{\oplus }_l{x}_l,y_r{\oplus }_r{x}_r]\\ =& \overline{y}⨁\overline{x}.\end{array}$$

(18)

• (2) By the associativity of ${\oplus }_l$ and ${\oplus }_r$, for any $\overline{x},\overline{y},\overline{z}\in I([0,\mathrm{\infty }])$, we have

  $$\begin{array}{rcl}(\overline{x}⨁\overline{y})⨁\overline{z}& =& [x_l{\oplus }_l{y}_l,x_r{\oplus }_r{y}_r]⨁[z_l,z_r]\\ =& [(x_l{\oplus }_l{y}_l){\oplus }_l{z}_l,(x_r{\oplus }_r{y}_r){\oplus }_r{z}_r]\\ =& [x_l{\oplus }_l(y_l{\oplus }_l{z}_l),x_r{\oplus }_r(y_r{\oplus }_r{z}_r)]\\ =& [x_l,x_r]⨁[y_l{\oplus }_l{x}_l,y_r{\oplus }_r{x}_r]\\ =& \overline{x}⨁(\overline{y}⨁\overline{z}).\end{array}$$

  (19)

• (3) By the commutativity of ${\odot }_l$ and ${\odot }_r$, for any $\overline{x},\overline{y}\in I([0,\mathrm{\infty }])$, we have

  $$\begin{array}{rcl}\overline{x}⨀\overline{y}& =& [x_l{\odot }_l{y}_l,x_r{\odot }_r{y}_r]\\ =& [y_l{\odot }_l{x}_l,y_r{\odot }_r{x}_r]\\ =& \overline{y}⨀\overline{x}.\end{array}$$

  (20)

• (4) By the associativity of ${\odot }_l$ and ${\odot }_r$ in Definition 2.1(2)(ii), for any $\overline{x},\overline{y},\overline{z}\in I([0,\mathrm{\infty }])$, we have

  $$\begin{array}{rcl}(\overline{x}⨀\overline{y})⨀\overline{z}& =& [x_l{\odot }_l{y}_l,x_r{\odot }_r{y}_r]⨀[z_l,z_r]\\ =& [(x_l{\odot }_l{y}_l){\odot }_l{z}_l,(x_r{\odot }_r{y}_r){\odot }_r{z}_r]\\ =& [x_l{\odot }_l(y_l{\odot }_l{z}_l),x_r{\odot }_r(y_r{\odot }_r{z}_r)]\\ =& [x_l,x_r]{⨀}_I[y_l{\odot }_l{x}_l,y_r{\odot }_r{x}_r]\\ =& \overline{x}⨀(\overline{y}⨀\overline{z}).\end{array}$$

  (21)

• (5) By the distributivity of ${\oplus }_s$ and ${\odot }_s$ for $s=l,r$ in Definition 2.1(2)(iv), for any $\overline{x},\overline{y},\overline{z}\in I([0,\mathrm{\infty }])$, we have

  $$\begin{array}{rcl}\overline{x}⨀(\overline{y}⨁\overline{z})& =& [x_l,x_r]⨀[y_l{\oplus }_l{z}_l,y_r{\oplus }_r{z}_r]\\ =& [x_l{\odot }_l(y_l{\oplus }_l{z}_l),x_r{\odot }_r(y_r{\oplus }_r{z}_r)]\\ =& [(x_l{\odot }_l{y}_l){\oplus }_l(x_l{\odot }_l{z}_l),(x_r{\odot }_r{y}_r){\oplus }_r(x_r{\odot }_r{z}_r)]\\ =& [x_l{\odot }_l{y}_l,x_r{\odot }_r{y}_r]⨁[x_l{\odot }_l{z}_l,x_r{\odot }_r{z}_r]\\ =& (\overline{x}⨀\overline{y})⨁(\overline{x}⨀\overline{z}).\end{array}$$

  (22)

By using a standard interval-valued pseudo-addition and a standard interval-valued pseudo-multiplication, we define the interval-valued $\overline{g}$-integral represented by its interval-valued generator $\overline{g}$. □

Definition 3.3 Let X be a set, two pseudo-additions ${\oplus }_l$ and ${\oplus }_r$ be representants of a standard interval-valued pseudo-addition ⨁, and two pseudo-multiplications ${\odot }_l$ and ${\odot }_r$ be representants of a standard interval-valued pseudo-multiplication ⨀.

• (1) An interval-valued function $\overline{f}:X\to I([0,\mathrm{\infty }))\setminus \{\mathrm{\varnothing }\}$ is said to be measurable if for any open set $O\subset [0,\mathrm{\infty })$,

  $${\overline{f}}^{-1}(O)=\{x\in X\mid \overline{f}(x)\cap O\ne \mathrm{\varnothing }\}\in \mathcal{A}.$$

  (23)

• (2) Let $g_s$ be a continuous strictly increasing surjective function for $s=l,r$ such that $g_l\le g_r$, $\overline{g}=[g_l,g_r]$, and $g_s(0)=0$ for $s=l,r$. The interval-valued $\overline{g}$-integral with respect to a fuzzy measure μ of a measurable interval-valued function $\overline{f}=[f_l,f_r]$ is defined by

  $${\int }_A^{⨁}\overline{f}⨀d\mu =[{\int }_A^{{\oplus }_l}{f}_l{\odot }_{l}d\mu ,{\int }_A^{{\oplus }_r}{f}_r{\odot }_{r}d\mu ]$$

  (24)

for all $A\in \mathcal{A}$.

• (3) $\overline{f}$ is said to be integrable on $A\in \mathcal{A}$ if

  $${\int }_A^{⨁}\overline{f}⨀d\mu \in I([0,\mathrm{\infty }]).$$

  (25)

Let $\mathfrak{IF}(X)$ be the set of all measurable interval-valued functions and ${\mathfrak{IF}}^{\ast }(X)$ be the set of all integrable interval-valued functions. Then, by Definition 3.3, we directly obtain the following theorem.

Theorem 3.2 If $g_s$ is a continuous strictly increasing surjective function for $s=l,r$ such that $g_l\le g_r$, $\overline{g}=[g_l,g_r]$, and $g_s(0)=0$ for $s=l,r$, two pseudo-additions ${\oplus }_l$ and ${\oplus }_r$ are representants of a standard interval-valued pseudo-addition ⨁, and two pseudo-multiplications ${\odot }_l$ and ${\odot }_r$ are representants of a standard interval-valued pseudo-multiplication ⨀, then we have

$${\int }_A^{⨁}\overline{f}⨀d\mu =[g_l^{-1}{\int }_A{g}_l(f_l(x))dx,g_r^{-1}{\int }_A{g}_r(f_r(x))dx].$$

(26)

Proof By Definition 2.3(1),

$${\int }_A^{{\oplus }_s}{f}_s{\odot }_{s}d\mu =g_s^{-1}{\int }_A{g}_s(f_s(x))dx$$

(27)

for $s=l,r$. By (27) and Definition 3.3, we have

$$\begin{array}{rcl}{\int }_A^{⨁}\overline{f}⨀d\mu & =& [{\int }_A^{{\oplus }_l}{f}_l{\odot }_{l}d\mu ,{\int }_A^{{\oplus }_r}{f}_r{\odot }_{r}d\mu ]\\ =& [g_l^{-1}{\int }_A{g}_l(f_l(x))dx,g_r^{-1}{\int }_A{g}_r(f_r(x))dx].\end{array}$$

(28)

By the definition of the interval-valued $\overline{g}$-integral, we directly obtain the following basic properties. □

Theorem 3.3 Let $g_s$ be a continuous strictly increasing surjective function for $s=l,r$ such that $g_l\le g_r$, $\overline{g}=[g_l,g_r]$, and $g_s(0)=0$ for $s=l,r$, two pseudo-additions ${\oplus }_l$ and ${\oplus }_r$ be representants of a standard interval-valued pseudo-addition ⨁, two pseudo-multiplications ${\odot }_l$ and ${\odot }_r$ be representants of a standard interval-valued pseudo-multiplication ⨀, and two pseudo-multiplications ${\otimes }_l$ and ${\otimes }_r$ be representants of a standard interval-valued pseudo-multiplication ⨂.

• (1) If $A\in \mathcal{A}$ and $\overline{f},\overline{h}\in {\mathfrak{IF}}^{\ast }(X)$ and $\overline{f}\le \overline{h}$, then we have

  $${\int }_A^{⨁}\overline{f}⨀d\mu \le {\int }_A^{⨁}\overline{h}⨀d\mu .$$

  (29)

• (2) If $A\in \mathcal{A}$ and $\overline{f},\overline{h}\in {\mathfrak{IF}}^{\ast }(X)$, then we have

  $${\int }_A^{⨁}(\overline{f}⨁\overline{h})⨀d\mu ={\int }_A^{⨁}\overline{f}⨀d\mu ⨁{\int }_A^{⨁}\overline{h}⨀d\mu .$$

  (30)

• (3) If $A\in \mathcal{A}$ and $\overline{c}=[c_l,c_r]\in I([0,\mathrm{\infty }))$, $\overline{h}\in {\mathfrak{IF}}^{\ast }(X)$, then we have

  $${\int }_A^{⨁}(\overline{c}⨂\overline{h})⨀d\mu =\overline{c}⨂{\int }_A^{⨁}\overline{h}⨀d\mu .$$

  (31)

Proof (1) Note that if $\overline{f},\overline{h}\in {\mathfrak{IF}}^{\ast }(X)$ and $\overline{f}\le \overline{h}$, then

$$f_s\le h_s$$

(32)

for $s=l,r$. Since $g_l$ and $g_r$ are strictly monotone increasing,

$$g_s\circ f_s\le g_s\circ h_s$$

(33)

for $s=l,r$. By (33) and Theorem 2.2(1),

$${\int }_A^{{\oplus }_s}{f}_s{\odot }_{s}d\mu \le {\int }_A^{{\oplus }_s}{h}_s{\odot }_{s}d\mu$$

(34)

for $s=l,r$. By (34) and Theorem 3.2,

$$\begin{array}{rcl}{\int }_A^{⨁}\overline{f}⨀d\mu & =& [{\int }_A^{{\oplus }_l}{f}_l{\odot }_{l}d\mu ,{\int }_A^{{\oplus }_r}{f}_r{\odot }_{r}d\mu ]\\ \le & [{\int }_A^{{\oplus }_l}{h}_l{\odot }_{l}d\mu ,{\int }_A^{{\oplus }_r}{h}_r{\odot }_{r}d\mu ]={\int }_A^{⨁}\overline{h}⨀d\mu .\end{array}$$

(35)

• (2) Note that if $\overline{f},\overline{h}\in {\mathfrak{IF}}^{\ast }(X)$, then

  $$\overline{f}⨁\overline{h}=[f_l{\oplus }_l{h}_l,f_r{\oplus }_r{h}_r].$$

  (36)

By Theorem 2.2(2),

$${\int }_A^{{\oplus }_s}(f_s{\oplus }_s{h}_s){\odot }_{s}d\mu ={\int }_A^{{\oplus }_s}{f}_s{\odot }_{s}d\mu {\oplus }_s{\int }_A^{{\oplus }_s}{h}_s{\odot }_{s}d\mu$$

(37)

for $s=l,r$. By (37) and Theorem 3.2,

$$\begin{array}{c}{\int }_A^{⨁}(\overline{f}⨁\overline{h})⨀d\mu \hfill \\ =[{\int }_A^{{\oplus }_l}(f_l{\oplus }_l{h}_l){\odot }_{l}d\mu ,{\int }_A^{{\oplus }_r}(f_r{\oplus }_r{h}_r){\odot }_{r}d\mu ]\hfill \\ =[{\int }_A^{{\oplus }_l}{f}_l{\odot }_{l}d\mu {\oplus }_l{\int }_A^{{\oplus }_l}{h}_l{\odot }_{l}d\mu ,{\int }_A^{{\oplus }_r}{f}_r{\odot }_{r}d\mu {\oplus }_r{\int }_A^{{\oplus }_r}{h}_r{\odot }_{r}d\mu ]\hfill \\ =[{\int }_A^{{\oplus }_l}{f}_l{\odot }_{l}d\mu ,{\int }_A^{{\oplus }_r}{f}_r{\odot }_{r}d\mu ]{⨁}_I[{\int }_A^{{\oplus }_l}{h}_l{\odot }_{l}d\mu ,{\int }_A^{{\oplus }_r}{h}_r{\odot }_{r}d\mu ]\hfill \\ ={\int }_A^{⨁}\overline{f}⨀d\mu ⨁{\int }_A^{⨁}\overline{h}⨀d\mu .\hfill \end{array}$$

(38)

• (3) Note that if $\overline{f}\in {\mathfrak{IF}}^{\ast }(X)$ and $\overline{c}\in I([0,\mathrm{\infty }))$, then

  $$\overline{c}⨂\overline{f}=[c_l{\otimes }_l{f}_l,c_r{\otimes }_r{f}_r].$$

  (39)

By Theorem 2.2(3),

$${\int }_A^{{\oplus }_s}(c_s{\otimes }_s{f}_s){\odot }_{s}d\mu =c_s{\otimes }_s{\int }_A^{{\oplus }_s}{f}_s{\odot }_{s}d\mu$$

(40)

for $s=l,r$. By (40) and Definition 3.3(2),

$$\begin{array}{c}{\int }_A^{⨁}(\overline{c}⨂\overline{f})⨀d\mu \hfill \\ =[{\int }_A^{{\oplus }_l}(c_l{\otimes }_l{f}_l){\odot }_{l}d\mu ,{\int }_A^{{\oplus }_r}(c_r{\otimes }_r{f}_r){\odot }_{r}d\mu ]\hfill \\ =[c_l{\otimes }_l{\int }_A^{{\oplus }_l}{f}_l{\odot }_{l}d\mu ,c_r{\otimes }_r{\int }_A^{{\oplus }_r}{f}_r{\odot }_{r}d\mu ]\hfill \\ =\overline{c}⨂{\int }_A^{⨁}\overline{f}⨀d\mu .\hfill \end{array}$$

(41)

 □

4 An interval-valued $\overline{g}$-convolution

In this section, by using the interval-valued $\overline{g}$-integral, we define the interval-valued $\overline{g}$-convolution of interval-valued functions in ${\mathfrak{IF}}^{\ast }(X)$.

Definition 4.1 If $\overline{g}$, ⨁, ⨀, and ⨂ satisfy the hypotheses of Theorem 3.2, then the interval-valued $\overline{g}$-convolution is defined by

$$(\overline{f}\star \overline{h})(t)={\int }_{[0,t]}^{⨁}[\overline{f}(t-u)⨂\overline{h}(u)]⨀d\mu (u)$$

(42)

for all $t\in [0,\mathrm{\infty })$.

From Definition 4.1, we directly obtain some characterization of an interval-valued $\overline{g}$-convolution by means of the interval-valued $\overline{g}$-integrals.

Theorem 4.1 If $\overline{g}$, ⨁, ⨀, and ⨂ satisfy the hypotheses of Theorem  3.2, then we have

$$\overline{f}\star \overline{h}=[f_l{\ast }_l{h}_l,f_r{\ast }_r{h}_r],$$

(43)

where $(f_s{\ast }_s{h}_s)(t)={\int }_{[0,t]}^{{\oplus }_s}{f}_s(t-u){\odot }_{s}d\mu$ for $s=l,r$.

Proof By Definition 2.4, we have

$$(f_s{\ast }_s{h}_s)(t)={\int }_{[0,t]}^{{\oplus }_s}{f}_s(t-u){\otimes }_s{h}_s(u){\odot }_{s}d\mu (u)$$

(44)

for $s=l,r$. By Theorem 3.2 and (44),

$$\begin{array}{rcl}(\overline{f}\star \overline{h})(t)& =& {\int }_{[0,t]}^{⨁}(\overline{f}(t-u)⨂\overline{h}(u))⨀d\mu (u)\\ =& {\int }_{[0,t]}^{⨁}[f_l(t-u){\otimes }_l{h}_l(u),f_r(t-u){\otimes }_r{h}_r(u)]⨀d\mu (u)\\ =& [{\int }_{[0,t]}^{{\oplus }_l}{f}_l(t-u){\otimes }_l{h}_l(u){\odot }_{l}d\mu ,{\int }_{[0,t]}^{{\oplus }_r}{f}_r(t-u){\otimes }_r{h}_r(u){\odot }_{r}d\mu ]\\ =& [(f_l{\ast }_l{h}_l)(t),(f_r{\ast }_r{h}_r)(t)].\end{array}$$

(45)

 □

From Theorem 4.1, we investigate the commutativity and the associativity of a standard interval-valued $\overline{g}$-convolution.

Theorem 4.2 If $\overline{g}$, ⨁, ⨀, and ⨂ satisfy the hypotheses of Theorem  3.2 and $\overline{f}$, $\overline{h}$, and $\overline{k}\in {\mathfrak{IF}}^{\ast }(X)$, then we have

• (1) $\overline{f}\star \overline{h}=\overline{h}\star \overline{f}$,

• (2) $(\overline{f}\star \overline{h})\star \overline{k}=\overline{f}\star (\overline{h}\star \overline{k})$.

Proof Let $\overline{f}=[f_l,f_r]$, $\overline{h}=[h_l,h_r]$, $\overline{k}=[k_l,k_r]\in {\mathfrak{IF}}^{\ast }(X)$. By (16), we have

$$f_l{\ast }_l{h}_l=h_l{\ast }_l{f}_l\text{and}{f}_r{\ast }_r{h}_r=h_r{\ast }_r{f}_r.$$

(46)

By Theorem 4.1 and (46), we have

$$\begin{array}{rcl}\overline{f}\star \overline{h}& =& [f_l{\ast }_l{h}_l,f_r{\ast }_r{h}_r]\\ =& [h_l{\ast }_l{f}_l,h_r{\ast }_r{f}_r]\\ =& \overline{h}\star \overline{f}.\end{array}$$

(47)

By (17), we have

$$(f_l{\ast }_l{h}_l){\ast }_l{k}_l=f_l{\ast }_l(h_l{\ast }_l{k}_l)\text{and}(f_r{\ast }_r{h}_r){\ast }_r{k}_r=f_r{\ast }_r(h_r{\ast }_r{k}_r).$$

(48)

By Theorem 4.1 and (48),

$$\begin{array}{rcl}(\overline{f}\star \overline{h})\star \overline{k}& =& [(f_l{\ast }_l{h}_l){\ast }_l{k}_l,(f_r{\ast }_r{h}_r){\ast }_r{k}_r]\\ =& [f_l{\ast }_l(h_l{\ast }_l{k}_l),f_r{\ast }_r(h_r{\ast }_r{k}_r)]\\ =& \overline{f}\star (\overline{h}\star \overline{k}).\end{array}$$

(49)

 □

Finally, we illustrate the following examples which are related with the interval-valued $\overline{g}$-integral and the interval-valued $\overline{g}$-convolution as follows.

Example 4.1 We give three examples of the interval-valued $\overline{g}$-integral.

• (1) If $g_l(x)=g_r(x)=x$ for all $x\in [0,\mathrm{\infty }]$ are the generators of ${\odot }_l$, ${\odot }_r$, ${\oplus }_l$, and ${\oplus }_r$, and $\overline{f}(x)=[\frac{e^{-x}}{2},e^{-x}]$ for all $x\in [0,\mathrm{\infty })$, and $A=[0,t]$ for all $t\in [0,\mathrm{\infty })$, then we have

  $$\begin{array}{rcl}{\int }_A^{⨁}\overline{f}⨀d\mu & =& [g_l^{-1}{\int }_0^t{g}_l(f_l(x))dx,g_r^{-1}{\int }_0^t{g}_r(f_r(x))dx]\\ =& [{\int }_0^t\frac{1}{2}{e}^{-x}dx,{\int }_0^t{e}^{-x}dx]\\ =& [\frac{1}{2}(1-e^{-t}),(1-e^{-t})].\end{array}$$

  (50)

• (2) If $g_l(x)=\frac{1}{2}x$, $g_r(x)=x$ for all $x\in [0,\mathrm{\infty }]$ are the generators of ${\odot }_l$, ${\odot }_r$, and $g_l(x)=g_r(x)=x$ for all $x\in [0,\mathrm{\infty }]$ are the generators of ${\oplus }_l$, ${\oplus }_r$, and $\overline{f}(x)=[\frac{e^{-x}}{2},e^{-x}]$ for all $x\in [0,\mathrm{\infty })$, and $A=[0,t]$ for all $t\in [0,\mathrm{\infty })$, then we have

  $$\begin{array}{rcl}{\int }_A^{⨁}\overline{f}⨀d\mu & =& [g_l^{-1}{\int }_0^t{g}_l(f_l(x))dx,g_r^{-1}{\int }_0^t{g}_r(f_r(x))dx]\\ =& [2{\int }_0^t\frac{1}{4}{e}^{-x}dx,{\int }_0^t{e}^{-x}dx]\\ =& [\frac{1}{2}(1-e^{-t}),(1-e^{-t})].\end{array}$$

  (51)

• (3) If $g_l(x)=x^2$, $g_r(x)=3x^2$ for all $x\in [0,\mathrm{\infty }]$ are the generators of ${\odot }_l$, ${\odot }_r$, and $g_l(x)=g_r(x)=x$ for all $x\in [0,\mathrm{\infty }]$ are the generators of ${\oplus }_l$, ${\oplus }_r$, and $\overline{f}(x)=[\frac{e^{-x}}{2},e^{-x}]$ for all $x\in [0,\mathrm{\infty })$, and $A=[0,t]$ for all $t\in [0,\mathrm{\infty })$, then we have

  $$\begin{array}{rcl}{\int }_A^{⨁}\overline{f}⨀d\mu & =& [g_l^{-1}{\int }_0^t{g}_l(f_l(x))dx,g_r^{-1}{\int }_0^t{g}_r(f_r(x))dx]\\ =& [\sqrt{{\int }_0^t\frac{1}{2}{e}^{-2x}dx},\sqrt{\frac{1}{3}{\int }_0^{t}3e^{-2x}dx}]\\ =& [\frac{1}{4}(1-e^{-2t}),\frac{1}{2}(1-e^{-t})].\end{array}$$

  (52)

Example 4.2 We give an example of the interval-valued $\overline{g}$-convolution.

If $g_l(x)=x^2$, $g_r(x)=3x^2$ for all $x\in [0,\mathrm{\infty }]$ are the generators of ${\odot }_l$, ${\odot }_r$, and $g_l(x)=g_r(x)=x$ for all $x\in [0,\mathrm{\infty }]$ are the generators of ${\oplus }_l$, ${\oplus }_r$, ${\otimes }_l$, ${\otimes }_r$, and $\overline{f}(x)=[\frac{e^{-x}}{2},e^{-x}]$ for all $x\in [0,\mathrm{\infty })$, $\overline{h}(x)=[\frac{1}{2}x,x]$ for all $x\in [0,\mathrm{\infty })$, and $A=[0,t]$ for all $t\in [0,\mathrm{\infty })$, then we have

$$\begin{array}{rcl}(\overline{f}\star \overline{h})(t)& =& {\int }_A^{⨁}[\overline{f}(t-u)⨂\overline{h}(u)]⨀d\mu (u)\\ =& [\sqrt{\frac{1}{2}{\int }_0^t{e}^{-2(t-u)}{e}^{-2u}du},\sqrt{3{\int }_0^t\frac{1}{2}{e}^{-2(x-u)}{e}^{-2u}du}]\\ =& [\frac{t}{4}{e}^{-2t},\frac{3t}{2}{e}^{-2t}].\end{array}$$

(53)

5 Conclusions

In this paper, we have considered the g-integral represented by its generating g, the pseudo-addition, the pseudo-multiplication (see Definition 2.3). This study was to define the g-convolution by means of the g-integral (see Definition 2.4) and to investigate some characterizations of the g-integral and the commutativity and the associativity of the g-convolution (see Theorems 2.2, 2.3, and 2.4).

We also defined the interval-valued $\overline{g}$-integral represented by its interval-valued generator $\overline{g}$. By using general notions of an interval-representable pseudo-multiplication (see Definition 3.2), we defined an interval-valued $\overline{g}$-integral (see Definition 3.3) and investigated some basic characterizations of them (see Theorems 3.2, 3.3).

From Definitions 2.3, 2.4, and Theorems 2.2, 2.3, we defined a standard interval-valued $\overline{g}$-convolution (see Definition 4.1). We also investigated some characterizations of a standard interval-valued $\overline{g}$-convolution of interval-valued functions by means of the interval-valued $\overline{g}$-integral including commutativity and associativity of an interval-representable convolution (see Theorems 4.1, 4.2).

In the future, we can study various inequalities of the interval-valued $\overline{g}$-integral and expect that the standard interval-valued $\overline{g}$-convolutions are used (i) to generalize the g-Laplace transform, Hamilton-Jacobi equation on the space of functions, such as in nonlinearity and optimization and such as in information theory (see [1, 14, 29]); (ii) to generalize the Stolasky-type inequality for the pseudo-integral of functions such as in economics, finance, decision making (see [2, 30]), etc.