## 3 Preliminaries  
Throughout this article, the symbol \(\mathbb{N}\) denotes the set of all natural numbers. For \(n\in\mathbb{N}\), the symbol \(\mathbb{R}^{n}\) refers to the \(n\)-dimensional Euclidean space. The symbol \(\mathbb{R}_{---}\) refers to the collection of all negative real numbers. The symbols \(\mathcal{O}\) and \(\mathcal{I}_{n}\) are employed to denote the empty set and the identity matrix of order \(n\times n\), respectively. For two matrices \(\mathcal{A},\mathcal{B}\in\mathbb{R}^{n\times n}\), the notation \(\mathcal{B}\leq\mathcal{A}\) is employed to denote that \(\mathcal{A}-\mathcal{B}\) is positive semidefinite. For any \(u\in\mathbb{R}^{n}\) and \(r>0\), the symbols \(B(u,r)\) and \(B[u,r]\) denote the open and closed balls of radius \(r\) centered at \(u\), respectively. For any non-empty set \(\mathfrak{X}\) and \(m\in\mathbb{N}\), the notation \(\mathfrak{X}^{m}\) represents the Cartesian product defined as  
\[\mathfrak{X}^{m}:=\underbrace{\mathfrak{X}\times\ldots\times\mathfrak{X}}_{(m \text{ times})}.\]  
Let \(\mathfrak{X}=[\underline{p},\overline{p}]\), \(\mathfrak{Z}:=[\underline{q},\overline{q}]\in\mathfrak{C}(\mathbb{R})\), and \(\mathfrak{\zeta}\in\mathbb{R}\). Corresponding to \(\mathfrak{X}\), \(\mathfrak{Z}\), and \(\mathfrak{\zeta}\), we define the following algebraic operations [28]:  
\[\mathfrak{X}\oplus\mathfrak{Z}:=[\underline{p}+\underline{q}, \overline{p}+\overline{q}],\] \[\mathfrak{X}\ominus\mathfrak{Z}:=[\underline{p}-\overline{q}, \overline{p}-\underline{q}],\] \[\mathfrak{\zeta}\ominus\mathfrak{X}= \begin{cases}[\xi\underline {p},\zeta\overline{p}],&\mathfrak{\zeta}\geq 0,\\ [\zeta\overline{p},\xi\underline{p}],&\mathfrak{\zeta}<0.\end{cases} \]  
The subsequent definition is from [37]. **Definition 1**. : _Consider an arbitrary set \(\mathcal{E}:=[\underline{e},\,\overline{e}]\in\mathfrak{C}(\mathbb{R})\). Then, the symbol \(\|\mathcal{E}\|_{I}\) represents the norm of \(\mathcal{E}\) and is defined as follows:_  
\[\|\mathcal{E}\|_{I}:=\max\{|\underline{e}|,|\underline{e}|\}.\]  
For \(\mathfrak{X}:=[\underline{p},\overline{p}]\) and \(\mathfrak{Z}:=[\underline{q},\overline{q}]\in\mathfrak{C}(\mathbb{R})\), we adopt the following notations throughout the article:  
\[\mathfrak{X}\leqneq_{LU}\mathfrak{Z}\iff\underline{p}\leq\mathfrak{q}\,\,\, \text{and}\,\,\,\overline{p}\leq\overline{q},\]  
\[\mathfrak{X}<_{LU}\mathfrak{Z}\iff\underline{p}<\mathfrak{q}\,\,\,\text{and} \,\,\,\overline{p}<\overline{q},\]  
\[\mathfrak{X}\leq_{LU}\mathfrak{Z}\iff\mathfrak{X}\leq_{LU}\mathfrak{Z}\,\, \,\text{and}\,\,\,\mathfrak{X}\
eq\mathfrak{Z}.\]  
Let \(\mathcal{E}=(\mathcal{E}_{1},\ldots,\mathcal{E}_{m})\), \(\mathfrak{R}=(\mathfrak{R}_{1},\ldots,\mathfrak{R}_{m})\in(\mathfrak{C}( \mathbb{R}))^{m}\). The ordered relations between \(\mathcal{E}\) and \(\mathfrak{R}\) are described as follows:  
\[\mathcal{E}\preceq_{m}\mathfrak{R}\iff\mathcal{E}_{k}\leq_{LU} \mathfrak{R}_{k},\quad\text{ for all }k\in\mathcal{I}_{m},\] \[\mathcal{E}\prec_{m}\mathfrak{R}\iff\mathcal{E}_{k}\leq_{LU} \mathfrak{R}_{k},\quad\text{ for all }k\in\mathcal{I}_{m}.\]  
The following definition is from [37]. **Definition 2**. : _For arbitrary intervals \(\mathfrak{X}:=[\underline{p},\overline{p}]\) and \(\mathfrak{Z}:=[\underline{q},\overline{q}]\in\mathfrak{C}(\mathbb{R})\), the symbol \(\mathfrak{X}\ominus_{g\overline{q}}\mathfrak{Z}\) represents the gH-difference between \(\mathfrak{X}\) and \(\mathfrak{Z}\) and is defined as follows:_  
\[\mathfrak{X}\ominus_{g\overline{q}}\mathfrak{Z}:=[\underline{p}-\underline{ q}\,\,\vee\,\,\overline{p}-\overline{q}].\]  
The notion of gH-continuity for \(\mathcal{G}:\mathbb{R}^{n}\to\mathfrak{C}(\mathbb{R})\) is recalled in the subsequent definition [37]. **Definition 3**.