# Efficient Solution Criteria for a Coupled Fractional Laplacian System on Some Infinite Domains
## 2 Preliminaries  
In this section, we introduce several essential backgrounds for this research. Consider \(\varphi\in\mathfrak{C}^{n}(\mathcal{T},\mathbb{R}^{+})\), which denotes a positive increasing function, and satisfies \(\varphi^{\prime}(\mathfrak{t})\
eq 0\), for each \(\mathfrak{t}\in\mathcal{T}\). Now, for \(\delta>0\), the \(\varphi\)-\(\mathbb{RL}\) fractional integral of order \(\delta\) is given as follows [30]:  
\[(\mathcal{I}_{c^{+}}^{\delta;\varphi}\mathfrak{g})(\mathfrak{t})=\frac{1}{ \Gamma(\delta)}\int_{c}^{\mathfrak{t}}\varphi^{\prime}(\mathfrak{v})\varphi^ {\delta-1}(\mathfrak{t};\mathfrak{v})\mathfrak{g}(\mathfrak{v})d\mathfrak{v },\;\mathfrak{t}>c.\]  
Also, for \(\delta\in(n-1,n]\), the \(\varphi\)-\(\mathbb{RL}\) fractional derivative of order \(\delta\) is given by the following [30]:  
\[(\mathcal{D}_{c^{+}}^{\delta;\varphi}\mathfrak{g})(\mathfrak{t}) = \left(\frac{1}{\varphi^{\prime}(\mathfrak{t})}\frac{d}{d \mathfrak{t}}\right)^{n}(\mathcal{I}_{c^{+}}^{n-\delta;\varphi}\mathfrak{g})( \mathfrak{t}),\mathfrak{t}>c,\]  
In addition, the following identities act as examples for the above fractional operators: \(\mathcal{I}_{c^{+}}^{\delta;\varphi}\varphi^{\gamma-1}(\mathfrak{t};c)= \frac{\Gamma(\gamma)}{\Gamma(\gamma+\delta)}\varphi^{\gamma+\delta-1}( \mathfrak{t};c),\delta>0\), and \(\mathcal{D}_{c^{+}}^{\delta;\varphi}\varphi^{\gamma-1}(\mathfrak{t};c)=\frac {\Gamma(\gamma)}{\Gamma(\gamma-\delta)}\varphi^{\gamma-\delta-1}(\mathfrak{t} ;c),n-1<\delta\leq n,\gamma>0\).  
Moreover, the fundamental fractional calculus of \(\varphi\)-\(\mathbb{RL}\) fractional derivative and integral are given as follows [31]:  
\[\mathcal{I}_{c^{+}}^{\delta;\varphi}\;\mathcal{D}_{c^{+}}^{\delta;\varphi} \mathfrak{g}(\mathfrak{t})=\mathfrak{g}(\mathfrak{t})-\sum_{k=1}^{n}a_{k} \varphi^{\delta-k}(\mathfrak{t};c),\;\;\forall\;\mathfrak{t}\in\mathcal{T}, \tag{2}\]  
for \(a_{k}\in\mathbb{R}\), \(\delta\in(n-1,n]\), and \(\mathfrak{g}^{n}\in\mathfrak{C}(\mathcal{T},\mathbb{R})\).  
Theorem 2.1 ([32]): _Let \((\mathcal{O},\mathbb{B})\) denote the generalized completed metric space, and \(\Xi\) be a contractive operator that maps \(\mathcal{O}\) into itself with the Lipschitz constant \(z<1\). If there is \(\ell\in\mathbb{N}\), where \(\mathbb{B}(\mathbb{E}^{\ell+1}\mathfrak{t},\mathbb{E}^{\ell}\mathfrak{t})<\infty\), for at least \(\mathfrak{t}\in\mathcal{O}\), then the following identities are fulfilled:_  
\((i)\)_\(\{\xi^{\ell}\mathfrak{t}\}\) be a convergence sequence tending to a fixed point \(\mathfrak{t}_{0}\in\mathcal{O}\);_  
\((ii)\)_\(\Xi\) owns one fixed point \(\mathfrak{t}_{0}\in\mathcal{O}^{*}=\{\mathfrak{s}\in\mathcal{O}|\mathbb{B}( \mathbb{E}^{\ell}\mathfrak{t},\mathfrak{s})<\infty\}\);_  
\((iii)\) _If \(\mathfrak{s}\in\mathcal{O}^{*}\),, then \(\mathbb{B}(\mathfrak{s},\mathfrak{t}_{0})\leq\frac{1}{1-z}\mathbb{B}(\Xi \mathfrak{s},\mathfrak{s})\)._  
Lemma 1 ([33]): _Consider \(\psi_{\mathfrak{p}}\) denotes the \(\mathfrak{p}\)-Laplacian function, so \((i)\) If \(|\mathcal{U}_{1}|,|\mathcal{U}_{2}|\geq\varrho>0\) and \(\mathcal{U}_{1},\mathcal{U}_{2}>0\), for every \(1<\mathfrak{p}\leq 2\), then_  
\[|\psi_{\mathfrak{p}}(\mathcal{U}_{1})-\psi_{\mathfrak{p}}(\mathcal{U}_{2})| \leq(\mathfrak{p}-1)\varrho^{\mathfrak{p}-2}\;|\mathcal{U}_{1}-\mathcal{U}_{2 }|.\]\((ii)\) _If \(|\mathcal{U}_{1}|,|\mathcal{U}_{2}|\leq\rho\), for every \(\mathfrak{p}>2\), then_  
\[|\psi_{\mathfrak{p}}(\mathcal{U}_{1})-\psi_{\mathfrak{p}}(\mathcal{U}_{2})|\leq( \mathfrak{p}-1)\rho^{\mathfrak{p}-2}\ |\mathcal{U}_{1}-\mathcal{U}_{2}|.\]