# Efficient Solution Criteria for a Coupled Fractional Laplacian System on Some Infinite Domains
## 3 Main Results
### Stability Results
In view of metric \(\mathrm{d}_{1}(\cdot,\cdot)\), and the assumptions \((\mathbf{A}\mathbf{S}_{1})-(\mathbf{A}\mathbf{S}_{4})\), for any \((\mathcal{U},\mathcal{V})\), \((\mathcal{U},\bar{\mathcal{V}})\in\mathbb{Y}_{1}\times\mathbb{Y}_{2}\), one finds  
\[\frac{\|(\mathbb{Y}_{1}\mathcal{V})(\mathrm{t})-(\mathbb{Y}_{1} \bar{\mathcal{V}})(\mathrm{t})\|}{(1+\varphi^{\theta_{1}}(\mathrm{t};c))} \] \[\leq\frac{(q-1)(\mathcal{A}_{2})^{q-2}\mathcal{H}\beta(\mathrm{t} )}{(1+\varphi^{\theta_{1}}(\mathrm{t};c))}\,\mathcal{I}_{c^{+}}^{\theta_{1}+ \theta_{1};\varphi}\Big{(}\mathcal{N}_{1}(\mathrm{t})(1+\varphi^{\theta_{1}}( \mathrm{t};c))\Big{)}\] \[\quad+\frac{|\Lambda_{1}|\varphi^{\theta_{1}-1}(\mathrm{t};c)}{(1 +\varphi^{\theta_{1}}(\mathrm{t};c))}(q-1)(\mathcal{A}_{2})^{q-2}\mathcal{H} \beta(\mathrm{t})\sum_{j=1}^{n}\eta_{j}\,\mathcal{I}_{c^{+}}^{\theta_{1}+ \theta_{1}-1;\varphi}\Big{(}\mathcal{N}_{1}(\epsilon_{j})(1+\varphi^{\theta_ {1}}(\epsilon_{j};c))\Big{)}\] \[\leq\mathcal{A}_{1}\mathcal{H}\beta(\mathrm{t}),\forall\mathrm{t }\in\mathcal{T},\quad 0\leq\mathcal{A}_{1}<1.\]  
Also  
\[\frac{\|(\mathcal{D}_{c^{+}}^{\gamma_{1},\varphi}\mathbb{Y}_{1} \mathcal{V})(\mathrm{t})-(\mathcal{D}_{c^{+}}^{\gamma_{1},\varphi}\mathbb{Y }_{1}\bar{\mathcal{V}})(\mathrm{t})\|}{(1+\varphi^{\theta_{1}}(\mathrm{t};c))}\] \[\leq\frac{(q-1)(\mathcal{A}_{2})^{q-2}\mathcal{H}\beta(\mathrm{t })}{(1+\varphi^{\theta_{1}}(\mathrm{t};c))}\,\mathcal{I}_{c^{+}}^{\theta_{1}- \gamma_{1}+\theta_{1};\varphi}\Big{(}\mathcal{N}_{1}(\mathrm{t})(1+\varphi^ {\theta_{1}}(\mathrm{t};c))\Big{)}\] \[\quad+\frac{|\Lambda_{1}|\Gamma(\theta_{1})\varphi^{\theta_{1}- \gamma_{1}-1}(\mathrm{t};c)}{\Gamma(\theta_{1}-\gamma_{1})(1+\varphi^{\theta _{1}}(\mathrm{t};c))}(q-1)(\mathcal{A}_{2})^{q-2}\mathcal{H}\beta(\mathrm{t}) \sum_{j=1}^{n}\eta_{j}\,\mathcal{I}_{c^{+}}^{\theta_{1}+\theta_{1}-1;\varphi} \Big{(}\mathcal{N}_{1}(\epsilon_{j})(1+\varphi^{\theta_{1}}(\epsilon_{j};c)) \Big{)}\] \[\left.\frac{\|\mathcal{V}\left(\mathrm{t}\  
_which suggests that the coupled \(\varphi\)-\(\mathbb{R}\mathbb{L}\) fractional Laplacian implicit differential system (1) allows for a stable \(\mathbb{U}\mathbb{H}\mathbb{R}\) type, and, as a consequence \(\mathbb{U}\mathbb{H}\) is stable. Given that_  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{1}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{1}(q-1)+1)}{\Gamma(\theta_{1}(q-1)+\alpha_{1}+1)} \varphi^{\theta_{1}(q-1)+\alpha_{1}}(\mathrm{t};c)\leq\mathcal{R}_{1}<\infty,\]0  
In the same manner, one has  
\[\left.\frac{\|\mathcal{V}\left(\mathrm{t}\  
_which suggests that the coupled \(\varphi\)-\(\mathbb{R}\mathbb{L}\) fractional Laplacian implicit differential system (1) allows for a stable \(\mathbb{U}\mathbb{H}\mathbb{R}\) type, and, as a consequence \(\mathbb{U}\mathbb{H}\) is stable. Given that_  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{1}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{1}(q-1)+1)}{\Gamma(\theta_{1}(q-1)+\alpha_{1}+1)} \varphi^{\theta_{1}(q-1)+\alpha_{1}}(\mathrm{t};c)\leq\mathcal{R}_{1}<\infty,\]1 \[\left.\frac{\|\mathcal{V}\left(\mathrm{t}\  
_which suggests that the coupled \(\varphi\)-\(\mathbb{R}\mathbb{L}\) fractional Laplacian implicit differential system (1) allows for a stable \(\mathbb{U}\mathbb{H}\mathbb{R}\) type, and, as a consequence \(\mathbb{U}\mathbb{H}\) is stable. Given that_  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{1}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{1}(q-1)+1)}{\Gamma(\theta_{1}(q-1)+\alpha_{1}+1)} \varphi^{\theta_{1}(q-1)+\alpha_{1}}(\mathrm{t};c)\leq\mathcal{R}_{1}<\infty,\]2  
Then, we obtain  
\[\left.\frac{\|\mathcal{V}\left(\mathrm{t}\  
_which suggests that the coupled \(\varphi\)-\(\mathbb{R}\mathbb{L}\) fractional Laplacian implicit differential system (1) allows for a stable \(\mathbb{U}\mathbb{H}\mathbb{R}\) type, and, as a consequence \(\mathbb{U}\mathbb{H}\) is stable. Given that_  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{1}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{1}(q-1)+1)}{\Gamma(\theta_{1}(q-1)+\alpha_{1}+1)} \varphi^{\theta_{1}(q-1)+\alpha_{1}}(\mathrm{t};c)\leq\mathcal{R}_{1}<\infty,\]3  
Now, using inequalities (22)-(23), we obtain  
\[\left.\frac{\|\mathcal{V}\left(\mathrm{t}\  
_which suggests that the coupled \(\varphi\)-\(\mathbb{R}\mathbb{L}\) fractional Laplacian implicit differential system (1) allows for a stable \(\mathbb{U}\mathbb{H}\mathbb{R}\) type, and, as a consequence \(\mathbb{U}\mathbb{H}\) is stable. Given that_  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{1}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{1}(q-1)+1)}{\Gamma(\theta_{1}(q-1)+\alpha_{1}+1)} \varphi^{\theta_{1}(q-1)+\alpha_{1}}(\mathrm{t};c)\leq\mathcal{R}_{1}<\infty,\]4 \[\left.\frac{\|\mathcal{V}\left(\mathrm{t}\  
_which suggests that the coupled \(\varphi\)-\(\mathbb{R}\mathbb{L}\) fractional Laplacian implicit differential system (1) allows for a stable \(\mathbb{U}\mathbb{H}\mathbb{R}\) type, and, as a consequence \(\mathbb{U}\mathbb{H}\) is stable. Given that_  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{1}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{1}(q-1)+1)}{\Gamma(\theta_{1}(q-1)+\alpha_{1}+1)} \varphi^{\theta_{1}(q-1)+\alpha_{1}}(\mathrm{t};c)\leq\mathcal{R}_{1}<\infty,\]5 \[\left.\frac{\|\mathcal{V}\left(\mathrm{t}\  
_which suggests that the coupled \(\varphi\)-\(\mathbb{R}\mathbb{L}\) fractional Laplacian implicit differential system (1) allows for a stable \(\mathbb{U}\mathbb{H}\mathbb{R}\) type, and, as a consequence \(\mathbb{U}\mathbb{H}\) is stable. Given that_  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{1}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{1}(q-1)+1)}{\Gamma(\theta_{1}(q-1)+\alpha_{1}+1)} \varphi^{\theta_{1}(q-1)+\alpha_{1}}(\mathrm{t};c)\leq\mathcal{R}_{1}<\infty,\]6  
and\[\left.\frac{\|\mathcal{V}\left(\mathrm{t}\  
_which suggests that the coupled \(\varphi\)-\(\mathbb{R}\mathbb{L}\) fractional Laplacian implicit differential system (1) allows for a stable \(\mathbb{U}\mathbb{H}\mathbb{R}\) type, and, as a consequence \(\mathbb{U}\mathbb{H}\) is stable. Given that_  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{1}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{1}(q-1)+1)}{\Gamma(\theta_{1}(q-1)+\alpha_{1}+1)} \varphi^{\theta_{1}(q-1)+\alpha_{1}}(\mathrm{t};c)\leq\mathcal{R}_{1}<\infty,\]7 \[\left.\frac{\|\mathcal{V}\left(\mathrm{t}\  
_which suggests that the coupled \(\varphi\)-\(\mathbb{R}\mathbb{L}\) fractional Laplacian implicit differential system (1) allows for a stable \(\mathbb{U}\mathbb{H}\mathbb{R}\) type, and, as a consequence \(\mathbb{U}\mathbb{H}\) is stable. Given that_  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{1}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{1}(q-1)+1)}{\Gamma(\theta_{1}(q-1)+\alpha_{1}+1)} \varphi^{\theta_{1}(q-1)+\alpha_{1}}(\mathrm{t};c)\leq\mathcal{R}_{1}<\infty,\]8 \[\left.\frac{\|\mathcal{V}\left(\mathrm{t}\  
_which suggests that the coupled \(\varphi\)-\(\mathbb{R}\mathbb{L}\) fractional Laplacian implicit differential system (1) allows for a stable \(\mathbb{U}\mathbb{H}\mathbb{R}\) type, and, as a consequence \(\mathbb{U}\mathbb{H}\) is stable. Given that_  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{1}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{1}(q-1)+1)}{\Gamma(\theta_{1}(q-1)+\alpha_{1}+1)} \varphi^{\theta_{1}(q-1)+\alpha_{1}}(\mathrm{t};c)\leq\mathcal{R}_{1}<\infty,\]9  
Analogously, we also obtain  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{2}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{2}(q-1)+1)}{\Gamma(\theta_{2}(q-1)+\alpha_{2}+1)} \varphi^{\theta_{2}(q-1)+\alpha_{2}}(\mathrm{t};c)\leq\mathcal{R}_{2}<\infty,\]0 \[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{2}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{2}(q-1)+1)}{\Gamma(\theta_{2}(q-1)+\alpha_{2}+1)} \varphi^{\theta_{2}(q-1)+\alpha_{2}}(\mathrm{t};c)\leq\mathcal{R}_{2}<\infty,\]1  
Then, the above inequalities imply that  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{2}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{2}(q-1)+1)}{\Gamma(\theta_{2}(q-1)+\alpha_{2}+1)} \varphi^{\theta_{2}(q-1)+\alpha_{2}}(\mathrm{t};c)\leq\mathcal{R}_{2}<\infty,\]2  
Thus, by employing Theorem 1, the mapping \(\mathbb{Y}\) admits a fixed point \((\tilde{\mathcal{U}},\tilde{\mathcal{V}})\), and it follows that  
\[\sup_{\mathrm{t}\in\mathcal{T}}\frac{1}{(1+\varphi^{\theta_{1}}( \mathrm{t};c))}\Big{(}\frac{\beta(\mathrm{t})}{\Gamma(\theta_{2}+1)}\Big{)}^{q -1}\frac{\Gamma(\theta_{2}(q-1)+1)}{\Gamma(\theta_{2}(q-1)+\alpha_{2}+1)} \varphi^{\theta_{2}(q-1)+\alpha_{2}}(\mathrm{t};c)\leq\mathcal{R}_{2}<\infty,\]3  
Hence, we infer that the coupled \(\varphi\)-\(\mathbb{R}\mathbb{L}\) fractional Laplacian implicit differential system (1) is UHR stable.