# Efficient Solution Criteria for a Coupled Fractional Laplacian System on Some Infinite Domains
## 3 Main Results
### Existences and Uniqueness Result
: According to the assumptions \((\mathbf{AS}_{2})\) and \((\mathbf{AS}_{4})\), for each \((\mathcal{U},\mathcal{V})\in\mathbb{M}\), and \(\mathfrak{t}\in\mathcal{T}\), one has  
\[\qquad+\frac{|\Lambda_{1}|\Gamma(\theta_{1})\varphi^{\theta_{1} -1}(\mathfrak{t};c)}{\Gamma(\alpha_{1})(1+\varphi^{\theta_{1}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{A}_{2})^{\mathfrak{q}-2}\sum_{j=1}^{n}\eta_{j} \,\mathcal{I}_{c^{+}}^{\theta_{1}+\theta_{1}-1;\varphi}\Big{(}\mathcal{N}_{1} (\mathfrak{e}_{j})(1+\varphi^{\theta_{1}}(\mathfrak{e}_{j};c))\Big{)}\Bigg{)} \leq\mathcal{A}_{1}<1,\]7 \[\qquad+\frac{|\Lambda_{1}|\Gamma(\theta_{1})\varphi^{\theta_{1} -1}(\mathfrak{t};c)}{\Gamma(\alpha_{1})(1+\varphi^{\theta_{1}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{A}_{2})^{\mathfrak{q}-2}\sum_{j=1}^{n}\eta_{j} \,\mathcal{I}_{c^{+}}^{\theta_{1}+\theta_{1}-1;\varphi}\Big{(}\mathcal{N}_{1} (\mathfrak{e}_{j})(1+\varphi^{\theta_{1}}(\mathfrak{e}_{j};c))\Big{)}\Bigg{)} \leq\mathcal{A}_{1}<1,\]8 \[\qquad+\frac{|\Lambda_{1}|\Gamma(\theta_{1})\varphi^{\theta_{1} -1}(\mathfrak{t};c)}{\Gamma(\alpha_{1})(1+\varphi^{\theta_{1}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{A}_{2})^{\mathfrak{q}-2}\sum_{j=1}^{n}\eta_{j} \,\mathcal{I}_{c^{+}}^{\theta_{1}+\theta_{1}-1;\varphi}\Big{(}\mathcal{N}_{1} (\mathfrak{e}_{j})(1+\varphi^{\theta_{1}}(\mathfrak{e}_{j};c))\Big{)}\Bigg{)} \leq\mathcal{A}_{1}<1,\]9 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Bigg{(}\frac{(\mathfrak{q}-1)(B_{2})^{\mathfrak{q}-2}}{(1+ \varphi^{\theta_{2}}(\mathfrak{t};c))}\,\mathcal{I}_{c^{+}}^{\mathfrak{t}_{ 2}+\theta_{2};\varphi}\Big{(}\mathcal{N}_{2}(\mathfrak{t})(1+\varphi^{ \theta_{2}}(\mathfrak{t};c))\Big{)}\]0 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Bigg{(}\frac{(\mathfrak{q}-1)(B_{2})^{\mathfrak{q}-2}}{(1+ \varphi^{\theta_{2}}(\mathfrak{t};c))}\,\mathcal{I}_{c^{+}}^{\mathfrak{t}_{ 2}+\theta_{2};\varphi}\Big{(}\mathcal{N}_{2}(\mathfrak{t})(1+\varphi^{ \theta_{2}}(\mathfrak{t};c))\Big{)}\]1 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Bigg{(}\frac{(\mathfrak{q}-1)(B_{2})^{\mathfrak{q}-2}}{(1+ \varphi^{\theta_{2}}(\mathfrak{t};c))}\,\mathcal{I}_{c^{+}}^{\mathfrak{t}_{ 2}+\theta_{2};\varphi}\Big{(}\mathcal{N}_{2}(\mathfrak{t})(1+\varphi^{ \theta_{2}}(\mathfrak{t};c))\Big{)}\]2 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Bigg{(}\frac{(\mathfrak{q}-1)(B_{2})^{\mathfrak{q}-2}}{(1+ \varphi^{\theta_{2}}(\mathfrak{t};c))}\,\mathcal{I}_{c^{+}}^{\mathfrak{t}_{ 2}+\theta_{2};\varphi}\Big{(}\mathcal{N}_{2}(\mathfrak{t})(1+\varphi^{ \theta_{2}}(\mathfrak{t};c))\Big{)}\]3 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Bigg{(}\frac{(\mathfrak{q}-1)(B_{2})^{\mathfrak{q}-2}}{(1+ \varphi^{\theta_{2}}(\mathfrak{t};c))}\,\mathcal{I}_{c^{+}}^{\mathfrak{t}_{ 2}+\theta_{2};\varphi}\Big{(}\mathcal{N}_{2}(\mathfrak{t})(1+\varphi^{ \theta_{2}}(\mathfrak{t};c))\Big{)}\]4 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Bigg{(}\frac{(\mathfrak{q}-1)(B_{2})^{\mathfrak{q}-2}}{(1+ \varphi^{\theta_{2}}(\mathfrak{t};c))}\,\mathcal{I}_{c^{+}}^{\mathfrak{t}_{ 2}+\theta_{2};\varphi}\Big{(}\mathcal{N}_{2}(\mathfrak{t})(1+\varphi^{ \theta_{2}}(\mathfrak{t};c))\Big{)}\]5 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Bigg{(}\frac{(\mathfrak{q}-1)(B_{2})^{\mathfrak{q}-2}}{(1+ \varphi^{\theta_{2}}(\mathfrak{t};c))}\,\mathcal{I}_{c^{+}}^{\mathfrak{t}_{ 2}+\theta_{2};\varphi}\Big{(}\mathcal{N}_{2}(\mathfrak{t})(1+\varphi^{ \theta_{2}}(\mathfrak{t};c))\Big{)}\]6 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Bigg{(}\frac{(\mathfrak{q}-1)(B_{2})^{\mathfrak{q}-2}}{(1+ \varphi^{\theta_{2}}(\mathfrak{t};c))}\,\mathcal{I}_{c^{+}}^{\mathfrak{t}_{ 2}+\theta_{2};\varphi}\Big{(}\mathcal{N}_{2}(\mathfrak{t})(1+\varphi^{ \theta_{2}}(\mathfrak{t};c))\Big{)}\]7  
\[\sup_{\mathfrak{t}\in\mathcal{T}} \Bigg{(}\frac{(\mathfrak{q}-1)(B_{2})^{\mathfrak{q}-2}}{(1+ \varphi^{\theta_{2}}(\mathfrak{t};c))}\,\mathcal{I}_{c^{+}}^{\mathfrak{t}_{ 2}+\theta_{2};\varphi}\Big{(}\mathcal{N}_{2}(\mathfrak{t})(1+\varphi^{ \theta_{2}}(\mathfrak{t};c))\Big{)}\]8 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Bigg{(}\frac{(\mathfrak{q}-1)(B_{2})^{\mathfrak{q}-2}}{(1+ \varphi^{\theta_{2}}(\mathfrak{t};c))}\,\mathcal{I}_{c^{+}}^{\mathfrak{t}_{ 2}+\theta_{2};\varphi}\Big{(}\mathcal{N}_{2}(\mathfrak{t})(1+\varphi^{ \theta_{2}}(\mathfrak{t};c))\Big{)}\]9 \[\qquad+\frac{|\Lambda_{2}|\Gamma(\theta_{2})\varphi^{\theta_{2}- 1}(\mathfrak{t};c)}{\Gamma(\alpha_{2})(1+\varphi^{\theta_{2}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{B}_{2})^{\mathfrak{q}-2}\sum_{i=1}^{m}\zeta_{i} \,\mathcal{I}_{c^{+}}^{\theta_{2}+\theta_{2}-1;\varphi}\Big{(}\mathcal{N}_{2 }(\mathfrak{e}_{i})(1+\varphi^{\theta_{2}}(\mathfrak{e}_{i};c))\Big{)}\Bigg{)} \leq\mathcal{B}_{1}<1,\]0 \[\qquad+\frac{|\Lambda_{2}|\Gamma(\theta_{2})\varphi^{\theta_{2}- 1}(\mathfrak{t};c)}{\Gamma(\alpha_{2})(1+\varphi^{\theta_{2}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{B}_{2})^{\mathfrak{q}-2}\sum_{i=1}^{m}\zeta_{i} \,\mathcal{I}_{c^{+}}^{\theta_{2}+\theta_{2}-1;\varphi}\Big{(}\mathcal{N}_{2 }(\mathfrak{e}_{i})(1+\varphi^{\theta_{2}}(\mathfrak{e}_{i};c))\Big{)}\Bigg{)} \leq\mathcal{B}_{1}<1,\]1 \[\qquad+\frac{|\Lambda_{2}|\Gamma(\theta_{2})\varphi^{\theta_{2}- 1}(\mathfrak{t};c)}{\Gamma(\alpha_{2})(1+\varphi^{\theta_{2}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{B}_{2})^{\mathfrak{q}-2}\sum_{i=1}^{m}\zeta_{i} \,\mathcal{I}_{c^{+}}^{\theta_{2}+\theta_{2}-1;\varphi}\Big{(}\mathcal{N}_{2 }(\mathfrak{e}_{i})(1+\varphi^{\theta_{2}}(\mathfrak{e}_{i};c))\Big{)}\Bigg{)} \leq\mathcal{B}_{1}<1,\]2 \[\qquad+\frac{|\Lambda_{2}|\Gamma(\theta_{2})\varphi^{\theta_{2}- 1}(\mathfrak{t};c)}{\Gamma(\alpha_{2})(1+\varphi^{\theta_{2}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{B}_{2})^{\mathfrak{q}-2}\sum_{i=1}^{m}\zeta_{i} \,\mathcal{I}_{c^{+}}^{\theta_{2}+\theta_{2}-1;\varphi}\Big{(}\mathcal{N}_{2 }(\mathfrak{e}_{i})(1+\varphi^{\theta_{2}}(\mathfrak{e}_{i};c))\Big{)}\Bigg{)} \leq\mathcal{B}_{1}<1,\]3 \[\qquad+\frac{|\Lambda_{2}|\Gamma(\theta_{2})\varphi^{\theta_{2}- 1}(\mathfrak{t};c)}{\Gamma(\alpha_{2})(1+\varphi^{\theta_{2}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{B}_{2})^{\mathfrak{q}-2}\sum_{i=1}^{m}\zeta_{i} \,\mathcal{I}_{c^{+}}^{\theta_{2}+\theta_{2}-1;\varphi}\Big{(}\mathcal{N}_{2 }(\mathfrak{e}_{i})(1+\varphi^{\theta_{2}}(\mathfrak{e}_{i};c))\Big{)}\Bigg{)} \leq\mathcal{B}_{1}<1,\]4 \[\qquad+\frac{|\Lambda_{2}|\Gamma(\theta_{2})\varphi^{\theta_{2}- 1}(\mathfrak{t};c)}{\Gamma(\alpha_{2})(1+\varphi^{\theta_{2}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{B}_{2})^{\mathfrak{q}-2}\sum_{i=1}^{m}\zeta_{i} \,\mathcal{I}_{c^{+}}^{\theta_{2}+\theta_{2}-1;\varphi}\Big{(}\mathcal{N}_{2 }(\mathfrak{e}_{i})(1+\varphi^{\theta_{2}}(\mathfrak{e}_{i};c))\Big{)}\Bigg{)} \leq\mathcal{B}_{1}<1,\]5 \[\qquad+\frac{|\Lambda_{2}|\Gamma(\theta_{2})\varphi^{\theta_{2}- 1}(\mathfrak{t};c)}{\Gamma(\alpha_{2})(1+\varphi^{\theta_{2}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{B}_{2})^{\mathfrak{q}-2}\sum_{i=1}^{m}\zeta_{i} \,\mathcal{I}_{c^{+}}^{\theta_{2}+\theta_{2}-1;\varphi}\Big{(}\mathcal{N}_{2 }(\mathfrak{e}_{i})(1+\varphi^{\theta_{2}}(\mathfrak{e}_{i};c))\Big{)}\Bigg{)} \leq\mathcal{B}_{1}<1,\]6 \[\qquad+\frac{|\Lambda_{2}|\Gamma(\theta_{2})\varphi^{\theta_{2}- 1}(\mathfrak{t};c)}{\Gamma(\alpha_{2})(1+\varphi^{\theta_{2}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{B}_{2})^{\mathfrak{q}-2}\sum_{i=1}^{m}\zeta_{i} \,\mathcal{I}_{c^{+}}^{\theta_{2}+\theta_{2}-1;\varphi}\Big{(}\mathcal{N}_{2 }(\mathfrak{e}_{i})(1+\varphi^{\theta_{2}}(\mathfrak{e}_{i};c))\Big{)}\Bigg{)} \leq\mathcal{B}_{1}<1,\]7 \[\qquad+\frac{|\Lambda_{2}|\Gamma(\theta_{2})\varphi^{\theta_{2}- 1}(\mathfrak{t};c)}{\Gamma(\alpha_{2})(1+\varphi^{\theta_{2}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{B}_{2})^{\mathfrak{q}-2}\sum_{i=1}^{m}\zeta_{i} \,\mathcal{I}_{c^{+}}^{\theta_{2}+\theta_{2}-1;\varphi}\Big{(}\mathcal{N}_{2 }(\mathfrak{e}_{i})(1+\varphi^{\theta_{2}}(\mathfrak{e}_{i};c))\Big{)}\Bigg{)} \leq\mathcal{B}_{1}<1,\]8 \[\qquad+\frac{|\Lambda_{2}|\Gamma(\theta_{2})\varphi^{\theta_{2}- 1}(\mathfrak{t};c)}{\Gamma(\alpha_{2})(1+\varphi^{\theta_{2}}(\mathfrak{t};c ))}(\mathfrak{q}-1)(\mathcal{B}_{2})^{\mathfrak{q}-2}\sum_{i=1}^{m}\zeta_{i} \,\mathcal{I}_{c^{+}}^{\theta_{2}+\theta_{2}-1;\varphi}\Big{(}\mathcal{N}_{2 }(\mathfrak{e}_{i})(1+\varphi^{\theta_{2}}(\mathfrak{e}_{i};c))\Big{)}\Bigg{)} \leq\mathcal{B}_{1}<1,\]9 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{1})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{1}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{1}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{A}_{2}<\infty,\]0  
Hence,  
\[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{1})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{1}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{1}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{A}_{2}<\infty,\]1  
Similarly, we obtain that  
\[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{1})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{1}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{1}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{A}_{2}<\infty,\]2 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{1})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{1}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{1}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{A}_{2}<\infty,\]3 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{1})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{1}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{1}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{A}_{2}<\infty,\]4 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{1})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{1}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{1}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{A}_{2}<\infty,\]5 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{1})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{1}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{1}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{A}_{2}<\infty,\]6and  
\[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{1})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{1}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{1}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{A}_{2}<\infty,\]7 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{1})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{1}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{1}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{A}_{2}<\infty,\]8 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{1})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{1}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{1}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{A}_{2}<\infty,\]9 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{2})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{2}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{2}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{B}_{2}<\infty,\]0 \[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{2})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{2}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{2}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{B}_{2}<\infty,\]1  
which proves that  
\[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{2})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{2}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{2}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{B}_{2}<\infty,\]2  
Therefore, we infer the following:  
\[\sup_{\mathfrak{t}\in\mathcal{T}} \Big{(}\frac{1}{\Gamma(\theta_{2})}\int_{c}^{\mathfrak{t}}\varphi^{ \prime}(\varphi)\varphi^{\theta_{2}-1}(\mathfrak{t};\mathfrak{v})\mathfrak{f }_{2}(\mathfrak{v})\,d\
u\Big{)}\leq\mathcal{B}_{2}<\infty,\]3  
and \(\mathbb{Y}_{1},\mathbb{Y}_{2}\) are continuous on \(\mathcal{T}\).