# Efficient Solution Criteria for a Coupled Fractional Laplacian System on Some Infinite Domains
## 3 Main Results
### Existences and Uniqueness Result
\begin{split}&\left\|\frac{(\mathcal{D}_{c^{+}}^{\gamma_{1} \varphi}\mathbb{Y}_{1}\mathcal{V})(\mathfrak{t})}{(1+\varphi^{\theta_{1}}( \mathfrak{t};c))}-\frac{(\mathcal{D}_{c^{+}}^{\gamma_{1}\varphi}\mathbb{Y}_{ 1}\bar{\mathcal{V}})(\mathfrak{t})}{(1+\varphi^{\theta_{1}}(\mathfrak{t};c))} \right\|\\ &\leq\frac{1}{\Gamma(\theta_{1}-\gamma_{1})(1+\varphi^{\theta_{1} }(\mathfrak{t};c))}\int_{c}^{\mathfrak{t}}\varphi^{\prime}(\mathsf{v}) \varphi^{\theta_{1}-\gamma_{1}-1}(\mathfrak{t};\mathsf{v})\big{\|}(\mathbb{Y }_{10}\mathcal{V})(\mathsf{v})-(\mathbb{Y}_{10}\bar{\mathcal{V}})(\mathsf{v} )\big{\|}d\mathsf{v}\\ &\quad+\frac{|\Lambda_{1}|\Gamma(\theta_{1})\varphi^{\theta_{1}- \gamma_{1}-1}(\mathfrak{t};c)}{\Gamma(\theta_{1}-\gamma_{1})(1+\varphi^{\theta _{1}}(\mathfrak{t};c))}\sum_{j=1}^{n}\eta_{j}\frac{1}{\Gamma(\theta_{1}-1)} \int_{c}^{\epsilon_{j}}\varphi^{\prime}(\mathsf{v})\varphi^{\theta_{1}-2}( \epsilon_{j};\mathsf{v})\big{\|}(\mathbb{Y}_{10}\mathcal{V})(\mathsf{v})-( \mathbb{Y}_{10}\bar{\mathcal{V}})(\mathsf{v})\big{\|}d\mathsf{v}\\ &\leq\frac{(\mathfrak{q}-1)(\mathcal{A}_{2})^{\mathfrak{q}-2} \big{\|}\mathcal{V}-\bar{\mathcal{V}}\big{\|}_{\mathbb{Y}_{1}}}{(1+\varphi^{ \theta_{1}}(\mathfrak{t};c))}\int_{c}^{\mathfrak{t}}\varphi^{\prime}(\mathsf{v}) \varphi^{\theta_{1}-\gamma_{1}-1}(\mathfrak{t};\mathsf{v})\,\mathcal{I}_{c^{+}}^{ \theta_{1};\varphi\prime}\Big{(}\mathcal{N}_{1}(\mathsf{v})(1+\varphi^{\theta_{1 }}(\mathsf{v};c))\Big{)}d\mathsf{v}\\ &\quad+\frac{|\Lambda_{1}|\Gamma(\theta_{1})\varphi^{\theta_{1}- \gamma_{1}-1}(\mathfrak{t};c)}{\Gamma(\theta_{1}-\gamma_{1})(1+\varphi^{\theta_{1 }}(\mathfrak{t};c))}\sum_{j=1}^{n}\eta_{j}\frac{(\mathfrak{q}-1)(\mathcal{A}_{2} )^{\mathfrak{q}-2}\big{\|}\mathcal{V}-\bar{\mathcal{V}}\big{\|}_{\mathbb{Y}_{1}}}{ \Gamma(\theta_{1}-1)}\\ &\qquad\qquad\qquad\times\int_{c}^{\epsilon_{j}}\varphi^{\prime}( \mathsf{v})\varphi^{\theta_{1}-2}(\epsilon_{j};\mathsf{v})\,\mathcal{I}_{c^{+}}^{ \theta_{1};\varphi\prime}\Big{(}\mathcal{N}_{1}(\mathsf{v})(1+\varphi^{\theta _{1}}(\mathsf{v};c))\Big{)}d\mathsf{v}\\ &\leq\frac{(\mathfrak{q}-1)(\mathcal{A}_{2})^{\mathfrak{q}-2} \big{\|}\mathcal{V}-\bar{\mathcal{V}}\big{\|}_{\mathbb{Y}_{1}}}{(1+\varphi^{ \theta_{1}}(\mathfrak{t};c))}\,\mathcal{I}_{c^{+}}^{\theta_{1}-\gamma_{1}+ \theta_{1};\varphi}\Big{(}\mathcal{N}_{1}(\mathsf{t})(1+\varphi^{\theta_{1}}( \mathfrak{t};c))\Big{)}\\ &\quad+\frac{|\Lambda_{1}|\Gamma(\theta_{1})\varphi^{\theta_{1}- \gamma_{1}-1}(\mathfrak{t};c)}{\Gamma(\theta_{1}-\gamma_{1})(1+\varphi^{\theta_{1 }}(\mathfrak{t};c))}(\mathfrak{q}-1)(\mathcal{A}_{2})^{\mathfrak{q}-2} \big{\|}\mathcal{V}-\bar{\mathcal{V}}\big{\|}_{\mathbb{Y}_{1}}\sum_{j=1}^{n}\eta_ {j}\frac{\mathcal{I}_{c^{+}}^{\theta_{1}+\theta_{1}-1;\varphi}\Big{(} \mathcal{N}_{1}(\epsilon_{j})(1+\varphi^{\theta_{1}}(\epsilon_{j};c))\Big{)} \Big{)}\\ &\leq\mathcal{A}_{1}\big{\|}\mathcal{V}-\bar{\mathcal{V}}\big{\|} _{\mathbb{Y}_{1}}.\end{split} \]Hence, we infer that  
\[\|\mathbb{Y}_{1}\mathcal{V}-\mathbb{Y}_{1}\bar{\mathcal{V}}\|_{\mathbb{Y}_{1}} \leq\mathcal{A}_{1}\big{\|}\mathcal{V}-\bar{\mathcal{V}}\big{\|}_{\mathbb{Y}_{1}}. \tag{19}\]  
Likewise, using inequality (18) and \((\mathbf{AS}_{2})\), we can also deduce that  
\[ \begin{split}&\left\|\frac{(\mathbb{Y}_{2}\mathcal{U})(\mathfrak{t})}{ (1+\varphi^{\theta_{2}}(\mathfrak{t};c))}-\frac{(\mathbb{Y}_{2}\mathcal{U})( \mathfrak{t})}{(1+\varphi^{\theta_{2}}(\mathfrak{t};c))}\right\|\leq\mathcal{ B}_{1}\big{\|}\mathcal{U}-\bar{\mathcal{U}}\big{\|}_{\mathbb{Y}_{2}},\\ &\left\|\frac{(\mathcal{D}_{c^{+}}^{\mathbb{Z}_{2}\varphi}\mathbb{ Y}_{2}d)(\mathfrak{t})}{(1+\varphi^{\theta_{2}}(\mathfrak{t};c))}-\frac{( \mathcal{D}_{c^{+}}^{\mathbb{Z}_{2}\varphi}\mathbb{Y}_{2}\mathcal{U})( \mathfrak{t})}{(1+\varphi^{\theta_{2}}(\mathfrak{t};c))}\right\|\leq\mathcal{ B}_{1}\big{\|}\mathcal{U}-\bar{\mathcal{U}}\big{\|}_{\mathbb{Y}_{2}},\end{split}