# Efficient Solution Criteria for a Coupled Fractional Laplacian System on Some Infinite Domains
## 1 Introduction
\begin{cases}\mathcal{D}_{c^{+}}^{q_{1}\varphi}\Phi_{\varphi}\left(\mathcal{D }_{c^{+}}^{q_{1}\varphi}\,\mathcal{U}(\mathfrak{t})\right)=\mathcal{W}_{1}( \mathfrak{t},\mathcal{V}(\mathfrak{t}),\mathcal{D}_{c^{+}}^{q_{1}\varphi}\, \mathcal{V}(\mathfrak{t})),&\mathfrak{t}\in\mathcal{T}=[c,\infty),\\ \mathcal{D}_{c^{+}}^{q_{2}\varphi}\[PERSON]_{\varphi}\left(\mathcal{D}_{c^{+}}^{q_ {2}\varphi}\,\mathcal{V}(\mathfrak{t})\right)=\mathcal{W}_{2}(\mathfrak{t}, \mathcal{U}(\mathfrak{t}),\mathcal{D}_{c^{+}}^{q_{2}\varphi}\,\mathcal{U}( \mathfrak{t})),&\mathfrak{t}\in\mathcal{T}=[c,\infty),\\ \mathcal{U}(c)=0,\ \mathcal{D}_{c^{+}}^{q_{1}\varphi}\,\mathcal{U}(c)=0,\ \lim_{t\to 0} \varphi^{-1}(\mathfrak{t};c)\,\mathcal{D}_{c^{+}}^{q_{1}\to \varphi}\mathcal{U}(\mathfrak{t})=\sum_{i=1}^{n}\eta\mathcal{D}_{c^{+}}^{q_{1} \varphi}\mathcal{U}(\mathfrak{t}_{i}),\ n\in\mathbb{N},\mathfrak{e}_{i}\in[c, \infty),\\ \mathcal{V}(c)=0,\ \mathcal{D}_{c^{+}}^{q_{1}\varphi}\,\mathcal{V}(c)=0,\ \lim_{t\to 0} \varphi^{-1}(\mathfrak{t};c)\,\mathcal{D}_{c^{+}}^{q_{1}\to \varphi}\mathcal{V}(\mathfrak{t})=\sum_{i=1}^{n}\zeta\mathcal{U}_{c^{+}}^{q_{ 1}\varphi}\mathcal{V}(\mathfrak{t}_{i}),\ m\in\mathbb{N},\mathfrak{e}_{i}\in[c,\infty),\end{cases}  \tag{1}\]  
where \(\mathcal{D}_{c^{+}}^{x,\varphi}\) is the \(\varphi\)-fractional \(\mathbb{RL}\) derivatives of order \(x\in\{\theta_{1},\theta_{2},\theta_{1},\theta_{2},\gamma_{1},\gamma_{2}\}\), such that \(\theta_{1},\theta_{2}\in(0,1]\), \(\theta_{1},\theta_{2}\in(1,2)\), \(1<\gamma_{1},\gamma_{2}<\theta_{1},\theta_{2}<2\); and \(\varphi(\mathfrak{t})\in\mathfrak{E}^{2}(\mathcal{T},\mathbb{R}^{+})\) be an increasing continuous positive function, with \(\varphi^{\prime}(\mathfrak{t})\
eq 0\), \(\forall\,\mathfrak{t}\in\mathcal{T}\). Furthermore, \(\mathcal{U},\mathcal{V}\in\mathfrak{C}(\mathcal{T},\mathbb{K})\), and \(\mathcal{W}_{1},\mathcal{W}_{2}\in\mathfrak{C}(\mathcal{T}\times\mathbb{K} \times\mathbb{K},\mathbb{K})\) are continuous functions, and \(\mathbb{K}\) is a real space; \(\eta_{j},\zeta_{i}\in\mathbb{K},c<\epsilon_{1}<\epsilon_{2}<\cdots<\epsilon_{ n}<\infty\), \(c<\epsilon_{1}<\epsilon_{2}<\cdots<\epsilon_{m}<\infty\), such that \((\theta_{1}-1)\sum_{j=1}^{n}\eta_{j}\varphi^{\theta_{1}-2}(\epsilon_{j};c)\
eq 0\), and \((\theta_{2}-1)\sum_{i=1}^{m}\zeta_{i}\varphi^{\theta_{2}-2}(\epsilon_{i};c)\
eq 0\), where we adopted the notation \(\varphi^{\delta}(\mathfrak{t};c)=\big{(}\varphi(\mathfrak{t})-\varphi(c)\big{)}^ {\delta}\). Moreover, the \(\mathfrak{p}\)-Laplacian operator \(\psi_{\mathfrak{p}}(\mathcal{U})\) is given as \(\psi_{\mathfrak{p}}(\mathcal{U})=|\mathcal{U}|^{p-2}\,\mathcal{U},\mathcal{U }\
eq 0,\psi_{\mathfrak{p}}(0)=0\), with the inverse operator defined by \(\psi_{\mathfrak{p}}^{-1}(\mathcal{U})=\psi_{\mathfrak{q}}(\mathcal{U})=| \mathcal{U}|^{q-2}\,\mathcal{U}\), such that \(\frac{1}{\mathfrak{p}}+\frac{1}{\mathfrak{q}}=1\). The main contributions of this work are as follows:  
1. A new coupled \(\varphi\)-\(\mathbb{RL}\) fractional implicit differential system is considered, with p-Laplacian operator and multi-point strip boundary conditions on unbounded domains. 2. An applicable Banach space is introduced to allow solutions to be defined on unbounded domains \([c,\infty)\). 3. The \(\mathbb{E}a\mathrm{U}\) of explicit iterative solution for the proposed system (1) is investigated by employing the Banach fixed point strategy. 4. Some types of the \(\mathbb{U}\mathbb{R}\) stabilities are established for the suggested coupled system (1). 5. The main results are applied to a coupled \(\varphi\)-\(\mathbb{RL}\) fractional turbulent flow model. 6. The proposed problem is more generalized than those existing in the literature. The remainder of this article is organized as follows: Section 2 introduces the fundamental concepts of fractional calculus. Section 3 explores the \(\mathbb{E}a\mathrm{U}\) outcome through explicit iterations solution and various forms of Ulam stability.