# Efficient Solution Criteria for a Coupled Fractional Laplacian System on Some Infinite Domains
## 1 Introduction  
The topic of fractional calculus has expanded from integer order to arbitrary order [1]. This has attracted the interest of many researchers [2, 3]. Fractional differential equations (\(FDEs\)) have been used in numerous areas of science, such as physics, engineering, and finance, to represent complex models that exhibit memory effects and nonlocal behavior [4, 5, 6, 7, 8]. The \(\mathfrak{p}\)-Laplacian operator is an essential tool for modeling nonlinear phenomena in various fields, and its study has led to significant progress in both applied and theoretical mathematics [9]. It has numerous applications in engineering, physics, and mathematics due to its ability to describe real phenomena [10]. Furthermore, its study has presented several mathematical challenges due to its nonlinearity [11]. The nonlinearity acts challenges in the literature studies, such as authors in [12] explored bilinear Backlund transformations and various wave solutions, including \(N\)-soliton, breather, fission/fusion, and hybrid solutions, for a specific \((3+1)\)-dimensional integrable wave equation relevant to fluid dynamics. Ref. [13] examined a complex \((3+1)\)-dimensional shallow water wave equation that evolves, with applications that are relevant to oceans and rivers. Ref. [14] investigated the hetero-Backlund transformation, bilinear forms, and multisoliton solutions of a \((2+1)\) dimensional, generalized, modified, dispersive water wave system. Many research works have investigated \(FDEs\), involving the p-Laplacian operator [15]. For example, [PERSON] [16] studied the global existence of solutions for the initial Caputo fractional Laplacian differential equations with singular points. [PERSON]. [17] applied the existence theorem to the \(\Phi\)-Caputo fractional Langevin differential system involving the p-Laplacian mapping. In addition, [PERSON] et al. [18] established some qualitative results for the Caputo hybrid \(FDEs\) with a p-Laplacian mapping. [PERSON]. [19] used the Avery-Peterson fixed point theorem to investigate the existence of positive solutions for the Caputo \(FDEs\) under some integral boundary conditions involving the p-Laplacian operator. Unbounded solutions often arise in nonlinear ordinary, partial, and fractional differential equations (\(FDEs\)), as well as multi-order nonlinear problems with excitable behavior [20; 21]. Their investigation is of great interest because it provides insights into the stability, limitations, and long-term behavior of dynamical systems [22]. [PERSON] et al. [23] studied unbounded solutions of nonlinear fractional \(q\)-\(\mathbb{RL}\) difference equations using classical fixed point theorems. [PERSON] and [PERSON] [21] investigated the existence of unbounded positive solutions for coupled nonlinear \(FDEs\) involving Hadamard derivatives by employing the Leggett-Williams and Guo-Krasnoselskii fixed point theorems. In 2024, [PERSON] and [PERSON] [24] explored iterative positive solutions for Hadamard \(FDEs\) with integral boundary conditions on the interval \((1,\infty)\). The authors of [25] discussed the \(\mathbb{E}a\mathbb{U}\) of an unbounded solution for coupled \(FDEs\). Furthermore, fixed point theorems have led to researchers studying integral equations and \(FDEs\); we refer readers to the works [26; 27; 28; 29]. In this study, motivated by the aforementioned works, we investigate the qualitative properties of an explicit iterative solution, such as the existence, uniqueness, and stability of various forms of the \(\mathbb{U}\mathbb{H}\) type, for the following novel coupled \(\varphi\)-\(\mathbb{RL}\) fractional Laplacian implicit differential system with multi-point strip boundary conditions on the unbounded domain \([c,\infty)\):  
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