Existence and uniqueness results of the sequential $(\kappa ,\Phi )$-Hilfer and $(\kappa ,\Phi )$-Caputo fractional system

This paper investigates the existence and uniqueness of positive solutions for a novel coupled system of sequential \((\kappa , \Phi )\) -Hilfer and \((\kappa , \Phi )\) -Caputo fractional differential equations, subject to generalized multistrip integral boundary conditions. We establish a robust mathematical framework that extends single equation models and standard fractional operators to a coupled system incorporating the more flexible \((\kappa , \Phi )\) -Hilfer and \((\kappa , \Phi )\) -Caputo operators, alongside dual algebraic and fractional integral constraints. By employing mixed monotone fixed point theory and the Banach contraction principle, we successfully analyze the system within a triple operator summation framework. A critical feature of our study is the distinct parameterization and the non-symmetric ordering of derivatives between the two equations of the system where the \((\kappa , \Phi )\) -Caputo derivative precedes the \((\kappa , \Phi )\) -Hilfer derivative in the first equation, and vice versa in the second. This configuration allows for the modeling of heterogeneous dynamical processes with varying memory and scaling properties. We provide sufficient conditions for the existence and uniqueness of positive solutions, demonstrating that our results are theoretically robust and invariant under parameter reduction.