This paper investigates the existence and uniqueness of implicit solutions in a coupled symmetry system of hybrid fractional order differential equations, along with hybrid integral boundary conditions in Banach Algebras. The methodology centers on a hybrid fixed-point theorem that involves mixed Lipschitz and Carathéodory conditions, serving to establish the existence of solutions. Moreover, it derives sufficient conditions for solution uniqueness and establishes the Hyers–Ulam types of solution stability. This study contributes valuable insights into complex hybrid fractional order systems and their practical implications.

Using variable-order fractional derivatives in differential equations is essential. It enables more precise modeling of complex phenomena with varying memory and long-range dependencies, improving our ability to describe real-world processes reliably. This study investigates the properties of solutions for a two-point boundary value problem associated with $\phi$-Caputo fractional derivatives of variable order. The primary objectives are to establish the existence and uniqueness of solutions, as well as explore their stability through the Ulam-Hyers concept. To achieve these goals, Banach’s and Krasnoselskii’s fixed point theorems are employed as powerful mathematical tools. Additionally, we provide numerical examples to illustrate results and enhance comprehension of theoretical findings. This comprehensive analysis significantly advances our understanding of variable-order fractional differential equations, providing a strong foundation for future research. Future directions include exploring more complex boundary value problems, studying the effects of varying fractional differentiation orders, extending the analysis to systems of equations, and applying these findings to real-world scenarios, all of which promise to deepen our understanding of Caputo fractional differential equations with variable order, driving progress in both theoretical and applied mathematics.

In this paper, we use a generalized form for the Jordan totient function in order to extend the Reciprocal power GCDQ matrices and power LCMQ matrices from the standard domain of natural integers to Euclidean domains. Structural theorems and determinantal arguments defined on both arbitrary and factor-closed q-ordered sets are presented over such domains. We illustrate our work in the case of Gaussian integers.

In this paper, we introduce and thoroughly examine new generalized $\psi$-conformable fractional integral and derivative operators associated with the auxiliary function $\psi \left(t\right)$. We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit $\psi$-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications.