Euler-Lagrange Equations with Generalized ABC and ABR Nabla Fractional Differences

Abstract

Background: Discrete fractional calculus has emerged as an important mathematical framework for modeling dynamical systems with memory and nonlocal behavior. In particular, nabla fractional differences with Mittag–Leffler kernels have been widely used in discrete fractional variational problems and their associated Euler–Lagrange equations.  Objectives: This study aims to derive new discrete fractional Euler–Lagrange equations involving generalized Atangana–Baleanu Caputo (ABC) and Atangana–Baleanu Riemann (ABR) nabla fractional differences with generalized Mittag–Leffler kernels. Methods: First, new summation-by-parts formulas for generalized ABC and ABR nabla fractional differences with a three-parameter Mittag–Leffler kernel are established. These formulas are then employed within the framework of discrete fractional calculus of variations to derive Euler–Lagrange equations for functionals containing generalized ABC nabla left and right fractional differences. Results: New discrete fractional Euler–Lagrange equations are obtained for variational problems involving generalized ABC nabla fractional differences with generalized Mittag–Leffler kernels. The proposed results extend previously known Euler–Lagrange formulations based on one-parameter kernels. Illustrative examples are presented to demonstrate the applicability of the theoretical findings. Conclusions: The results provide a generalized framework for discrete fractional variational problems and contribute to the development of Euler–Lagrange theory with generalized ABC and ABR nabla fractional operators, opening new directions for future research in discrete fractional calculus.