Generalized Lyapunov-and Hartman–Wintner-type Inequalities for Mixed Fractional Differential Equations with Advanced Fractional Operators
Sukalwad Umesh Ramrao *
Late Babasaheb Deshmukh Gorthekar College, Umri, India.
R. N. Ingle
Bahirji Smarak Mahavidyalaya, Basmathnagar, India.
*Author to whom correspondence should be addressed.
Abstract
Fractional differential equations model systems with memory and nonlocal effects, with applications across many scientific fields. Lyapunov and Hartman–Wintner type inequalities help analyze solution existence, eigenvalues, and oscillatory behavior, but their extension to modern fractional operators remains limited. Current research lacks a unified framework for these inequalities across different advanced operators. In this paper, we investigate Lyapunov- and Hartman–Wintner-type inequalities for a class of fractional boundary value problems involving non-singular kernels. The analysis is carried out using Caputo–Fabrizio and Atangana–Baleanu fractional operators. We establish new bounds under suitable assumptions and derive conditions for the existence of nontrivial solutions. The obtained inequalities generalize several known results in the literature. Illustrative examples are provided to validate the theoretical findings and to demonstrate the applicability of the proposed approach.
Keywords: Fractional boundary value problems, lyapunov inequality, hartman–wintner inequality, tempered derivative, atangana–baleanu derivative, caputo–fabrizio derivative, variable-order operator, green’s function, fixed-point theorem