q-Analogues of Lyapunov-type inequalities involving Riemann–Liouville fractional derivatives

Abstract

In this article, new q-analogues of Lyapunov-type inequalities are presented for two-point fractional boundary value problems involving the Riemann–Liouville fractional q-derivative with well-posed q-boundary conditions. The study relies on the properties of the q-Green’s function, which is constructed to solve such problems and allows for the analytical derivation of the inequalities. These inequalities find application in two directions: establishing precise lower bounds for the eigenvalues of corresponding q-fractional spectral problems and formulating criteria for the absence of real zeros in q-analogues of Mittag-Leffler functions. The obtained results generalize classical and fractional Lyapunov inequalities, offering new perspectives for the analysis of stability and spectral properties of q-fractional differential systems. The relevance of the work is driven by the growing interest in q-calculus in discrete models, such as viscoelastic systems or quantum circuits, where discrete dynamics play a key role. The convenience of closed-form analytical expressions makes the results practically applicable. The research lays the foundation for further generalizations, including Caputo derivatives or multidimensional q-systems, which may stimulate new discoveries in discrete fractional analysis.