We consider positivity, monotonicity, and convexity results for discrete fractional operators with exponential kernels. Our results cover both the sequential and nonsequential cases, and we demonstrate both similarities and dissimilarities between the exponential kernel case and fractional differences with other types of kernels. This shows that the qualitative information gleaned in the exponential kernel case is not precisely the same as in other cases.

In this article we establish a few Lyapunov-type inequalities for twopoint discrete fractional boundary value problems involving Riemann-Liouville type backward differences. To illustrate the applicability of established results, we obtain criteria for the nonexistence of nontrivial solutions and estimate lower bounds for eigenvalues of the corresponding eigenvalue problems. We also apply these inequalities to deduce criteria for the nonexistence of real zeros of certain discrete Mittag-Leffler functions.

We investigate the Hyers-Ulam stability, the generalized Hyers-Ulam stability, and the Fα-Hyers-Ulam stability of a linear fractional nabla difference equation using discrete Laplace transform. We provide a few examples to illustrate the applicability of established results.

In this paper, we study the coupled system of nonlinear Langevin equations involving Caputo–Hadamard fractional derivative and subject to nonperiodic boundary conditions. The existence, uniqueness, and stability in the sense of Ulam are established for the proposed system. Our approach is based on the features of the Hadamard fractional derivative, the implementation of fixed point theorems, and the employment of Urs's stability approach. An example is introduced to facilitate the understanding of the theoretical findings.

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In this article, we consider a particular class of nabla fractional boundary value problems with general boundary conditions, and establish sufficient conditions on existence and uniqueness of its solutions. We present two examples to illustrate the applicability of established results.

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In this article, we consider a family of two-point Riemann-Liouville type nabla fractional boundary value problems involving a fractional difference boundary condition. We construct the corresponding Green's function and deduce its ordering property. Then, we obtain a Lyapunov-type inequality using the properties of the Green's function, and illustrate a few of its applications.