A Lyapunov-type inequality for a fractional boundary value problem

Abstract

In this work we obtain a Lyapunov-type inequality for a fractional differential equation subject to Dirichlet-type boundary conditions. Moreover, we apply this inequality to deduce a criteria for the nonexistence of real zeros of a certain Mittag-Leffler function.

A Lyapunov-type inequality for a fractional q-difference boundary value problem

“…The obtained inequality generalizes several existing results in the literature including the standard Lyapunov inequality (1.1), Hartman and Wintner inequality [14], a Lyapunov-type inequality due to Ferreira [13], etc. We then use that inequality to obtain an interval, where a certain Mittag-Leffler function has no real zeros.…”

“…Such problem was considered recently by Ferreira in [13]. In that work, he obtained the following result.…”

“…We then use that inequality to obtain an interval, where a certain Mittag-Leffler function has no real zeros. We show that the obtained interval is better than that obtained in [13]. We also present another application of the obtained Lyapunovtype inequality, where we prove that existence implies uniqueness for a certain class of fractional q-difference boundary value problems.…”

“…In [22], Lyapunov established the following result: This result has found several applications in various problems related with differential equations and, since then, there have been several results to improve and generalize Theorem 1.1 in many directions (see [2,5,6,7,8,13,14,15,20,22,24,25,26,30] and the references therein).…”

Lyapunov-type Inequalities for Certain Higher Order Fractional Differential Equations

“…They include continuous and discrete as special cases [2][3][4][5]. However, the Lyapunov-type inequality for the higher order fractional differential equation has not been studied.…”