Recent Developments of Lyapunov–Type Inequalities for Fractional Differential Equations

Abstract

A survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions is presented. This includes Dirichlet, mixed, Robin, fractional, Sturm-Liouville, integral, nonlocal, multi-point, anti-periodic, conjugate, right-focal and impulsive conditions. Furthermore, our study includes Riemann-Liouville, Caputo, Hadamard, Prabhakar, Hilfer and conformable type fractional derivatives. Results for boundary value problems involving fractional p-Laplacian, fractional operators with nonsingular Mittag-Leffler kernels, q-difference, discrete, and impulsive equations, are also taken into account.Definition 2. (Riemann-Liouville fractional derivative) The Riemann-Liouville fractional derivative of order α ≥ 0 is defined by (D 0 f )(t) = f (t) andwhere m is the smallest integer greater than or equal to α.Definition 3. (Caputo fractional derivative) The Caputo fractional derivative of order α ≥ 0 is defined by ( C D 0 f )(t) = f (t) and

Lyapunov-Type Inequalities for Riemann-Liouville Type Fractional Boundary Value Problems with Fractional Boundary Conditions

“…Assume that y is a nontrivial solution of the Riemann-Liouville type fractional eigenvalue problem (11) where y(t) = 0 for each t ∈ (a, b). Then,…”

“…For the first time, Ferreira [5] obtained a Lyapunov-type inequality for a two-point Riemann-Liouville type fractional boundary value problem associated with Dirichlet boundary conditions as follows: Theorem 1.1. [5] If the fractional boundary value problem Recently, Ntouyas et al [11] presented a survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions. This article shows a gap in the literature on Lyapunov-type inequalities for two-point Riemann-Liouville type fractional boundary value problems associated with fractional boundary conditions.…”

Lyapunov-type inequality for a Riemann-Liouville type fractional boundary value problem with anti-periodic boundary conditions

“…Recently, Ntouyas et al [7] presented a survey of results on Lyapunovtype inequalities for fractional differential equations associated with a variety of boundary conditions. This article shows that a Lyapunov-type inequality for a two-point Riemann-Liouville type fractional boundary value problem associated with anti-periodic boundary conditions is not yet reported.…”

Lyapunov-type inequalities for differential equation with Caputo–Hadamard fractional derivative under multipoint boundary conditions

“…The Lyapunov inequality (1.2) is a useful tool in various branches of mathematics, including disconjugacy, oscillation theory, and eigenvalue problems. Many improvements and generalizations of inequality (1.2) have appeared in the literature; see [2][3][4][5][6][7][8][9][10][11][12][13] and references therein.…”