On Lyapunov-type inequalities for boundary value problems of fractional Caputo-Fabrizio derivative

Abstract

In this study, Lyapunov-type inequalities for fractional boundary value problems involving the fractional Caputo Fabrizio differential equation with mixed boundary conditions when the fractional order of β ∈ (1, 2] and Dirichlet-type boundary condition when the fractional order of σ ∈ (2, 3] have been derived. Some consequences of the results related to the fractional Sturm-Liouville eigenvalue problems have also been given. Additionally, we examine the nonexistence of the solution of the fractional boundary value problem.

On The Solutions of Fractional Boundary Value Problems for a Combined Caputo Fractional Differential Equations

“…(1) T has a fixed point in O or (2) There is u ∈ ∂O and λ ∈ (0, 1) such that u = λTu. Lemma 2.9 [32] The fractional differential equation ( 3) is equivalent to the following fractional integral equations given as…”

“…In [32], the existence of solutions for fractional boundary value problems (FBVP) of the combined Caputo fractional differential equation has been discussed and presented. In this paper, we will present the positive solution of the following FBVP (4)…”

A Survey on Recent Results on Lyapunov-Type Inequalities for Fractional Differential Equations

“…Next, we give Lyapunov-type inequalities for the Sturm-Liouville-Hadamard fractional boundary value problem (22).…”

“…The following theorem contains a Lyapunov-type inequality for the fractional boundary value problem (28). In 2020, Toprakseven [22] studied Lyapunov-type inequalities for fractional boundary value problems with mixed boundary conditions and involving the fractional Caputo-Fabrizio fractional derivative. He established a Lyapunov-type inequality for the following boundary value problem:…”

Hartman-Wintner and Lyapunov-type inequalities for high order fractional boundary value problems

“…Some generalizations and applications of Lyapunov-type inequalities for the fractional boundary value problems with different boundary conditions have been studied in the literature. The problem (7) subject to the fractional boundary condition in [14], a Robin boundary condition in [6], a mixed boundary condition in [7] has been considered and a class of fractional boundary value problems has been studied in [11], [15].…”