Lyapunov and Wirtinger inequalities

Hartman-Wintner-Type Inequality for a Fractional Boundary Value Problem via a Fractional Derivative with respect to Another Function

“…Many other generalizations and extensions of inequality (3) exist in the literature; see, for instance, [7,[13][14][15][16][17][18][19][20][21][22] and references therein. Due to the positive impact of fractional calculus on several applied sciences (see, for instance, [23]), several authors investigated Lyapunov-type inequalities for various classes of fractional boundary value problems.…”

A Refinement of Lyapunov-Type Inequality for a Class of Nonlinear Systems

“…Hence, the inequalities in (5) reduce to the inequalities in the following Corollary 1. (6) such that x(a) = x(b) = 0 and x(t) = 0 ∀ t ∈ (a, b), then we have…”

“…In [5], Hartman obtained the following inequality: Over the past few decades, there have been many new proofs and generalizations of the inequality (2). It has been generalized to nonlinear second order equations Yong-In Kim [3,10,11], to delay differential equations [2], to higher order differential equations [9,13,14], to discrete linear Hamiltonian systems [4], and so on [6,7,8,12].…”

On an integral inequality of the Wirtinger type

“…The purpose of this paper is to develop further this method and to demonstrate that new integral inequalities of the Wirtinger type can be obtained by considering an underlying nonlinear second-order differential inequality of the form In the case α = β = 1, the results of this work reduce to the well-known results from [2]. The more general ''mixed linear-half-linear'' case α > 0 and β = 1 was studied in [3], but the paper contains several errors and the results contained in Theorems 5 and 6 are surely not true. In particular, from the differential identity used in the proof it is clear that the integrand on the left-hand side of (15) and (17) should be (qv α−2 + λ 0 r)u 2 and not (q + λ 0 )u 2 v α−2 .…”