In this work, first we give a survey of the most basic results on Lyapunov-type inequality, and next we sketch some recent developments related to this type of inequalities. AMS Subject Classifications: 39A11, 34C15.

This paper investigates stability of interconnection of integral input-to-state stable (iISS) systems. A new tool to verify the stability is proposed by focusing on Lyapunov inequalities each system is to satisfy in accordance with a small-gain-type condition. The purpose of this paper is to extend the technique of “flexible Lyapunov inequalities” developed previously for input-to-state stable(...

We prove two kinds of Lyapunov type inequalities for pseudo-integrals. One discusses pseudo-integrals where pseudo-operations are given by a monotone and continuous function g. The other one focuses on the pseudo-integrals based on a semiring 0; 1 ½ Š; sup; ð Þ , where the pseudo-multiplication is generated. Some examples are given to illustrate the validity of these inequalities. As a generali...

Let us consider the linear boundary value problem u′′(x) + a(x)u(x) = 0, x ∈ (0, L), u′(0) = u′(L) = 0, (0.1) where a ∈ Λ0 and Λ0 is defined by Λ0 = {a ∈ L∞(0, L) \ {0} : Z L 0 a(x) dx ≥ 0, (0.1) has nontrivial solutions}. Classical Lyapunov inequality states that Z L 0 a(x) dx > 4/L for any function a ∈ Λ0, where a(x) = max{a(x), 0}. The constant 4/L is optimal. Let us note that Lyapunov inequ...

where q : [a,b] → R is a continuous function, and the zeros a and b of every solution y(t) are consecutive. Since then, many generalizations of the Lyapunov inequality have appeared in the literature (see [–] and the references therein). Recently, the research of Lyapunov-type inequalities for fractional boundary value problem has begun. In [], Ferreira investigated a Lyapunov-type inequali...

We present some new nonlinear integral inequalities Bellman-Bihari type with delay for discontinuous functions integro-sum inequalities; impulse integral inequalities . Some applications of the results are included: conditions of boundedness uniformly , stability by Lyapunov uniformly , practical stability by Chetaev uniformly for the solutions of impulsive differential and integrodifferential ...