Anti-periodic solutions of Liénard equations with state dependent impulses

Fink type conjecture on affine-periodic solutions and Levinson's conjecture to Newtonian systems

“…which shows that x(t, ϕ 0 ) is a (Q, T ) affine-periodic solution of system (2). Finally, we have to show that x(t, ϕ 0 ) is globally equi-asymptotically stable, which implies the uniqueness of the (Q, T ) affine-periodic solutions.…”

“…In this section, we will discuss the existence of (Q, T ) affine-periodic solutions and the asymptotic stability for system (2). That is, for functional differential equations with finite delay, we hope to establish a result corresponding to ordinary differential equations.…”

“…2) Fink [9] proved that for a periodic system, the exponential uniform asymptotic stability implies the existence of bounded solutions, thus it follows the existence of periodic solutions, and he conjectured that for an almost periodic system the exponential uniform asymptotic stability implies the existence of almost periodic solutions, which was proved by Lin [15] when the system satisfies the Lipschitz condition and is uniformly asymptotically stable. For some periodic solution theory, see [5,4,11,7,2] and references therein.…”

“…As a result, we give corresponding Lyapunov's methods, in which the assumptions on the directional derivative are quite weak, and the negative definiteness is removed. Next in Section 3, we give an analog for functional differential equations (2). All the arguments are based on the modular degree theory due to Zabreiko and Krasnoselskii [23].…”

Anti-periodicity on high-order inertial Hopfield neural networks involving mixed delays

“…On the other hand, in neural networks dynamics involving fields such as communication, economics, biology or ecology, the relevant state variables are usually considered as proteins and molecules, light intensity levels or electric charges, which are naturally antiperiodic [22][23][24]. Considering this factor, many recurrent neural network models involv-ing time-varying delays and anti-periodic environments have been widely investigated in [16,17,[24][25][26].…”