Integral manifolds of impulsive differential equations

Abstract

ABSTPCTThe present paper is concerned with the existence of integral manifolds of impulsive differential equations as t-,+oo. Under the assumption of exponential trichotomy on the linear part of the right-hand side of the equation, it is proved that if the nonlinear perturbation is small enough, then there exist integral manifolds as t-/ oo for the perturbed equations.

A Zabczyk type method for the study of the exponential trichotomy of discrete dynamical systems

“…Starting with the pioneering works of Elaydi and Hájek (see [12,13]), over the past decades various concepts of exponential trichotomy were introduced and studied (see [2,3,[5][6][7]10,11,14,15,20,[22][23][24][25]29,30]). The investigations were focused on exponential trichotomies from diverse perspectives among we mention robustness (see [2,5,29]), existence (see [14,15,20,[22][23][24][25]), invariant manifolds and regularity (see [3,6,7,30]), providing significant applications in the qualitative theory of dynamical systems. A special attention was devoted to the investigation of the exponential trichotomy of discrete dynamical systems (see [2,[5][6][7]10,11,14,22,25]).…”

A Reduction Theorem for Systems of Differential Equations with Impulse Effect in a Banach Space

“…This problem involving the impulse effect was first conw x sidered by the author and L. Sermone in 20᎐23 and D. D. Bainov, S. I. w x Kostadinov, and Nguyen Van Minh in 24,25 . In the present paper, a reduction theorem for systems of differential equations with impulses in a Banach space is proven assuming that the system splits into two parts.…”

Dicotomias em equações diferenciais ordinárias generalizadas e aplicações

“…Para equações diferença, a literaturaé mais esparsa, mas Co↵man e Schäfer [6] são pioneiros aqui. Para equações diferenciais impulsivas (EDIs), a teoria de dicotomia exponencial pode ser encontrada em Bainov et al [4], por exemplo. Na teoria de sistemas dinâmicos não autônomos, a importância da dicotomia exponencial deve-se ao fato dela ser muito utilizada para resolver problemas não lineares como perturbações de problemas lineares (veja Henry [19] e Sakamoto [32], por exemplo).…”