Fixed points, stability, and harmless perturbations

Abstract

Much has been written about systems in which each constant is a solution and each solution approaches a constant. It is a small step to conjecture that functions promoting such behavior constitute harmless perturbations of stable equations. That idea leads to a new way of avoiding delay terms in a functional-differential equation. In this paper we use fixed point theory to show that such a conjecture is valid for a set of classical equations.

Fixed points and stability of neutral stochastic delay differential equations

“…However, there exist a number of difficulties encountered in the study of stability by means of Lyapunov's direct method. Recently, Burton and his co-authors have applied fixed point theory to investigate the stability, which shows that some of these difficulties vanish when applying fixed point theory [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]25,26]. Lyapunov theory is now more than one hundred years old and it has been a very fruitful area.…”

Sunflower model: time-dependent coefficients and topology of the periodic solutions set

“…More recently, Liu and Kalmár-Nagy [20] computed limit cycle amplitudes and frequencies for (1). Other meaningful results related to this equation can be found e.g., in [2,8,10,19,22,31]. Our investigation follows a different path: We extend Eq.…”

Soluções periódicas para versões generalizadas do modelo de produção de células sanguíneas (hematopoiese) de Mackey e Glass

“…As equações diferenciais funcionais com retardamento (EDFR) constituem uma ferramenta muito importante na modelagem matemática, pois determinam modelos mais realísticos, e por essa razão o estudo sobre o comportamento de suas soluções se faz necessário. Ao longo do século XX, os funcionais de Liapunov foram a principal ferramenta para o estudo de propriedades qualitativas de soluções de EDFR, mas a dificuldade em determinar tais funcionais impulsionaram o estudo de estabilidade através da teoria de ponto fixo, ferramenta com a qual tal dificuldade tende a desaparecer, como se pode constatar em [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], por exemplo. No texto apresentamos a motivação para o problema e realizamos com detalhes a demonstração dos resultados essenciais para garantir a existência de soluções periódicas para o modelo proposto.…”