Equivalent forms of the Brouwer fixed point theorem II

The Brouwer fixed point theorem and periodic solutions of differential equations

“…Then, either f has a zero x ∈ ∂C or F (x, λ) = 0 for all x ∈ ∂C and λ ∈ [0, 1]. In this last case, by using the properties of the Brouwer degree d B in the open, bounded set Ω = int(C) containing zero (note that ∂Ω = ∂C since C is convex), we obtain For a more detailed study on the connections between the Brouwer fixed point theorem and the existence of zeros for certain functions the reader is referred to [13,19,20].…”

“…satisfies the assumptions of Theorem 1.1 for each T > 0. A sequence of T kperiodic solutions of ( 5) with T k → 0 as k → ∞ provides an equilibrium of ( 5) and hence a fixed point of g. Despite of the impressive list of equivalent reformulations of Brouwer fixed point theorem in the literature (see for instance [1,5,12,13,17,21]), the equivalence between the Brouwer fixed point theorem and the existence of periodic solutions for some differential equation seems to have been unnoticed. Although Poincaré had used a topological statement equivalent to the Brouwer fixed point theorem (the so-called Poincaré-Miranda theorem) to study periodic solutions in celestial mechanics as early as 1883, and furthermore had shown how to reduce the existence of periodic solutions of differential systems to the fixed points of the Poincaré map, it is not known if Poincaré heard about the Brouwer fixed point theorem, published shortly before his dead.…”

Equilibrium points, periodic solutions and the Brouwer fixed point theorem for convex and non-convex domains

“…As clearly explained in the recent book of Dinca and Mawhin [16] and developed in the content of the book, as well as in the survey articles [41,48] and the references therein, Brouwer theorem can be seen as a "core result" from which several important theorems about the existence of fixed points and periodic points can be proved. Further connections between the Brouwer fixed point theorem and other results (including theorems of combinatorial nature) can be found in [24,50] and [33,34,51]. From this perspective, an interesting line of research, already pursued by several authors, is to establish different results…”

Equilibrium points, periodic solutions and the Brouwer fixed point theorem for convex and non-convex domains

“…As clearly explained in the recent book of Dinca and Mawhin [16] and developed in the content of the book, as well as in the survey articles [41,48] and the references therein, Brouwer theorem can be seen as a "core result" from which several important theorems about the existence of fixed points and periodic points can be proved. Further connections between the Brouwer fixed point theorem and other results (including theorems of combinatorial nature) can be found in [24,50] and [33,34,51]. From this perspective, an interesting line of research, already pursued by several authors, is to establish different results of Nonlinear Analysis and its application to differential equations which can be obtained, more or less directly, from this theorem.…”