Retracts, fixed point index and differential equations

Abstract

Abstract. Some problems in differential equations evolve towards Topology from an analytical origin. Two such problems will be discussed: the existence of solutions asymptotic to the equilibrium and the stability of closed orbits of Hamiltonian systems. The theory of retracts and the fixed point index have become useful tools in the study of these questions. Retractos,índice de punto fijo y ecuaciones diferencialesResumen. Algunos problemas de las ecuaciones diferenciales evolucionan hacia la Topología desde un origen analítico. Se discutirán dos problemas de este tipo: la existencia de soluciones asintóticas al equilibrio y la estabilidad de lasórbitas cerradas de los sistemas Hamiltonianos. La teoría de retractos y elíndice de punto fijo se han convertido en herramientas muyútiles para el estudio de estas cuestiones.

Periodic Solutions of a Forced Relativistic Pendulum via Twist Dynamics

“…For the proof of the instability result in Theorem 1 we use a theorem by Ortega [20], together with the Poincaré-Hopf Theorem [8].…”

“…For the special case of dimension two, there exists a relation between the stability of a fixed point and its fixed point index. In fact it was proved in [20] that if a continuous one-to-one map S which is also orientation and area preserving has a stable fixed point p then either S = Id in some neighbourhood of p or there exists a sequence of Jordan curves {Γ n } converging to p such that, for each n,…”

Smooth linearisation of planar periodic maps

“…These points are also fixed under h 2 = h•h and so x * is a non-isolated fixed point of h 2 . Corollary 1 in [18] can be applied 2 to the map h 2 : W → R 2 . It implies that h 2 = Id in W.…”

Waźewski Topological Principle and V-bounded Solutions of Nonlinear Systems

“…It should be noted that the Waźewski topological principle was successfully exploited for proving the existence of bounded solutions to some boundary value problems in [6] and to quasihomogeneous systems in [7,8] (see also a discussion in [9]).…”

“…Basically topological method of guiding functions, which was developed by M. A. Krasnosel'ski and A. I. Perov, is an effective tool for proving the existence of bounded solutions of essentially nonlinear systems too (see the bibliography in [10,11]). But, except [9,13], in all papers known to us only independent of time guiding functions were used.…”