On the development of qualitative methods for solving nonlinear equations and some consequences

Aim. The aim of the paper is investigation of the development of the fixed-point method and mapping degree theory associated with the names of P. Bohl, L. Brouwer, K. Borsuk, S. Ulam and others and its application to study of the trajectories of dynamical systems behavior and stable states of ordered media. Method. The study is based on an analysis of the fundamental works of the mentioned mathematicians 1900-1930ś, as well as later results of N. Levinson, G. Volovik, V. Mineev, J. Toland and H. Hofer of an applied nature. Results. Brouwer made an essential contribution to the solvability theory of nonlinear equations of the form $f(x) = x$ in a finite-dimensional statement. This was preceded by the study of singular points of vector fields undertaken by H. Poincaré, as well as the proof of Bohl theorem on the impossibility of mapping a disk onto its boundary. The first mathematician who used the fixed point method in the study of systems of differential equations was Bohl. This theme was continued 40 years later in the works of Levinson, who showed the existence at least one periodic solution in deterministic dissipative dynamical systems. The fundamental concept of the mapping degree ($\mathrm{deg} f$) introduced by Brouwer "began to play" in the most unexpected situations. Investigations of Volovik and Mineev revealed a direct dependence of ordered media defects on the topological invariant $\mathrm{deg} f$, characterizing the transformation$ f$ of a neighborhood of a singular point onto the sphere. Another non-standard application of the mapping degree was discovered by Toland and Hofer in the study of some Hamiltonian systems. Calculating $\mathrm{deg} f$ for mappings of a special kind helped them to prove the existence of periodic, homoclinic, and heteroclinic trajectories of these systems. Discussion. The fixed point method and mapping degree are the basic tools of qualitative methods for solving nonlinear equations. They proved to be in demand not only within the framework of mathematics, but also in applications, and this trend, apparently, will persist even in the transition to the infinite-dimensional case.