ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

qualitative methods

About the history of nonlinear integral equations

The work is dedicated to the history of the theory of nonlinear integral equations, covering a period before the start of the 1930s. By analyzing the specifics of the initial period, authors emphasize that the integral equations (in particular, nonlinear equations) is independent object of research with their own problems, requiring its own system of concepts and own language. As a starting point here A.M.

Legacy of Alexander Mikhailovich Lyapunov and nonlinear dynamics

Aim. The aim of the work is to study the scientific heritage of A.M. Lyapunov from the standpoint of nonlinear physics. Fundamental importance Lyapunov’s contribution is determined not only by the methods he created, which became the basis of the mathematical apparatus in the study of nonlinear phenomena, but his ideas and concepts introduced by him contributed to the formation of concepts and principles of nonlinear dynamics. Method. The study is based on an analysis of Lyapunov’s original works with the involvement of existing literature on his scientific heritage.

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 1900–1930’s, as well as later results of N. Levinson, G. Volovik, V. Mineev, J. Toland and H. Hofer of an applied nature. Results.