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. Lyapunov’s and A.Poincare’s works about the figures of equilibrium of rotating fluids were taken (in these works non-linear integral equations first appeared and qualitative methods originated). As a continuation, corresponding results of some their followers (E. Schmidt, T. Lalesku and G. Bratu) are discussed. It is noted that by the end of 1920s-beginning of 1930s the old ideological framework – «equation-solution», dominated in mathematics in XVIII–XIX centuries, is exhausted itself. For the further progress new ideas and new approaches were needed. The authors attributed this period to the next stage of development, when it became involved topological and functional-analytic methods and began to build a consistent deductive theory, based on strict definitions and common structures. In this context, the contribution to the development of the theory of nonlinear integral equations of the European mathematicians – L. Lichtenstein and A. Hammerstein and domestic mathematicians – P.S. Urysohn and A.I. Nekrasov is analyzed. The influence of the theory of nonlinear integral equations on the creation and establishment of functional analysis is also estimated.