On the development of nonlinear integral equations. First steps, becoming and new ideas

The emergence of the theory of nonlinear integral equations was a natural consequence of the development of all mathematics in the 18th-19th centuries. At the same time, the growing interest in nonlinear problems in the late 19th and early 20th centuries had a strong motivating effect. The study of specific nonlinear integral equations was prompted by the urgent problem of the equilibrium figures of rotating liquid masses, which attracted the attention of a significant number of leading mathematicians, starting with Newton. In the first decades of the development of the theory of nonlinear integral equations, traditional approaches were cultivated, which were used to study differential and algebraic equations according to the equation-solution scheme. The complexity and originality of nonlinear problems immediately revealed the relevance of the issues of existence and uniqueness of their solutions, which made it necessary to attract new ideas and methods, primarily from the field of functional analysis. The report will present the results of R. Rado, A.M. Lyapunov, E. Schmidt, P.S. Urysohn, A.I. Nekrasov, A. Hammerstein, L. Lichtenstein, Yu. Schauder, W. Orlich, V.V. Nemytskii, M.A. Krasnoselskii, Ya.B. Rutitskii and some other mathematicians.
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