On the development of nonlinear integral equations at the early stage and the contribution of domestic mathematics

The paper considers the preconditions and the origin of the theory of nonlinear integral equations. The appearance of this theory was a natural consequence of the development of all mathematics of the XVIII-XIX cc. At the same time, the growing interest in nonlinear problems in the late XIX and early XX centuries had a strong motivating effect. The direct investigation of specific nonlinear integral equations was triggered by an urgent applied problem on the equilibrium figures of rotating liquid masses, which has attracted a significant number of major mathematicians since 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. That is, the foreground was the calculation and assessment of its accuracy. The complexity and originality of nonlinear problems immediately revealed the relevance of questions of the existence and uniqueness of their solutions, which made it necessary to involve other, just emerging areas of mathematics. The theory of integral equations in general was one of the origins of functional analysis. Moreover, both theories were closely intertwined and mutually stimulated each other in their evolution. This fully applies to nonlinear integral equations, for which qualitative methods have become of paramount importance. At the stage considered in this work, there was a parallel development and mixing of traditional methods for studying equations and new approaches of a qualitative nature. In the next phase, new approaches came to the fore, merging with functional analysis and topology.