Nekrasov integral equation

A non-linear integral equation of the form

$$\phi(x)=\lambda\int\limits_a^b[\phi(y)+R(\lambda,y,\phi(y))]K(x,y)\,dy,\label{*}\tag{*}$$

where $R$ and $K$ are known functions, $K$ being symmetric, $\phi$ is the unknown function, and $\lambda$ is a numerical parameter. Integral equations of this type were obtained by A.I. Nekrasov (see [1]) in the solution of problems arising in the theory of waves on the surface of a fluid. Under certain conditions Nekrasov has constructed a solution of \eqref{*} in the form of a series in powers of a small parameter; its convergence has been proved by the method of majorants.

Sometimes an equation of the type \eqref{*} is called a Hammerstein equation, although Nekrasov [2] published his investigations before A. Hammerstein [3].

References

[1]	A.I. Nekrasov, "Collected works" , 1 , Moscow (1961) (In Russian)
[2]	A.I. Nekrasov, Izv. Ivanovo-Vozn. Politekhn. Inst. , 6 (1922) pp. 155–171
[3]	A. Hammerstein, "Nichtlineare Integralgleichungen nebst Anwendungen" Acta Math. , 54 (1930) pp. 117–176

References

[a1]	P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian)