View source for Nekrasov integral equation

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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 [[#References|[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|Hammerstein equation]], although Nekrasov [[#References|[2]]] published his investigations before A. Hammerstein [[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Nekrasov,   "Collected works" , '''1''' , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Nekrasov,   ''Izv. Ivanovo-Vozn. Politekhn. Inst.'' , '''6'''  (1922)  pp. 155–171</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Hammerstein,   "Nichtlineare Integralgleichungen nebst Anwendungen"  ''Acta Math.'' , '''54'''  (1930)  pp. 117–176</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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)</TD></TR></table>