View source for Hammerstein equation

A non-linear integral equation of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h0462901.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h0462902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h0462903.png" /> are given functions, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h0462904.png" /> is the unknown function. Named after A. Hammerstein [[#References|[1]]], who considered the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h0462905.png" /> is a symmetric and positive Fredholm kernel, i.e. all its eigen values are positive. If, in addition, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h0462906.png" /> is continuous and satisfies the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h0462907.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h0462908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h0462909.png" /> are positive constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h04629010.png" /> is smaller than the first eigen value of the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h04629011.png" />, the Hammerstein equation has at least one continuous solution. If, on the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h04629012.png" /> happens to be a non-decreasing function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h04629013.png" /> for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h04629014.png" /> from the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h04629015.png" />, Hammerstein's equation cannot have more than one solution. This property holds also if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h04629016.png" /> satisfies the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h04629017.png" /></td> </tr></table>

where the positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h04629018.png" /> is smaller than the first eigen value of the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046290/h04629019.png" />. A solution of the Hammerstein equation may be constructed by the method of successive approximation (cf. [[Sequential approximation, method of|Sequential approximation, method of]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Hammerstein,   "Nichtlineare Integralgleichungen nebst Anwendungen"  ''Acta Math.'' , '''54'''  (1930)  pp. 117–176</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.G. Tricomi,   "Integral equations" , Dover, reprint  (1985)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Vainberg,   "Variational methods for the study of nonlinear operators" , Holden-Day  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.A. Krasnosel'skii,   "Topological methods in the theory of nonlinear integral equations" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.S. Smirnov,   "Introduction to the theory of integral equations" , Moscow-Leningrad  (1936)  (In Russian)</TD></TR></table>