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A non-linear integral equation of the type
 
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>
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$$\phi(x)+\int\limits_a^bK(x,s)f[s,\phi(s)]ds=0,\quad a\leq x\leq b,$$
  
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
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where $K(x,s)$ and $f(x,s)$ are given functions, while $\phi(x)$ is the unknown function. Named after A. Hammerstein [[#References|[1]]], who considered the case where $K(x,s)$ is a symmetric and positive Fredholm kernel, i.e. all its eigen values are positive. If, in addition, the function $f(x,s)$ 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>
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$$|f(x,s)|\leq C_1|s|+C_2,$$
  
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
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where $C_1$ and $C_2$ are positive constants and $C_1$ is smaller than the first eigen value of the kernel $K(x,s)$, the Hammerstein equation has at least one continuous solution. If, on the other hand, $f(x,s)$ happens to be a non-decreasing function of $s$ for any fixed $x$ from the interval $(a,b)$, Hammerstein's equation cannot have more than one solution. This property holds also if $f(x,s)$ 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>
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$$|f(x,s_1)-f(x,s_2)|\leq C|s_1-s_2|,$$
  
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]]).
+
where the positive constant $C$ is smaller than the first eigen value of the kernel $K(x,s)$. 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====
 
====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>
 
<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>

Latest revision as of 11:07, 24 August 2014

A non-linear integral equation of the type

$$\phi(x)+\int\limits_a^bK(x,s)f[s,\phi(s)]ds=0,\quad a\leq x\leq b,$$

where $K(x,s)$ and $f(x,s)$ are given functions, while $\phi(x)$ is the unknown function. Named after A. Hammerstein [1], who considered the case where $K(x,s)$ is a symmetric and positive Fredholm kernel, i.e. all its eigen values are positive. If, in addition, the function $f(x,s)$ is continuous and satisfies the condition

$$|f(x,s)|\leq C_1|s|+C_2,$$

where $C_1$ and $C_2$ are positive constants and $C_1$ is smaller than the first eigen value of the kernel $K(x,s)$, the Hammerstein equation has at least one continuous solution. If, on the other hand, $f(x,s)$ happens to be a non-decreasing function of $s$ for any fixed $x$ from the interval $(a,b)$, Hammerstein's equation cannot have more than one solution. This property holds also if $f(x,s)$ satisfies the condition

$$|f(x,s_1)-f(x,s_2)|\leq C|s_1-s_2|,$$

where the positive constant $C$ is smaller than the first eigen value of the kernel $K(x,s)$. A solution of the Hammerstein equation may be constructed by the method of successive approximation (cf. Sequential approximation, method of).

References

[1] A. Hammerstein, "Nichtlineare Integralgleichungen nebst Anwendungen" Acta Math. , 54 (1930) pp. 117–176
[2] F.G. Tricomi, "Integral equations" , Dover, reprint (1985)
[3] M.M. Vainberg, "Variational methods for the study of nonlinear operators" , Holden-Day (1964) (Translated from Russian)
[4] M.A. Krasnosel'skii, "Topological methods in the theory of nonlinear integral equations" , Pergamon (1964) (Translated from Russian)
[5] N.S. Smirnov, "Introduction to the theory of integral equations" , Moscow-Leningrad (1936) (In Russian)
How to Cite This Entry:
Hammerstein equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hammerstein_equation&oldid=33125
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article