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, Volume 29, Issue 1, pp 49–84 | Cite as

On a class of Hammerstein integral equations

  • Horst R. Thieme


By a monotone representation of the nonlinearity we derive sufficient (and partly necessary) conditions for the unique existence of positive solutions of the Hammerstein integral equation
$$\begin{array}{*{20}c} {u(x) = \int\limits_D {f(y,u(y)) k(x,y) dy ,} } & {x \in D,} \\ \end{array} $$
and for the convergence of successive approximations towards the solution. Further we study the corresponding nonlinear eigenvalue problem. Essentially we assume that the integral kernel k satisfies appropriate positivity conditions and that, for the nonlinearity f and any y ∈ D, rf(y,r) strictly monotone increases and f(y,r)/r strictly monotone decreases as r>0 increases.


Integral Equation Eigenvalue Problem Number Theory Algebraic Geometry Topological Group 
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Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Horst R. Thieme
    • 1
  1. 1.Mathematisches Institut der Westfälischen Wilhelms-UniversitätMünsterBundesrepublik Deutschland

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