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Solutions of Hammerstein integral equations via a variational principle. (English) Zbl 1060.45006
The authors study solutions of the Hammerstein equation \[ u(x)= \int_\Omega k(x,y)f \bigl(y,u(y)\bigr)\,dy \] as critical points of the corresponding energy functional on a suitable Hilbert space. The main tool is a new variational principle of B. Ricceri [J. Comput. Appl. Math. 113, No. 1–2, 401–410 (2000; Zbl 0946.49001)] which also applies to kernel functions \(k\) of changing sign. The abstract result is illustrated by means of an application to the polyharmonic equation \((-\Delta)^mu(x)= f(x,u(x))\).

MSC:
45G10 Other nonlinear integral equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
49J22 Optimal control problems with integral equations (existence) (MSC2000)
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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