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The diagonalization and computation of some nonlinear integral operators. (English) Zbl 0815.47063

The author studies a nonlinear integral equation of the form \[ Au(x)= u(x)- \int_ D k(x,y) f(u(y))dy= v(x) \] where \(D\) is a compact domain in \(\mathbb{R}^ n\), \(k: D\times D\to \mathbb{R}\) is a positive and symmetric kernel, \(k\in L_ 3 (D\times D)\). The assumptions for the nonlinearity \(f: \mathbb{R}\to \mathbb{R}\) are modelled on A. Ambrosetti and G. Prodi [Ann. Mat. Pura Appl., IV. Ser. 93, 231-246 (1972; Zbl 0288.35020)]; essentially: \(f\) is convex and the values of \(f'\) are majorized in terms of the reciprocals of the first and second eigenvalue of the linear operator \(Ku(x)= \int_ D k(x,y) u(y) dy\).
The author proves that \(A\) is then a global fold, in the sense that after conjugation by homeomorphisms it takes the form \(\widetilde {A}: \mathbb{R} \times E\to \mathbb{R}\times E\), \(\widetilde {A}: (t,v)\to (t^ 2,v)\). Therefore the equation \(Au=v\) may have one or two solutions, or no solution, according to the value of \(v\).
Reviewer: L.Rodino (Torino)

MSC:

47G10 Integral operators
47J05 Equations involving nonlinear operators (general)

Citations:

Zbl 0288.35020
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References:

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