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A nonlinear boundary problem. (English) Zbl 0591.30038

The author gives a solution of the nonlinear Hilbert-Gakhov problem

[Φ + (x)] α +[Φ - (x)] α =f,0<x<1,0<α<1,

where f has a derivative satisfying a Hölder condition on [0,1] with the possible exception of the endpoints 0,1. The method of the investigation is based on the reduction to a generalized Abel equation. Reviewer’s remark. The main part of the paper is a solution of the generalized Abel type equation. This equation in detail was investigated in many papers (during the years 1970-74 years) and in the monograph ”Hypersingular integrals and their applications” (1984; Zbl 0577.42016) by S. G. Samko which is unknown for the author.

Reviewer: N.K.Karapetianc
MSC:
30E25Boundary value problems, complex analysis
45G05Singular nonlinear integral equations