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Existence and uniqueness theorem for a hammerstein nonlinear integral equation

Khachatryan A. Kh., Khachatryan Kh. A.

Opuscula Mathematica

2011

Vol. 31, no. 3

393-398

The existence of a solution, as well as some properties of the obtained solution for a Hammerstein type nonlinear integral equation have been investigated. For a certain class of functions the uniqueness theorem has also been proved.

The reactance wave diffraction problem by a strip in a scale of Bessel potential spaces

Castro L. P., Natroshvili D.

Opuscula Mathematica

2006

Vol. 26, no. 2

289-303

We consider a boundary-transmission problem for the Helmholtz equation, in a Bessel potential space setting, which arises within the context of wave diffraction theory. The boundary under consideration consists of a strip, and certain reactance conditions are assumed on it. Operator theoretical methods are used to deal with the problem and, as a consequence, several convolution type operators are constructed and associated to the problem. At the end, the well-posedness of the problem is shown for a range of regularity orders of the Bessel potential spaces, and for a set of possible reactance numbers (dependent on the wave number).