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The existence of a solution of a
nonlinear perturbation of an elliptic convolution equation of Wiener-Hopf
type in a bounded region G of Rn is proved. More explicitly, let A be an
elliptic convolution operator on G of order α,α > 0;Aj the principal part of
A in a local coordinate system and Ãj(xj,ξ) be the symbol of Aj with a
factorization with respect to ξn of the form: Ãj(xj,ξ) = Aj+(xj,ξ)Ãj−(xj,ξ) for
xnj = 0.Aj+,Ãj− are homogeneous of orders 0,α in ξ respectively; the first
admitting an analytic continuation in Imξn > 0, the second in Imξn ≦ 0. Let
Tk,k = 0,⋯,[α] − 1 be bounded linear operators from H+k(G) into L2(G)
where H+k(G),k ≧ 0 are the Sobolev-Slobo detskii spaces of generalized
functions.
The purpose of the paper is to prove the solvability of: Au++λαu+ = f(x,T0u+,⋯,T[α]−1u+)
on G;u+ in H+α(G) for large |λ| and on a ray arg λ = 𝜃 such that Ãj + λα≠0 for
|ξ| + |λ|≠0 and for all j. f(x,ζ0,⋯,ζα−1) has at most a linear growth in (ζ0,⋯,ζα−1)
and is continuous in all the variables.
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