Vol. 24, No. 3, 1968

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Nonlinear elliptic convolution equations of Wiener-Hopf type in a bounded region

Bui An Ton

Vol. 24 (1968), No. 3, 577–587

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) be the symbol of Aj with a factorization with respect to ξn of the form: Ãj(xj) = Aj+(xj)Ãj(xj) for xnj = 0.Aj+,Ãj are homogeneous of orders 0in 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 + λα0 for |ξ| + |λ|0 and for all j. f(x,ζ0,α1) has at most a linear growth in (ζ0,α1) and is continuous in all the variables.

Mathematical Subject Classification
Primary: 45.30
Secondary: 35.00
Received: 20 March 1967
Published: 1 March 1968
Bui An Ton