By Dinh Nho Hao, Tran Duc Van

This publication is dedicated to the idea of infinite-order linear and nonlinear differential operators with a number of actual arguments and their purposes to difficulties of partial differential equations and numerical research. half I develops the speculation of pseudodifferential operators with actual analytic symbols, the neighborhood representatives of that are linear differential operators of limitless order appearing within the areas of easy and generalized services according to the duality of the areas of genuine analytic services and functionals. purposes to numerous difficulties of PDEs and numerical research are given. half II is dedicated to the speculation of Sobolev-Orlicz areas of countless order and the solvability of nonlinear partial differential equations with arbitrary nonlinearities.

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L()m U-/ l5)F! -F=(4*) mS 2m — , 1, ,x, 0F 2 otherwise. - (m-l)! 14) that I t 2y/FtJ*> *,m {x_ K 1 ^d*»V(x) ™\^ dxim s)ds ' = e X p (-^)^ ( a ; ) ' as required. 12) we find that -s)ds. 7 e"^'(iM-s)ds. 1 < e(t, * ' *■)M-)>= * > >='2T7W»L 2(v^Fi)T -» 2 /(4«) v ? 11) we obtain 2 •x /4 X p 6(x)-^ ) == eexp(( -AA) ) ( 2 ^ e _C l 2 / 4 (2^)" The lemma is completely proved. ■=—. 4. The algebra of pseudo-differential operators with analytic symbols. Let G be an open set of H ? 1. The space of test functions Wjt°°(]Rn) is the set of functions / ( x ) satisfying the following conditions: i) f(x) admits analytic continuation as an entire function to (Tf, ii) / ( ( ) is the Fourier-Laplace transform of an analytic functional u £ A'[IRn], with supp u C K C G, where K is a compact set.

1. Suppose that L(£) and £ _ 1 (£) are analytic in some domain G C K?. Then for any h € W Q ° ° ( I R " ) Eq. 1. We consider the Helmholtz equation (u> is a complex parameter) Au(x)+u2u = h{x), xeJrT. 3) Note that the symbol of the Helmholtz operator and the inverse operator / / ( A + u 2 I ) are analytic for £2 ^ w 2 , £ € IRn. Hence, the whole space IRj? is the common domain of analyticity of the symbols of both operators, except when u> is real. The space Wj[E°(IRn) consists of all functions h(x) that admit an analytic continuation as entire functions and are the Fourier-Laplace transform of h € A'[IR"].

A function / : IR1 —> d? is said to be semifinitary if f(x) = 0 for x < a. 1. (Shilov [1]). ,v in the lower = Imz, or, equivalently, l/WI < r*(i + |*|)-V for all r] < 0, where Tk,vk are constants depending only on k. Then f{z) is a holo morphic extension of the Fourier transform of some semifinitary function fix). 20. Weakly nonlinear equations. 1) where the operator A (which is generally speaking nonlinear) possesses two prop erties: coercivity and weak compactness. We call such equations weakly nonlinear 29 PRELIMINARIES equations.