Fundamental operator-functions of singular differential-operator mappings in Banach spaces.
by
Michail Falaleev

We consider the differential operator B \fracd^Ndt^N - A where B, A - closed linear operators from E₁ in E₂,   E₁,   E₂ - Banach spaces. Let B   - is Fredholm operator, [`(D(A) \cap D(B))] = E<sub>1</sub>, D(B) subset D(A), then for operator B exists bounded operator Schmidt \Gamma. Let dimN(B) = dimN(B <sup>*</sup> ) = n end B has a complete A   - Jordan set { j<sup>(j)</sup><sub>i</sub>, i = [`1, n],   j = [`(1, p<sub>i</sub>)] }, then B <sup>*</sup> has a complete A <sup>*</sup> - Jordan set { \psi<sup>(j)</sup><sub>i</sub>,   i = [`1, n],   j = [`(1, p_i)] }, in this case generalized operator-function

EN (t) = \GammaU(A \Gammat)[I - n å i=1  pi å j=1  < ·, \psi(j)i > A j(pi + 1 - j)i] \theta(t) -

- n å i=1  [ pi - 1 å k=0  { pi - k å j=1  < ·, \psi(j)i > j(pi -k + 1 - j)i }\delta(k ·pi) (t)]

is fundamental for differential mapping B \fracd^Ndt^N - A, where

U(A \Gammat) \equiv \infty å i=1  (A \Gamma)i - 1 \fracti N - 1(i N - 1)!.

With the help of operator-function E_N (t) may be reconstructed continuous and generalized solutions of corresponding Cauchy problems.

Date received: March 23, 1999