By Myroslav L. Gorbachuk
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Additional resources for Boundary Value Problems for Operator Differential Equations
11) (F€-::"'xl) X3 is fixed. The value offunction u at point x is added from and X3. 3). 4) type. We will briefly discuss Neumann problem and plan ways for its solution. We refer to the some results of paper . 5. 12) x' where u+ E H 1/ 2 (r+) is the solution of the Wiener-Hopf equation Xr+BkU+ = h, Bk is a pseudodifferential operator with symbol We introduce the operators (e? 6. 3) is given by formula - -1 -1 U = -sgn X3 WO lIB; lh, where the operator Wo = ITBkBi11 R(fI) is iverted by a Neumann series and is absent in case k = 1, lh is arbitrary continuation of h E H- 1/ 2 (L,1) onto H_ 1/ 2 (JR2).
Then F-1(B=u_) = b * u_. 3 of convolution Cn. where u_(x - y) is considered as a function on y (x is fixed), and notation bey) means that functional b acts on y - variable. Let us show that (b * u_)(x) = 0 under x E C:;.. Consider two cases: y E -C:;' and y tJ. -C:;.. :. y E C+ and, thus u_(x - y) = 0 because suppu_(x - y) C JR2 \ C:;.. In the second case (b * u_) vanishes because y tJ. supp b. ).