Analytic Functions On The Two Types Of Infinite Straight Line Boundary Value Problem Solution Stability

In this paper, we mainly study the following two problems:In the chapter two, we discuss solutions of Hilbert boundary value problem and its stability under the condition of perturbation of infinite line. We also put forward a conception of perturbation quasi-solvability and quasi-solutions when the index of problem is less then zero.Zhang Hongmei considered the stability of solutions of Riemann boundary value problem when the smooth perturbation of infinite line occurred, which required the value of perturbationω( x) at infinity to be zero. In this section we don't require the value of perturbationω( x) at infinity to be zero, in other words any point on the infinite line can"move".SupposeΦω(σ) is solutions of Hilbert boundary value problem when the smooth perturbationω( x) of infinite line X occurred Re {[ aω( x ) + ibω( x ) ]Φω+ ( x )} = cω( x ),x∈X (1) Theorem 1. Let ( )ω∈Bρ0, ifκ≥0,then Hilbert boundary value problem(1)is resolvable without any conditions and solutions of them satisfy(?)Whenκ< 0, if and only if satisfy: (?)Hilbert boundary value problem(1)is perturbation quasi-solvability and solutions of them satisfy:(?)Chapter three considers the stability of solutions of periodic Hilbert boundary value problem on infinite line. It makes use of the mapζ= tan z, which takes strip S 0 conformably into the planeζ.so it is difficult to get solutions of periodic Hilbert boundary value problem, more difficult to the stability. In the end, we find perturbationω( x) need to meet the condition, which can keep solutions existent and stable.SupposeΦω(σ) are solutions of periodic Hilbert boundary value problem when the smooth perturbationω( x) of infinite line X occurred Re {[ aω(σ) + ibω(σ) ]Φω+ (σ)} = cω(σ),σ∈X (2) Theorem 2. Letω∈B(ρ0),ifκ≥0,then periodic Hilbert boundary value problem(2)is resolvable without any conditions and solutions of them satisfy(?)periodic Hilbert boundary value problem ( 2 ) is perturbation quasi-solvability and solutions of them satisfy(?).