具有极点和正则点的非线性迭代方程的解析解

本文主要研究具有极点和正则点的非线性迭代方程G(z)x′(z)=x(αz+βx(z))+F(x(z))的解析解.在第二章和第三章中通过把已知方程转化为不含未知函数迭代的辅助方程ψ(λz)-αψ(z)]λψ′(λz)-αψ′(z)]G(ψ(z))=ψ(z)ψ(λz)-αψ(z)]ψ(λ2z)-αψ(λz)]ψ′(z)+β2ψ(z)ψ′(z)F(1/β(ψ(λz)-αψ(z))),z∈C.和G(g(z))γg′(γz)-αg′(z)]=b(γ2z)-αg(γz)]g′(z)+βg′(z)F(1/β(g(γz)-αg(z))).从而得到原方程在极点和正则点处的解析解x(z)=1/βψ(λψ-1(z))-αz,x(z)=1/βg(γg—1(z))-αz].

THE EXISTENCE OF ANALYTIC SOLUTIONS OF A NONLINEAR ITERATIVE EQUATIONS NEAR POLES AND REGULAR POINTS

In this paper,analytic solutions of a nonlinear iterative equation G(z)x′(z) =x(αz + βx(z)) + F(x(z))is studied when given functions have poles and regular points.By reducing the equation to another functional equation without iteration of the unknown function ψ(λz)-αψ(z)]λψ′(λz)-αψ′(z)](G)(ψ(z))=ψ(z)ψ(λz)-αψ(z)]ψ(λ2z)-αψ(λz)]ψ′(z)+β2ψ(z)ψ′(z)(F)(1/β(ψ(λz)-αψ(z))),z ∈C,and G(g(z))γg′(γz)-αg′(z)] =g(γ2z)-αg(γz)]g′(z) + βg′(z)F(1/β(g(γz) αg(z))),the invertible analytic solutions of the form x(z) =1/βψ(λψ-1(z))-αz] and x(z) =g(γg-1(z))-αz] for the original iterative functional equation are obtained near poles and regular points