The Existence Of Local Analytic Solutions Of An Extended Jabotinsky Functional Differential Equation

This paper we study the existence of local analytic solutions of an extended Jabotinsky functional differential equation G(z)·F'(z)=G[F(z)]+H[z,F(z),…,Fm.(z)]. We obtain results of analytic solutious in the case of Brjuno condition.By Davie's Lemma we study the power series solutions of auxiliary equation under different conditions,we obtain an analytic solutions of the original equation.The structure of the thesis as follows:In Chapter 1,we concepts and developpings of small divisor,iterative func-tional equation and the third Jabotinsky equation.In Chapter 2,we study the existence of analytic solutions of G(z)·F'(z)=G[F(z)]+H[z,F(z),…,Fm.(z)].We will distinguish three different cases onα:(C1) 0<|α|<l;(C2)α=e2πi.θ∈R\Q andθis a B rjuno number([3],[4]):B(θ)= (?)<∞,where {Pn/qn} denotes the sequence of partion of the continued fraction expansion ofθ: (C3)α=e2πiq/p for some integer p∈N with p≥2 and q∈Z\{0},andα≠e2πiξ/v for all≤v≤p-1 andξ∈Z\{0}.In Chapter 3.we state the exstence of local analytic solutions of the initial Jahotinsky differential equation G(z)·F'(z)=G[F(z)]+H[z,F(z),…,Fm.(z)].