Analytic Solutions of a Nonlinear Functional Differential Equation With Proportional Delays

Abstract

Abstract. This paper is concerned with a nonlinear functional differential equation of the form m ¿=i By means of the method of majorant series conditions are given for existence of analytic solutions of the above equation.It is well-known that the third Jabotinsky equationplays an important role in iteration theory. This equation has been studied by many authors [1][2][3]. In this paper, we will be concerned with a more general class of equation of the formwhere pj and qj are complex numbers. It is clear that (2) includes equation (1) as a special case. In the linear case, such equations with proportional delays also have been studied to some extent by many authors [4][5][6]. By means of the method of majorant series, we will discuss the existence of analytic solutions of equation (2) in a neighborhood of the origin. Some ideas in the proof of our main results are similar to that used in [3,7,8].We now state and prove our main results in this note.1991 Mathematics Subject Classification: 34K25.