The existence of analytic solutions of an iterative functional differential equation is studied when the given functions are all analytic and when the given functions have regular points. By reducing the equation to another functional equation without iteration of the unknown function an existence theorem is established for analytic solutions of the original equation.

Functional differential equations with state-dependent delay have attracted the attentions of many authors in the last few years (see [

In this paper, analytic solutions of nonlinear iterative functional differential equations are investigated. Existence of locally analytic solutions and their construction is given in the case that all given functions exist regular points. As well as in previous work [

In this section we assume that both

If there exists a complex constant

The indeterminate constant

Take notations

A change of variable further transforms (

Suppose that (I1) holds, then (

Since

Without loss of generality, we can assume that

Consider a solution

If

The sequence

In what follows we need to prove that the series (

This implies that there exists a constant

Therefore, from (

In order to construct a majorant series of (

Define the function

Choosing

Moreover, it is easy to see from (

It follows that the power series

Furthermore, one has

Thus

We observe that

It is easy to show that

Let

Let

Let

there is a universal constant

Now we state and prove the following theorem under Brjuno condition.

Suppose that (I2) holds. Then (

As in the proof of Theorem

To construct a governing series of (

Note that the series (

Now, we can deduce, by induction, that

Note that

The next theorem is devoted to the case of (I3), where

Suppose that (I3) holds and

If

Analogously to the proof of Theorem

It is easy to check that there exists a constant

By inequality (

The following theorem shows that each analytic solution of (

Suppose that (H1) holds and that

Let

If

From (

Now we show the convergence of series (

Note that since

By (

Let

Then we have

This shows that

In the case (H2) we obtain similarly an analogue to Theorem

Suppose that (H2) holds and that

In the case (H3) we also obtain similarly an analogue to Theorem

Suppose that (H3) holds,

If

Having known analytic solutions of the auxiliary equation (

Suppose that the conditions of Theorems

In view of Theorems

This shows that

Under the hypothesis of Theorem

When

Under one of the conditions in Theorems

In Theorems

Therefore,

Consider the equation

It is in the form of (

Consider the equation

It is in the form of (

This work was supported by the Natural Science Foundation of Shandong Province (ZR2012AM017) and the Natural Science Foundation of Shandong Province (2011ZRA07006).