Analytic solutions of differential-functional equations

Abstract

We consider the general differential-functional equations

$$\frac{\begin{gathered} \_ \hfill \\ dy(z) \hfill \\ \end{gathered} }{{dz}} = \overline F _{[ - \Delta _{1,} \Delta _z ]} (\overline y (z + \xi )),$$

$\xi \in R|,\Delta _1 ,\Delta _2 > 0, z \in \mathbb{C}, \bar y(z):\mathbb{C} \to \mathbb{C}^n , \bar F_{[ - \Delta _1 ,\Delta _2 ]} (\bar \varphi (\xi ))$ is a vector-valued functional taking$\overline \varphi (\xi ) \in C [ - \Delta _1 , \Delta _2 ]$ into ℂⁿ. We obtain existence theorems and theorems giving the number of analytic solutions represented by Dirichlet series. A study is made of their properties.