The fundamental theory of existence, uniqueness and continuous differentiability of Lp-solutions for Neutral Functional Differential Equations is presented. Also, the spectrum of the solution operator of general autonomous linear NFDEs is described. Finally, an extension of Hartman Grobman Theorem on local conjugacy near a hyperbolic equilibrium is proved.

This work deals with the analysis pertaining some dynamic behavior of vector solutions of first order two-dimensional neutral delay differential systems of the form $$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}u\left(t\right)+pu(t-\tau )\\ v\left(t\right)+pv(t-\tau )\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}u(t-\alpha )\\ v(t-\beta )\end{array}\right].$$ The effort has been made to study $$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}x\left(t\right)-p\left(t\right)h_1\left(x(t-\tau )\right)\\ y\left(t\right)-p\left(t\right)h_2\left(y(t-\tau )\right)\end{array}\right]+\left[\begin{array}{cc}a\left(t\right)& b\left(t\right)\\ c\left(t\right)& d\left(t\right)\end{array}\right]\left[\begin{array}{c}{f}_1\left(x(t-\alpha )\right)\\ f_2\left(y(t-\beta )\right)\end{array}\right]=0,$$ where $p,a,b,c,d,h_1,h_2,f_1,f_2\in C(ℝ,ℝ)$; $\alpha ,\beta ,\tau \in {ℝ}^{+}$. We verify our results with the examples.