View source for Neutral differential equation

A differential equation with distributed argument (cf. [[Differential equations, ordinary, with distributed arguments|Differential equations, ordinary, with distributed arguments]]) in which the highest derivative occurs for more than one value of the argument, among them a basic (untransformed) one, and this latter value is the largest of those present in the equation. For example, the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066450/n0664501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

is a neutral differential equation when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066450/n0664502.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066450/n0664503.png" />.

For a neutral differential equation the initial value problem is solvable; thus, if for (*) with increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066450/n0664504.png" /> one gives

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066450/n0664505.png" /></td> </tr></table>

then for

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066450/n0664506.png" /></td> </tr></table>

there exists a (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066450/n0664507.png" /> unique) piecewise-smooth solution, which belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066450/n0664508.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066450/n0664509.png" /> compatibility conditions hold, that is, conditions of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066450/n06645010.png" /></td> </tr></table>

Neutral differential equations are one of the most thoroughly studied classes of equations with distributed arguments. They occur naturally in applied problems that contain in their statement some recurrence property.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.K. Hale,   "Theory of functional differential equations" , Springer  (1977)  pp. Chapt. 12</TD></TR></table>