View source for Differential equations, ordinary, with distributed arguments

''ordinary differential equations with deviated arguments, ordinary differential equations with deviating arguments''

Ordinary differential equations connecting the argument, the unknown function and its derivatives generally taken for different values of the argument. Examples are:

<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/d/d032/d032100/d0321001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<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/d/d032/d032100/d0321002.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d0321003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d0321004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d0321005.png" /> are given; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d0321006.png" /> in equation (1) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d0321007.png" /> in equation (2) are the deviations, retardations or lags of the arguments. There are also more complicated differential equations with a large number of deviations of the argument, which may represent given functions (in particular, if they are constant, the equation is often referred to as a differential-difference equation) or may even depend on the solution sought. There were also sporadic studies of differential equations with deviated arguments in which the unknown function depends on more than one argument. Differential equations with deviated arguments first appeared in connection with the formal solution of partial differential equations, and were subsequently studied for their own sake, in the context of geometrical problems, and later in various applications, mainly in [[Automatic control theory|automatic control theory]]. A systematic construction of the theory of ordinary differential equations with deviated arguments was begun in 1949.

The definition of differential equations with deviated arguments permits any superpositions of the sought solution (of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d0321008.png" />) and its integrals; formally the class of ordinary differential equations with deviated arguments includes all equations of mathematical analysis. Nevertheless, ordinary differential equations with deviated arguments are usually understood to mean some natural class of ordinary differential equations in which a deviation of the argument, permitting the construction of a meaningful theory, has been introduced. Several properties of such equations are directly analogous to those of ordinary differential equations, while other properties are principally new.

An equation (or a system of equations)

<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/d/d032/d032100/d0321009.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

(for a system, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210011.png" /> are vectors), where all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210012.png" />, is said to be an equation (system) of retarded (lag), neutral, or leading type if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210015.png" />, respectively. Equations of other forms are classified in this way on the base of a transformation into the form (3) by the substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210017.png" /> is an increasing function; for instance, equation (1) is of retarded type if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210018.png" /> and is of leading type (by the substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210019.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210020.png" />. If the deviations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210021.png" /> depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210022.png" />, equation (3) may change type; then equation (2) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210023.png" /> is of retarded type if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210024.png" /> and is of leading type if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210026.png" /> depends on the solution sought, equation (3) may be of different types for different solutions. The theory of differential equations with deviated arguments of retarded type is developed in most detail, that of neutral type is developed to a lesser extent, while that of leading type has not been studied to any significant extent.

The following is one of the simplest classes of differential equations with deviated arguments:

<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/d/d032/d032100/d03210027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

The following fundamental initial value problem is formulated for this class of equations: Given an initial point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210028.png" />, an initial function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210030.png" />, and a value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210031.png" />; the solution of this problem for equation (4) is understood to be a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210033.png" />, such that equation (4) holds identically, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210036.png" /> is to be substituted for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210037.png" /> in the right-hand side of the equation. The problem may be solved by the method of steps: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210038.png" />, one solves the initial value problem for equation (4) in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210039.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210040.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210041.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210042.png" />, i.e. one has already constructed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210043.png" />, etc. Thus, each step consists in solving a [[Cauchy problem|Cauchy problem]] for an equation without a deviating argument. If the two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210045.png" /> are continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210046.png" />, the solution of the problem exists on some interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210047.png" /> and may be extended in the usual way, and if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210048.png" /> satisfies a Lipschitz condition with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210049.png" />, this solution is unique and depends continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210052.png" />. If, in addition, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210053.png" /> is sufficiently smooth, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210054.png" /> is continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210056.png" /> is continuous if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210057.png" />, etc. (the smoothing property).

The initial value problem is formulated and the solution is constructed for systems of equations of the form (4) and for equations of higher orders in a completely similar manner. If several lags are present, the smallest one is taken as the step. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210059.png" /> should be defined on an entire initial set of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210061.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210062.png" /> has zeros the method of steps cannot be employed, but it is possible to prove the solvability of the initial value problem, similar to the one stated above, if simple approximation or iteration methods are employed. The numerical methods of solution are the same, in principle, as for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210063.png" />. If the given functions are discontinuous or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210064.png" />, the concept of a solution can be naturally extended.

The solution of the formulated initial value problem is constructed in the increasing direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210065.png" /> only. Another feature of it is that the manifold of solutions for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210066.png" /> is usually infinite-dimensional. (Equations without pre-history, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210067.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210068.png" />, such as equation (2) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210070.png" />, are an exception to this rule.) This is a significant difference between the theory of differential equations with deviated arguments and the theory of differential equations without deviated arguments.

In equation (4) the lag is concentrated. There are also equations with deviated lags, the right-hand side of which contains the integrals

<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/d/d032/d032100/d03210071.png" /></td> </tr></table>

(these are Volterra-type integro-differential equations) or, in the combined case,

<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/d/d032/d032100/d03210072.png" /></td> </tr></table>

etc. The most common type of first-order differential equation with deviated arguments of retarded type is the differential-functional Volterra-type 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/d/d032/d032100/d03210073.png" /></td> </tr></table>

where the right-hand side is a functional for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210074.png" />. The initial value problem is solvable for such equations as well.

For differential equations with deviated arguments of neutral type, for example

<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/d/d032/d032100/d03210075.png" /></td> </tr></table>

the formulation and the properties of the initial value problems resemble those given above, but there is no smoothing property; complexities may occur if the variable lag <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210076.png" /> has zeros. The initial value problem is ill-posed for differential equations with deviated arguments of leading type.

Linear autonomous differential equations with deviated arguments (i.e. equations in which the coefficients and the deviations of the argument are constant) have been studied in most detail. The equation (without constant term)

<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/d/d032/d032100/d03210077.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

(all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210078.png" />) has particular solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210079.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210080.png" /> satisfies the [[Characteristic equation|characteristic 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/d/d032/d032100/d03210081.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210082.png" /> is a quasi-polynomial; the solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210083.png" /> of equation (5) correspond to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210084.png" />-fold root of equation (6). If even a single <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210085.png" />, equation (6) has an infinite number of roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210086.png" />. A necessary and sufficient condition for equation (5) to be of retarded (neutral) type is

<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/d/d032/d032100/d03210087.png" /></td> </tr></table>

In such a case each solution of equation (5) has a series expansion in the particular solutions given above, and methods of ordinary operational calculus may be employed in solving the initial value problem for equation (5) and the corresponding inhomogeneous equation. Systems of equations and equations with deviated lags display similar properties.

The usual conditions for stability of a solution are directly extended to differential equations with deviated arguments of retarded and neutral types. The condition

<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/d/d032/d032100/d03210088.png" /></td> </tr></table>

is necessary and sufficient for the asymptotic stability of solutions of equation (5). If this condition is met, the zero solution of non-linear autonomous equations, for which equation (5) is a linear approximation, is also asymptotically stable.

The method of Lyapunov functions for the study of the stability of differential equations with deviated arguments was extended by N.N. Krasovskii [[#References|[5]]]. He suggested the use of the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210089.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210090.png" />; the total derivative of such a functional  "along the solution"  of a given equation of the type (4) may be computed by virtue of this equation, and it represents a functional of the same type, i.e.

<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/d/d032/d032100/d03210091.png" /></td> </tr></table>

The analogue of Lyapunov's stability theorem is as follows: If

<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/d/d032/d032100/d03210092.png" /></td> </tr></table>

<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/d/d032/d032100/d03210093.png" /></td> </tr></table>

<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/d/d032/d032100/d03210094.png" /></td> </tr></table>

then the trivial solution of equation (4) is stable. The main stability theorems, expressed in such terms, have a converse.

Several results have been obtained for periodic differential equations with deviated arguments. Thus, if

<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/d/d032/d032100/d03210095.png" /></td> </tr></table>

the fixed points of the operator

<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/d/d032/d032100/d03210096.png" /></td> </tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210097.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210098.png" /> is the solution of equation (4) for the initial function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d03210099.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d032100100.png" /> is a natural number) determine the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d032100101.png" />-periodic solutions of equation (4). Owing to this and to a number of similar facts, it is possible to apply methods of the theory of non-linear operators to find periodic solutions and to clarify their stability. For the case of linear homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d032100102.png" />-periodic differential equations with deviated arguments of retarded type, it proved possible to approximate each solution to a desired degree of accuracy in the scale of exponents by linear combinations of solutions of the form

<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/d/d032/d032100/d032100103.png" /></td> </tr></table>

Other trends in the study of differential equations with deviated arguments include: a detailed study of asymptotic and oscillation properties of equations

<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/d/d032/d032100/d032100104.png" /></td> </tr></table>

and analogous equations of the second order; obtaining asymptotic expressions for the solutions of systems with small deviations of the argument or with a small non-linearity; the extension of the asymptotic Krylov–Bogolyubov methods to differential equations with deviated arguments (cf. [[Krylov–Bogolyubov method of averaging|Krylov–Bogolyubov method of averaging]]); a similar extension of L.S. Pontryagin's method of optimal control (cf. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]); the study of boundary value problems; the study of partial differential evolution equations with time lag (cf. [[Evolution equation|Evolution equation]]); the study of stochastic differential equations with deviated arguments (cf. [[Stochastic differential equation|Stochastic differential equation]]), etc.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.E. El'sgol'ts,   S.B Norkin,   "Introduction to the theory and application of differential equations with deviating arguments" , Acad. Press  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Pinney,   "Ordinary difference-differential equations" , Univ. California Press  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Bellman,   C.L. Cooke,   "Differential-difference equations" , Acad. Press  (1963)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.D. Myskis,   "Linear differential equations with retarded argument" , Moscow  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.N. Krasovskii,   "Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay" , Stanford Univ. Press  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.B. Norkin,   "Differential equations of the second-order with retarded argument" , Amer. Math. Soc.  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.P. Rubanik,   "Oscillations of quasi-linear systems with retardment" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  M.N. Oĝuztöreli,   "Time-lag control systems" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Halanay,   "Differential equations: stability, oscillations, time-lags" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J.K. Hale,   "Functional differential equations" , Springer  (1971)</TD></TR></table>



====Comments====
Another, rather striking, difference between differential equations (without constant term) with delays and ordinary differential equations is that the former may have non-zero solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d032100105.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d032100106.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d032100107.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d032100108.png" />. This can happen, for instance, for systems of delay equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d032100109.png" /> when the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032100/d032100110.png" /> is singular and non-zero.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.-N. Chow,   J.K. Hale,   "Methods of bifurcation theory" , Springer  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.-O. Peitgen (ed.)  H.-O. Walther (ed.) , ''Functional differential equations and approximation of fixed points'' , ''Lect. notes in math.'' , '''730''' , Springer  (1979)</TD></TR></table>