A Source Book in Classical Analysis

Date: 1876


Lipschitz on the Lipschitz Condition


The most important advances in the theory of systems of differential equations since the work of Jacobi have stemmed from the development of complex analysis. For each system of differential equations, the first essential question is . . . whether one can determine a system of functions of the independent variable which satisfy the equations and which assume prescribed values for a given value of that variable. This question has been studied when the system of differential equations permits one to regard the variable and [its values] as [complex] quantities . . . . Since every function of a complex variable can be expanded in a power series [in

], except at certain [singular] points of its domain, the question posed evidently depends on this: can one find series, convergent in some domain of the independent variable, which, when substituted for the unknown functions, satisfy the differential equations? . . .

If, on the contrary, the expressions which enter into the system of differential equations are . . . real and do not permit an immediate extension to complex values, one may no longer assume that the unknown functions can be expanded in [such power] series . . . . One must use different methods to establish conditions for complete integrability. No work having precisely this aim has appeared to my knowledge; it is the object of the investigations which follow.2

Suppose that the given system of differential equations, where x is the independent variable and

arc unknown functions, can be put into the following form:

where j takes on the values

The functions fj are [assumed] given on a continuously connected set of values of the variables
this set of values will be called the domain ["champ"] G. For all values . . . in this domain, the n functions fj are assumed to be single-valued, continuous, and uniformly bounded. Besides, they must be such that, given two points ["systèmes de valeurs"]
where x remains the same, the inequality3

is satisfied: the quantities

are positive constants, and, here and below, the symbol
represents the absolute value of w. The continuity condition imposed requires that, for two systems of values
the difference
can be made arbitrarily small when the difference
tends to zero: hence if one takes into account the in- equality (2), one sees that this condition [implies] that one can take the difference
so small that the inequality

is satisfied, for any preassigned

The system (1) will be completely integrated if one determines a system of functions

satisfying equations (1), and which for
satisfy the equations4

Let the point

lie in the interior of the domain G, at a finite distance from its boundary. Then one can determine positive quantities A, Bj, such that all points satisfying the inequalities
lie in the interior of G; hence, there exist positive finite constants Cj such that, for these same values, one always has

If one chooses the positive quantity M so that

the domain determined by the inequalities

will be entirely in the interior of G; we shall call it H.

Under these conditions, there always exists a unique system of n functions

satisfying the differential equations (1), varying continuously in the interior of the domain H0 as x ranges from
and satisfying the equations

To prove this theorem, it suffices to consider x in the interval

since the interval
can be treated similarly. Hence let us imagine a sequence of intermediate values
between x0 and
such that

and let us determine n quantities

by the following n equations:

These equations would coincide with the given system of differential equations (1) if one there replaced dx and dyj in the left side by

and x, yj in the right side by x0,
By virtue of the inequalities (4) and (4α), equations (5) imply the inequalities

hence the point

lies in the domain H. One can form similarly a sequence of points
successively in the equation

All these points will certainly be in the domain H. We subdivide the interval further by intercalculating between xr and

increasing numbers

From this new partition of the interval

one obtains a sequence of points, beginning with
all of which will be in the domain H0: this new sequence of points
is obtained if in the equation

r is replaced by

μr by the numbers
and one finally takes
If we regard the first quantities
as fixed, and let the integers q increase and the intervals
decrease indefinitely according to some law, the problem now is to establish that the values
which correspond to . . .
converge to a fixed limit . . . . This proof shows that, under the hypotheses stated above, it is always possible to choose
so that for arbitrary r and σ,

If in equation (6) one successively sets

and adds, one obtains the equation

which, by (4), gives

This inequality expresses the fact that if r is held fixed and the subscript μr is allowed to range from 0 to

the point
remains in a domain K0 whose [diameter] can be made arbitrarily small by taking the differences
sufficiently small. Since, by hypothesis, the function fj is continuous in the domain D, the difference
can always be taken small enough so that the values of fj are less than any fixed positive constant ε, no matter how small. Supposing that the difference
has been determined in this way, the number p is very large, and ε is some proper fraction, equation (7) will give for

Subtracting equation (5j) from this equation, it becomes

but, because of (2), one has

Hence, if one sets

equations (9) give the sequence of inequalities

One also has

Now it is clear that, if one forms a sequence of quantities

by means of the equations

where the subscript r ranges from 0 to

one will always have

Now for our proof we must show that if we take λ sufficiently small, the quantities

stay arbitrarily small: our goal will be attained if we prove the same thing for the quantities
or for larger quantities.

Let c be a positive quantity greater than the largest of the n2 constants

if one defines the quantities
by the equations

one will evidently have, for

Now the first equation (12j) gives

and, by virtue of the second, one has

Hence equation (3) can be written


and so one has

Now the product

where nc is positive, itself has a positive value smaller than

hence one has

Comparison of this inequality with the inequalities (13) and (13j) gives

and, since the factor

is bounded, one concludes from this that the difference
can be made arbitrarily small; for λ is an arbitrarily small quantity depending only on the choice of the intervals
As the differences
can be taken arbitrarily small, the quantities
which correspond to the fixed value
of the variable, converge to a determined limit, independent of the law of increase of the number qr and of the law of decrease of the new intervals. By equations (8), these limiting values define a system of solutions of the differential equations, a system for which the functions yj equal
The existence of a system of solutions satisfying the given initial conditions is therefore established, and the first part of our program has been completed.

[Lipschitz concludes his paper with a rather sketchy and confused proof of uniqueness: "there exists no other solution of the system (1) satisfying the stated conditions."]

1 Lipschitz, "Sur la possibilité d’intégrer complètement un système . . . ," Bull. Sci. Math. 10 (1876), 149–159.

2 The author evidently does not know the works of Cauchy, which have been summarized in an incomplete manner by M. the Abbé Moigno, in his Traité de calcul intégral, and those of Coriolis, in the Journal de Liouville. [Note by the editor of the Bulletin des Sciences Mathématiques.]

3 The inequality (2) is a form of the now classic "Lipschitz condition."

4 Lipschitz used superscripts where we have used subscripts,

where we have used ηi, and
where we have used


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Chicago: "Lipschitz on the Lipschitz Condition," A Source Book in Classical Analysis in A Source Book in Classical Analysis, ed. Garrett Birkhoff (Cambridge: Harvard University Press, 1973), 246–250. Original Sources, accessed July 20, 2024, http://www.originalsources.com/Document.aspx?DocID=DBQZGF7946EAWVU.

MLA: . "Lipschitz on the Lipschitz Condition." A Source Book in Classical Analysis, Vol. 10, in A Source Book in Classical Analysis, edited by Garrett Birkhoff, Cambridge, Harvard University Press, 1973, pp. 246–250. Original Sources. 20 Jul. 2024. http://www.originalsources.com/Document.aspx?DocID=DBQZGF7946EAWVU.

Harvard: , 'Lipschitz on the Lipschitz Condition' in A Source Book in Classical Analysis. cited in 1973, A Source Book in Classical Analysis, ed. , Harvard University Press, Cambridge, pp.246–250. Original Sources, retrieved 20 July 2024, from http://www.originalsources.com/Document.aspx?DocID=DBQZGF7946EAWVU.