Mathematics
Lipschitz on the Lipschitz Condition
^{1}
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
f_{j} 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
f_{j} are assumed to be single-valued, continuous, and uniformly bounded. Besides, they must be such that, given two points ["systèmes de valeurs"]
and
where
x remains the same, the inequality
^{3}
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
and
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 equations
^{4}
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,
B_{j}, such that all points satisfying the inequalities
lie in the interior of
G; hence, there exist positive finite constants
C_{j} 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
H_{0} as
x ranges from
to
and satisfying the equations
at
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
x_{0} 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 dy_{j} in the left side by
and
x, y_{j} in the right side by
x_{0},
By virtue of the inequalities (4) and (4
^{α}), equations (5) imply the inequalities
hence the point
lies in the domain
H. One can form similarly a sequence of points
setting
successively in the equation
All these points will certainly be in the domain H. We subdivide the interval further by intercalculating between x_{r} and
[any]
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
H_{0}: this new sequence of points
is obtained if in the equation
r is replaced by
μ_{r} 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 μ_{r} is allowed to range from 0 to
the point
remains in a domain
K_{0} whose [diameter] can be made arbitrarily small by taking the differences
sufficiently small. Since, by hypothesis, the function
f_{j} is continuous in the domain
D, the difference
can always be taken small enough so that the values of
f_{j} 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 (5^{j}) 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 n^{2} constants
if one defines the quantities
by the equations
and
one will evidently have, for
Now the first equation (12^{j}) gives
and, by virtue of the second, one has
Hence equation (3) can be written
or
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 (13^{j}) 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
q_{r} 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
y^{j} equal
when
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
η_{i}, and
where we have used