1550078481-Ordinary_Differential_Equations__Roberts_

Systems of First-Order Differential Equations 319

The following theorem for the existence and uniqueness of a solution to a
system initial value problem is analo gous to the fundamental existence and
uniqueness theorem for the scalar initial value problem: y' = f(x, y); y(c) = d.
In this theorem, R denotes a generali zed rectangle in XY1Y2 · · · Yn-space.

FUNDAMENTAL EXISTENCE AND UNIQUENESS THEOREM

FOR SYSTEM INITIAL VALUE PROBLEMS

Let R = {(x, Y1, Y2, ... ) Yn) I a < x < f3 and Ii < Yi < oi} where

a, {3, ri and oi are all finite real constants. If each of the n functions

fi(x, yi, y2, ... , Yn), i = 1, 2, ... , n is a continuous function of x, y 1 , y2,

... , and Yn in R, if each of the n^2 partial derivatives 8 Ji/ 8y 1 , i, j =

1, 2, ... , n is a continuous function of x, Y1, y2, ... , and Yn in R, and if

(c, d 1 , d2, ... , dn) E R, then there exists a unique solution to the system

initial value problem

(lla)

(llb)

Y~ =Ji (x, Y1, Y2, · · ·, Yn)

Y~ = f2(x,y1,Y2,. · .,yn)

on some interval I= (c - h, c + h) where I is a subinterval of (a, {3).

The hypotheses of the fundamental existence and uniqueness theorem are
sufficient conditions and guarantee the existence of a unique solution to the
initial value problem on some interval of length 2h. An expression for cal-
culating a value for h is not specified by the theorem and so the interval of
existence and uniqueness of the solution may be very small or very large.
Furthermore, the conditions stated in the hypotheses are not necessary con-
ditions. Therefore, if some condition stated in the hypotheses is not fulfilled
(perhaps 8 Ji/ 8y 1 is not continuous for a single value of i and j and in only
one particular variable), we cannot conclude a solution to the initial value
problem does not exist or is not unique. It might exist and be unique. It
might exist and not be unique. Or, it might not exist. We essentially have
no information with respect to solving the initial value problem, if the hy-
potheses of the fundamental theorem are not satisfied and little information
concerning the interval of existence and uniqueness, if the hypotheses are sat-
isfied. Finally, observe that the hypotheses of the theorem do not include any
requirements on the functions 8fi/8x, ah/ax, ... , 8fn/8x.