A.5 Well-Posed Problems and Mathematical Models

In addition to uniqueness, another consequence of the inequality in (45) is the fact that solutions of the differential equation

dxdt=f(x,t) (47)

depend continuously on the initial value x(a); that is, if x1(t) and x2(t) are two solutions of (47) on the interval atT such that the initial values x1(a) and x2(a) are sufficiently close to one another, then the values of x1(t) and x2(t) remain close to one another. In particular, if |x1(a)x2(a)|δ, then (45) implies that

|x1(t)x2(t)|δek(Ta)=ϵ (48)

for all t with atT. Obviously, we can make ϵ as small as we wish by choosing 1 sufficiently close to zero.

This continuity of solutions of (47) with respect to initial values is important in practical ...

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