2.4 CONTINUOUS DEPENDENCE ON A PARAMETER OR THE INITIAL CONDITION

It should be evident that if in place of the ODE (2.2.1) with the IC (2.2.2), we consider the problem

where ? is a parameter, the existence and uniqueness of the solution is again guaranteed as long as f is Lipschitz continuous in the variable y. We now show that besides these properties ensured for fixed values of ? and y ₀, the solution possesses the additional feature of being continuously dependent on the parameter ? and the initial value y ₀.

For this, consider the two problems

where f satisfies a Lipschitz condition in y:

Then eq. (2.2.4) applied to eqs. (2.4.1) and (2.4.2) leads to

and

Subtracting gives

We again need to consider separately the regions x > x ₀ and x < x ₀. For x > x ₀, we obtain

Let us assume that the function f depends continuously on the parameter. Thus

where ? is small if ? ? ? is small. With this and eq. (2.4.3), we have

Let us denote

Then

and (2.4.6) becomes the differential inequality

with the IC

by definition of E, and p( x) > 0.

Following the same approach employed in section 2.2, while solving for ?( x) defined by eq. (2.2.18), we obtain

Differentiating both sides with respect to x gives

Since p