The Computable Differential Equation Lecture ... - Bruce E. Shapiro

32 CHAPTER 2. SUCCESSIVE APPROXIMATIONS
g(a) = f(a) − a > 0 (2.20)
g(b) = f(b) − b < 0 (2.21)
Hence by the intermediate value theorem, g has a root r ∈ (a, b), where g(r) = 0. ⇐=
Then
f(r) = r (2.22)
i.e., r is a fixed point of f.
Review: Theorems about Functions
Theorem 2.2. Rolle’s Theorem. Suppose that f ∈ C[a, b] and f ∈ D(a, b). If
f(a) = f(b) then there exists some number c ∈ (a, b) such that f ′ (c) = 0.
Theorem 2.3. Mean Value Theorem. Suppose that f ∈ C[a, b] and f ∈ D(a, b).
If f(a) = f(b) then there exists some number c ∈ (a, b) such that
f ′ (c) =
f(b) − f(a)
b − a
(2.23)
Theorem 2.4. Extreme Value Theorem. If f ∈ C[a, b] then there exist numbers
c 1 , c 2 ∈ [a, b] such that f(c 1 ) are f(c 2 ) are the global minimum and maximum values
of f(x) on [a, b]. In other words, f has a maximum and minimum value in [a, b].
Furthermore, if f ∈ D(a, b) then each of c 1 and c 2 occur at either an endpoint or
[a, b] or where f ′ (t) = 0.
Theorem 2.5. Generalized Rolle’s Theorem. Suppose that f ∈ C n [a, b], and
that f(t) has n + 1 distinct zeroes
t 0 , t 1 , . . . , t n (2.24)
in [a, b]. Then there is some number c ∈ (a, b) such that f (n) (c) = 0.
Theorem 2.6. Intermediate Value Theorem. Suppose that f ∈ C[a, b] and that
K is some number between f(a) and f(b). Then there exists some number c ∈ [a, b]
such that f(c) = K. In other words, f takes on every value between the values at
the endpoints.
Theorem 2.7 (Condition for a unique fixed point). Suppose that
(a) f ∈ C[a, b] maps its domain into a subset of itself.
(b) There exists some K > 0, K < 1, such that
|f ′ (t)| ≤ K, ∀t ∈ [a, b] (2.25)
Then f(t) has a unique fixed point in [a, b].
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007