Theorem 3.18: The Contraction Lemma Let (X, d) be a complete metric space and f: X→ X a contractive function. Then there is exactly one point x in X for which f(x) = x. x is called a fixed point of f Cany). Proof: To show the existence of such a point, choose a point x; in X and define x2 = f(x), x; = f(x2), ..., X, = f(x-1),. n2 2. The fact that f is a contractive function insures that {xn}%-1 is a Cauchy sequence. By completeness, this sequence has a limit x in X. v - Use Thus f(x) = x. To show the required uniqueness property, suppose that y is a second point satisfying f(y) = y. Then d(x, y) = d(f(x), f(y)) s ad(x, y). Since a < 1, this relation cannot hold unless d(x, y) = 0 and x = y.

Theorem 3.18: The Contraction Lemma Let (X, d) be a complete metric space and f: X→ X a contractive function. Then there is exactly one point x in X for which f(x) = x. x is called a fixed point of f Cany). Proof: To show the existence of such a point, choose a point x; in X and define x2 = f(x), x; = f(x2), ..., X, = f(x-1),. n2 2. The fact that f is a contractive function insures that {xn}%-1 is a Cauchy sequence. By completeness, this sequence has a limit x in X. v - Use Thus f(x) = x. To show the required uniqueness property, suppose that y is a second point satisfying f(y) = y. Then d(x, y) = d(f(x), f(y)) s ad(x, y). Since a < 1, this relation cannot hold unless d(x, y) = 0 and x = y.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as:

Question

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Banach F.P. Th
Theorem 3.18: The Contraction Lemma Let (X, d) be a complete metric
space and f: X → X a contractive function. Then there is exactly one point x in X
for which f(x) = x.
x is called a
fixed point of f
(any).
Proof: To show the existence of such a point, choose a point x, in X and define
x2 = f(x1), x3 = f(x2), ..., Xn = f(xp-1),. n2 2.
The fact that f is a contractive function insures that {xn} is a Cauchy sequence.
By completeness, this sequence has a limit x in X. A V
w- wnOSE m Thus f(x) = x.
To show the required uniqueness property, suppose that y is a second point
satisfying f(y) = y. Then
d(x, y) = d(f(x), f(y)) s ad(x, y).
Since a < 1, this relation cannot hold unless d(x, y) = 0 and x = y.
Show dian, tnt ) * d(x,%2)
use seguence definition
Exoma
HWl
Showw d(xn, Xntk
t..
Xuti
L Use O to bound the right-side of O
(&
& shows Hhat { Zu } is Cauchy
(we a<!}
(2)
ntk)
Use
M=
use

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