(b) Let a > 0, and let X = C (|-a,a];IR) be the space of continuous real-valued functions defined on [-a, a] equipped with the supremum metric d, defined by dm(f,8) = sup If (x) – 8(x)|, for all f,8 € X. Using the Contraction Mapping Theorem, prove that the integral equation S(x) = 1 + ; 27 has a unique solution f in X. Justify any assertions that you make.
(b) Let a > 0, and let X = C (|-a,a];IR) be the space of continuous real-valued functions defined on [-a, a] equipped with the supremum metric d, defined by dm(f,8) = sup If (x) – 8(x)|, for all f,8 € X. Using the Contraction Mapping Theorem, prove that the integral equation S(x) = 1 + ; 27 has a unique solution f in X. Justify any assertions that you make.
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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![(b) Let a > 0, and let X = C(|-a,a];R) be the space of continuous real-valued functions
defined on [-a, a] equipped with the supremum metric d, defined by
d»(f,8) = sup \f (x) – 8(x)|,
x€l-a,a)
for all
f,8 € X.
Using the Contraction Mapping Theorem, prove that the integral equation
1
f6) = 1+ 2 권 f0) dh,
1+(x-t) -
x€ (-a,a)
has a unique solution f in X. Justify any assertions that you make.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00806677-d2bc-4666-a1ab-1cc08d4fb4c2%2Fe23ad827-a487-459b-b96b-ceb5c7d7222f%2Fjqprum8_processed.png&w=3840&q=75)
Transcribed Image Text:(b) Let a > 0, and let X = C(|-a,a];R) be the space of continuous real-valued functions
defined on [-a, a] equipped with the supremum metric d, defined by
d»(f,8) = sup \f (x) – 8(x)|,
x€l-a,a)
for all
f,8 € X.
Using the Contraction Mapping Theorem, prove that the integral equation
1
f6) = 1+ 2 권 f0) dh,
1+(x-t) -
x€ (-a,a)
has a unique solution f in X. Justify any assertions that you make.
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