A metric space is a set M together with a distance function p(r, y) that represents the "distance" between elements x and y of the set A. The distance function must satisfy 0 and p(z,y) 0 if and only if x y; (i) ρ(z,y) (iii) y) a(z,z) + (s,y) for all z, y,s in M. Let (M.p) be a metric space. A mapping T from M into M is called a contraction if for some constant o with o sac1, and all r and y in M Let C(0, 1] denote the set of continuous functions f 0,1]R with metric p(f.g) max If (t) - g(t tejo.l Using f.g in Cl0. 1] given by fe) -1 and gt) 0 show that the mapping Clo. 1] → C[0, 1] defined by (Ta)(t) 1+r(s) ds is not a contraction mapping. (a) Showy that the mapping T : С 0.1/2 -C 0 1/21 defined by (1) is a contraction (b) Prom the contraction mapping theorem Chapter 3, Theorem 3.1], T has a fixed 4. mapping point x for which Tz # r. Find the fixed point function r(t)
A metric space is a set M together with a distance function p(r, y) that represents the "distance" between elements x and y of the set A. The distance function must satisfy 0 and p(z,y) 0 if and only if x y; (i) ρ(z,y) (iii) y) a(z,z) + (s,y) for all z, y,s in M. Let (M.p) be a metric space. A mapping T from M into M is called a contraction if for some constant o with o sac1, and all r and y in M Let C(0, 1] denote the set of continuous functions f 0,1]R with metric p(f.g) max If (t) - g(t tejo.l Using f.g in Cl0. 1] given by fe) -1 and gt) 0 show that the mapping Clo. 1] → C[0, 1] defined by (Ta)(t) 1+r(s) ds is not a contraction mapping. (a) Showy that the mapping T : С 0.1/2 -C 0 1/21 defined by (1) is a contraction (b) Prom the contraction mapping theorem Chapter 3, Theorem 3.1], T has a fixed 4. mapping point x for which Tz # r. Find the fixed point function r(t)
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![A metric space is a set M together with a distance function p(r, y) that represents the
"distance" between elements x and y of the set A. The distance function must satisfy
0 and p(z,y)
0 if and only if x
y;
(i) ρ(z,y)
(iii) y) a(z,z) + (s,y) for all z, y,s in M.
Let (M.p) be a metric space. A mapping T from M into M is called a contraction if
for some constant o with o sac1, and all r and y in M
Let C(0, 1] denote the set of continuous functions f 0,1]R with metric
p(f.g) max If (t) - g(t
tejo.l
Using f.g in Cl0. 1] given by fe) -1 and gt) 0 show that the mapping
Clo. 1] → C[0, 1] defined by
(Ta)(t) 1+r(s) ds
is not a contraction mapping.
(a) Showy that the mapping T : С 0.1/2 -C 0 1/21 defined by (1) is a contraction
(b) Prom the contraction mapping theorem Chapter 3, Theorem 3.1], T has a fixed
4.
mapping
point x for which Tz # r. Find the fixed point function r(t)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4aa5341f-3bcd-401e-8b0d-7704b9391f07%2F8be340a6-0c8c-4f13-bbda-2e1f700d1171%2Fhh04ha8.jpeg&w=3840&q=75)
Transcribed Image Text:A metric space is a set M together with a distance function p(r, y) that represents the
"distance" between elements x and y of the set A. The distance function must satisfy
0 and p(z,y)
0 if and only if x
y;
(i) ρ(z,y)
(iii) y) a(z,z) + (s,y) for all z, y,s in M.
Let (M.p) be a metric space. A mapping T from M into M is called a contraction if
for some constant o with o sac1, and all r and y in M
Let C(0, 1] denote the set of continuous functions f 0,1]R with metric
p(f.g) max If (t) - g(t
tejo.l
Using f.g in Cl0. 1] given by fe) -1 and gt) 0 show that the mapping
Clo. 1] → C[0, 1] defined by
(Ta)(t) 1+r(s) ds
is not a contraction mapping.
(a) Showy that the mapping T : С 0.1/2 -C 0 1/21 defined by (1) is a contraction
(b) Prom the contraction mapping theorem Chapter 3, Theorem 3.1], T has a fixed
4.
mapping
point x for which Tz # r. Find the fixed point function r(t)
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