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Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) *Give appropriate graphs and required codes. * Make use of inequalities if you think that required. *You are supposed to use kreszig for reference. Holder inequality: Ins j=1 (Eur)' (En)" where p > 1 and 1 1 + P Σε Cauchy-Schwarz inequality: Σ&P j=1 Minkowski inequality: + where p > 1. q 1. + ΣΙ m=1 Problem 2: Metric Spaces and Fixed Point Theorems Problem Statement: Let (M,d) be a complete metric space, and let f: MM be a contraction mapping, i.e., there exists a constant 0

Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) *Give appropriate graphs and required codes. * Make use of inequalities if you think that required. *You are supposed to use kreszig for reference. Holder inequality: Ins j=1 (Eur)' (En)" where p > 1 and 1 1 + P Σε Cauchy-Schwarz inequality: Σ&P j=1 Minkowski inequality: + where p > 1. q 1. + ΣΙ m=1 Problem 2: Metric Spaces and Fixed Point Theorems Problem Statement: Let (M,d) be a complete metric space, and let f: MM be a contraction mapping, i.e., there exists a constant 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 13E
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Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) *Give appropriate graphs and required codes. * Make use of inequalities if you think that required. *You are supposed to use kreszig for reference. Holder inequality: Ins j=1 (Eur)' (En)" where p > 1 and 1 1 + P Σε Cauchy-Schwarz inequality: Σ&P j=1 Minkowski inequality: + where p > 1. q 1. + ΣΙ m=1 Problem 2: Metric Spaces and Fixed Point Theorems Problem Statement: Let (M,d) be a complete metric space, and let f: MM be a contraction mapping, i.e., there exists a constant 0<c< 1 such that Tasks: d(f(x), f(y)) c-d(x,y) Vr,yЄM. a) Banach Fixed Point Theorem: State the Banach Fixed Point Theorem and use it to prove that f has a unique fixed point in M. b) Convergence Rate: Show that for any initial point zo Є M, the iterative sequence {n} defined by n+1 = f(x) converges to the fixed point. Estimate the rate of convergence in terms of c. c) Visualization: Consider MR with the standard metric and f(x)=2. Plot the function f along with the line y = and illustrate the convergence of iterations +1 = f(x) to the fixed point.
Transcribed Image Text:Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) *Give appropriate graphs and required codes. * Make use of inequalities if you think that required. *You are supposed to use kreszig for reference. Holder inequality: Ins j=1 (Eur)' (En)" where p > 1 and 1 1 + P Σε Cauchy-Schwarz inequality: Σ&P j=1 Minkowski inequality: + where p > 1. q 1. + ΣΙ m=1 Problem 2: Metric Spaces and Fixed Point Theorems Problem Statement: Let (M,d) be a complete metric space, and let f: MM be a contraction mapping, i.e., there exists a constant 0<c< 1 such that Tasks: d(f(x), f(y)) c-d(x,y) Vr,yЄM. a) Banach Fixed Point Theorem: State the Banach Fixed Point Theorem and use it to prove that f has a unique fixed point in M. b) Convergence Rate: Show that for any initial point zo Є M, the iterative sequence {n} defined by n+1 = f(x) converges to the fixed point. Estimate the rate of convergence in terms of c. c) Visualization: Consider MR with the standard metric and f(x)=2. Plot the function f along with the line y = and illustrate the convergence of iterations +1 = f(x) to the fixed point.
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