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Banach Contraction Principle

Banach Contraction Principle Let (X,d) be a complete metric space, then each contraction map f : X → X has a unique fixed point. Let h be a contraction constant of the mapping f . We will explicitly construct a sequence converging to the fixed point. Let x0 be an arbitrary but fixed element in X. Define a sequence of iterates {xn} in X by xn = f(xn−1) (= f n(x0)), for all n ≥ 1. Since f is a contraction, we have d(xn, xn+1) = d(f(xn−1), f(xn)) ≤ hd(xn−1, xn), for any n ≥ 1. Thus, we obtain d(xn, xn+1) ≤ hnd(x0, x1), for all n ≥ 1. Hence, for any m > n, we have d(xn, xm) ≤  hn +hn+1 +···+hm−1 d(x0, x1) ≤ hn 1−h d(x0, x1). We deduce that {xn} is Cauchy sequence in a complete space X. Let xn → p ∈ X. Now using the  continuity of the map f , we get p = lim n→∞ xn = lim n→∞ f(xn−1) = f(p). Finally, to show f has at most one fixed point in X, let p and q be fixed points of f . Then, d(p,q) = d(f(p), f(q)) ≤ hd(p,q). Since h < 1, we must have p = q The proof of the BCP yields the foll
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SIMPLE HARMONIC MOTION

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THERMAL PROPERTIES

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GROUP

The concept of a binary operation on a nonempty set has already been explained in the previous. One may recall that a binary operation on a nonempty set A is just a function *:A x A → A so, for each (a, b) in A x A, * associates an element *(a, b) of A. We shall denote *(a, b) by a * b. If A is a nonempty set with a binary operation *, then A is said to be closed under *. DEFINAITIONS AND EXAMPLES A pair (G, *), where G is a nonempty set and * is a binary operation on G, is called a group if the following conditions called axioms of a group, are satisfied in G: (1) The binary operation * is associative. That is, (a * b) * c = a *(b*c) for all a, b, c, are belong to G (2) There is an element e in G such that a*e = e*a = a for all a belong to G e is called the identity element of G. (3) For each a belong to G, there is an a. Note: Use of word the before identity element and inverse of an e

HEAT AND THERMODYNAMICS

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SETS

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PROBABILITY

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CURVE FITTING BY LEAST SQARES

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TIME SERIES ANALYSIS

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