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
SIMPLE HARMONIC MOTION A body is said to be vibrating if it moves back and forth or to and fro about a point. Another term for vibration is oscillation. A special kind of vibratory or oscillatory motion is called the simple harmonic motion A spider detects its prey due to vibration produced in the web. MOTION OF MASS ATTACHED TO A SPRING One of the simplest types of oscillatory motion is that of horizontal mass-spring system . If the spring is stretched or compressed through a small displacement x from its mean position, it exerts a force F on the mass. According to Hooke’s law this force is directly proportional to the change in length x of the spring. F = - k x where x is the displacement of the mass from its mean position O, and k is a constant called the spring constant defined as k = - F The value of k is a measure of the stiffness of the spring. Stiff springs have