Question

2. बनाक का संकुचन सिद्धांत का कथन लिखकर सिद्ध कीजिये।
State and prove Banach contraction principle.​

Answer 1

Banach Contraction Principle

Step-by-step explanation:

If X is a complete metric space and f : X → X is a contraction mapping, then f has a unique fixed point p, and for any x in X, the sequence f n(x) converges to p. In fact,

ρ(f n(x),p) ≤ Kn/1 − K ρ(x,f(x)).

The importance of this latter inequality is as follows. Suppose we are willing to accept an error of ϵ, i.e., instead of the actual fixed point p of f we will be satisfied with a point p` of X satisfying ρ(p,p`) < ϵ, and suppose also that we start our iteration at some point x in X. Then from the inequality it is easy to specify an integer N so that p = f^N (x) will be a satisfactory answer. Since we want ρ(f^N (x),p) ≤ ϵ, we just have to pick N so large that K^N/1−K ρ(x,f(x)) < ϵ.

Now the quantity d = ρ(x,f(x)) is something that we can compute after the first iteration and we can then compute how large N has to be by taking the log of the above inequality and solving for N (remembering that log(K) is negative).

If d = ρ(x,f(x)) and

N > [log(ϵ) + log(1 − K) − log(d)]/ log(K)

then ρ(f^N (x),p) < ϵ.