Define $T:\mathbb R^3→\mathbb R^3$, $(x,y,z)\mapsto\left(\dfrac12\cos y +1,\dfrac23\sin z,\dfrac34x\right)$. I have checked that this example is a contraction and now I am trying to apply Banach contraction principle on it, but whatever the initial guess I am taking to make an iterative sequence for it, I am not getting a sequence i.e. I am unable to generalize its sequence.
Kindly help me with this. Thanks in advance.
So if you know it's a contraction, there is a unique fixed point for it.
The theorem gives you the iterative method, which means you have to program a computer (or good calculator) to approximate the fixed point. You know that it will converge.
The theorem does not give you any clues about finding exact answers. Do some programming instead. You cannot work exactly (i.e. in formulas) but use floating point numbers in a computer.
I get
(1.4175200046951226, 0.5825908262245517, 1.063140003521342)from my simple Python program, as a good approximation of the fixed point.