Suppose that $T: M \to M$ is a self map of a nonempty closed set $M$ in a complete metric Space $(X,d)$. Suppose further that $$d(Tx,Ty) \le k(a,b)d(x,y)$$ for all $x,y \in M$ with $0 \lt a \le d(x,y) \le b$ and arbitrary numbers $a,b$.Here $0 \le k(a,b) \lt 1$. Then show that $T$ has […]

This question is from Foundations of mathematical analysis by Richard Johnsonbaugh The thing with this question is that there is a question that seems to prove the opposite claim Prove the map has a fixed point – someone look into this How should one go about dealing with this question?

I’ve recently discovered that modifying the standard Newton-Raphson iteration by “squashing” $\frac{f (t)}{\dot{f} (t)}$ with the hyperbolic tangent function so that the iteration function is $$N_f (t) = t – \tanh \left( \frac{f (t)}{\dot{f} (t)} \right)$$ results in much larger (Fatou) convergence domains, also known as (immediate) basins of attraction. The convergence theorems of the […]

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