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Suppose $f$ is differentiable on $(-\infty , \infty)$. We say that $x$ is a fixed point if $f(x) = x$. If there is a constant $A$ such that $0 <A <1$ and $|f'(x)| \leq A$ for all real $x$. Then there is a fixed point $x$ such that $x = \lim_{n\to \infty} x_n$ , where $x_1$ is any arbitrary real number and $x_{n+1} = f(x_n)$ for all $n= 1 , 2, \dots$

  • But my doubt is that if $f(x) = x + (1 + e^x)^{-1}$ and $ 0 <f'(x) <1$ and $f(x)$ has no fixed point .Both statement are contradict here.

Please clear my doubt. Thank you

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The derivative of your function is always less then 1, but the least upper bound is equal to 1. Since A equals 1, the theorem does not apply.

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