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For orientation, the root function <math>\ f\ </math> and the fixed-point functions are simply related by <math>\ F(x) = x + \varepsilon f(x)\ ,</math> where <math>\ \varepsilon\ </math> is some scalar constant small enough in magnitude to make <math>\ F\ </math> stable under iteration, but large enough for the [[non-linearity]] of the function <math>\ f\ </math> to be appreciable. All issues of a more general [[Banach space]] vs. basic [[real numbers]] being momentarily ignored for the sake of the comparison.
This method for finding fixed points of a real-valued function has been generalized for functions <math>\ F : X \to X\ </math> that map a [[Banach space]] <math>\ X\ </math> onto itself or even more generally <math>\ F : X \to Y\ </math> that map from one [[Banach space]] <math>X </math> into another [[Banach space]] <math>\ Y ~.</math> The generalized method assumes that a [[Indexed family|family]] of [[Bounded set|bounded]] [[linear operators]] <math>\ \bigl\{\ G(u,v): u, v \in X\ \bigr\}\ </math> associated with <math>\ u\ </math> and <math>\ v\ </math> can be devised that (locally) satisfies the condition<ref name=Johnson-Scholz-1968/>
{{NumBlk|:|<math> F\left( u \right) - F\left( v \right) = G\left( u, v \right)\ \bigl(\ u - v\ \bigr) \quad</math>|{{EquationRef|1}}}}
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