An implicit function theorem without differentiability

Abstract

We combine a “global” version of the classical inverse function theorem with Schauder’s fixed point theorem to investigate the existence and continuity properties of a function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper F comma x right-parenthesis right-arrow eta left-parenthesis upper F comma x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi>η<!-- η --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(F,x) \to \eta (F,x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="eta left-parenthesis upper F comma x right-parenthesis equals upper F left-parenthesis eta left-parenthesis upper F comma x right-parenthesis comma x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>η<!-- η --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>η<!-- η --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\eta (F,x) = F(\eta (F,x),x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

• Proceedings of the American Mathematical Society - View - PDF