Gâteaux and Hadamard Differentiability via Directional Differentiability
Abstract
<jats:p> Let <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be a separable Banach space, <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">Y</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> a Banach space and <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f: X \to Y</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> an arbitrary mapping. Then the following implication holds at each point <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">x\in X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> except a <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\sigma</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -directionally porous set: If the one-sided Hadamard directional derivative <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f'_{H+}(x,u)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>+</mml:mo> </mml:mrow> <mml:mo mathvariant="normal" lspace="0em" rspace="0em">′</mml:mo> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo separator="true">,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> exists in all directions <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">u</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> from a set <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">S_x \subset X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> whose linear span is dense in <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , then <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is Hadamard differentiable at <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">x</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . This theorem improves and generalizes a recent result of A. D. Ioffe, in which the linear span of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">S_x</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> equals <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">Y = \mathbb{R}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . An analogous theorem, in which <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is pointwise Lipschitz, and which deals with the usual one-sided derivatives and Gâteaux differentiability is also proved. It generalizes a result of D. Preiss and the author, in which <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow>