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Fréchet differentiability via partial Fréchet differentiability

Luděk Zajíček (2023)

Commentationes Mathematicae Universitatis Carolinae

Let X 1 , , X n be Banach spaces and f a real function on X = X 1 × × X n . Let A f be the set of all points x X at which f is partially Fréchet differentiable but is not Fréchet differentiable. Our results imply that if X 1 , , X n - 1 are Asplund spaces and f is continuous (respectively Lipschitz) on X , then A f is a first category set (respectively a σ -upper porous set). We also prove that if X , Y are separable Banach spaces and f : X Y is a Lipschitz mapping, then there exists a σ -upper porous set A X such that f is Fréchet differentiable at every...

Fréchet directional differentiability and Fréchet differentiability

John R. Giles, Scott Sciffer (1996)

Commentationes Mathematicae Universitatis Carolinae

Zaj’ıček has recently shown that for a lower semi-continuous real-valued function on an Asplund space, the set of points where the function is Fréchet subdifferentiable but not Fréchet differentiable is first category. We introduce another variant of Fréchet differentiability, called Fréchet directional differentiability, and show that for any real-valued function on a normed linear space, the set of points where the function is Fréchet directionally differentiable but not Fréchet differentiable...

Functions of finite fractional variation and their applications to fractional impulsive equations

Dariusz Idczak (2017)

Czechoslovak Mathematical Journal

We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak σ -additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a σ -additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.

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