Some Gruss type inequalities for Frechet differentiable mappings

Caracterisation d'une norme Frechet differentiable pour les duaux Banach complexes

Let $X$ be a complex Banach space; in this work we characterize the property of Frechet diffrentiability for the dual space $X^*$.

Frechet differentiability via partial Frechet differentiability

Let $X_1, \dots, X_n$ be Banach spaces and $f$ a real function on $X=X_1 \times\dots \times X_n$. Let $A_f$ be the set of all points $x \in X$ at which $f$ is partially Fr\' echet differentiable but is not Fr\' echet differentiable. Our results imply that if $X_1, \dots, X_{n-1}$ are Asplund spaces and $f$ is continuous (resp. Lipschitz) on $X$, then $A_f$ is a first category set (resp. a $σ$-upper porous set). We also prove that if $X$, $Y$ are separable Banach spaces and $f:X \to Y$ is a Lipschitz mapping, then the set of all points $x \in X$ at which $f$ is G\^ ateaux differentiable, is Fr\…

An Inverse Function Theorem in Frechet Spaces

I present an inverse function theorem for differentiable maps between Frechet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C^2, or even C^1, or even Frechet-differentiable.

Necessary and sufficient conditions for $n$-times Fréchet differentiability on ${\mathcal S}^p,$ $1 <p<\infty.$

Let $1<p<\infty$ and let $n\geq 1$. It is proved that a function $f:{\mathbb R}\to {\mathbb C}$ is $n$-times Fréchet differentiable on ${\mathcal S}^p$ at every self-adjoint operator if and only if $f$ is $n$-times differentiable, $f',f'',\ldots,f^{(n)}$ are bounded and $f^{(n)}$ is uniformly continuous.

复对偶Banach空间的Frechet可微范数刻画

Caracterisation of Frechet differentiability norm for dual complex Banach spaces

本文刻画了复Banach空间对偶空间的Frechet可微性，并证明若X的对偶空间Gateaux可微，则Lp(X)的对偶空间对所有p同样具备该性质。

点态Lipschitz映射的Gateaux可微性

On Gateaux differentiability of pointwise Lipschitz mappings

我们证明了对于每个函数$f:X\to Y$，其中$X$是可分的Banach空间，$Y$是具有RNP的Banach空间，存在一个集合$A\in\tilde\mcA$使得$f$在所有$x\in S(f)\setminus A$处都是Gateaux可微的，其中$S(f)$是$f$是逐点Lipschitz的点的集合。这改善了Bongiorno的结果。作为推论，我们得到了可分Banach空间上的每一个$K$ -单调函数在属于$\tilde\mcC$的集合之外都是Hadamard可微的，这改进了Borwein和Wang的一个结果.另一个推论是如果$X$是Asplund，$f:X\to\R$锥单调，$g:X\to\R$连续凸的，则在$X$中存在一个点，其中$f$是Hadamard可微的，$g$是Frechet可微的.