Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.05313

Partial Gateaux and Frechet Derivatives and Applications to Variational Analysis

Jinlu Li
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we define partial Gateaux and Frechet derivatives for multivariable and single-valued mappings between Banach spaces. We will prove some properties of partial Gateaux and Frechet derivatives. By these definitions, we find the explicit formulas for some polynomial type operators with two variables from lp by lp to lp with respect to each variable. Then, we will introduce the concepts of generalized partially critical points and partial ordered extrema in partially ordered Banach spaces. By these concepts, we will investigate the connection between generalized partially critical points and partial ordered-extrema of two-variable and single-valued mappings in partially ordered Banach spaces. These results extend the connection between critical points and extrema of real valued functions in calculus. We will give some applications of partial Gateaux and Frechet derivatives to ordered optimizations and variational inequality problems between partially ordered Banach spaces. Finally, we study the connection between partial Gateaux and Frechet derivatives and ordered monotone of two-variable and single-valued mappings in partially ordered Banach spaces.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jinlu Li. 2026-07-06. Partial Gateaux and Frechet Derivatives and Applications to Variational Analysis. https://arxiv.org/abs/2607.05313

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Conditional expectation operators on $C(X)$

At the COSAEF conference in 2021, several participants asked the question whether a conditional expectation operator in the sense of Kuo, Labaushagne and Watson could be constructed in vector lattices other than $\mathcal{L}_p$ spaces and in particular in $C(X)$. This work answers positively to this question and participates in an old discussion on integrals in $C(X)$ space.

math.FA↗
arXiv2026

Hypercontractivity for a family of quantum Ornstein-Uhlenbeck semigroups

We show that a family of quantum Ornstein-Uhlenbeck semigroups is hypercontractive. We also obtain the optimal order of the optimal time up to a constant. The main ingredient of our proof is Meixner polynomials. The goal of this paper is twofold: to provide more examples of hypercontractive quantum Markov semigourps on non-tracial von Neumann algebras, and to determine the optimal order of the optimal time for quantum Ornstein-Uhlenbeck semigroups.

math.FA↗
arXiv2026

Fixed Point Rigidity of the Operator $Γ_pΠ_p^\ast$ and the LYZ Conjecture

We characterize the fixed points of the operator $Γ_pΠ_p^\ast$ for $n\geq 3$ and $1 0$ if and only if $K$ is an origin-centered ellipsoid, thereby settling the Lutwak--Yang--Zhang fixed-point conjecture in this range. Our proof is based on a variational analysis along linear reflection shadow systems. To address the nonlinear structure of the $L_p$ setting, we introduce the $L_p$-Projection Rolodex, which provides a dimensional reduction of the volume of the polar $L_p$-projection body to weighted lower-dimensional sectional functionals. A suitable change of variables, together with Ball's harmonic Prékopa--Leindler inequality, yields the convexity needed along the shadow system. Under the fixed-point condition, a first-variation identity then forces $\operatorname{vol}_n(Π_p^\ast K_t)$ to remain constant throughout the deformation. The rigidity statement follows from the equality characterization under Steiner symmetrization.

math.FA↗