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Differentiable Functions on Normed Linear Spaces

Yasunari Shidama (2012)

Formalized Mathematics

In this article, we formalize differentiability of functions on normed linear spaces. Partial derivative, mean value theorem for vector-valued functions, continuous differentiability, etc. are formalized. As it is well known, there is no exact analog of the mean value theorem for vector-valued functions. However a certain type of generalization of the mean value theorem for vector-valued functions is obtained as follows: If ||ƒ'(x + t · h)|| is bounded for t between 0 and 1 by some constant M, then...

Differential conditions to verify the Jacobian Conjecture

Ludwik M. Drużkowski, Halszka K. Tutaj (1992)

Annales Polonici Mathematici

Let F be a polynomial mapping of ℝ², F(O) = 0. In 1987 Meisters and Olech proved that the solution y(·) = 0 of the autonomous system of differential equations ẏ = F(y) is globally asymptotically stable provided that the jacobian of F is everywhere positive and the trace of the matrix of the differential of F is everywhere negative. In particular, the mapping F is then injective. We give an n-dimensional generalization of this result.

Differentiation bases for Sobolev functions on metric spaces.

Petteri Harjulehto, Juha Kinnunen (2004)

Publicacions Matemàtiques

We study Lebesgue points for Sobolev functions over other collections of sets than balls. Our main result gives several conditions for a differentiation basis, which characterize the existence of Lebesgue points outside a set of capacity zero.

Differentiation of Vector-Valued Functions on n -Dimensional Real Normed Linear Spaces

Takao Inoué, Noboru Endou, Yasunari Shidama (2010)

Formalized Mathematics

In this article, we define and develop differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [16] and [17]).

Differentiation through starlike sets in $R^{m}$

Calixto Calderón (1973)

Studia Mathematica

Dimension and Structure of Typical Compact Sets, Continua and Curves.

Peter M. Gruber (1989)

Monatshefte für Mathematik

Directions De Majoration D'une Fonction Quasiconvexe Et Applications

Amara, Charki (1998)

Serdica Mathematical Journal

We introduce the convex cone constituted by the directions of majoration of a quasiconvex function. This cone is used to formulate a qualification condition ensuring the epiconvergence of a sequence of general quasiconvex marginal functions in finite dimensional spaces.

Discrete convexity.

Danilov, V.I., Koshevoj, G.A. (2004)

Zapiski Nauchnykh Seminarov POMI

Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation

Dariusz Idczak (1998)

Czechoslovak Mathematical Journal

We give characterizations of the distributional derivatives $D^{1, 1}$ , $D^{1, 0}$ , $D^{0, 1}$ of functions of two variables of locally finite variation. Then we use these results to prove the existence theorem for the hyperbolic equation with a nonhomogeneous term containing the distributional derivative determined by an additive function of an interval of finite variation. An application of the above theorem to a hyperbolic equation with an impulse effect is also given.