Differentiability on semilinear spaces
type: Item
instance of: scholarly article
title: Differentiability on semilinear spaces
zbMATH Open document ID: 1206.49016
author: George N. Galanis
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-08-21T00:00:00Z
review text: The aim of this short note is to introduce a natural differentiability concept between semilinear spaces that may preserve the basic properties of the Gâteaux and Hukuhara differentials and may be efficiently used in applications on stochastic manifolds, set-valued and fuzzy differential equations, etc. More precisely, if \(S_1, S_2\) are topological semilinear spaces, \(A\subset S_1\) is open then a continuous mapping \(f(\cdot):A\to S_2\) is said to be differentiable at \(x_0\in A\) if there exist the neighborhoods \(U\subset A\) of \(x_0\) and \(I\subset\mathbb R\) of \(0\) and the continuous mappings \(df(\cdot,\cdot):U_0\times S_1\to S_2\), \(\varphi(\cdot,\cdot,\cdot):I\times U_0\times S_1\to S_2\) such that \(df(x,\cdot), \;x\in U_0\) are semilinear and satisfy: \[ \begin{gathered} f(x+t.y)=f(x)+t.df(x,y)+t.\varphi(t,x,y), \\ \lim_{t\to 0+}\varphi(t,x,y)=0, \;t\in I, \;x\in U_0, \;y\in S_1.\end{gathered} \] Several simple propositions containing the basic properties of the new differential, \(df(\cdot,\cdot),\) are proved.
Mathematics Subject Classification ID: 49J50
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 26E25
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 5597870
zbMATH Keywords: differentiation on semilinear spaces
zbMATH Keywords: Gateaux differentiability
zbMATH Keywords: Hukuhara differentiability
zbMATH Keywords: semilinear space
reviewed by: Ştefan Mirică
MaRDI profile type: Publication
full work available at URL: j.na.2009.03.045
OpenAlex ID: W2083255439
cites work: Existence theorems of set optimization with set-valued maps
cites work: Set valued functions in Fréchet spaces: continuity, Hukuhara differentiability and applications to set differential equations
cites work: Set Differential Equations in Fréchet Spaces
cites work: Differentiability and convexity of fuzzy mappings
cites work: Directional derivatives and subdifferential of convex fuzzy mappings and application in convex fuzzy programming
cites work: Q5543500
cites work: Q5484956
cites work: Q4520889
cites work: First order linear fuzzy differential equations under generalized differentiability
cites work: Boundaries of Semilinear Spaces and Semialgebras
DOI: 10.1016/J.NA.2009.03.045
description: scientific article; zbMATH DE number 5597870