TY - JOUR
T1 - On the range of the derivative of a smooth mapping between Banach spaces
AU - Deville, Robert
N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
PY - 2005
Y1 - 2005
N2 - We survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of ( X,Y) for the existence of a Fréchet differentiable mapping f from X into Y so that f′ (X) =A . Whenever f is only assumed Gteaux differentiable, new phenomena appear: for instance,there exists a mapping f from 1 () into 2, which is bounded, Lipschitz-continuous, and so that for all x,y 1 (), if xy, then f′ (x) f′ (y) >1 .
AB - We survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of ( X,Y) for the existence of a Fréchet differentiable mapping f from X into Y so that f′ (X) =A . Whenever f is only assumed Gteaux differentiable, new phenomena appear: for instance,there exists a mapping f from 1 () into 2, which is bounded, Lipschitz-continuous, and so that for all x,y 1 (), if xy, then f′ (x) f′ (y) >1 .
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U2 - 10.1155/AAA.2005.499
DO - 10.1155/AAA.2005.499
M3 - Article
SN - 1085-3375
VL - 2005
SP - 499
EP - 507
JO - Abstract and Applied Analysis
JF - Abstract and Applied Analysis
IS - 5
ER -