scholarly journals Directional upper derivatives and the chain rule formula for locally Lipschitz functions on Banach spaces

2015 ◽  
Vol 368 (7) ◽  
pp. 4685-4730 ◽  
Author(s):  
Olga Maleva ◽  
David Preiss
Keyword(s):  
Banach Spaces ◽  
Lipschitz Functions ◽  
Chain Rule ◽  
Locally Lipschitz Functions ◽  
Locally Lipschitz

scholarly journals Continuity characterisations of differentiability of locally Lipschitz functions

1990 ◽  
Vol 41 (3) ◽  
pp. 371-380 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Equivalent Norm ◽  
Lipschitz Functions ◽  
Distance Functions ◽  
Locally Lipschitz Functions ◽  
Locally Lipschitz ◽  
Differentiability Properties

Recently David Preiss contributed a remarkable theorem about the differentiability of locally Lipschitz functions on Banach spaces which have an equivalent norm differentiable away from the origin. Using his result in conjunction with Frank Clarke's non-smooth analysis for locally Lipschitz functions, continuity characterisations of differentiability can be obtained which generalise those for convex functions on Banach spaces. This result gives added information about differentiability properties of distance functions.

The Multidirectional Mean Value Theorem in Banach Spaces

1997 ◽  
Vol 40 (1) ◽  
pp. 88-102 ◽  
Author(s):  
M. L. Radulescu ◽  
F. H. Clarke
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Continuous Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Locally Lipschitz Functions ◽  
Bump Function ◽  
Lipschitz Continuous ◽  
Lower Semi ◽  
Locally Lipschitz

AbstractRecently, F. H. Clarke and Y. Ledyaev established a multidirectional mean value theorem applicable to lower semi-continuous functions on Hilbert spaces, a result which turns out to be useful in many applications. We develop a variant of the result applicable to locally Lipschitz functions on certain Banach spaces, namely those that admit a C1-Lipschitz continuous bump function.

scholarly journals Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces

1993 ◽  
Vol 47 (2) ◽  
pp. 205-212 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Lipschitz Function ◽  
Separable Banach Space ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz Functions ◽  
Dini Derivative ◽  
Locally Lipschitz ◽  
Set Of Points ◽  
Clarke Derivative

For a locally Lipschitz function on a separable Banach space the set of points of Gâteaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gâteaux but not strictly differentiable is of the first category.

Equivalence of Some Multi-Valued Elliptic Variational Inequalities and Variational-Hemivariational Inequalities

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Siegfried Carl
Keyword(s):  
Variational Inequalities ◽  
Lipschitz Functions ◽  
Hemivariational Inequalities ◽  
Locally Lipschitz Functions ◽  
Generalized Gradient ◽  
Elliptic Variational Inequalities ◽  
Comparison Results ◽  
Locally Lipschitz ◽  
Additional Regularity ◽  
Definition Of

AbstractFirst, we prove existence and comparison results for multi-valued elliptic variational inequalities involving Clarke’s generalized gradient of some locally Lipschitz functions as multi-valued term. Only by applying the definition of Clarke’s gradient it is well known that any solution of such a multi-valued elliptic variational inequality is also a solution of a corresponding variational-hemivariational inequality. The reverse is known to be true if the locally Lipschitz functions are regular in the sense of Clarke. Without imposing this kind of regularity the equivalence of the two problems under consideration is not clear at all. The main goal of this paper is to show that the equivalence still holds true without any additional regularity, which will fill a gap in the literature. Existence and comparison results for both multi-valued variational inequalities and variational-hemivariational inequalities are the main tools in the proof of the equivalence of these problems.

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