A limiting example

for the Local ``fuzzy'' sum rule

in nonsmooth analysis

Authors:
Jon Vanderwerff and Qiji J. Zhu

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2691-2697

MSC (1991):
Primary 26B05, 49J50, 49J52

MathSciNet review:
1451834

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that assuming all the summand functions to be lower semicontinuous is not sufficient to ensure a (strong) fuzzy sum rule for subdifferentials in any infinite dimensional Banach space. From this we deduce that additional assumptions are also needed on functions for chain rules, multiplier rules for constrained minimization problems and Clarke-Ledyaev type mean value inequalities in the infinite dimensional setting.

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Robert R. Phelps,*Convex functions, monotone operators and differentiability*, 2nd ed., Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1993. MR**1238715****[Z1]**Q. J. Zhu, Subderivatives and their applications, Proceedings of the International Conference on Dynamic Systems, Springfield, MO, June 1996.**[Z2]**Q. J. Zhu, Clarke-Ledyaev mean value inequality in smooth Banach spaces, CECM Research Report 96-78 (1996),*Nonlinear Analysis: TMA*, to appear.

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Additional Information

**Jon Vanderwerff**

Affiliation:
Department of Mathematics, Walla Walla College, College Place, Washington 99324

Email:
vandjo@wwc.edu

**Qiji J. Zhu**

Affiliation:
Department of Mathematics & Statistics, Western Michigan University, Kalamazoo, Michigan 49008

Email:
zhu@math-stat.wmich.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-98-04365-2

Received by editor(s):
January 30, 1997

Additional Notes:
The first author’s research was partially supported by a Walla Walla College Faculty Development Grant.

The second author’s work was partially supported by a grant from the Faculty Research and Creative Activities Support Fund, Western Michigan University.

Research for this note was completed while the authors were visiting Simon Fraser University. The authors thank J.M. Borwein and the CECM for their hospitality.

Communicated by:
Dale Alspach

Article copyright:
© Copyright 1998
American Mathematical Society