| Title:
|
A characterization of $C^{1,1}$ functions via lower directional derivatives (English) |
| Author:
|
Bednařík, Dušan |
| Author:
|
Pastor, Karel |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
134 |
| Issue:
|
2 |
| Year:
|
2009 |
| Pages:
|
217-221 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The notion of $\tilde {\ell }$-stability is defined using the lower Dini directional derivatives and was introduced by the authors in their previous papers. In this paper we prove that the class of $\tilde {\ell }$-stable functions coincides with the class of C$^{1,1}$ functions. This also solves the question posed by the authors in SIAM J. Control Optim. 45 (1) (2006), pp.\ 383--387. (English) |
| Keyword:
|
second-order derivative |
| Keyword:
|
$C^{1,1}$ function |
| Keyword:
|
$\ell $-stable function |
| Keyword:
|
$\tilde {\ell }$-stability |
| MSC:
|
26B05 |
| MSC:
|
49K10 |
| idZBL:
|
Zbl 1212.49043 |
| idMR:
|
MR2535149 |
| DOI:
|
10.21136/MB.2009.140656 |
| . |
| Date available:
|
2010-07-20T17:59:23Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140656 |
| . |
| Reference:
|
[1] Bednařík, D., Pastor, K.: Second-order sufficient condition for $\tilde{\ell}$-stable functions.Acta Univ. Palacki. Olomuc., Fac. Rer. Nat., Mathematica 46 (2007), 7-18. MR 2387488 |
| Reference:
|
[2] Bednařík, D., Pastor, K.: Errata: Elimination of strict convergence in optimization.SIAM J. Control Optim. 45 (2006), 383-387. MR 2225311, 10.1137/050636309 |
| Reference:
|
[3] Bruckner, A. M.: Differentiation of Real Functions, 2nd ed.Amer. Math. Soc., Providence, Rhode Island (1994). MR 1274044 |
| Reference:
|
[4] Cannarsa, P., Sinestrari, C.: Semiconave Functions, Hamilton-Jacobi Equations, and Optimal Control.Progress in Nonlinear Differential Equations 58, Birkäuser, Boston, MA (2004). MR 2041617 |
| Reference:
|
[5] Ginchev, I., Guerraggio, A., Rocca, M.: From scalar to vector optimization.Appl. Math. 51 (2006), 5-36. Zbl 1164.90399, MR 2197320, 10.1007/s10492-006-0002-1 |
| Reference:
|
[6] Thompson, B. S.: Real Analysis.Springer, Berlin (1985). |
| . |
