A characterization of C 1 , 1 functions via lower directional derivatives

Dušan Bednařík; Karel Pastor

Mathematica Bohemica (2009)

  • Volume: 134, Issue: 2, page 217-221
  • ISSN: 0862-7959

Abstract

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The notion of ℓ ˜ -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 ℓ ˜ -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.

How to cite

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Bednařík, Dušan, and Pastor, Karel. "A characterization of $C^{1,1}$ functions via lower directional derivatives." Mathematica Bohemica 134.2 (2009): 217-221. <http://eudml.org/doc/38088>.

@article{Bednařík2009,
abstract = {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.},
author = {Bednařík, Dušan, Pastor, Karel},
journal = {Mathematica Bohemica},
keywords = {second-order derivative; $C^\{1,1\}$ function; $\ell $-stable function; $\tilde\{\ell \}$-stability; -stable function; -stability},
language = {eng},
number = {2},
pages = {217-221},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A characterization of $C^\{1,1\}$ functions via lower directional derivatives},
url = {http://eudml.org/doc/38088},
volume = {134},
year = {2009},
}

TY - JOUR
AU - Bednařík, Dušan
AU - Pastor, Karel
TI - A characterization of $C^{1,1}$ functions via lower directional derivatives
JO - Mathematica Bohemica
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 134
IS - 2
SP - 217
EP - 221
AB - 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.
LA - eng
KW - second-order derivative; $C^{1,1}$ function; $\ell $-stable function; $\tilde{\ell }$-stability; -stable function; -stability
UR - http://eudml.org/doc/38088
ER -

References

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  1. Bednařík, D., Pastor, K., Second-order sufficient condition for ℓ ˜ -stable functions, Acta Univ. Palacki. Olomuc., Fac. Rer. Nat., Mathematica 46 (2007), 7-18. (2007) MR2387488
  2. Bednařík, D., Pastor, K., 10.1137/050636309, SIAM J. Control Optim. 45 (2006), 383-387. (2006) MR2225311DOI10.1137/050636309
  3. Bruckner, A. M., Differentiation of Real Functions, 2nd ed, Amer. Math. Soc., Providence, Rhode Island (1994). (1994) MR1274044
  4. Cannarsa, P., Sinestrari, C., Semiconave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations 58, Birkäuser, Boston, MA (2004). (2004) MR2041617
  5. Ginchev, I., Guerraggio, A., Rocca, M., 10.1007/s10492-006-0002-1, Appl. Math. 51 (2006), 5-36. (2006) Zbl1164.90399MR2197320DOI10.1007/s10492-006-0002-1
  6. Thompson, B. S., Real Analysis, Springer, Berlin (1985). (1985) 

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