# A characterization of ${C}^{1,1}$ functions via lower directional derivatives

Mathematica Bohemica (2009)

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

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topBednařík, 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

top- 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. (2007) MR2387488
- Bednařík, D., Pastor, K., 10.1137/050636309, SIAM J. Control Optim. 45 (2006), 383-387. (2006) MR2225311DOI10.1137/050636309
- Bruckner, A. M., Differentiation of Real Functions, 2nd ed, Amer. Math. Soc., Providence, Rhode Island (1994). (1994) MR1274044
- Cannarsa, P., Sinestrari, C., Semiconave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations 58, Birkäuser, Boston, MA (2004). (2004) MR2041617
- 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
- Thompson, B. S., Real Analysis, Springer, Berlin (1985). (1985)

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