Differential
Inclusions, Control and Optimization 21 (2001) 81-95

doi: 10.7151/dmdico.1018

Libor Jüttner

*Department of Mathematics Analysis
Faculty of Science, Palacký University
Tomkova 40, 779 00 Olomouc-Hejcín, Czech Republic*

The problem of linearity of a multivalued derivative and consequently the problem of necessary and sufficient conditions for derivo-periodic multifunctions are investigated. The notion of a derivative of multivalued functions is understood in various ways. Advantages and disadvantages of these approaches are discussed.

**Keywords and phrases:** differential of multivalued functions, multivalued
differential, contingent derivative, linearity of contingent derivative, periodic
multivalued functions, derivo-periodic multivalued functions.

**2000 Mathematics Subject Classification:** 26E25, 58F25.

[1] | J. Andres, Derivo-periodic boundary value problems for nonautonomous
ordinary differential equations, Riv. Mat. Pura Appl. 13 (1993), 63-90. |

[2] | J. Andres, Nonlinear rotations, Nonlin. Anal. 30 (1)
(1997), 495-503. |

[3] | J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin 1984. |

[4] | J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston 1990. |

[5] | H.T. Banks and M.Q. Jacobs, A differential calculus for multifunctions,
J. Math. Anal. Appl. 29 (1970), 246-272. |

[6] | F.S. De Blasi, On the differentiability of multifunctions, Pacific
J. Math. 66 (1) (1976), 67-81. |

[7] | M. Farkas, Periodic Motions, Springer, Berlin 1994. |

[8] | J.S. Cook, W.H. Louisell and W.H. Yocom, Stability of an electron beam
on a slalom orbit, J. Appl. Phys. 29 (1958), 583-587. |

[9] | G. Fournier and D. Violette, A fixed point theorem for a class of
multi-valued continuously differentiable maps, Anal. Polon. Math. 47 (1987),
381-402. |

[10] | M. Martelli and A. Vignoli, On differentiability of multi-valued maps,
Bollettino U.M.I. 10 (4) (1974), 701-712. |

[11] | J. Mawhin, From Tricomi's equation for synchronous motors to the
periodically forced pendulum, In Tricomi's Ideas and Contemporary Applied Mathematics,
Atti Conv. Lincei 147, Accad. Naz. Lincei (Roma), (1998), 251-269. |

[12] | P. Meystre, Free-electron Lasers, An Introduction, ``Laser Physics (D.F. Walls and J.D. Harvey, ed.)'', Academic Press, Sydney-New York-London-Toronto-San Francisco 1980. |

Received 9 June 2000