Aequationes mathematicae

, Volume 81, Issue 1–2, pp 97–108 | Cite as

On multivalued iteration semigroups

  • Magdalena Piszczek
Open Access


We will give a necessary and sufficient condition for the family {F t t ≥ 0} of multifunctions \({F_t(x) = \sum_{i=0}^{\infty} \frac{t^i}{i!}G^i(x)}\), where G is a continuous and additive multifunction, to be an iteration semigroup.

Mathematics Subject Classification (2000)

26E25 39B12 47D03 


Iteration semigroup Hukuhara’s derivative Riemann integral of multifunctions 


Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Hukuhara M.: Intégration des application mesurables dont la valeur est un compact convexe. Funkcial. Ekvac. 10, 205–223 (1967)MathSciNetMATHGoogle Scholar
  2. 2.
    Nadler S.B. Jr: Multivalued contraction mappings. Pacific J. Math. 30, 475–488 (1969)MathSciNetMATHGoogle Scholar
  3. 3.
    Piszczek M.: On multivalued cosine families. J. Appl. Anal. 13, 57–76 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Piszczek M.: Second Hukuhara derivative and cosine family of linear set-valued functions. Ann. Acad. Pead. Cracoviensis. Studia Math. 5, 87–98 (2006)MathSciNetGoogle Scholar
  5. 5.
    Rådström H.: An embedding theorem for spaces of convex sets. Proc. Am. Math. Soc. 3, 165–169 (1952)MATHCrossRefGoogle Scholar
  6. 6.
    Smajdor A.: Hukuhara’s differentiable iteration semigroup of linear set-valued functions. Ann. Polon. Math. 83(1), 1–10 (2004)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Smajdor A.: Hukuhara’s derivative and concave iteration semigrups of linear set-valued functions. J. Appl. Anal. 8, 297–305 (2002)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Smajdor A.: Increasing iteration semigroups of Jensen set-valued functions. Aequationes Math. 56, 131–142 (1998)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Smajdor A.: On a multivalued differential problem. Int. J. Bifur. Chaos Appl. Sci. Eng. 13, 1877–1882 (2003)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Smajdor A.: On concave iteration semigroups of linear set-valued functions. Aequationes Math. 75, 149–162 (2008)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Smajdor A.: On regular multivalued cosine families. Ann. Math. Sil. 13, 271–280 (1999)MathSciNetMATHGoogle Scholar
  12. 12.
    Smajdor W.: Superadditive set-valued functions and Banach-Steinhaus Theorem. Rad. Mat. 3, 203–214 (1987)MathSciNetMATHGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Institute of MathematicsPedagogical UniversityKrakówPoland

Personalised recommendations