Aequationes mathematicae

, Volume 90, Issue 6, pp 1129–1145 | Cite as

On Hukuhara’s differentiable iteration semigroups of linear set-valued functions



In this paper, we investigate the uniform convergence of continuous linear set-valued functions on compact sets. We also consider conditions under which the family of continuous linear extensions of a differential iteration semigroup of continuous linear set-valued functions is a differentiable iteration semigroup. In particular, since the cones and normed spaces are not supposed to be complete our main results generalize some recent results on Hukuhara’s derivative of set-valued functions.


Hukuhara’s derivative Iterations Linear set-valued functions Cauchy problem for a set-valued differential equation Riemann integral for set-valued functions 

Mathematics Subject Classification

Primary 46G05 Secondary 39B12 54C60 


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  1. 1.
    Aghajani M., Nourouzi K.: On the regular cosine family of linear correspondences. Aequationes Math. 8(33), 215–221 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aliprantis C.D., Border K.C.: Infinite Dimentional Analysis, A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)MATHGoogle Scholar
  3. 3.
    Dinghas A.: Zum Minkowskischen Integralbegriff abgeschlossener Mengen. Math. Zeitschrift 66, 173–188 (1956)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Hukuhara M.: Intégration des applications measurables dont la valeur est un compact convexe. Funcial. Ekvac. 10, 205–223 (1967)MathSciNetMATHGoogle Scholar
  5. 5.
    Piszczek M.: Integral representations of convex and concave set-valued functions. Demonstr. Math. 35, 727–742 (2002)MathSciNetMATHGoogle Scholar
  6. 6.
    Rädström H.: An embedding theorem for space of convex sets. Proc. Am. Math. Soc. 3, 165–169 (1952)CrossRefGoogle Scholar
  7. 7.
    Smajdor A.: Hukuhara’s derivative and concave iteration semigroups of linear set-valued functions. J. Appl. Anal. 8(2), 297–305 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Smajdor A.: Hukuhara’s differentiable iteration semigroups of linear set-valued functions. Ann. Polon. Math. 83(1), 1–10 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Smajdor A.: On a multivalued differential problem. Int. J. Bifurc. Chaos 13(7), 1877–1882 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Smajdor A.: On regular multivalued cosine families, European Conference on Iteration Theory (Muszyna-Zlockie, 1998). Ann. Math. Sil. 13, 271–280 (1999)MathSciNetGoogle Scholar
  11. 11.
    Smajdor W.: Superadditive set-valued functions and Banach-Steinhause Theorem. Rad. Mat. 3, 203–214 (1987)MathSciNetMATHGoogle Scholar
  12. 12.
    Szczawińska J.: On some families of set-valued functions. Aequationes Math. 78, 157–166 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceShahid Rajaee Teacher Training UniversityTehranIran
  2. 2.Faculty of MathematicsK. N. Toosi University of TechnologyTehranIran

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