Set-Valued Analysis

, 14:263 | Cite as

Semifixed Sets of Maps in Hyperspaces with Application to Set Differential Equations

Article

Abstract

For maps φ on hyperspaces the existence of semifixed sets, i.e., of sets A satisfying one of the relations Aφ(A), Aφ(A), Aφ(A) ≠ ∅, is considered. An application to set differential equations is also presented.

Key words

hyperspace compact convex set multifunction semifixed set Hukuhara's derivative set differential equation 

Mathematics Subject Classifications (2000)

Primary 47H04 Secondary 34A60 

References

  1. 1.
    Artstein, Z.: A calculus for set-valued maps and set-valued evolution equations, Set-Valued Anal. 3 (1995), 213–261.CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Brandão Lopes Pinto, A. J., de Blasi, F. S. and Iervolino, F.: Uniqueness and existence theorems for differential equations with compact convex valued solutions, Boll. Un. Mat. Ital. 3(4) (1970), 47–54.MathSciNetGoogle Scholar
  3. 3.
    de Blasi, F. S. and Georgiev, P. G.: Kakutani–Fan's fixed point theorem in hyperspaces, Tokyo J. Math. 24 (2001), 331–342.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    de Blasi, F. S. and Iervolino, F.: Equazioni differenziali con soluzioni a valore compatto convesso, Boll. Un. Mat. Ital. 2(4) (1969), 491–501.MATHGoogle Scholar
  5. 5.
    de Blasi, F. S. and Pianigiani, G.: Approximate selections in α-convex metric spaces and topological degree, Topol. Methods Nonlinear Anal. 24 (2004), 347–375.MathSciNetMATHGoogle Scholar
  6. 6.
    Górniewicz, L.: Topological Fixed Point Theory of Multivalued Mappings, Kluwer, Dordrecht, 1999.MATHGoogle Scholar
  7. 7.
    Hukuhara, M.: Intégration des applications mesurables dont la valeur est un compact convexe, Funkc. Ekvac. 10 (1967), 205–223.MathSciNetMATHGoogle Scholar
  8. 8.
    Hu, S. and Papageorgiou, N. S.: Handbook of Multivalued Analysis, Kluwer, Dordrecht, 1997.MATHGoogle Scholar
  9. 9.
    Kisielewicz, M.: Description of a class of differential equations with set-valued solutions, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 58(8) (1975), 158–162.MathSciNetMATHGoogle Scholar
  10. 10.
    Lakshmikantham, V., Leela, S. and Vatsala, A. S.: Interconnection between set and fuzzy differential equations, Nonlinear Anal. 54 (2003), 351–360.CrossRefMathSciNetGoogle Scholar
  11. 11.
    Lakshmikantham, V. and Tolstonogov, A. N.: Existence and interrelation between set and fuzzy differential equations, Nonlinear Anal. 55 (2003), 255–268.CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Nadler, S. B.: Multivalued contraction mappings, Pacific J. Math. 30 (1969), 475–488.MathSciNetMATHGoogle Scholar
  13. 13.
    Plotnikov, A. V.: Averaging differential inclusions with the Hukuhara derivative, Ukrainian Math. J. 41 (1989), 112–115.CrossRefMathSciNetGoogle Scholar
  14. 14.
    Plotnikov, A. V. and Tumbzukaki, A. V.: Integrodifferential inclusions with Hukuhara’s derivative, Nelīnīĭnī Koliv. 8 (2005), 80–88.MATHGoogle Scholar
  15. 15.
    Plotnikov, V. A. and Melnik, T. A.: A generalization of a theorem of A.N. Tikhonov for quasidifferential equations, Differential Equations 33 (1997), 1036–1040.MathSciNetMATHGoogle Scholar
  16. 16.
    Tolstonogov, A. N.: Differential Inclusions in a Banach Space, Kluwer, Dordrecht, 2000.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly

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