On Continuous Selections of Set-valued Mappings with Almost Convex Values

  • R. A. KhachatryanEmail author
Real and Complex Analysis


In this paper, it is proved that through each point of the graph of a continuous setvalued mapping with almost convex and star-like values can be passed a continuous selection of that mapping.


Set-valued mapping star-like set almost convexity selector 

MSC2010 numbers

26E25 49J52 46J05 


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  1. 1.
    Yu.G. Borisovich, B.D. Gel’man, A.D. Myshkis, V.V. Obukhovskii, Introduction to set–valued mappings and differential embeddings (KomKniga,Moscow, 2005).Google Scholar
  2. 2.
    F. P. Vasilyev, Methods of solution of extremal problems (Nauka, Moscow, 1981).Google Scholar
  3. 3.
    M.A. Krasnosselsky, “Sur un critére pour qu’un domaine soit étoilé”,Mat. Sb., 19(61) (2), 309–310, 1946.MathSciNetzbMATHGoogle Scholar
  4. 4.
    E. S. Polovinkin, Multivalued analysis and differential embeddings (Fizmatlit, Moscow, 2014).Google Scholar
  5. 5.
    L.S. Pontryagin, “Linear differential games of pursuit”, Mat. Sb., 112 (3), 307–330, 1980.MathSciNetzbMATHGoogle Scholar
  6. 6.
    J.–P. Aubin and I. Ekeland, Applied nonlinear analysis (Mir,Moscow, 1988).zbMATHGoogle Scholar
  7. 7.
    V.V. Ostapenko, “Approximate solution of the approach–deviation problem”, Preprint–82–16, Inst. Kibernetiki AN USSR, Kiev, 1982.Google Scholar
  8. 8.
    V.V. Ostapenko, “On an almost convexity condition”, Ukr.Mat. J., 35 (2), 169–172, 1983.zbMATHGoogle Scholar
  9. 9.
    P. V. Semenov, “On paraconvexity of starlike sets”, Sib. Mat. Zh., 37 (2), 399–405, 1996.CrossRefzbMATHGoogle Scholar
  10. 10.
    D. Repovs, P.V. Semenov, “Michael’s theory of continuous selections. Development and applications”, UMN, 49 (6), 151–188, 1994.MathSciNetzbMATHGoogle Scholar
  11. 11.
    R.A. Khachatryan, “On directional derivatives of selections of set–valued mappings”, Journal of Contemporary Mathem. Analysis (Armenian Academy of Sciences), 51 (3), 65–84, 2016.MathSciNetzbMATHGoogle Scholar
  12. 12.
    R.A. Khachatryan, “On existence of continuous and smooth selections of set–valued mappings”, Journal of ContemporaryMathem. Analysis (Armenian Academy of Sciences), 37 (2), 65–76, 2002.Google Scholar
  13. 13.
    F. Bernard, L. Thibold, N. Zlateva “Characterizations of prox–regular sets in uniformly convex Banach spaces”, Journal Convex Analysis, 13 (3/4), 525–559, 2006.MathSciNetzbMATHGoogle Scholar
  14. 14.
    F.H. Clarke, R.J. Stern, P.R. Wolenski, “Proximal Smoothness and the Lower–C2 Property”, Journal of Convex Analysis, 2 (1/2), 117–144, 1995.MathSciNetzbMATHGoogle Scholar
  15. 15.
    E. Michael, “Continuous selections 1”, Ann.Math., 63, 361–381, 1956.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    H. Heremes, “On continuos and measurables selections and the existence of solutions of generalized differential equations”, Proc.Amer.Math. Sci., 29 (3), 535–542, 1971.CrossRefGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Yerevan State UniversityYerevanArmenia

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