On the Stability of the Directional Regularity

  • Radek Cibulka
  • Marius Durea
  • Marian Panţiruc
  • Radu StrugariuEmail author


In this paper we select two tools of investigation of the classical metric regularity of set-valued mappings, namely the Ioffe criterion and the Ekeland Variational Principle, which we adapt to the study of the directional setting. In this way, we obtain in a unitary manner new necessary and/or sufficient conditions for directional metric regularity. As an application, we establish stability of this property at composition and sum of set-valued mappings. In this process, we introduce directional tangent cones and the associated generalized primal differentiation objects and concepts. Moreover, we underline several links between our main assertions by providing alternative proofs for several results.


Directional regularity Ioffe criterion Directional openness stability Primal conditions 

Mathematics Subject Classification (2010)

47J22 49K40 90C29 


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Radek Cibulka was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports. The work of Marius Durea was supported by a grant of Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0188, within PNCDI III. The work of Marian Panţiruc and Radu Strugariu was supported by a research grant of TUIASI, project number TUIASI-GI-2018-0647.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.NTIS – New Technologies for the Information Society and Department of Mathematics, Faculty of Applied SciencesUniversity of West BohemiaPilsenCzech Republic
  2. 2.Faculty of Mathematics“Alexandru Ioan Cuza” UniversityIaşiRomania
  3. 3.“Octav Mayer” Institute of Mathematics of the Romanian AcademyIaşiRomania
  4. 4.Department of Mathematics“Gh. Asachi” Technical UniversityIaşiRomania

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