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condensation of the singularities; equicontinuity; generalized convex set-valued mappings; closed convex processes
In the present paper we establish an abstract principle of condensation of singularities for families consisting of set-valued mappings. By using it as a basic tool, the condensation of the singularities and the equicontinuity of certain families of generalized convex set-valued mappings are studied. In particular, a principle of condensation of the singularities of families of closed convex processes is derived. This principle immediately yields the uniform boundedness theorem stated in [1, Theorem 2.3.1].
[1] Aubin J.P., Frankowska H.: Set-Valued Analysis. Birkhäuser Boston (1990). MR 1048347 | Zbl 0713.49021
[2] Breckner W.W.: Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen. Publ. Inst. Math. (Beograd) 23 (37) (1978), 13-20. MR 0508122 | Zbl 0416.46029
[3] Breckner W.W.: A principle of condensation of singularities for set-valued functions. Mathematica-Rev. Anal. Numér. Théor. Approx., Sér. L'Anal. Numér. Théor. Approx. 12 (1983), 101-111. MR 0743128 | Zbl 0527.54015
[4] Breckner W.W.: Continuity of generalized convex and generalized concave set-valued functions. Rev. Anal. Numér. Théor. Approx. 22 (1993), 39-51. MR 1621546 | Zbl 0799.54016
[5] Breckner W.W., Trif T.: On the singularities of certain families of nonlinear mappings. Pure Math. Appl. 6 (1995), 121-137. MR 1430264 | Zbl 0852.47034
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