Multivalued mappings

Abstract

Gel'man, Obukhovs kii UDC 517.988.52The present paper is a survey of the contemporary state of art in the theory of multivalued mappings. In it one considers different forms of continuity of multivalued mappings, one investigates differentiable and measurable multivalued mappings, one considers single-valued continuous approximations and sections of multivalued mappings, one studies fixed points of multivalued mappings and other questions of this theory. One gives references to the literature regarding applications to the theory of games, mathematical economics, the theory of differential inclusions and generalized dynamical systems. The paper contains an extensive bibliography.

On depth and deep points: a calculus

“…See Nikaido (1968), Theorem 4.8, page 72, or Borisovich, Gel'man, Myshkis andObukhovskii (1982), Theorem 1.3.21 on page 138.…”

Lower Semicontinuity Concepts, Continuous Selections, and Set Valued Metric Projections

“…The statements (1), (2), and (3) seem fairly well known. Statement (5) can be proved using the same idea as in the proof of Saks's lemma [11, p. 308].…”

“…For a set valued mapping F: X Q 2 Y between topological spaces X and Y, Brown [8] defined the derived mapping FOE of F and a transfinite sequence of higher derived mappings. The derived mapping appears, without the name and in a particular context, in [7]; see also [2]. Przesławski and Rybiń ski [23], in LOWER SEMICONTINUITY CONCEPTS a narrower context, defined weak lower semicontinuity and some related notions.…”

On the Connection of Degree Theory and 0-epi Maps

“…This class was apparently first introduced in [18]; the theory was further developed even in the multivalued context by Obukhovskiȋ and others. We refer to the surveys [2][3][4] and the monographs [1,9,10]. We will consider a further generalization from [15].…”