Multivalued maps as a tool in modeling and rigorous numerics

Abstract

Applications of the fixed point theory of multivalued maps can be classified into several areas: (1) Game theory and mathematical economics;(2) Discontinuous differential equations, differential inclusions, and optimal control; (3) Computing homology of maps; (4) Computer assisted proofs in dynamics; (5) Digital imaging. We give an overview of the most classical and well developed areas of applications (1) and (2), where a multivalued map is used as a generalization of a single-valued continuous map, and we survey more recent applications (3), (4), and (5), where multivalued maps play the role of a numerical tool. (2000). Primary 54C60; Secondary 37B30, 47H04, 54-02, 54H25, 55-04, 55M20, 58K05, 68U10, 91A10. Mathematics Subject Classification

Ascent, essential ascent, descent and essential descent for a linear relation in a linear space

“…A systematic bibliography on these applications including references to other and more recent contributions can be found in [10]. In view of the above remarks the attempt to generalize the existing results concerning the spectra of operators to general context of linear relations appears as natural.…”

Common fixed point iterations for a class of multi-valued mappings in $$\mathbf {CAT}(0)$$ spaces

“…The theory of fixed point for multi-valued mappings has become of great interest to many researchers, see, for instance, [3,4,8,11,13,15] and references therein. This concept contributes significantly in convex approximation, fractals, optimal control, digital imaging, and economics.…”

Quasi-Fredholm linear relations in Hilbert spaces

“…Applications of the fixed point theory of linear relations in: Game theory and Mathematical Economics; Discontinuous differential equations which occur in the Biological Sciences (for example, population in dynamics and epidemiology); Optimal control and Digital Imaging. (See [8] and the references therein).…”