Multivalued Function - Applications

Applications

Multifunctions arise in optimal control theory, especially differential inclusions and related subjects as game theory, where the Kakutani fixed point theorem for multifunctions has been applied to prove existence of Nash equilibria (note: in the context of game theory, a multivalued function is usually referred to as a correspondence.) This amongst many other properties loosely associated with approximability of upper hemicontinuous multifunctions via continuous functions explains why upper hemicontinuity is more preferred than lower hemicontinuity.

Nevertheless, lower hemicontinuous multifunctions usually possess continuous selections as stated in the Michael selection theorem which provides another characterisation of paracompact spaces (see: E. Michael, Continuous selections I" Ann. of Math. (2) 63 (1956), and D. Repovs, P.V. Semenov, Ernest Michael and theory of continuous selections" arXiv:0803.4473v1). Other selection theorems, like Bressan-Colombo directional continuous selection, Kuratowski—Ryll-Nardzewski measurable selection, Aumann measurable selection, Fryszkowski selection for decomposable maps are important in optimal control and the theory of differential inclusions.

In physics, multivalued functions play an increasingly important role. They form the mathematical basis for Dirac's magnetic monopoles, for the theory of defects in crystal and the resulting plasticity of materials, for vortices in superfluids and superconductors, and for phase transitions in these systems, for instance melting and quark confinement. They are the origin of gauge field structures in many branches of physics.