Multivalued function

In mathematics, a multivalued function (short form: multifunction; other names: many-valued function, set-valued function, set-valued map, point-to-set map, multi-valued map, multimap, correspondence, carrier) is a left-total relation (that is, every input is associated with at least one output) in which at least one input is associated with multiple (two or more) outputs.

In the strict sense, a well-defined function associates one, and only one, output to any particular input. The term "multivalued function" is, therefore, a misnomer because functions are single-valued. Multivalued functions often arise as inverses of functions that are not injective. Such functions do not have an inverse function, but they do have an inverse relation. The multivalued function corresponds to this inverse relation.

Examples

• Every real number greater than zero has two real square roots. The square roots of 4 are in the set {+2,−2}. The square root of 0 is 0.
• Each complex number except zero has two square roots, three cube roots, and in general n nth roots. The nth root of 0 is 0.
• The complex logarithm function is multiple-valued. The values assumed by $\log(a+bi)$ for real numbers $a$ and $b$ are $\log{\sqrt{a^2 + b^2}} + i\arg (a+bi) + 2 \pi n i$ for all integers $n$.
• Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic. We have

$\tan\left({\textstyle\frac{\pi}{4}}\right) = \tan\left({\textstyle\frac{5\pi}{4}}\right) = \tan\left({\textstyle\frac{-3\pi}{4}}\right) = \tan\left({\textstyle\frac{(2n+1)\pi}{4}}\right) = \cdots = 1.$

As a consequence, arctan(1) is intuitively related to several values: π/4, 5π/4, −3π/4, and so on. We can treat arctan as a single-valued function by restricting the domain of tan x to -π/2 < x < π/2 – a domain over which tan x is monotonically increasing. Thus, the range of arctan(x) becomes -π/2 < y < π/2. These values from a restricted domain are called principal values.

• The indefinite integral can be considered as a multivalued function. The indefinite integral of a function is the set of functions whose derivative is that function. The constant of integration follows from the fact that the derivative of a constant function is 0.

These are all examples of multivalued functions that come about from non-injective functions. Since the original functions do not preserve all the information of their inputs, they are not reversible. Often, the restriction of a multivalued function is a partial inverse of the original function.

Multivalued functions of a complex variable have branch points. For example, for the nth root and logarithm functions, 0 is a branch point; for the arctangent function, the imaginary units i and −i are branch points. Using the branch points, these functions may be redefined to be single-valued functions, by restricting the range. A suitable interval may be found through use of a branch cut, a kind of curve that connects pairs of branch points, thus reducing the multilayered Riemann surface of the function to a single layer. As in the case with real functions, the restricted range may be called principal branch of the function.

Set-valued analysis

Set-valued analysis is the study of sets in the spirit of mathematical analysis and general topology.

Instead of considering collections of only points, set-valued analysis considers collections of sets. If a collection of sets is endowed with a topology, or inherits an appropriate topology from an underlying topological space, then the convergence of sets can be studied.

Much of set-valued analysis arose through the study of mathematical economics and optimal control, partly as a generalization of convex analysis; the term "variational analysis" is used by authors such as R. T. Rockafellar and Roger Wets, Jon Borwein and Adrian Lewis, and Boris Mordukhovich. In optimization theory, the convergence of approximating subdifferentials to a subdifferential is important in understanding necessary or sufficient conditions for any minimizing point.

There exist set-valued extensions of the following concepts from point-valued analysis: continuity, differentiation, integration, implicit function theorem, contraction mappings, measure theory, fixed-point theorems, optimization, and topological degree theory.

Equations are generalized to inclusions.

Types of multivalued functions

One can differentiate many continuity concepts, primarily closed graph property and upper and lower hemicontinuity. (One should be warned that often the terms upper and lower semicontinuous are used instead of upper and lower hemicontinuous reserved for the case of weak topology in domain; yet we arrive at the collision with the reserved names for upper and lower semicontinuous real-valued function). There exist also various definitions for measurability of multifunction.

History

The practice of allowing function in mathematics to mean also multivalued function dropped out of usage at some point in the first half of the twentieth century. Some evolution can be seen in different editions of A Course of Pure Mathematics by G. H. Hardy, for example. It probably persisted longest in the theory of special functions, for its occasional convenience.

The theory of multivalued functions was fairly systematically developed for the first time in Claude Berge's Topological spaces (1963).

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 among 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.^[1][2] Other selection theorems, like Bressan-Colombo directional continuous selection, Kuratowski and Ryll-Nardzewski measurable selection theorem, 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 crystals 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.

Contrast with

• Bijection
• Injection
• Surjection

References

1. Ernest Michael (Mar 1956). "Continuous Selections. I" (PDF). Annals of Mathematics. Second Series 63 (2): 361—382.
2. Dušan Repovš; P.V. Semenov (2008). "Ernest Michael and theory of continuous selections". Topology Appl. 155 (8): 755—763. arXiv:0803.4473v1.

• C. D. Aliprantis and K. C. Border, Infinite dimensional analysis. Hitchhiker's guide, Springer-Verlag Berlin Heidelberg, 2006
• J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer Academic Publishers, 2003
• J.-P. Aubin and A. Cellina, Differential Inclusions, Set-Valued Maps And Viability Theory, Grundl. der Math. Wiss. 264, Springer - Verlag, Berlin, 1984
• J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Basel, 1990
• K. Deimling, Multivalued Differential Equations, Walter de Gruyter, 1992
• A. Geletu, Introduction to Topological Spaces and Set-Valued Maps (Lecture notes), Ilmenau University of Technology, 2006
• H. Kleinert, Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation, World Scientific (Singapore, 2008) (also available online)
• H. Kleinert, Gauge Fields in Condensed Matter, Vol. I: Superflow and Vortex Lines, 1–742, Vol. II: Stresses and Defects, 743–1456, World Scientific, Singapore, 1989 (also available online: Vol. I and Vol. II)
• D. Repovš and P.V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Academic Publishers, Dordrecht 1998
• E. U. Tarafdar and M. S. R. Chowdhury, Topological methods for set-valued nonlinear analysis, World Scientific, Singapore, 2008

See also

• Partial function
• correspondence
• Fat link, a one-to-many hyperlink
• Interval finite element
• Hans Rådström