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Multivalued Maps and Differential Inclusions: An Exploratory Study

Dylan Mitchell

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Published in Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications

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Multivalued maps and differential inclusions are important tools in the study of nonlinear analysis and control theory. They arise naturally in a variety of applications, including the modeling of physical systems, economic systems, and biological systems.

In this article, we will provide an overview of the theory of multivalued maps and differential inclusions. We will begin by introducing the basic definitions and concepts. We will then discuss some of the fundamental properties of multivalued maps and differential inclusions. Finally, we will present some applications of multivalued maps and differential inclusions to nonlinear analysis and control theory.

A multivalued map is a function that maps a set of points from one space to a set of points in another space. In other words, a multivalued map is a function that has multiple values for each input value.

Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications

by Pam Halter

4.5 out of 5

Language	:	English
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A differential inclusion is a differential equation that involves a multivalued map. In other words, a differential inclusion is an equation that describes the evolution of a system over time, where the system's dynamics are governed by a multivalued map.

Here are some of the basic definitions and concepts related to multivalued maps and differential inclusions:

Domain and range: The domain of a multivalued map is the set of points for which the map is defined. The range of a multivalued map is the set of all values that the map can take on.
Graph: The graph of a multivalued map is the set of all pairs of points (x, y) such that y is in the range of the map and x is in the domain of the map.
Inverse: The inverse of a multivalued map is a multivalued map that maps the range of the original map to its domain.
Fixed point: A fixed point of a multivalued map is a point that is mapped to itself by the map.
Solution: A solution to a differential inclusion is a function that satisfies the differential equation.

Multivalued maps and differential inclusions have a number of interesting and useful properties. Here are some of the most important properties:

Closedness: A multivalued map is closed if its graph is a closed set.
Compactness: A multivalued map is compact if its range is a compact set.
Convexity: A multivalued map is convex if its graph is a convex set.
Local Lipschitz continuity: A multivalued map is locally Lipschitz continuous if there exists a constant L such that for all x and y in the domain of the map, the distance between the values of the map at x and y is less than or equal to L times the distance between x and y.
Measurability: A multivalued map is measurable if its graph is a measurable set.

Multivalued maps and differential inclusions have a wide range of applications in nonlinear analysis and control theory. Here are some of the most important applications:

Fixed point theory: Multivalued maps can be used to study the existence and uniqueness of fixed points of nonlinear operators.
Topological degree theory: Multivalued maps can be used to study the topological degree of nonlinear operators.
Control theory: Differential inclusions can be used to model and analyze nonlinear control systems.
Economic theory: Multivalued maps can be used to model and analyze nonlinear economic systems.
Biological systems: Differential inclusions can be used to model and analyze nonlinear biological systems.

Multivalued maps and differential inclusions are important tools in the study of nonlinear analysis and control theory. They have a wide range of applications in a variety of fields. In this article, we have provided an overview of the theory of multivalued maps and differential inclusions. We have introduced the basic definitions and concepts, discussed some of the fundamental properties, and presented some applications.

Multivalued Maps And Differential Inclusions: Elements Of Theory And Applications

by Pam Halter

4.5 out of 5