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Covering mappings in mathematical optimisation

Many problems arising in optimisation and set-valued analysis lead to the investigation of covering properties of mappings. EU-funded mathematicians looked into mappings in metric and functional spaces in the search for solutions to equations and inequalities.
Covering mappings in mathematical optimisation
The properties of mappings, together with metric regularity and Lipschitzian continuity, offer one of the main tools for classifying spaces. In particular, the property of covering is used to investigate fixed and coincidence points of mappings and to obtain extremum conditions for optimisation problems.

In the first part of the project COVMAPS (Covering mappings and their applications in functional equations, difference equations and optimization), mathematicians identified sufficient conditions for single- and set-valued mappings to be locally covering and have a continuous inverse mapping.

Properties of covering mappings were then investigated in generalised metric spaces. To derive solvability conditions in terms of covering mappings, the team considered the coincidence point problem for two set-valued mappings. The results of this problem indicated how to confirm the existence of double fixed points.

The second part of COVMAPS focused on difference and functional equations and systems of equations subject to constraints. The mathematicians searched for the solvability conditions in generalised metric spaces and investigated their solutions' properties in the case of continuous, measurable and convex functions.

Following, project members examined an optimal control problem in which ordinary differential equations had been replaced by Volterra integral equations and mixed constraints. This new system of equations was reduced to a problem of double fixed points of two set-valued mappings.

In other words, the COVMAPS team obtained sufficient conditions to solve the considered control problem and estimate its solutions. Similar problems naturally appear in modern engineering where designs based on numerical models frequently require optimisation.

COVMAPS results are in the realm of mathematical analysis. Still, the newly developed mathematical tools are of interest to the engineering research community as well.

Related information


Covering mappings, mathematical optimisation, COVMAPS, functional equations, difference equations
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