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Covering mappings and their applications in functional equations, difference equations and optimization

Final Report Summary - COVMAPS (Covering mappings and their applications in functional equations, difference equations and optimization)

The project was aimed at the investigation of certain type of equations and inequalities in abstract metric and functional spaces using the covering mapping theory. There were declared two main objects of the research: properties of covering mappings and solvability problems for certain class of equations and inclusions in abstract metric and functional spaces. The objectives of the Project for the reporting period consists in the successful termination of the following tasks
1. The properties of covering mappings. The task includes: to investigate the properties of covering mappings in generalized metric spaces, to obtain sufficient solvability conditions for inclusions defined by conditionally covering set-valued mappings in metric spaces, to obtain uniform estimates of the distance from points of the domain of two set-valued mappings to the coincidence points set of these mappings.
2. Difference equations, functional equations and inequalities. The task includes: to consider implicit difference equations, to obtain new solvability conditions, to investigate the question on equilibrium existence and stability, to consider a certain type of equations in space of Lebesgue measurable essentially bounded functions, to derive solvability conditions for these equations, to consider the system of convex inequalities in the space of measurable essentially bounded functions, to derive necessary and sufficient solvability conditions for this system.
3. Implicit control systems with mixed constraints. The task includes: to consider control systems defined by Volterra equations and differential inclusions with mixed constraints defined by inclusions, to obtain global solvability condition for this control system, to obtain solution estimates.
In the framework of the project the following results were obtained.
1. There were investigated the properties of covering mappings in generalized metric spaces. In order to derive solvability conditions for inclusions defined by conditionally covering set-valued mappings there were considered the coincidence point problem for two-set-valued mappings. The known results for this problem state that under additional assumptions a locally covering and a Lipschitz-like in a neighborhood of an initial point set-valued mappings have a coincidence point, and provide a linear estimate from the initial point to the coincidence points set. There arises a natural question if this estimate is valid for any point from a neighborhood of initial point, i.e. if a uniform estimate holds. It was shown that under the same assumptions as in known coincidence points theorems the uniform estimate holds. There were obtained exact estimate for the radius of the neighborhood in which the uniform estimates are valid. This result allowed to derive solvability conditions for inclusions defined by conditionally covering set-valued mappings and new sufficient conditions for double fixed points existence.
2. There were considered implicit difference equations. For these equations there were obtained solvability conditions in terms of covering mappings, the conditions for equilibrium existence and stability. There were studied equations in space of Lebesgue measurable essentially bounded functions. The application of covering mapping theory allowed to derive solvability conditions for these equations and solution estimates. A system of strict and non-strict convex inequalities in the space of essentially bounded Lebesgue measurable functions was considered. The goal was to derive necessary and sufficient solvability conditions for this class of systems in the form of the theorem of the alternative analogous to the Gordan theorem, Motzkin theorem, and Ky Fan theorem for finite dimensional linear and convex systems. The application of certain results from nonlinear analysis, including covering mappings theory and the results on uniform estimates allowed the researcher to obtain a very general regularity conditions for the considered systems of inequalities. As a consequence, the desired solvability condition in the form of the theorem of the alternative was obtained.
3. The control system with the dynamics defined by an implicit differential inclusion and mixed constraints was considered. This system was reduced to the problem of double fixed points of two set-valued mappings acting in the spaces of measurable essentially bounded functions. The subsequent application of a theorem on double fixed point existence allowed to derive the solution to the initial problem. Namely, there were obtained the sufficient conditions for solvability of the considered control system and solution estimates. An analogous results were obtained for control system with the Volterra-type dynamics and mixed constraints.
The results obtained may have application in various areas of mathematics. The theorems on uniform estimates can be used to investigate differential inclusions, control systems, difference equations, etc. An example of such applications was demonstrated by the researcher, when he derived solvability conditions for inclusions, difference equations, functional equations, control systems and convex inequalities. The mentioned results on convex systems solvability can be employed in various investigations for establishing properties of systems of linear and convex inequalities and for deriving optimality criteria for extremal problems. In particular, they can be used to obtain optimality criteria for a certain type of convex extremal problem in functional spaces, to develop the duality theory for a certain type of optimal control problems with linear and convex phase constraints, to develop controllability criterion for some classes of control systems, to prove optimality criterions for a certain type vector optimization problems. The mentioned problems naturally appear in modern engineering practice since it frequently involves optimization. So, the obtained results are useful and interesting not only from the mathematical point of view.

As it was mentioned above the results of the COVMAPS project are in the realm of mathematical analysis with relevance to control and optimization, and, therefore, they are of interest to the mathematical and engineering research communities. Their pertinence is explained by the challenges arising in the rapid development of the modern technologies as they require increasingly complicated mathematical tools. The results obtained in the project reinforce the mathematical apparatus that is used in various applications.