| Description |
This dissertation consists of two parts that focus on two interrelated areas of Applied Mathematics. The first part explores fundamental properties and applications of functions with values in L-spaces. The second part is connected to Approximation Theory and dives deeper into the analysis of functions with values in specific classes of L-spaces (in particular, L-spaces of sets). In the first project devoted to the theory and numerical methods for the solution of integral equations, we explore linear Volterra and Fredholm integral equations for functions with values in L-spaces (which are generalizations of set-valued and fuzzy-valued functions). In this study, we prove the existence and uniqueness of the solution for such equations, suggest algorithms for finding approximate solutions, and study their convergence. The exploration of these equations is of great importance given the wide variety of their applications in biology (population modeling), physics (heat conduction), and engineering (feedback systems), among others. We extend the aforementioned results of existence and uniqueness to nonlinear equations. In addition, we study the dependence of solutions of such equations on variations in the data. In order to be able to better analyze the convergence of the suggested algorithms for the solutions of integral equations, we develop new results on the approximation of functions with values in L-spaces by adapted linear positive operators (Bernstein, Schoenberg, modified Schoenberg operators, and piecewise linear interpolation). The second project is devoted to problems of interpolation by generalized polynomials and splines for functions whose values lie in a specific L-space, namely a space of sets. Because the structure of such a space is richer than the structure of a general L-space, we have additional tools available (e.g., the support function of a set) which allow us to obtain deeper results for the approximation and interpolation of set-valued functions. We are working on defining various methods of approximation based on the support function of a set. Questions related to error estimates of the approximation of set-valued functions by those novel methods are also investigated. |
