Applications of Functional Analysis to the resolution of Nonlinear Problems, Convex Optimization and Lineability

PIs (subproject 1): Genaro López Acedo, Rafael Espínola

PI (subproject 2): Jesús García Falset

Abstract: This Project takes up several problems where the application of tools originally provided by Functional Analysis is fundamental for the approaches we follow. A common feature to these problems is the relevance of the geometry of the environment spaces as well as the properties of the operators under consideration. 
We focus on some of the most relevant problems concerning the existence of fixed points of nonexpansive mappings defined on subsets of metric spaces. Among these, we consider problems such as that of stability of the Fixed Point Property, the study of the Fixed Point Property for nonreflexive Banach spaces and the existence of fixed points on unbounded domains. This theory requires of very deep Functional Analysis tools and usually gives an interplay between the geometry of Banach spaces and the existence of fixed points. A very relevant area of research of our group has also been the study in metric spaces of geometric properties associated to the existence of fixed points in Banach spaces. The most suitable metric spaces for such developments are hyperbolic and geodesic metric spaces which include, in particular, Hadamard and Hilbert manifolds. 
These results provide with sufficient conditions to guarantee the existence  of a solution for integral and differential equations, as well as a better understanding of the asymptotic behaviour of such solutions. Existence and approximation to fixed point results find a natural field of application in optimization problems too. Even though these questions have traditionally been stated in the framework of Banach spaces, one of the goals of our research is to prove that a powerful theory of functions and convex sets may also be developed on geodesic spaces. 
We also deal with the study of lineability of families of mathematical objects with a given chaotic behaviour. In this topic we find problems such as the search of linear and/or algebraic structures within sets which do not enjoy these properties themselves. The great development of Functional Analysis enables us to apply a large number of techniques in several contexts. Our goal here is to investigate topological, linear and algebraic structures of families of hypercyclic  functions with respect to operators defined on spaces of analytic functions, of sets of holomorphic functions which admit no prolongation outside the boundary of the domain, of families of infinitely differentiable functions with given zeros, of families of measurable and surjective functions in extreme degree, and of sets of surjective holomorphic functions all around the boundary. 
The Project falls within the area of Fundamental Mathematics. However its results find applications in different fields of mathematics as differential and integral equations, theory of approximation, game theory, as well as in some other knowledge areas as economy, biology, image recovering,… 
The research team consists of 15 doctors from the Universities of León, Seville and Valencia.