SCHEDULE

Thursday 28th:

    16:00-16:45. Victoria Martín (Universidad de Sevilla)

    16:45-17:30. Shoham Sabach (The Technion - Israel Institute of Technology). Bregman firmly nonexpansive operators

                       Abstract: Nonexpansive operator theory and monotone mapping theory have transpired to be crucial in both the algorithmic design and analysis of optimization problems. It turns out that nonexpansive fixed point theory can be applied to the solution of diverse problems such as finding zeroes of monotone operators and solutions to certain evolution equations, and solving convex feasibility (CFP), variational inequality (VIP) and equilibrium problems (EP). In this talk I will talk on the well- known class of Bregman firmly nonexpansive operators and present results about the fixed points of these operators. In addition, I will consider a new result about the connection between the concepts of Bregman firmly nonexpansive operators and T-monotone mappings.

    17:30-18:00. Coffee break

    18:00-18:45. Aurora Fernández León (Universidad de Sevilla). Open fixed point problems on geodesic spaces

    18:45-19:30. Adriana Nicolae (Bobes-Bolyai University, Romania). On Ptolemy geodesic spaces 

                         Abstract: In a metric space \((X,d)\), the Ptolemy inequality says that

\[d(x,y)d(z,p) \le d(x,z)d(y,p) + d(x,p)d(y,z) \mbox{ for every } x,y,z,p \in X.\]

Ptolemy geometry falls very close to CAT(0)-like ones under certain regularity conditions. Our aim is to determine the weakest conditions under this is still possible. In this talk we will show that geodesic Ptolemy spaces with a continuous midpoint map are strictly convex. Moreover, we show that geodesic Ptolemy spaces with a uniformly continuous midpoint map are reflexive (from the metric point of view) and that in such a setting bounded sequences have unique asymptotic centers. 

Friday 29th:  

    9:30-10:15. David Ariza Ruiz (Universidad de Sevilla). Some results on weakly Zamfirescu mappings

                        Abstract: In this short talk we introduce the concept of weakly Zamfirescu maps and we study the existing independence between some types such as contractions maps, Kannan maps, Chatterjea maps and its respective weak concepts. Moreover, we extend Zamfirescu’s fixed point theorem to the class of weakly Zamfirescu maps and then we prove a continuation method, for this class of maps, extending various known results. Also, we obtain for weakly Zamfirescu mappings a simple expression of Cauchy modulus and modulus of convergence.

   10:15-11:00. Eyvind Martol Briseid (Technische Universität Darmstadt). On nonstandard methods and proof mining

                        Abstract: In later years one has been able to obtain improved versions of known theo- rems in e.g. metric fixed point theory and nonlinear ergodic theory by so-called "proof mining". This involves using logical tools from proof theory both to pre- dict that improved theorems exist, and as a guide when trying to prove these new theorems. Work in proof mining involves an interplay between concrete applications and theoretical groundwork – the latter amounting to strengthen- ing the logical metatheorems with which one can "mine" proofs in nonlinear analysis. Such metatheorems typically have the form:

“Suppose we can prove in a theory \(A\) a theorem of the form \(\forall x\exists y ∈ \mathbb{N} \phi (x, y)\). Then from the proof we can construct an explicit \(F\) such that \(\forall x\exists y ≤ F (x) \phi (x, y)\).”

A wide range of theorems in analysis are of the required form, and much effort has gone into making the logical theories A to which the metatheorems apply as strong as possible – since then the potential stock of theorems to consider is correspondingly larger. We will here discuss ongoing work with the aim of es- tablishing useful metatheorems for theories covering typical uses of nonstandard analysis, and we will present partial results in this direction.

    11:00-11:30. Coffee break

    11:30-12:15. Carlos A. Hernández Linares (Universidad de Sevilla). Renormings on \(\ell_1\): Fixed Point Property and Stability

                         Abstract: Let \((X,\|\cdot\|)\) be a Banach space and \(C\) a subset of \(X\).  We say that \(T\) is non-expansive if \(\|Tx - Ty\| \leq \|x - y\|\) for all \(x, y \in C\). A set \(C\) has the fixed point property if for every non-expansive mapping \(T: C \to C\) there exists \(x \in C\) such that \(Tx = x\).  It is said that a Banach space \(X\) satisfies the fixed point property (FPP) if every closed convex bounded set $C \subset X$ has the fixed point property.  

It is not difficult to show that the Banach spaces \(\ell_1\) and \(c_0\) endowed with their usual norms do not have the FPP.  A long time open question was whether all Banach spaces with the FPP were reflexive.  In 2008 P.K. Lin proved the following result:  Let \(\gamma_k = \frac{8^k}{1 + 8^k}\) for all \(k \in \mathbf{N}\) and define for \(x = (t_n) \in \ell_1\) the norm \(\||x\||=\sup_k \gamma_k \sum_{n=k}^\infty |t_k|\). Then the Banach space \((\ell_1,\||\cdot\||)\) has the FPP. This shows that \(\ell_1\) can be renormed to have the FPP and \((\ell_1, \|| \cdot \||)\) turned out to be the first known Banach space with the FPP and non-reflexive.

From P. K. Lin's  renorming it can be deduced that the FPP is not invariant under isomorphisms.  However for  many classical reflexive Banach spaces, the fixed point property is preserved under renormings whenever the Banach-Mazur distance between the original space and the renormed one is less than a certain constant. This is known as the problem of the stability and it can be formulated as follows: let \((X,\|\cdot\|)\) be a Banach space with the FPP and let \(\|| \cdot \||\) be an equivalent norm on \(X\). Does there exist some constant \(K>1\) such that if \(d((X,\|\cdot\|),(X,\||\cdot\||))<K\), then \((X,\||\cdot\||)\) has the FPP?

In this talk we will discover new renormings on \(\ell_1\) with the FPP and we will study if these renormings and the one given by P. K. Lin produce stability of the FPP.

    12:15-13:00. Elena Moreno Gálvez (Universidad Católica de Valencia). Fixed point results and examples for some generalized nonexpansive mappings

                        Abstract: A class of generalized nonexpansive mappings will be discussed in order to determine which known classes of mappings lie inside this new class and to obtain fixed point results. Some examples separating the studied classes will be given. As a particular case of such study, we will show that the normal structure implies the FPP for mappings satisfying condition (C) of Suzuki.