Best proximity points for proximal contractions

In this paper we improve and extend some best proximity point results concerning the so- called proximal contractions. Specifically, compactness assumptions under the sets A and B are removed to consider completeness conditions instead

Best Proximity Pair Theorems for Noncyclic Mappings in Banach and Metric Spaces

Let A and B be two nonempty subsets of a metric space X. A mapping T : A[B ! A[B is said to be noncyclic if T(A) A and T(B) B. For such a mapping, a pair (x; y) 2 A B such that Tx = x, Ty = y and d(x; y) = dist(A;B) is called a best proximity pair. In this paper we give some best proximity pair results for noncyclic mappings under certain contractive conditions

On best proximity points in metric and Banach spaces

In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We do it from two different approaches, leading each one of them to different results which complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT(0) spaces where we study the particular behavior of these spaces regarding the problems we are concerned with

CAT(k)-spaces, weak convergence and fixed points

In this paper we show that some of the recent results on ¯xed point for CAT(0) spaces still hold true for CAT(1) spaces, and so for any CAT(k) space, under natural boundedness conditions. We also introduce a new notion of convergence in geodesic spaces which is related to the ¢-convergence and applied to study some aspects on the geometry of CAT(0) spaces. At this point, two recently posed questions in [12] (W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (12) (2008), 3689-3696) are answered in the negative. The work ¯nishes with the study of the Lif¸sic characteristic and property (P) of Lim-Xu to derive ¯xed point results for uniformly lipschitzian mappings in CAT(k) spaces. A conjecture raised in [4] (S. Dhompongsa, W.A. Kirk and B. Sims, Fixed points of uniformly lipschitzian mappings, Nonlinear Anal., 65 (2006), 762{772) on the Lif¸sic characteristic function of CAT(k) spaces is solved in the positive

Fixed points of single- and set-valued mappings in uniformly convex metric spaces with no metric convexity

We study the existence of fixed points and convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces. We also study the existence of fixed points for setvalued nonexpansive mappings in the same class of spaces. Our results do not assume convexity of the metric which makes a big difference when studying the existence of fixed points for set-valued mappings.Ministerio de Ciencia y TecnologíaJunta de Antalucí

A CASE STUDY ON HOW PRIMARY-SCHOOL IN-SERVICE TEACHERS CONJECTURE AND PROVE: AN APPROACH FROM THE MATHEMATICAL COMMUNITY

This paper studies how four primary-school in-service teachers develop the mathematical practices of conjecturing and proving. From the consideration of professional development as the legitimate peripheral participation in communities of practice, these teachers’ mathematical practices have been characterised by using a theoretical framework (consisting of categories of activities) that describes and explains how a research mathematician develops these two mathematical practices. This research has adopted a qualitative methodology and, in particular, a case study methodological approach. Data was collected in a working session on professional development while the four participants discussed two questions that invoked the development of the mathematical practices of conjecturing and proving. The results of this study show the significant presence of informal activities when the four participants conjecture, while few informal activities have been observed when they strive to prove a result. In addition, the use of examples (an informal activity) differs in the two practices, since examples support the conjecturing process but constitute obstacles for the proving process. Finally, the findings are contrasted with other related studies and several suggestions are presented that may be derived from this work to enhance professional development

Caracterizando cómo conjeturan los investigadores en matemáticas: un estudio de caso

Este trabajo se enmarca en la rama de investigación en educación matemática que estudia las actividades matemáticas que los investigadores en matemáticas desarrollan al construir conocimiento matemático. En concreto, tiene por objeto avanzar en la caracterización de la práctica matemática de conjeturar de esta comunidad de profesionales. Para ello, se analiza qué usa y qué crea (según Rasmussen et al., 2005) una investigadora concreta del área de análisis matemático cuando construye conjeturas en su investigación. La metodología cualitativa que se sigue es el estudio de caso. Los resultados de este estudio muestran el relevante papel que juegan los ejemplos en la dimensión horizontal de la práctica matemática de conjeturar, destacando cómo se crean esos ejemplos y en qué momentos de la actividad investigadora se usan y se crean los ejemplos

Characterising pre-service primary teachers’ discursive activity when defining

2021/FQM-226, Junta de Andalucía; PPIIV.4/2021/005, Universidad de Sevilla

Aspectos del discurso de estudiantes universitarios cuando construyen definiciones matemáticas

En este trabajo estudiamos el proceso de definir de estudiantes universitarios para profesor a través del análisis del discurso matemático. Concretamente, nos centraremos en los resultados que hemos obtenido al analizar dicho discurso cuando los estudiantes definen cuerpos geométricos tridimensionales. Nuestra investigación adopta una perspectiva sociocultural. En particular, el marco teórico que utilizamos es el de la comognición (commognition, unión de communication y cognition), propuesto por Sfard (2008), que considera que el aprendizaje matemático es un cambio en el discurso matemátic