Quasi-Metric Properties of the Dual Cone of an Asymmetric Normed Space

[EN] We obtain some quasi-metric properties of the dual cone of an asymmetric normed space. Thus, we prove that it is balanced, and hence its topology is completely regular. We also prove that it is complete in the sense of D. Doitchinov. These results generalize those obtained by Romaguera et al. in [18] because, in our study, the asymmetric normed space does not necessarily satisfy the T1 axiom. Moreover, we provide a class of asymmetric normed spaces whose dual cones are right K-sequentially complete. Finally, we represent an arbitrary asymmetric normed space as a function space by using the unit ball of its dual space.Alegre Gil, MC. (2022). Quasi-Metric Properties of the Dual Cone of an Asymmetric Normed Space. Results in Mathematics. 77(4):1-10. https://doi.org/10.1007/s00025-022-01720-611077

The weak topology in finite dimensional asymmetric normed spaces

[EN] In this paper we show that, in contrast to what happens in a normed space, the weak topology of a finite dimensional asymmetric normed space is not necessarily the same as the topology of the asymmetric norm. We provide a class of finite dimensional asymmetric normed spaces where both topologies coincide. We also prove that the weak topology of an infinite dimensional asymmetric normed space is strictly coarser than the topology of the asymmetric norm. (C) 2019 Elsevier B.V. All rights reserved.Alegre Gil, MC. (2019). The weak topology in finite dimensional asymmetric normed spaces. Topology and its Applications. 264:455-461. https://doi.org/10.1016/j.topol.2019.06.047S45546126

A note on phi-contractions in probabilistic and fuzzy metric spaces

[EN] In a recent paper Fang (2015) [1], J.X. Fang generalized a crucial fixed point theorem for probabilistic phi-contractions on complete Menger spaces due to Jachymski (2010) [3]. In this note we show that actually Fang s theorem is an easy consequence of Jachymski s theorem. We also observe that the proof of a fixed point theorem for complete metric spaces deduced by Fang from his main result is not correct and present a new proof of it.Alegre Gil, MC.; Romaguera Bonilla, S. (2017). A note on phi-contractions in probabilistic and fuzzy metric spaces. Fuzzy Sets and Systems. 313:119-121. https://doi.org/10.1016/j.fss.2016.06.014S11912131

A Caristi fixed point theorem for complete quasi-metric spaces by using mw-distances

[EN] In this paper we give a quasi-metric version of Caristi's fixed point theorem by using mw-distances. Our theorem generalizes a recent result obtained by Karapinar and Romaguera in [7]The authors thank the support of the Ministry of Economy and Competitiveness of Spain, Grants MTM2012-37894-C02-01 and MTM2015-64373-P (MINECO, FEDER,UE).Alegre Gil, MC.; Marín Molina, J. (2018). A Caristi fixed point theorem for complete quasi-metric spaces by using mw-distances. Fixed Point Theory. 19(1):25-32. https://doi.org/10.24193/fpt-ro.2018.1.03S253219

Revisiting Bianchini and Grandolfi Theorem in the Context of Modified omega-Distances

[EN] In this paper, we establish a proof for Bianchini and Grandolfi Theorem in the context of quasi-metric spaces via modified omega-distances. As consequences of our main results, we derive several existing fixed point theorems in the literature. Various examples are presented to illustrate our obtained results.Alegre Gil, MC.; Karapinar, E.; Marín Molina, J.; Tirado Peláez, P. (2019). Revisiting Bianchini and Grandolfi Theorem in the Context of Modified omega-Distances. Results in Mathematics. 74(4):1-9. https://doi.org/10.1007/s00025-019-1074-zS19744Alegre, C., Marín, J., Romaguera, S.: A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces. Fixed Point Theory Appl. 2014, 40 (2014)Alegre, C., Marín, J.: Modified w-distances on quasi-metric spaces and a fixed point theorem on complete quasi-metric spaces. Topol. Appl. 203, 32–41 (2016)Al-Homidan, S., Ansari, Q.H., Yao, J.C.: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Anal. Theory Methods Appl. 69, 126–139 (2008)Alsulami, H., Gulyaz, S., Karapinar, E., Erhan, I.M.: Fixed point theorems for a class of $$\alpha$$-admissible contractions and applications to boundary value problem. Abstr. Appl. Anal. Article ID 187031 (2014)Bianchini, R.M., Grandolfi, M.: Trasformazioni di tipo contrattivo generalizatto in uno spacio metrico. Atti della Accademia Nazionale dei Lincei 45, 212–216 (1969)Cobzas, S.: Functional Analysis in Asymmetric Normed Spaces. Birkhauser, Basel (2013)Kada, O., Suzuki, T., Takahashi, W.: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jpn. 44, 381–391 (1996)Karapınar, E., Romaguera, S., Tirado, P.: Contractive multivalued maps in terms of $$Q$$-functions on complete quasimetric spaces. Fixed Point Theory Appl. 2014, 53 (2014)Karapinar, E., Samet, B.: Generalized $$(\alpha -\psi )$$ contractive type mappings and related fixed point theorems with applications, Abstr. Appl. Anal. Article ID 793486 (2012)Künzi, H.P.A.: Nonsymmetric distances and their associated topologies: About the origins of basic ideas in the area of asymmetric topology. In: Aull, C.E., Lowen, R. (eds.) Handbook of the History of General Topology, vol. 3, pp. 853–968. Kluwer, Dordrecht (2001)Marín, J., Romaguera, S., Tirado, P.: Generalized contractive set-valued maps on complete preordered quasi-metric spaces. J. Funct. Spaces Appl. Article ID 269246, p. 6 (2013)Marín, J., Romaguera, S., Tirado, P.: Q-functons on quasimetric spaces and fixed points for multivalued maps. Fixed Point Theory Appl. Article ID 603861 (2011)Marín, J., Romaguera, S., Tirado, P.: Weakly contractive multivalued maps and w-distances on complete quasi-metric spaces. Fixed Point Theory Appl. 1, 1–9 (2011)Park, S.: On generalizations of the Ekeland-type variational principles. Nonlinear Anal. Theory Methods Appl. 39, 881–889 (2000)Proinov, P.D.: A generalization of the Banach contraction principle with high order of convergence of successive approximations nonlinear analysis. Theory Methods Appl. 67, 2361–2369 (2007)Rus, I.A.: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca (2001

The uniform boundedness theorem in asymmetric normed spaces

We obtain a uniform boundedness type theorem in the frame of asymmetric normed spaces. The classical result for normed spaces follows as a particular case.The authors are very grateful to the referee for many observations and comments that have allowed to improve the quality of the paper. The first two authors acknowledge the support of the Spanish Ministry of Science and Innovation under Grant MTM2009-12872-C02-01. The first author also acknowledges the support of the Spanish Ministry of Science and Innovation under Grant MTM2009-14483-C02-02.Alegre Gil, MC.; Romaguera Bonilla, S.; Veeramani, P. (2012). The uniform boundedness theorem in asymmetric normed spaces. Abstract and Applied Analysis. 2012:1-8. https://doi.org/10.1155/2012/809626S182012García-Raffi, L. M., Romaguera, S., & Sánchez-Pérez, E. A. (2002). Sequence spaces and asymmetric norms in the theory of computational complexity. Mathematical and Computer Modelling, 36(1-2), 1-11. doi:10.1016/s0895-7177(02)00100-0García-Raffi, L. M., & Sanchez-Pérez, R. (2003). The Dual Space of an Asymmetric Normed Linear Space. Quaestiones Mathematicae, 26(1), 83-96. doi:10.2989/16073600309486046Alegre, C., Ferrer, J., & Gregori, V. (1999). Acta Mathematica Hungarica, 82(4), 325-330. doi:10.1023/a:1006692309917Ferrer, J., Gregori, V., & Alegre, C. (1993). Quasi-uniform Structures in Linear Lattices. Rocky Mountain Journal of Mathematics, 23(3), 877-884. doi:10.1216/rmjm/1181072529Romaguera, S., & Sanchis, M. (2000). Semi-Lipschitz Functions and Best Approximation in Quasi-Metric Spaces. Journal of Approximation Theory, 103(2), 292-301. doi:10.1006/jath.1999.3439Alimov, A. R. (2001). Functional Analysis and Its Applications, 35(3), 176-182. doi:10.1023/a:1012370610709Rodríguez-López, J., Schellekens, M. P., & Valero, O. (2009). An extension of the dual complexity space and an application to Computer Science. Topology and its Applications, 156(18), 3052-3061. doi:10.1016/j.topol.2009.02.009Romaguera, S., Sánchez-Pérez, E. A., & Valero, O. (2006). The Dual Complexity Space as the Dual of a Normed Cone. Electronic Notes in Theoretical Computer Science, 161, 165-174. doi:10.1016/j.entcs.2006.04.031Romaguera, S., & Schellekens, M. P. (2002). Duality and quasi-normability for complexity spaces. Applied General Topology, 3(1), 91. doi:10.4995/agt.2002.2116Cobzaş, S. (2004). Separation of Convex Sets and Best Approximation in Spaces with Asymmetric Norm. Quaestiones Mathematicae, 27(3), 275-296. doi:10.2989/16073600409486100Alegre, C. (2008). Continuous operators on asymmetric normed spaces. Acta Mathematica Hungarica, 122(4), 357-372. doi:10.1007/s10474-008-8039-0Alegre, C., Ferrando, I., García-Raffi, L. M., & Sánchez Pérez, E. A. (2008). Compactness in asymmetric normed spaces. Topology and its Applications, 155(6), 527-539. doi:10.1016/j.topol.2007.11.004Borodin, P. A. (2001). Mathematical Notes, 69(3/4), 298-305. doi:10.1023/a:1010271105852García-Raffi, L. M. (2005). Compactness and finite dimension in asymmetric normed linear spaces. Topology and its Applications, 153(5-6), 844-853. doi:10.1016/j.topol.2005.01.014García-Raffi, L. M., Romaguera, S., & Sánchez-Pérez, E. A. (2009). The Goldstine Theorem for asymmetric normed linear spaces. Topology and its Applications, 156(13), 2284-2291. doi:10.1016/j.topol.2009.06.001Howes, N. R. (1995). Modern Analysis and Topology. Universitext. doi:10.1007/978-1-4612-0833-4Alegre, C., Ferrer, J., & Gregori, V. (1998). On a class of real normed lattices. Czechoslovak Mathematical Journal, 48(4), 785-792. doi:10.1023/a:102249992548

A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces

We obtain a fixed point theorem for generalized contractions on complete quasi-metric spaces, which involves w-distances and functions of Meir-Keeler and Jachymski type. Our result generalizes in various directions the celebrated fixed point theorems of Boyd and Wong, and Matkowski. Some illustrative examples are also given.The authors are grateful to the referees for several useful suggestions. They also thank the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.Alegre Gil, MC.; Marín Molina, J.; Romaguera Bonilla, S. (2014). A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces. Fixed Point Theory and Applications. 2014(40):1-8. https://doi.org/10.1186/1687-1812-2014-40S1820144

Adaptación de álgebra al EEES

[ES] Durante el curso 2010-11 se ha implantado la titulación de Grado en Ingeniería Informática en la Escuela Técnica Superior de Ingeniería Informática (ETSINF) de la Universidad Politécnica de Valencia (UPV). Álgebra es una de las asignaturas básicas de primer curso de este nuevo grado y sus contenidos constituyen una parte importante de la base de otras asignaturas dentro del nuevo plan de estudios. Dada la importancia que los conceptos de álgebra tienen en la formación de un graduado en informática, al programar la adaptación de la asignatura al EEES hemos considerado prioritario adecuar las metodologías docentes y las estrategias de evaluación para mejorar la adquisición de las competencias que aporta nuestra asignatura. En este trabajo presentamos las acciones realizadas durante el curso 2010-11: preparar material docente adecuado, introducir la evaluación a través de PoliformaT, realizar proyectos donde se resalta la parte práctica de la asignatura, a fin de mejorar el aprendizaje de nuestros alumnos.Sanabria-Codesal, E.; Alegre Gil, MC.; Bravo Villar, MP.; Esteban Romero, R.; Fuster Capilla, R.; Gasso Matoses, MT.; Martínez-Pastor, A.... (2011). Adaptación de álgebra al EEES. Instituto de Ciencias de la Educación de la Universidad de Alicante. 921-931. http://hdl.handle.net/10251/178210S92193