A characterization of Smyth complete quasi-metric spaces via Caristi's fixed point theorem

We obtain a quasi-metric generalization of Caristi's fixed point theorem for a kind of complete quasi-metric spaces. With the help of a suitable modification of its proof, we deduce a characterization of Smyth complete quasi-metric spaces which provides a quasi-metric generalization of the well-known characterization of metric completeness due to Kirk. Some illustrative examples are also given. As an application, we deduce a procedure which allows to easily show the existence of solution for the recurrence equation of certain algorithms.The authors are grateful to the reviewers for several suggestions which have allowed to improve the first version of the paper. This research is supported by the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.Romaguera Bonilla, S.; Tirado Peláez, P. (2015). A characterization of Smyth complete quasi-metric spaces via Caristi's fixed point theorem. Fixed Point Theory and Applications. 2015:183. https://doi.org/10.1186/s13663-015-0431-1S2015:183Cobzaş, S: Functional Analysis in Asymmetric Normed Spaces. Springer, Basel (2013)Künzi, HPA: Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology. In: Aull, CE, Lowen, R (eds.) Handbook of the History of General Topology, vol. 3, pp. 853-968. Kluwer Academic, Dordrecht (2001)Reilly, IL, Subrhamanyam, PV, Vamanamurthy, MK: Cauchy sequences in quasi-pseudo-metric spaces. Monatshefte Math. 93, 127-140 (1982)Künzi, HPA, Schellekens, MP: On the Yoneda completion of a quasi-metric spaces. Theor. Comput. Sci. 278, 159-194 (2002)Romaguera, S, Valero, O: Domain theoretic characterisations of quasi-metric completeness in terms of formal balls. Math. Struct. Comput. Sci. 20, 453-472 (2010)Künzi, HPA: Nonsymmetric topology. In: Proc. Szekszárd Conf. Bolyai Society of Math. 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Fixed point results for generalized cyclic contraction mappings in partial metric spaces

Rus (Approx. Convexity 3:171–178, 2005) introduced the concept of cyclic contraction mapping. P˘acurar and Rus (Nonlinear Anal. 72:1181–1187, 2010) proved some fixed point results for cyclic φ-contraction mappings on a metric space. Karapinar (Appl. Math. Lett. 24:822–825, 2011) obtained a unique fixed point of cyclic weak φ- contraction mappings and studied well-posedness problem for such mappings. On the other hand, Matthews (Ann. New York Acad. Sci. 728:183–197, 1994) introduced the concept of a partial metric as a part of the study of denotational semantics of dataflow networks. He gave a modified version of the Banach contraction principle, more suitable in this context. In this paper, we initiate the study of fixed points of generalized cyclic contraction in the framework of partial metric spaces. We also present some examples to validate our results.S. Romaguera acknowledges the support of the Ministry of Science and Innovation of Spain, grant MTM2009-12872-C02-01.Abbas, M.; Nazir, T.; Romaguera Bonilla, S. (2012). Fixed point results for generalized cyclic contraction mappings in partial metric spaces. Revista- Real Academia de Ciencias Exactas Fisicas Y Naturales Serie a Matematicas. 106(2):287-297. https://doi.org/10.1007/s13398-011-0051-5S2872971062Abdeljawad T., Karapinar E., Tas K.: Existence and uniqueness of a common fixed point on partial metric spaces. Appl. Math. Lett. 24(11), 1894–1899 (2011). doi: 10.1016/j.aml.2011.5.014Altun, I., Erduran A.: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl. article ID 508730 (2011). doi: 10.1155/2011/508730Altun I., Sadarangani K.: Corrigendum to “Generalized contractions on partial metric spaces” [Topology Appl. 157 (2010), 2778–2785]. Topol. 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Q-functions on quasimetric spaces and fixed points for multivalued maps

[EN] We discuss several properties of Q-functions in the sense of Al-Homidan et al.. In particular, we prove that the partial metric induced by any T0 weighted quasipseudometric space is a Q-function and show that both the Sorgenfrey line and the Kofner plane provide signi¿cant examples of quasimetric spaces for which the associated supremum metric is a Q-function. In this context we also obtain some ¿xed point results for multivalued maps by using Bianchini-Grandol¿ gauge functions.The authors thank one of the reviewers for suggesting the inclusion of a concrete example to which Theorem 3.3 applies. They acknowledge the support of the Spanish Ministry of Science and Innovation, Grant no. MTM2009-12872-C02-01.Marín Molina, J.; Romaguera Bonilla, S.; Tirado Peláez, P. (2011). Q-functions on quasimetric spaces and fixed points for multivalued maps. Fixed Point Theory and Applications. 2011:1-10. https://doi.org/10.1155/2011/603861S1102011Ekeland, I. (1979). Nonconvex minimization problems. Bulletin of the American Mathematical Society, 1(3), 443-475. doi:10.1090/s0273-0979-1979-14595-6Al-Homidan, S., Ansari, Q. H., & Yao, J.-C. (2008). Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Analysis: Theory, Methods & Applications, 69(1), 126-139. doi:10.1016/j.na.2007.05.004Alegre, C. (2008). Continuous operators on asymmetric normed spaces. Acta Mathematica Hungarica, 122(4), 357-372. doi:10.1007/s10474-008-8039-0ALI-AKBARI, M., HONARI, B., POURMAHDIAN, M., & REZAII, M. M. (2009). The space of formal balls and models of quasi-metric spaces. Mathematical Structures in Computer Science, 19(2), 337-355. doi:10.1017/s0960129509007439Cobzaş, S. (2009). Compact and precompact sets in asymmetric locally convex spaces. Topology and its Applications, 156(9), 1620-1629. doi:10.1016/j.topol.2009.01.004García-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.2009.06.001García-Raffi, L. M., Romaguera, S., & Schellekens, M. P. (2008). Applications of the complexity space to the General Probabilistic Divide and Conquer Algorithms. Journal of Mathematical Analysis and Applications, 348(1), 346-355. doi:10.1016/j.jmaa.2008.07.026Heckmann, R. (1999). Applied Categorical Structures, 7(1/2), 71-83. doi:10.1023/a:1008684018933Romaguera, S., & Schellekens, M. (2005). Partial metric monoids and semivaluation spaces. Topology and its Applications, 153(5-6), 948-962. doi:10.1016/j.topol.2005.01.023Romaguera, S., & Tirado, P. (2011). The complexity probabilistic quasi-metric space. Journal of Mathematical Analysis and Applications, 376(2), 732-740. doi:10.1016/j.jmaa.2010.11.056ROMAGUERA, S., & VALERO, O. (2009). A quantitative computational model for complete partial metric spaces via formal balls. Mathematical Structures in Computer Science, 19(3), 541-563. doi:10.1017/s0960129509007671ROMAGUERA, S., & VALERO, O. (2010). Domain theoretic characterisations of quasi-metric completeness in terms of formal balls. Mathematical Structures in Computer Science, 20(3), 453-472. doi:10.1017/s0960129510000010Schellekens, M. P. (2003). A characterization of partial metrizability: domains are quantifiable. Theoretical Computer Science, 305(1-3), 409-432. doi:10.1016/s0304-3975(02)00705-3Waszkiewicz, P. (2003). Applied Categorical Structures, 11(1), 41-67. doi:10.1023/a:1023012924892WASZKIEWICZ, P. (2006). Partial metrisability of continuous posets. Mathematical Structures in Computer Science, 16(02), 359. doi:10.1017/s0960129506005196Proinov, P. D. (2007). A generalization of the Banach contraction principle with high order of convergence of successive approximations. Nonlinear Analysis: Theory, Methods & Applications, 67(8), 2361-2369. doi:10.1016/j.na.2006.09.008Reilly, I. L., Subrahmanyam, P. V., & Vamanamurthy, M. K. (1982). Cauchy sequences in quasi-pseudo-metric spaces. Monatshefte f�r Mathematik, 93(2), 127-140. doi:10.1007/bf01301400Rakotch, E. (1962). A note on contractive mappings. Proceedings of the American Mathematical Society, 13(3), 459-459. doi:10.1090/s0002-9939-1962-0148046-1Proinov, P. D. (2010). New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems. Journal of Complexity, 26(1), 3-42. doi:10.1016/j.jco.2009.05.00

Localic completion of uniform spaces

We extend the notion of localic completion of generalised metric spaces by Steven Vickers to the setting of generalised uniform spaces. A generalised uniform space (gus) is a set X equipped with a family of generalised metrics on X, where a generalised metric on X is a map from the product of X to the upper reals satisfying zero self-distance law and triangle inequality. For a symmetric generalised uniform space, the localic completion lifts its generalised uniform structure to a point-free generalised uniform structure. This point-free structure induces a complete generalised uniform structure on the set of formal points of the localic completion that gives the standard completion of the original gus with Cauchy filters. We extend the localic completion to a full and faithful functor from the category of locally compact uniform spaces into that of overt locally compact completely regular formal topologies. Moreover, we give an elementary characterisation of the cover of the localic completion of a locally compact uniform space that simplifies the existing characterisation for metric spaces. These results generalise the corresponding results for metric spaces by Erik Palmgren. Furthermore, we show that the localic completion of a symmetric gus is equivalent to the point-free completion of the uniform formal topology associated with the gus. We work in Aczel's constructive set theory CZF with the Regular Extension Axiom. Some of our results also require Countable Choice.Comment: 39 page

Computable decision making on the reals and other spaces via partiality and nondeterminism

Though many safety-critical software systems use floating point to represent real-world input and output, programmers usually have idealized versions in mind that compute with real numbers. Significant deviations from the ideal can cause errors and jeopardize safety. Some programming systems implement exact real arithmetic, which resolves this matter but complicates others, such as decision making. In these systems, it is impossible to compute (total and deterministic) discrete decisions based on connected spaces such as $\mathbb{R}$. We present programming-language semantics based on constructive topology with variants allowing nondeterminism and/or partiality. Either nondeterminism or partiality suffices to allow computable decision making on connected spaces such as $\mathbb{R}$. We then introduce pattern matching on spaces, a language construct for creating programs on spaces, generalizing pattern matching in functional programming, where patterns need not represent decidable predicates and also may overlap or be inexhaustive, giving rise to nondeterminism or partiality, respectively. Nondeterminism and/or partiality also yield formal logics for constructing approximate decision procedures. We implemented these constructs in the Marshall language for exact real arithmetic.Comment: This is an extended version of a paper due to appear in the proceedings of the ACM/IEEE Symposium on Logic in Computer Science (LICS) in July 201

The connected Vietoris powerlocale

The connected Vietoris powerlocale is defined as a strong monad Vc on the category of locales. VcX is a sublocale of Johnstone's Vietoris powerlocale VX, a localic analogue of the Vietoris hyperspace, and its points correspond to the weakly semifitted sublocales of X that are “strongly connected”. A product map ×:VcX×VcY→Vc(X×Y) shows that the product of two strongly connected sublocales is strongly connected. If X is locally connected then VcX is overt. For the localic completion of a generalized metric space Y, the points of are certain Cauchy filters of formal balls for the finite power set with respect to a Vietoris metric. \ud Application to the point-free real line gives a choice-free constructive version of the Intermediate Value Theorem and Rolle's Theorem. \ud \ud The work is topos-valid (assuming natural numbers object). Vc is a geometric constructio

The extensional realizability model of continuous functionals and three weakly non-constructive classical theorems

We investigate wether three statements in analysis, that can be proved classically, are realizable in the realizability model of extensional continuous functionals induced by Kleene's second model $K_2$. We prove that a formulation of the Riemann Permutation Theorem as well as the statement that all partially Cauchy sequences are Cauchy cannot be realized in this model, while the statement that the product of two anti-Specker spaces is anti-Specker can be realized