A new model based on a fuzzy quasi-metric type Baire applied to analysis of complexity

[EN] We analyze the complexity of an expoDC algorithm by deducing the existence of solution for the recurrence inequation associated to this algorithm by means of techniques of Denotational Semantics in the context of fuzzy quasi-metric spaces. The fuzzy quasi-metrics provide an additional parameter "t" such that a suitable use of this ingredient gives rise to extra information on the involved computational process. This analysis is done by means of a fuzzy quasi-metric version of the Banach contraction principle on a space of partial functions endowed by a suitable adaptation of the Baire quasi-metric.This research is supported by the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01 and by Universitat Politecnica de Valencia, Grant PAID-06-12-SP20120471.Tirado Peláez, P. (2014). A new model based on a fuzzy quasi-metric type Baire applied to analysis of complexity. Journal of Intelligent and Fuzzy Systems. 27:2545-2550. https://doi.org/10.3233/IFS-141228S254525502

Contractive Maps and Complexity Analysis in Fuzzy Quasi-Metric Spaces

En los últimos años se ha desarrollado una teoría matemática con propiedades robustas con el fin de fundamentar la Ciencia de la Computación. En este sentido, un avance significativo lo constituye el establecimiento de modelos matemáticos que miden la "distancia" entre programas y entre algoritmos, analizados según su complejidad computacional. En 1995, M. Schellekens inició el desarrollo de un modelo matemático para el análisis de la complejidad algorítmica basado en la construcción de una casi-métrica definida en el espacio de las funciones de complejidad, proporcionando una interpretación computacional adecuada del hecho de que un programa o algoritmo sea más eficiente que otro en todos su "inputs". Esta información puede extraerse en virtud del carácter asimétrico del modelo. Sin embargo, esta estructura no es aplicable al análisis de algoritmos cuya complejidad depende de dos parámetros. Por tanto, en esta tesis introduciremos un nuevo espacio casi-métrico de complejidad que proporcionará un modelo útil para el análisis de este tipo de algoritmos. Por otra parte, el espacio casi-métrico de complejidad no da una interpretación computacional del hecho de que un programa o algoritmo sea "sólo" asintóticamente más eficiente que otro. Los espacios casi-métricos difusos aportan un parámetro "t", cuya adecuada utilización puede originar una información extra sobre el proceso computacional a estudiar; por ello introduciremos la noción de casi-métrica difusa de complejidad, que proporciona un modelo satisfactorio para interpretar la eficiencia asintótica de las funciones de complejidad. En este contexto extenderemos los principales teoremas de punto fijo en espacios métricos difusos , utilizando una determinada noción de completitud, y obtendremos otros nuevos. Algunos de estos teoremas también se establecerán en el contexto general de los espacios casi-métricos difusos intuicionistas, de lo que resultarán condiciones de contracción menos fuertes. Los resultados obtTirado Peláez, P. (2008). Contractive Maps and Complexity Analysis in Fuzzy Quasi-Metric Spaces [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/2961Palanci

Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results

[EN] With the help of C-contractions having a fixed point, we obtain a characterization of complete fuzzy metric spaces, in the sense of Kramosil and Michalek, that extends the classical theorem of H. Hu (see "Am. Math. Month. 1967, 74, 436-437") that a metric space is complete if and only if any Banach contraction on any of its closed subsets has a fixed point. We apply our main result to deduce that a well-known fixed point theorem due to D. Mihet (see "Fixed Point Theory 2005, 6, 71-78") also allows us to characterize the fuzzy metric completeness.This research was partially funded by Ministerio de Ciencia, Innovacion y Universidades, under grant PGC2018-095709-B-C21 and AEI/FEDER, UE funds.Romaguera Bonilla, S.; Tirado Peláez, P. (2020). Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results. Mathematics. 8(2):1-7. https://doi.org/10.3390/math8020273S1782Connell, E. H. (1959). Properties of fixed point spaces. Proceedings of the American Mathematical Society, 10(6), 974-979. doi:10.1090/s0002-9939-1959-0110093-3Hu, T. K. (1967). On a Fixed-Point Theorem for Metric Spaces. The American Mathematical Monthly, 74(4), 436. doi:10.2307/2314587Subrahmanyam, P. V. (1975). Completeness and fixed-points. Monatshefte f�r Mathematik, 80(4), 325-330. doi:10.1007/bf01472580Kirk, W. A. (1976). Caristi’s fixed point theorem and metric convexity. Colloquium Mathematicum, 36(1), 81-86. doi:10.4064/cm-36-1-81-86Caristi, J. (1976). Fixed point theorems for mappings satisfying inwardness conditions. Transactions of the American Mathematical Society, 215, 241-241. doi:10.1090/s0002-9947-1976-0394329-4Suzuki, T., & Takahashi, W. (1996). Fixed point theorems and characterizations of metric completeness. Topological Methods in Nonlinear Analysis, 8(2), 371. doi:10.12775/tmna.1996.040Suzuki, T. (2007). A generalized Banach contraction principle that characterizes metric completeness. Proceedings of the American Mathematical Society, 136(05), 1861-1870. doi:10.1090/s0002-9939-07-09055-7Romaguera, S., & Tirado, P. (2019). A Characterization of Quasi-Metric Completeness in Terms of α–ψ-Contractive Mappings Having Fixed Points. Mathematics, 8(1), 16. doi:10.3390/math8010016Samet, B., Vetro, C., & Vetro, P. (2012). Fixed point theorems for -contractive type mappings. Nonlinear Analysis: Theory, Methods & Applications, 75(4), 2154-2165. doi:10.1016/j.na.2011.10.014Abbas, M., Ali, B., & Romaguera, S. (2015). Multivalued Caristi’s type mappings in fuzzy metric spaces and a characterization of fuzzy metric completeness. Filomat, 29(6), 1217-1222. doi:10.2298/fil1506217aCastro-Company, F., Romaguera, S., & Tirado, P. (2015). On the construction of metrics from fuzzy metrics and its application to the fixed point theory of multivalued mappings. Fixed Point Theory and Applications, 2015(1). doi:10.1186/s13663-015-0476-1Radu, V. (1987). Some fixed point theorems probabilistic metric spaces. Lecture Notes in Mathematics, 125-133. doi:10.1007/bfb0072718Sehgal, V. M., & Bharucha-Reid, A. T. (1972). Fixed points of contraction mappings on probabilistic metric spaces. Mathematical Systems Theory, 6(1-2), 97-102. doi:10.1007/bf01706080Ćirić, L. (2010). Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 72(3-4), 2009-2018. doi:10.1016/j.na.2009.10.00

Contraction mappings in fuzzy quasi-metric spaces and [0,1]-fuzzy posets

[EN] It is well known that each bounded ultraquasi-metric on a set induces, in a natural way, an [0,1]-fuzzy poset. On the other hand, each [0,1]-fuzzy poset can be seen as a stationary fuzzy ultraquasi-metric space for the continuous t-norm Min. By extending this construction to any continuous t-norm, a stationary fuzzy quasi-metric space is obtained. Motivated by these facts, we present several contraction principles on fuzzy quasi-metric spaces that are applied to the class of spaces described above. Some illustrative examples are also given. Finally, we use our approach to deduce in an easy fashion the existence and uniqueness of solution for the recurrence equations typically associated to the analysis of Probabilistic Divide and Conquer Algorithms.The author thanks the support of the Spanish Ministry of Science and Innovation, grand MTM2009-12872-C02-01. The author also thanks the referees because their suggestions and remarks have allowed to improve the first version of this paper.Tirado Peláez, P. (2012). Contraction mappings in fuzzy quasi-metric spaces and [0,1]-fuzzy posets. Fixed Point Theory. 13(1):273-283. http://hdl.handle.net/10251/56871S27328313

The Meir-Keeler Fixed Point Theorem for Quasi-Metric Spaces and Some Consequences

[EN] We obtain quasi-metric versions of the famous Meir¿Keeler fixed point theorem from which we deduce quasi-metric generalizations of Boyd¿Wong¿s fixed point theorem. In fact, one of these generalizations provides a solution for a question recently raised in the paper ¿On the fixed point theory in bicomplete quasi-metric spaces¿, J. Nonlinear Sci. Appl. 2016, 9, 5245¿5251. We also give an application to the study of existence of solution for a type of recurrence equations associated to certain nonlinear difference equationsPedro Tirado acknowledges the support of the Ministerio de Ciencia, Innovación y Universidades, under grant PGC2018-095709-B-C21Romaguera Bonilla, S.; Tirado Peláez, P. (2019). The Meir-Keeler Fixed Point Theorem for Quasi-Metric Spaces and Some Consequences. Symmetry (Basel). 11(6):1-10. https://doi.org/10.3390/sym11060741S110116Alegre, C., Dağ, H., Romaguera, S., & Tirado, P. (2016). On the fixed point theory in bicomplete quasi-metric spaces. Journal of Nonlinear Sciences and Applications, 09(08), 5245-5251. doi:10.22436/jnsa.009.08.10Boyd, D. W., & Wong, J. S. W. (1969). On nonlinear contractions. Proceedings of the American Mathematical Society, 20(2), 458-458. doi:10.1090/s0002-9939-1969-0239559-9Meir, A., & Keeler, E. (1969). A theorem on contraction mappings. Journal of Mathematical Analysis and Applications, 28(2), 326-329. doi:10.1016/0022-247x(69)90031-6Aydi, H., & Karapinar, E. (2012). A Meir-Keeler common type fixed point theorem on partial metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-26Chen, C.-M. (2012). Fixed point theory for the cyclic weaker Meir-Keeler function in complete metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-17Chen, C.-M. (2012). Fixed point theorems for cyclic Meir-Keeler type mappings in complete metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-41Chen, C.-M., & Karapınar, E. (2013). Fixed point results for the α-Meir-Keeler contraction on partial Hausdorff metric spaces. Journal of Inequalities and Applications, 2013(1). doi:10.1186/1029-242x-2013-410Choban, M. M., & Berinde, V. (2017). Multiple fixed point theorems for contractive and Meir-Keeler type mappings defined on partially ordered spaces with a distance. Applied General Topology, 18(2), 317. doi:10.4995/agt.2017.7067Di Bari, C., Suzuki, T., & Vetro, C. (2008). Best proximity points for cyclic Meir–Keeler contractions. Nonlinear Analysis: Theory, Methods & Applications, 69(11), 3790-3794. doi:10.1016/j.na.2007.10.014Jachymski, J. (1995). Equivalent Conditions and the Meir-Keeler Type Theorems. Journal of Mathematical Analysis and Applications, 194(1), 293-303. doi:10.1006/jmaa.1995.1299Karapinar, E., Czerwik, S., & Aydi, H. (2018). (α,ψ)-Meir-Keeler Contraction Mappings in Generalized b-Metric Spaces. Journal of Function Spaces, 2018, 1-4. doi:10.1155/2018/3264620Mustafa, Z., Aydi, H., & Karapınar, E. (2013). Generalized Meir–Keeler type contractions on G-metric spaces. Applied Mathematics and Computation, 219(21), 10441-10447. doi:10.1016/j.amc.2013.04.032Nashine, H. K., & Romaguera, S. (2013). Fixed point theorems for cyclic self-maps involving weaker Meir-Keeler functions in complete metric spaces and applications. Fixed Point Theory and Applications, 2013(1). doi:10.1186/1687-1812-2013-224Park, S., & Bae, J. S. (1981). Extensions of a fixed point theorem of Meir and Keeler. Arkiv för Matematik, 19(1-2), 223-228. doi:10.1007/bf02384479Piątek, B. (2011). On cyclic Meir–Keeler contractions in metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 74(1), 35-40. doi:10.1016/j.na.2010.08.010Rhoades, B. ., Park, S., & Moon, K. B. (1990). On generalizations of the Meir-Keeler type contraction maps. Journal of Mathematical Analysis and Applications, 146(2), 482-494. doi:10.1016/0022-247x(90)90318-aSamet, B. (2010). Coupled fixed point theorems for a generalized Meir–Keeler contraction in partially ordered metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 72(12), 4508-4517. doi:10.1016/j.na.2010.02.026Samet, B., Vetro, C., & Yazidi, H. (2013). A fixed point theorem for a Meir-Keeler type contraction through rational expression. Journal of Nonlinear Sciences and Applications, 06(03), 162-169. doi:10.22436/jnsa.006.03.02Schellekens, M. (1995). The Smyth Completion. Electronic Notes in Theoretical Computer Science, 1, 535-556. doi:10.1016/s1571-0661(04)00029-5Romaguera, S., & Schellekens, M. (1999). Quasi-metric properties of complexity spaces. Topology and its Applications, 98(1-3), 311-322. doi:10.1016/s0166-8641(98)00102-3García-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.2008.07.026Mohammadi, Z., & Valero, O. (2016). A new contribution to the fixed point theory in partial quasi-metric spaces and its applications to asymptotic complexity analysis of algorithms. Topology and its Applications, 203, 42-56. doi:10.1016/j.topol.2015.12.074Romaguera, 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., & Tirado, P. (2015). A characterization of Smyth complete quasi-metric spaces via Caristi’s fixed point theorem. Fixed Point Theory and Applications, 2015(1). doi:10.1186/s13663-015-0431-1Stevo, S. (2002). The recursive sequence xn+1 = g(xn, xn−1)/(A + xn). Applied Mathematics Letters, 15(3), 305-308. doi:10.1016/s0893-9659(01)00135-

Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points

[EN] Involving w-distances we prove a fixed point theorem of Caristi-type in the realm of (non -necessarily T-1) quasi-metric spaces. With the help of this result, a characterization of quasi-metric completeness is obtained. Our approach allows us to retrieve several key examples occurring in various fields of mathematics and computer science and that are modeled as non-T-1 quasi-metric spaces. As an application, we deduce a characterization of complete G-metric spaces in terms of a weak version of Caristi's theorem that involves a G-metric version of w-distances.Karapinar, E.; Romaguera Bonilla, S.; Tirado Peláez, P. (2022). Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points. Demonstratio Mathematica (Online). 55(1):939-951. https://doi.org/10.1515/dema-2022-017793995155

The bicompletion of fuzzy quasi-metric spaces

Extending the well-known result that every fuzzy metric space, in the sense of Kramosil and Michalek, has a completion which is unique up to isometry, we show that every KM-fuzzy quasi-metric space has a bicompletion which is unique up to isometry, and deduce that for each KM-fuzzy quasi-metric space, the completion of its induced fuzzy metric space coincides with the fuzzy metric space induced by its bicompletion. © 2010 Elsevier B.V.Supported by the Spanish Ministry of Science and Innovation, under Grant MTM2009-12872-C02-01.Castro Company, F.; Romaguera Bonilla, S.; Tirado Peláez, P. (2011). The bicompletion of fuzzy quasi-metric spaces. Fuzzy Sets and Systems. 166(1):56-64. https://doi.org/10.1016/j.fss.2010.12.004S5664166

On the domain of formal balls of the Sorgenfrey quasi-metric space

[EN] We show that the poset of formal balls of the Sorgenfrey quasi-metric space is an omega-continuous domain, and deduce that it is also a computational model, in the sense of R.C. Flagg and R. Kopperman, for the Sorgenfrey line. Furthermore, we study its structure of quantitative domain in the sense of P. Waszkiewicz. (C) 2016 Elsevier B.V. All rights reserved.Supported by the Ministry of Economy and Competitiveness of Spain, under grant MTM2012-37894-C02-01.Romaguera Bonilla, S.; Schellekens, M.; Tirado Peláez, P.; Valero Sierra, Ó. (2016). On the domain of formal balls of the Sorgenfrey quasi-metric space. Topology and its Applications. 203:177-187. https://doi.org/10.1016/j.topol.2015.12.086S17718720

A Discussion on p-Geraghty Contraction on mw-Quasi-Metric Spaces

[EN] In this paper we consider a kind of Geraghty contractions by using mw-distances in the setting of complete quasi-metric spaces. We provide fixed point theorems for this type of mappings and illustrate with some examples the results obtained.This research was partially supported by the Spanish Ministry of Science, Innovation and Universities. Grant number PGC2018-095709-B-C21 and AEI/FEDER, UE funds.Alegre Gil, MC.; Fulga, A.; Karapinar, E.; Tirado Peláez, P. (2020). A Discussion on p-Geraghty Contraction on mw-Quasi-Metric Spaces. Mathematics. 8(9):1-10. https://doi.org/10.3390/math8091437S11089Geraghty, M. A. (1973). On contractive mappings. Proceedings of the American Mathematical Society, 40(2), 604-604. doi:10.1090/s0002-9939-1973-0334176-5Gupta, V., Shatanawi, W., & Mani, N. (2016). Fixed point theorems for $$(\psi , \beta )$$ ( ψ , β ) -Geraghty contraction type maps in ordered metric spaces and some applications to integral and ordinary differential equations. Journal of Fixed Point Theory and Applications, 19(2), 1251-1267. doi:10.1007/s11784-016-0303-2Cho, S.-H., Bae, J.-S., & Karapınar, E. (2013). Fixed point theorems for α-Geraghty contraction type maps in metric spaces. Fixed Point Theory and Applications, 2013(1). doi:10.1186/1687-1812-2013-329Alegre, C., & Marín, J. (2016). Modified w-distances on quasi-metric spaces and a fixed point theorem on complete quasi-metric spaces. Topology and its Applications, 203, 32-41. doi:10.1016/j.topol.2015.12.073Alegre Gil, C., Karapınar, 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). doi:10.1007/s00025-019-1074-zAlegre, C., Marín, J., & Romaguera, S. (2014). A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces. Fixed Point Theory and Applications, 2014(1). doi:10.1186/1687-1812-2014-40Park, S. (2000). On generalizations of the Ekeland-type variational principles. Nonlinear Analysis: Theory, Methods & Applications, 39(7), 881-889. doi:10.1016/s0362-546x(98)00253-

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