Pedro Tirado

Contraction Maps on Ifqm-spaces with Application to Recurrence Equations of Quicksort

Recently, C. Alaca, D. Turkoglu and C. Yildiz [Chaos, Solitons and Fractals, 2006], have proved intuitionistic fuzzy versions of the celebrated Banach fixed point theorem and Edelstein fixed point theorem respectively, by means of a notion of intuitionistic fuzzy metric space which is based on the concept of fuzzy metric space due to I. Kramosil and J. Michalek [Kybernetika, 1975]. In this paper we generalize the notions of intuitionistic fuzzy metric space by Alaca, Turkoglu and Yildiz to the quasi-metric setting and we present an intuitionistic fuzzy quasi-metric version of the Banach contraction principle. We apply this approach to deduce the existence of solution for the recurrence equations associated to the analysis of Quicksort algorithm in the framework of intuitionistic fuzzy quasi-metric spaces (ifqm-spaces, in short).

New results on the mathematical foundations of asymptotic complexity analysis of algorithms via complexity spaces

Schellekens [The Smyth completion: A common foundation for denotational semantics and complexity analysis, Electron. Notes Theor. Comput. Sci. 1 (1995), pp. 211–232.] introduced the theory of complexity (quasi-metric) spaces as a part of the development of a topological foundation for the asymptotic complexity analysis of programs and algorithms in 1995. The applicability of this theory to the asymptotic complexity analysis of divide and conquer algorithms was also illustrated by Schellekens in the same paper. In particular, he gave a new formal proof, based on the use of the Banach fixed-point theorem, of the well-known fact that the asymptotic upper bound of the average running time of computing of Mergesort belongs to the asymptotic complexity class of nlog 2 n. Recently, Schellekens’ method has been shown to be useful in yielding asymptotic upper bounds for a class of algorithms whose running time of computing leads to recurrence equations different from the divide and conquer ones reported in Cerdà-Uguet et al. [The Baire partial quasi-metric space: A mathematical tool for the asymptotic complexity analysis in Computer Science, Theory Comput. Syst. 50 (2012), pp. 387–399.]. However, the variety of algorithms whose complexity can be analysed with this approach is not much larger than that of algorithms that can be analysed with the original Schellekens method. In this paper, on the one hand, we extend Schellekens’ method in order to yield asymptotic upper bounds for a certain class of recursive algorithms whose running time of computing cannot be discussed following the techniques given by Cerdà-Uguet et al. and, on the other hand, we improve the original Schellekens method by introducing a new fixed-point technique for providing, contrary to the case of the method introduced by Cerdà-Uguet et al., lower asymptotic bounds of the running time of computing of the aforementioned algorithms and those studied by Cerdà-Uguet et al. We illustrate and validate the developed method by applying our results to provide the asymptotic complexity class (asymptotic upper and lower bounds) of the celebrated algorithms Quicksort, Largetwo and Hanoi.

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.

Weakly contractive multivalued maps and w- distances on complete quasi-metric spaces

We obtain versions of the Boyd and Wong fixed point theorem and of the Matkowski fixed point theorem for multivalued maps and w-distances on complete quasi-metric spaces. Our results generalize, in several directions, some well-known fixed point theorems.

Complete partial metric spaces have partially metrizable computational models

We show that the domain of formal balls of a complete partial metric space (X, p) can be endowed with a complete partial metric that extends p and induces the Scott topology. This result, that generalizes well-known constructions of Edalat and Heckmann [A computational model for metric spaces, Theoret. Comput. Sci. 193 (1998), pp. 53–73] and Heckmann [Approximation of metric spaces by partial metric spaces, Appl. Cat. Struct. 7 (1999), pp. 71–83] for metric spaces and improves a recent result of Romaguera and Valero [A quantitative computational model for complete partial metric spaces via formal balls, Math. Struct. Comput. Sci. 19 (2009), pp. 541–563], motivates a notion of a partially metrizable computational model which allows us to characterize those topological spaces that admit a compatible complete partial metric via this model.

Complete partial metric spaces have partially metrizable computational models-CMMSE2010

We show that the domain of formal balls of a complete partial metric space (X, p) can be endowed with a complete partial metric that extends p and induces the Scott topology. This result, that generalizes well-known constructions of Edalat and Heckmann [A computational model for metric spaces, Theoret. Comput. Sci. 193 (1998), pp. 53–73] and Heckmann [Approximation of metric spaces by partial metric spaces, Appl. Cat. Struct. 7 (1999), pp. 71–83] for metric spaces and improves a recent result of Romaguera and Valero [A quantitative computational model for complete partial metric spaces via formal balls, Math. Struct. Comput. Sci. 19 (2009), pp. 541–563], motivates a notion of a partially metrizable computational model which allows us to characterize those topological spaces that admit a compatible complete partial metric via this model.

Contractive multivalued maps in terms of Q-functions on complete quasimetric spaces

AbstractIn this paper we prove the existence of a fixed point for multivalued maps satisfying a contraction condition in terms of Q-functions, and via Bianchini-Grandolfi gauge functions, for complete T0-quasipseudometric spaces. Our results extend, improve, and generalize some recent results in the literature. We present some examples to validate and illustrate our results. MSC:54H25, 47H10, 54E50.

A fixed point theorem for preordered complete fuzzy quasi-metric spaces and an application

We obtain a fixed point theorem for a type of generalized contractions on preordered complete fuzzy quasi-metric spaces which is applied to deduce, among other results, a procedure to show in a direct and easy fashion the existence of solution for the recurrence equations that are typically associated to Quicksort and Divide and Conquer algorithms, respectively.MSC:47H10, 54H25, 06A06, 68Q25.

Some classes of t-norms and fuzzy metric spaces

Recently, V. Gregori, S. Morillas and A. Sapena have discussed [On a class of completable fuzzy metric spaces, Fuzzy Sets and Systems 161 (2010), 2193-2205] the so-called strong fuzzy metrics when looking for a class of completable fuzzy metric spaces in the sense of A. George and P. Veeramani and state the question of finding a non-strong fuzzy metric space for a continuous t-norm different from the minimum. Later on, J. Gutierrez Garcia and S. Romaguera solved this question [Examples of non-strong fuzzy metrics, Fuzzy Sets and Systems 162 (2011) 91-93] by means of two examples for the product and the Lukasiewicz t-norm, respectively. In this direction they posed to find further examples of non-strong fuzzy metrics for continuous t-norm that are greater than the product but different from minimum. In this paper we obtain, among others facts on t-norms and fuzzy metric spaces, an example of this kind.

Functions on Quasimetric Spaces and Fixed Points for Multivalued Maps

We discuss several properties of -functions in the sense of Al-Homidan et al.. In particular, we prove that the partial metric induced by any weighted quasipseudometric space is a -function and show that both the Sorgenfrey line and the Kofner plane provide significant examples of quasimetric spaces for which the associated supremum metric is a -function. In this context we also obtain some fixed point results for multivalued maps by using Bianchini-Grandolfi 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.