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$$Q$$-functions on quasimetric spaces and fixed points for multivalued maps. (English) Zbl 1215.49016
Summary: We discuss several properties of $$Q$$-functions in the sense of Al-Homidan et al. We prove that the partial metric induced by any $$T_0$$ weighted quasipseudometric space is a $$Q$$-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 $$Q$$-function. In this context we also obtain some fixed point results for multivalued maps by using Bianchini-Grandolfi gauge functions.

##### MSC:
 49J40 Variational inequalities 47H10 Fixed-point theorems
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##### References:
 [1] Kada, O; Suzuki, T; Takahashi, W, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Mathematica Japonica, 44, 381-391, (1996) · Zbl 0897.54029 [2] Caristi, J; Kirk, WA, Geometric fixed point theory and inwardness conditions, No. 490, 74-83, (1975), Berlin, Germany · Zbl 0315.54052 [3] Ekeland, I, Nonconvex minimization problems, Bulletin of the American Mathematical Society, 1, 443-474, (1979) · Zbl 0441.49011 [4] Takahashi, W; Théra, MA (ed.); Baillon, JB (ed.), Existence theorems generalizing fixed point theorems for multivalued mappings, No. 252, 397-406, (1991), Harlow, UK · Zbl 0760.47029 [5] Al-Homidan, S; Ansari, QH; Yao, J-C, Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory, Nonlinear Analysis: Theory, Methods & Applications, 69, 126-139, (2008) · Zbl 1142.49005 [6] Hussain, N; Shah, MH; Kutbi, MA, Coupled coincidence point theorems for nonlinear contractions in partially ordered quasi-metric spaces with a [inlineequation not available: see fulltext.]-function, No. 2011, 21, (2011) [7] Latif, A; Al-Mezel, SA, Fixed point results in quasimetric spaces, No. 2011, 8, (2011) · Zbl 1207.54061 [8] Alegre, C, Continuous operators on asymmetric normed spaces, Acta Mathematica Hungarica, 122, 357-372, (2009) · Zbl 1199.54165 [9] Ali-Akbari, M; Honari, B; Pourmahdian, M; Rezaii, MM, The space of formal balls and models of quasi-metric spaces, Mathematical Structures in Computer Science, 19, 337-355, (2009) · Zbl 1168.54012 [10] Cobzaş, S, Compact and precompact sets in asymmetric locally convex spaces, Topology and its Applications, 156, 1620-1629, (2009) · Zbl 1185.46001 [11] García-Raffi, LM; Romaguera, S; Sánchez-Pérez, EA, The goldstine theorem for asymmetric normed linear spaces, Topology and its Applications, 156, 2284-2291, (2009) · Zbl 1185.54028 [12] García-Raffi, LM; Romaguera, S; Schellekens, MP, Applications of the complexity space to the general probabilistic divide and conquer algorithms, Journal of Mathematical Analysis and Applications, 348, 346-355, (2008) · Zbl 1149.68080 [13] Heckmann, R, Approximation of metric spaces by partial metric spaces, Applied Categorical Structures, 7, 71-83, (1999) · Zbl 0993.54029 [14] Matthews SG: Partial metric topology. In Proceedings of the 14th Summer Conference on General Topology and Its Applications, 1994, Annals of the New York Academy of Sciences. Volume 728. The New York Academy of Sciences, New York, NY, USA; 183-197. · Zbl 0911.54025 [15] Romaguera, S, On computational models for the hyperspace, 277-294, (2009), New York, NY, USA [16] Romaguera, S; Schellekens, M, Partial metric monoids and semivaluation spaces, Topology and its Applications, 153, 948-962, (2005) · Zbl 1084.22002 [17] Romaguera, S; Tirado, P, The complexity probabilistic quasi-metric space, Journal of Mathematical Analysis and Applications, 376, 732-740, (2011) · Zbl 1227.54035 [18] Romaguera, S; Valero, O, A quantitative computational model for complete partial metric spaces via formal balls, Mathematical Structures in Computer Science, 19, 541-563, (2009) · Zbl 1172.06003 [19] Romaguera, S; Valero, O, Domain theoretic characterisations of quasi-metric completeness in terms of formal balls, Mathematical Structures in Computer Science, 20, 453-472, (2010) · Zbl 1193.54016 [20] Schellekens, M, The smyth completion: a common foundation for denotational semantics and complexity analysis, No. 1, 535-556, (1995), Amsterdam, The Netherlands [21] Schellekens, MP, A characterization of partial metrizability: domains are quantifiable, Theoretical Computer Science, 305, 409-432, (2003) · Zbl 1043.54011 [22] Waszkiewicz, P, Quantitative continuous domains, Applied Categorical Structures, 11, 41-67, (2003) · Zbl 1030.06005 [23] Waszkiewicz, P, Partial metrisability of continuous posets, Mathematical Structures in Computer Science, 16, 359-372, (2006) · Zbl 1103.06004 [24] Proinov, PD, A generalization of the Banach contraction principle with high order of convergence of successive approximations, Nonlinear Analysis: Theory, Methods & Applications, 67, 2361-2369, (2007) · Zbl 1130.54021 [25] Fletcher P, Lindgren WF: Quasi-Uniform Spaces, Lecture Notes in Pure and Applied Mathematics. Volume 77. Marcel Dekker, New York, NY, USA; 1982:viii+216. [26] Künzi, H-PA, Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology, No. 3, 853-968, (2001), Dordrecht, The Netherlands · Zbl 1002.54002 [27] Reilly, IL; Subrahmanyam, PV; Vamanamurthy, MK, Cauchy sequences in quasipseudometric spaces, Monatshefte für Mathematik, 93, 127-140, (1982) · Zbl 0472.54018 [28] Rakotch, E, A note on contractive mappings, Proceedings of the American Mathematical Society, 13, 459-465, (1962) · Zbl 0105.35202 [29] Bianchini, RM; Grandolfi, M, Trasformazioni di tipo contrattivo generalizzato in uno spazio metrico, Atti della Accademia Nazionale dei Lincei, 45, 212-216, (1968) · Zbl 0205.27202 [30] Proinov, PD, New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems, Journal of Complexity, 26, 3-42, (2010) · Zbl 1185.65095 [31] Chang, TH, Common fixed point theorems for multivalued mappings, Mathematica Japonica, 41, 311-320, (1995) · Zbl 0840.47041
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