5,073 research outputs found

    Fixed point results for generalized cyclic contraction mappings in partial metric spaces

    Full text link
    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. Appl. 158, 1738–1740 (2011)Altun I., Simsek H.: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud. 1, 1–8 (2008)Altun I., Sola F., Simsek H.: Generalized contractions on partial metric spaces. Topol. Appl. 157, 2778–2785 (2010)Aydi, H.: Some fixed point results in ordered partial metric spaces. arxiv:1103.3680v1 [math.GN](2011)Boyd D.W., Wong J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)Bukatin M., Kopperman R., Matthews S., Pajoohesh H.: Partial metric spaces. Am. Math. Monthly 116, 708–718 (2009)Bukatin M.A., Shorina S.Yu. et al.: Partial metrics and co-continuous valuations. In: Nivat, M. (eds) Foundations of software science and computation structure Lecture notes in computer science vol 1378., pp. 125–139. Springer, Berlin (1998)Derafshpour M., Rezapour S., Shahzad N.: On the existence of best proximity points of cyclic contractions. Adv. Dyn. Syst. Appl. 6, 33–40 (2011)Heckmann R.: Approximation of metric spaces by partial metric spaces. Appl. Cat. Struct. 7, 71–83 (1999)Karapinar E.: Fixed point theory for cyclic weak ϕ{\phi} -contraction. App. Math. Lett. 24, 822–825 (2011)Karapinar, E.: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. 2011,4 (2011). doi: 10.1186/1687-1812-2011-4Karapinar E.: Weak φ{\varphi} -contraction on partial metric spaces and existence of fixed points in partially ordered sets. Math. Aeterna. 1(4), 237–244 (2011)Karapinar E., Erhan I.M.: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett. 24, 1894–1899 (2011)Karpagam S., Agrawal S.: Best proximity point theorems for cyclic orbital Meir–Keeler contraction maps. Nonlinear Anal. 74, 1040–1046 (2011)Kirk W.A., Srinavasan P.S., Veeramani P.: Fixed points for mapping satisfying cylical contractive conditions. Fixed Point Theory. 4, 79–89 (2003)Kosuru, G.S.R., Veeramani, P.: Cyclic contractions and best proximity pair theorems). arXiv:1012.1434v2 [math.FA] 29 May (2011)Matthews S.G.: Partial metric topology. in: Proc. 8th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci. 728, 183–197 (1994)Neammanee K., Kaewkhao A.: Fixed points and best proximity points for multi-valued mapping satisfying cyclical condition. Int. J. Math. Sci. Appl. 1, 9 (2011)Oltra S., Valero O.: Banach’s fixed theorem for partial metric spaces. Rend. Istit. Mat. Univ. Trieste. 36, 17–26 (2004)Păcurar M., Rus I.A.: Fixed point theory for cyclic ϕ{\phi} -contractions. Nonlinear Anal. 72, 1181–1187 (2010)Petric M.A.: Best proximity point theorems for weak cyclic Kannan contractions. Filomat. 25, 145–154 (2011)Romaguera, S.: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. (2010, article ID 493298, 6 pages).Romaguera, S.: Fixed point theorems for generalized contractions on partial metric spaces. Topol. Appl. (2011). doi: 10.1016/j.topol.2011.08.026Romaguera S., Valero O.: A quantitative computational model for complete partial metric spaces via formal balls. Math. Struct. Comput. Sci. 19, 541–563 (2009)Rus, I.A.: Cyclic representations and fixed points. Annals of the Tiberiu Popoviciu Seminar of Functional equations. Approx. Convexity 3, 171–178 (2005), ISSN 1584-4536Schellekens M.P.: The correspondence between partial metrics and semivaluations. Theoret. Comput. Sci. 315, 135–149 (2004)Valero O.: On Banach fixed point theorems for partial metric spaces. Appl. Gen. Top. 6, 229–240 (2005)Waszkiewicz P.: Quantitative continuous domains. Appl. Cat. Struct. 11, 41–67 (2003

    Convexity and boundedness relaxation for fixed point theorems in modular spaces

    Full text link
    [EN] Although fixed point theorems in modular spaces have remarkably applied to a wide variety of mathematical problems, these theorems strongly depend on some assumptions which often do not hold in practice or can lead to their reformulations as particular problems in normed vector spaces. A recent trend of research has been dedicated to studying the fundamentals of fixed point theorems and relaxing their assumptions with the ambition of pushing the boundaries of fixed point theory in modular spaces further. In this paper, we focus on convexity and boundedness of modulars in fixed point results taken from the literature for contractive correspondence and single-valued mappings. To relax these two assumptions, we seek to identify the ties between modular and b-metric spaces. Afterwards we present an application to a particular form of integral inclusions to support our generalized version of Nadler’s theorem in modular spaces.The authors gratefully acknowledge the reviewer and the editor for their useful observations and recommendations.Lael, F.; Shabanian, S. (2021). Convexity and boundedness relaxation for fixed point theorems in modular spaces. Applied General Topology. 22(1):91-108. https://doi.org/10.4995/agt.2021.13902OJS91108221M. Abbas, F. Lael and N. Saleem, Fuzzy b-metric spaces: Fixed point results for ψ-contraction correspondences and their application, Axioms 9, no. 2 (2020), 1-12. https://doi.org/10.3390/axioms9020036A. Ait Taleb and E. Hanebaly, A fixed point theorem and its application to integral equations in modular function spaces, Proceedings of the American Mathematical Society 128 (1999), 419-426. https://doi.org/10.1090/S0002-9939-99-05546-XM. R. Alfuraidan, Fixed points of multivalued mappings in modular function spaces with a graph, Fixed Point Theory and Applications 42 (2015), 1-14. https://doi.org/10.1186/s13663-015-0292-7A. H. Ansari, T. Dosenovic, S. Radenovic, N. Saleem, V. Sesum-Cavic and J. Vujakovic, C-class functions on some fixed point results in ordered partial metric spaces via admissible mappings, Novi Sad Journal of Mathematics 49, no. 1 (2019), 101-116. https://doi.org/10.30755/NSJOM.07794A. H. Ansari, J. M. Kumar and N. Saleem, Inverse-C-class function on weak semi compatibility and fixed point theorems for expansive mappings in G-metric spaces, Mathematica Moravica 24, no. 1 (2020), 93-108. https://doi.org/10.5937/MatMor2001093HA. Aghajani, M. Abbas and J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca 64, no. 4 (2014), 941-960. https://doi.org/10.2478/s12175-014-0250-6I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal., Unianowsk, Gos. Ped. Inst. 30 (1989), 26-37.S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181M. Berziga, I. Kédimb and A. Mannaic, Multivalued fixed point theorem in b-metric spaces and its application to differential inclusions, Filomat 32 no. 8 (2018), 2963-2976. https://doi.org/10.2298/FIL1808963BR. K. Bishta, A remark on asymptotic regularity and fixed point property, Filomat 33 no. 14 (2019), 4665-4671. https://doi.org/10.2298/FIL1914665BM. Boriceanu, Strict fixed point theorems for multivalued operators in b-metric spaces, Int. J. Mod. Math. 4 (2009), 285-301.M. Bota, A. Molnar and C. Varga, On Ekeland's variational principle in b-metric spaces, Fixed Point Theory 12, no. 2 (2011), 21-28.N. Bourbaki, Topologie Generale; Herman, Paris, France, 1974.M. S. Brodskii and D. P. Milman, On the center of a convex set, Doklady Acad. N. S. 59 (1948), 837-840.S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1 (1993), 5-11.S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Semin. Mat. Fis. Univ. Modena 46 (1998), 263-276.T. Dominguez-Benavides, M. A. Khamsi and S. Samadi, Asymptotically regular mappings in modular function spaces, Scientiae Mathematicae Japonicae 2 (2001), 295-304. https://doi.org/10.1016/S0362-546X(00)00117-6S. Dhompongsa, T. D. Benavides, A. Kaewcharoen and B. Panyanak, Fixed point theorems for multivalued mappings in modular function spaces, Sci. Math. Japon. (2006), 139-147.Y. Feng, S. Liu, Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings, J. Math. Anal. Appl. 317 (2006), 103-112. https://doi.org/10.1016/j.jmaa.2005.12.004K. Fallahi, K. Nourouzi, Probabilistic modular spaces and linear operators. Acta Appl. Math. 105 (2009), 123-140. https://doi.org/10.1007/s10440-008-9267-6N. Hussain, V. Parvaneh, J. R. Roshan and Z. Kadelburg, Fixed points of cyclic weakly (ψ, φ , L, A, B)-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl. 2013 (2013), 256. https://doi.org/10.1186/1687-1812-2013-256M. A. Japon, Some geometric properties in modular spaces and application to fixed point theory, J. Math. Anal. Appl. 295 (2004), 576-594. https://doi.org/10.1016/j.jmaa.2004.02.047M. A. Japon, Applications of Musielak-Orlicz spaces in modern control systems, Teubner-Texte Math. 103 (1988), 34-36.W. W. Kassu, M. G. Sangago and H. Zegeye, Convergence theorems to common fixed points of multivalued ρ-quasi-nonexpansive mappings in modular function spaces, Adv. Fixed Point Theory 8 (2018), 21-36.M. A. Khamsi, A convexity property in modular function spaces, Math. Japonica 44, no. 2 (1996), 269-279.M. A. Khamsi, W. K. Kozlowski and C. Shutao, Some geometrical properties and fixed point theorems in Orlicz spaces, J. Math. Anal. Appl. 155 (1991), 393-412. https://doi.org/10.1016/0022-247X(91)90009-OM. A. Khamsi, W. M. Kozlowski and S. Reich, Fixed point theory in modular function spaces, Nonlinear Analysis, Theory, Methods and Applications 14 (1990), 935-953. https://doi.org/10.1016/0362-546X(90)90111-SM. S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30, no. 1 (1984), 1-9. https://doi.org/10.1017/S0004972700001659S. H. Khan, Approximating fixed points of (λ, ρ)-firmly nonexpansive mappings in modular function spaces, arXiv:1802.00681v1, 2018. https://doi.org/10.1007/s40065-018-0204-xN. Kir and H. Kiziltunc, On some well known fixed point theorems in b-metric spaces, Turk. J. Anal. Number Theory 1, no. 1 (2013), 13-16. https://doi.org/10.12691/tjant-1-1-4D. Klim and D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007), 132-139. https://doi.org/10.1016/j.jmaa.2006.12.012W. M. Kozlowski, Modular Function Spaces, Marcel Dekker, 1988.P. Kumam and W. Sintunavarat, The existence of fixed point theorems for partial q-set-valued quasicontractions in b-metric spaces and related results, Fixed Point Theory Appl. 2014 (2014), 226. https://doi.org/10.1186/1687-1812-2014-226M. A. Kutbi and A. Latif, Fixed points of multivalued maps in modular function spaces, Fixed Point Theory and Applications 2009 (2009), 786357. https://doi.org/10.1155/2009/786357F. Lael and K. Nourouzi, On the fixed points of correspondences in modular spaces, International Scholarly Research Network ISRN Geometry 2011 (2011), 530254. https://doi.org/10.5402/2011/530254A. Lukács and S. Kajántó, Fixed point theorems for various types of F-contractions in complete b-metric spaces, Fixed Point Theory 19, no. 1 (2018), 321-334. https://doi.org/10.24193/fpt-ro.2018.1.25J. Markin, A fixed point theorem for set valued mappings, Bull. Am. Math. Soc. 74 (1968), 639-640. https://doi.org/10.1090/S0002-9904-1968-11971-8K. Mehmet and K. Hukmi, On some well known fixed point theorems in b-metric space, Turkish Journal of Analysis and Number Theory 1 (2013), 13-16. https://doi.org/10.12691/tjant-1-1-4R. Miculescu and A. Mihail, New fixed point theorems for set-valued contractions in bb-metric spaces, J. Fixed Point Theory Appl. 19 (2017), 2153-2163. https://doi.org/10.1007/s11784-016-0400-2J. Musielak and W. Orlicz, On modular spaces, Studia Mathematica 18 (1959), 49-65. https://doi.org/10.4064/sm-18-1-49-65J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034, Lecture Notes in Mathematics, Springer-Verlag, 1983. https://doi.org/10.1007/BFb0072210S. B. Nadler, Multi-valued contraction mappings, Pacific Journal of Mathematics 30 (1969), 475-488. https://doi.org/10.2140/pjm.1969.30.475H. Nakano, Modular Semi-Ordered Linear Spaces, Maruzen, Tokyo, Japan, 1950.F. Nikbakht Sarvestani, S. M. Vaezpour and M. Asadi, A characterization of the generalization of the generalized KKM mapping via the measure of noncompactness in complete geodesic spaces, J. Nonlinear Funct. Anal. 2017 (2017), 8.K. Nourouzi and S. Shabanian, Operators defined on n-modular spaces, Mediterranean Journal of Mathematics 6 (2009), 431-446. https://doi.org/10.1007/s00009-009-0016-5W. Orlicz, Über eine gewisse klasse von Raumen vom Typus B, Bull. Acad. Polon. Sci. A (1932), 207-220.W. Orlicz, Über Raumen LM, Bull. Acad. Polon. Sci. A (1936), 93-107.M. O. Olatinwo, Some results on multi-valued weakly jungck mappings in b-metric space, Cent. Eur. J. Math. 6 (2008), 610-621. https://doi.org/10.2478/s11533-008-0047-3M. Pacurar, Sequences of almost contractions and fixed points in b-metric spaces, Analele Univ. Vest Timis. Ser. Mat. Inform. XLVIII 3 (2010), 125-137.S. Radenovic, T. Dosenovic, T. A. Lampert and Z. Golubovíc, A note on some recent fixed point results for cyclic contractions in b-metric spaces and an application to integral equations, Applied Mathematics and Computation 273 (2016), 155-164. https://doi.org/10.1016/j.amc.2015.09.089N.Saleem, I. Habib and M. Sen, Some new results on coincidence points for multivalued Suzuki-type mappings in fairly?? complete spaces, Computation 8, no. 1 (2020), 17. https://doi.org/10.3390/computation8010017N. Saleem, M. Abbas, B. Ali, and Z. Raza, Fixed points of Suzuki-type generalized multivalued (f, θ, L)-almost contractions with applications, Filomat 33, no. 2 (2019), 499-518. https://doi.org/10.2298/FIL1902499SN. Saleem, M. Abbas, B. Bin-Mohsin and S. Radenovic, Pata type best proximity point results in metric spaces,?? Miskolac Notes 21, no. 1 (2020), 367-386. https://doi.org/10.18514/MMN.2020.2764N. Saleem, I. Iqbal, B. Iqbal, and S. Radenovic, Coincidence and fixed points of multivalued F-contractions in generalized metric space with application, Journal of Fixed Point Theory and Applications 22 (2020), 81. https://doi.org/10.1007/s11784-020-00815-3S. Shabanian and K. Nourouzi, Modular Space and Fixed Point Theorems, thesis (in persian), 2007, K.N.Toosi University of Technology.W. Shan He, Generalization of a sharp Hölder's inequality and its application, J. Math. Anal. Appl. 332, no. 1 (2007), 741-750. https://doi.org/10.1016/j.jmaa.2006.10.019S. L. Singh and B. Prasad, Some coincidence theorems and stability of iterative procedures, Comput. Math. Appl. 55, no. 11 (2008), 2512-2520. https://doi.org/10.1016/j.camwa.2007.10.026W. Sintunavarat, S. Plubtieng and P. Katchang, Fixed point result and applications on b-metric space endowed with an arbitrary binary relation, Fixed Point Theory Appl. 2013 (2013), 296. https://doi.org/10.1186/1687-1812-2013-296T. Van An, L. Quoc Tuyen and N. Van Dung, Stone-type theorem on b-metric spaces and applications, Topology and its Applications 185-186 (2015), 50-64. https://doi.org/10.1016/j.topol.2015.02.00

    Fixed points of α-Θ-Geraghty type and Θ-Geraghty graphic type contractions

    Full text link
    [EN] In this paper, by using the concept of the α-Garaghty contraction, we introduce the new notion of the α-Θ-Garaghty type contraction and prove some fixed point results for this contraction in partial metric spaces. Also, we give some examples and applications to illustrate the main results.The first author would like to thank the Research Professional Development Project Under the Science Achievement Scholarship of Thailand (SAST) for the Master’s degree Program at KMUTT. This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT.Onsod, W.; Kumam, P.; Cho, YJ. (2017). Fixed points of α-Θ-Geraghty type and Θ-Geraghty graphic type contractions. Applied General Topology. 18(1):153-171. https://doi.org/10.4995/agt.2017.6694SWORD153171181T. Abdeljawad, Meir-Keeler α-contractive fixed and common fixed point theorems, Fixed Point Theory Appl. 19 (2013). https://doi.org/10.1186/1687-1812-2013-19T. Abdeljawad and D. Gopal, Erratum to Meir-Keeler alphaalpha-contractive fixed and common fixed point theorems, Fixed Point Theory Appl. 110 (2013). H. Alikhani, D. Gopal, M. A. Miandaragh, Sh. Rezapour and N. Shahzad, Some endpoint results for β-generalized weak contractive multifunctions, The Scientific World Journal (2013), Article ID 948472.Beg, I., Butt, A. R., & Radojević, S. (2010). The contraction principle for set valued mappings on a metric space with a graph. Computers & Mathematics with Applications, 60(5), 1214-1219. doi:10.1016/j.camwa.2010.06.003A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. (Debr.) 57 (2000), 31-37.Cho, 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), 329. doi:10.1186/1687-1812-2013-329Chandok, S. (2015). Some fixed point theorems for (α, β)-admissible Geraghty type contractive mappings and related results. Mathematical Sciences, 9(3), 127-135. doi:10.1007/s40096-015-0159-4Geraghty, M. A. (1973). On contractive mappings. Proceedings of the American Mathematical Society, 40(2), 604-604. doi:10.1090/s0002-9939-1973-0334176-5GOPAL, D., ABBAS, M., PATEL, D. K., & VETRO, C. (2016). Fixed points of α -type F -contractive mappings with an application to nonlinear fractional differential equation. Acta Mathematica Scientia, 36(3), 957-970. doi:10.1016/s0252-9602(16)30052-2Gordji, M., Ramezani, M., Cho, Y., & Pirbavafa, S. (2012). A generalization of Geraghty’s theorem in partially ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory and Applications, 2012(1), 74. doi:10.1186/1687-1812-2012-74Hussain, N., Karapinar, E., Salimi, P., & Akbar, F. (2013). alpha-Admissible mappings and related Fixed point Theorems. Journal of Inequalities and Applications, 2013(1), 114. doi:10.1186/1029-242x-2013-114Jachymski, J. (2007). The contraction principle for mappings on a metric space with a graph. Proceedings of the American Mathematical Society, 136(04), 1359-1373. doi:10.1090/s0002-9939-07-09110-1X. D. Liu, S. S. Chang, Y. Xiao and L. C. Zhao, Existence of fixed points for Θ-type contraction and Θ-type Suzuki contraction in complete metric spaces, Fixed Point Theory Appl. 8 (2016). J. Martinez-Moreno, W. Sintunavarat and Y. J. Cho, Common fixed point theorems for Geraghty's type contraction mappings using the monotone property with two metrics, Fixed Point Theory Appl. 174 (2015).S. G. Mathews, Partial metric topology, in Proceedings of the 11th Summer Conference on General Topology and Applications 728 (1995), 183-197, The New York Academy of Sci. C. Mongkolkehai, Y. J. Cho and P. Kumam, Best proximity points for Geraghty's proximal contraction mappings, Fixed Point Theory Appl. 180 (2013).W. Onsod and P. Kumam, Common fixed point results for φ-ψ-weak contraction mappings via f-α-admissible Mappings in intuitionistic fuzzy metric spaces, Communications in Mathematics and Applications 7 (2016), 167-178.V. L. Rosa and P. Vetro, Fixed point for Geraghty-contractions in partial metric spaces, J. Nonlinear Sci. Appl. 7 (2014), 1-10.Samet, 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.2011.10.01

    Dynamic Processes, Fixed Points, Endpoints, Asymmetric Structures, and Investigations Related to Caristi, Nadler, and Banach in Uniform Spaces

    Get PDF
    Research ArticleIn uniform spaces (...) with symmetric structures determined by the D-families of pseudometrics which define uniformity in these spaces, the new symmetric and asymmetric structures determined by the J-families of generalized pseudodistances on (...) are constructed; using these structures the set-valued contractions of two kinds of Nadler type are defined and the new and general theorems concerning the existence of fixed points and endpoints for such contractions are proved. Moreover, using these new structures, the single-valued contractions of two kinds of Banach type are defined and the new and general versions of the Banach uniqueness and iterate approximation of fixed point theorem for uniform spaces are established. Contractions defined and studied here are not necessarily continuous. One of the main key ideas in this paper is the application of our fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies in uniform spaces with structures determined by J-families. Results are new also in locally convex and metric spaces. Examples are provided
    corecore