Abstract
In this survey we present an exposition of the development during the last decade of metric fixed point theory on hyperconvex metric spaces. Therefore we mainly cover results where the conditions on the mappings are metric. We will recall results about proximinal nonexpansive retractions and their impact into the theory of best approximation and best proximity pairs. A central role in this survey will be also played by some recent developments on \({\mathbb{R}}\)-trees. Finally, some considerations and new results on the extension of compact mappings will be shown.
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Aksoy A.G., Khamsi M.A.: A selection theorem in metric trees. Proc. Am. Math. Soc. 134, 2957–2966 (2006)
Aksoy A.G., Khamsi M.A.: Fixed points of uniformly Lipschitzian mappings in metric trees. Sci. Math. Jpn. 65, 31–41 (2007)
Ambrosio L., Tilli P.: Topics on Analysis in Metric Spaces. Oxford University Press, Oxford (2004)
Amini-Harandi A., Farajzadeh A.P.: A best approximation theorem in hyperconvex metric spaces. Nonlinear Anal. 70, 2453–2456 (2009)
Amini-Harandi A., Farajzadeh A.P.: Best approximation, coincidence and fixed point theorems for set-valued maps in R-trees. Nonlinear Anal. 71, 1649–1653 (2009)
Amini-Harandi, A.; Farajzadeh, A.P.; O’Regan, D.; Agarwal, R.P.: Coincidence point, best approximation, and best proximity theorems for condensing set-valued maps in hyperconvex metric spaces. Fixed Point Theory and Appl. Article ID 543158 (2008)
Amini-Harandi A., Farajzadeh A.P., O’Regan D., Agarwal R.P.: Fixed point theory for α-condensing set valued maps in hyperconvex metric spaces. Commun. Appl. Nonlinear Anal. 15, 39–46 (2008)
Aronszajn N., Panitchpakdi P.: Extensions of uniformly continuous transformations and hyperconvex metric spaces. Pac. J. Math. 6, 405–439 (1956)
Baillon J.B.: Nonexpansive mappings and hyperconvex spaces. Contemp. Math. 72, 11–19 (1988)
Ball K.: Markov chains, Riesz transforms and Lipschitz maps. Geom. Funct. Anal. 2, 137–172 (1992)
Borkowski M., Bugajewski D.: On fixed point theorems of Leray–Schauder type. Proc. Am. Math. Soc. 136, 973–980 (2008)
Borkowski, M.; Bugajewski, D.; Phulara, D.: On some properties of hyperconvex spaces. Fixed Point Theory Appl. Article ID 213812 (2010)
Bridson M.R., Haefliger A.: Metric Spaces of Non-Positive Curvature. Springer-Verlag, Berlin (1999)
Bugajewski D., Espínola R.: Measure of nonhyperconvexity and fixed-point theorems. Abstr. Appl. Anal. 2003, 111–119 (2003)
Chang T.-H., Chen C.-M., Peng C.Y.: Generalized KKM theorems on hyperconvex metric spaces and some applications. Nonlinear Anal. 69, 530–535 (2008)
Dhompongsa S., Kirk W.A., Sims B.: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 65, 762–772 (2006)
Dhompongsa S., Kirk W.A., Panyanak B.: Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear Convex Anal. 8, 35–45 (2007)
Dress A., Scharlau R.: Gated sets in metric spaces. Aequationes Math. 34, 112–120 (1987)
Espínola R.: Darbo-Sadovski’s theorem in hyperconvex metric spaces, Supplemento ai Rendiconti del Circolo Matematico di Palermo. Serie II 40, 129–137 (1996)
Espínola R.: On selections of the metric projection and best proximity pairs in hyperconvex spaces. Ann. Univ. Mariae Curie-Skłodowska Sect. A 59, 9–17 (2005)
Espínola R., Khamsi M.A.: Introduction to hyperconvex spaces. In: Kirk, W.A., Sims, B. (eds.) Handbook of Metric Fixed Point Theory, pp. 391–435. Kluwer Academic Publishers, Dordrecht (2001)
Espínola R., Kirk W.A.: Fixed point theorems in \({\mathbb{R}}\)-trees with applications to graph theory. Topol. Appl. 153, 1046–1055 (2006)
Espínola R., Kirk W.A., López G.: Nonexpansive retracts in hyperconvex spaces. J. Math. Anal. Appl. 251, 557–570 (2000)
Espínola R., López G.: Extension of compact mappings and \({\aleph_0}\)-hyperconvexity. Nonlinear Anal. 49, 1127–1135 (2002)
Espínola R., Lorenzo P., Nicolae A.: Fixed points, selections and common fixed points for nonexpansive-type mappings. J. Math. Anal. Appl. 382, 503–515 (2011)
Goebel K., Kirk W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)
Goebel K., Kirk W.A.: A fixed point theorem for transformations whose iterates have uniform Lipschitz constant. Studia Math. 47, 135–140 (1973)
Goebel K., Reich S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Marcel Dekker, New York (1984)
Grünbaum B.: On some covering and intersection properties in Minkowski spaces. Pac. J. Math. 9, 487–494 (1959)
Heinonen J.: Lectures on Analysis on Metric Spaces. Springer, Berlin (2001)
Heinonen J.: Nonsmooth calculus. Bull. Am. Math. Soc. (N.S.) 44, 163–232 (2007)
Isbell J.R.: Injective envelopes of of Banach spaces are rigidly attached. Bull. Am. Math. Soc. 70, 727–729 (1964)
Khamsi M.A.: KKM and Ky Fan Theorems in hyperconvex metric spaces. J. Math. Anal. Appl. 204, 298–306 (1996)
Khamsi M.A.: Sadovskii’s fixed point theorem without convexity. Nonlinear Anal. 53, 829–837 (2003)
Khamsi M.A., Lin M., Sine R.: On the fixed points of commuting nonexpansive maps in hyperconvex spaces. J. Math. Anal. Appl. 168, 372–380 (1992)
Khamsi M.A., Kirk W.A.: An Introduction to Metric Spaces and Fixed Point Theory. Wiley Interscience, New York (2001)
Khamsi M.A., Kirk W.A., Martínez Yáñez C.: Fixed point and selection theorems in hyperconvex spaces. Proc. Am. Math. Soc. 128, 3275–3283 (2000)
Khan A.R., Hussain N., Thaheem A.B.: Some generalizations of Ky Fan’s best approximation theorem. Anal. Theory Appl. 20, 189–198 (2004)
Khamsi M.A., Knaust H., Nguyen N.T., O’Neill M.D.: Lambda-hyperconvexity in metric spaces. Nonlinear Anal. 43, 21–31 (2001)
Kirk W.A.: Hyperconvexity of \({\mathbb{R}}\)-trees. Fundamenta Mathematicae 156, 67–72 (1998)
Kirk, W.A.: A note on geodesically bounded \({\mathbb{R}}\)-trees. Fixed Point Theory Appl. Article ID 393470 (2010)
Kirk W.A.: Fixed point theorems in CAT(0) spaces and \({\mathbb{R}}\)-trees. Fixed Point Theory Appl. 4, 309–316 (2004)
Kirk W.A.: Krasnoselskii’s iteration process in hyperbolic space. Numer. Funct. Anal. Optimiz. 4, 371–381 (1981–1982)
Kirk W.A., Panyanak B.: Best approximation in \({\mathbb{R}}\)-trees. Numer. Funct. Anal. Optim. 28, 681–690 (2007)
Kirk W.A., Panyanak B.: Remarks on best approximation in \({\mathbb{R}}\)-trees. Ann. Univ. Mariae Curie-Skłodowska Sect. A 63, 133–138 (2009)
Kirk W.A., Shin S.S.: Fixed point theorems in hyperconvex spaces. Houst. J. Math. 23, 175–187 (1997)
Kirk W.A., Reich S., Veeramani P.: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Opt. 24, 851–862 (2003)
Lacey H.E.: The Isometric Theory of Classical Banach Spaces. Springer-Verlag, New York (1974)
Lancien G., Randrianantoanina B.: On the extension of Hölder maps with values in spaces of continuous functions. Isr. J. Math. 147, 75–92 (2005)
Lin M., Sine R.: On the fixed point set of order preserving maps. Math. Zeit. 203, 227–234 (1990)
Lindenstrauss J.: Extension of compact operators. Mem. Am. Math. Soc. AMS 48, 1–112 (1964)
Markin J.T.: A best approximation theorem for nonexpansive set-valued mappings in hyperconvex metric spaces. Rocky Mt. J. Math. 35, 2435–2441 (2009)
Markin J.T.: Fixed points, selections, and best approximation for multivalued mappings in \({\mathbb{R}}\)-trees. Nonlinear Anal. 67, 2712–2716 (2007)
Markin J.T., Shahzad N.: Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces. Nonlinear Anal. 70, 2435–2441 (2009)
Nadler S.B. Jr.: Multi-valued contraction mappings. Pac. J. Math. 30, 2059–2063 (2005)
Nowakowski R., Rival I.: Fixed-edge theorem for graphs with loops. J. Graph Theory 3, 339–350 (1979)
Pia¸tek B.: Best approximation of coincidence points in metric trees. Ann. Univ. Mariae Curie-Skłodowska Sect. A 62, 113–121 (2008)
Pia¸tek, B.; Espínola, R.: Fixed points and non-convex sets in CAT(0) spaces. Topol. Meth. Nonlinear Anal (to appear)
Razani A., Salahifard H.: Invariant approximation for CAT(0) spaces. Nonlinear Anal. 72, 2421–2425 (2010)
Sine R.: On nonlinear contraction semigroups in sup norm spaces. Nonlinear Anal. 3, 885–890 (1979)
Sine R.: Hyperconvexity and approximate fixed points. Nonlinear Anal. 13, 863–869 (1989)
Sine R.: Hyperconvexity and nonexpansive multifunctions. Trans. Am. Math. Soc. 315, 755–767 (1989)
Sine R.: Hyperconvexity and approximate fixed points. Nonlinear Anal. 13, 863–869 (1989)
Soardi P.: Existence of fixed points of nonexpansive mappings in certain Banach lattices. Proc. Am. Math. Soc. 73, 25–29 (1979)
Suzuki T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 340, 1088–1095 (2008)
Young G.S.: The introduction of local connectivity by change of topology. Am. J. Math. 68, 479–494 (1946)
Acknowledgments
The first author wishes to thank The University of Tabuk for its kind invitation to take part in The International Mathematical Workshop (Fixed Point Theory and its Applications) May 26–27, 2012. This work is a contribution to this event. This research is partially supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 and Grant P08-FQM-03543, and by MEC Grant MTM2009-10696-C02-01.
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Espínola, R., Lorenzo, P. Metric fixed point theory on hyperconvex spaces: recent progress. Arab. J. Math. 1, 439–463 (2012). https://doi.org/10.1007/s40065-012-0044-z
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DOI: https://doi.org/10.1007/s40065-012-0044-z