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Metric fixed point theory on hyperconvex spaces: recent progress. (English) Zbl 1273.54052
A metric space \(M\) is said to be hyperconvex if given any family \(\{x_{\alpha}\}\) of points of \(M\) and any family \(\{r_{\alpha}\}\) of real numbers satisfying \( d(x_{\alpha}, x_{\beta}) \leq r_{\alpha}+r_{\beta}\), then \(\bigcap_{\alpha} B(x_{\alpha},r_{\alpha}) \neq \varnothing\). In this survey paper, the authors review the development of metric fixed point theory on hyperconvex metric spaces. In Section 4, they discuss the problem of characterizing proximal nonexpansive retracts of hyperconvex spaces and its connections to several problems in best approximation and best proximity pairs. In Section 5, recent developments on \(\mathbb{R}\) trees and metric fixed point theory are treated. In the last section, some recent advances on the theory of extensions of Hölder maps and their relation to extensions of uniformly continuous mappings under \(\chi_0\) hyperconvex conditions are presented. Some new results on the extension of compact mappings are also given.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
54-02 Research exposition (monographs, survey articles) pertaining to general topology
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[1] Aksoy, A.G.; Khamsi, M.A., A selection theorem in metric trees, Proc. Am. Math. Soc., 134, 2957-2966, (2006) · Zbl 1102.54022
[2] Aksoy, A.G.; Khamsi, M.A., Fixed points of uniformly Lipschitzian mappings in metric trees, Sci. Math. Jpn., 65, 31-41, (2007) · Zbl 1145.54039
[3] Ambrosio L., Tilli P.: Topics on Analysis in Metric Spaces. Oxford University Press, Oxford (2004) · Zbl 1080.28001
[4] Amini-Harandi, A.; Farajzadeh, A.P., A best approximation theorem in hyperconvex metric spaces, Nonlinear Anal., 70, 2453-2456, (2009) · Zbl 1179.90251
[5] 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) · Zbl 1179.54050
[6] 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) · Zbl 1158.54317
[7] 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) · Zbl 1161.54021
[8] Aronszajn, N.; Panitchpakdi, P., Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pac. J. Math., 6, 405-439, (1956) · Zbl 0074.17802
[9] Baillon, J.B., Nonexpansive mappings and hyperconvex spaces, Contemp. Math., 72, 11-19, (1988)
[10] Ball, K., Markov chains, Riesz transforms and Lipschitz maps, Geom. Funct. Anal., 2, 137-172, (1992) · Zbl 0788.46050
[11] Borkowski, M.; Bugajewski, D., On fixed point theorems of Leray-Schauder type, Proc. Am. Math. Soc., 136, 973-980, (2008) · Zbl 1136.54026
[12] Borkowski, M.; Bugajewski, D.; Phulara, D.: On some properties of hyperconvex spaces. Fixed Point Theory Appl. Article ID 213812 (2010) · Zbl 1213.54037
[13] Bridson M.R., Haefliger A.: Metric Spaces of Non-Positive Curvature. Springer-Verlag, Berlin (1999) · Zbl 0988.53001
[14] Bugajewski, D.; Espínola, R., Measure of nonhyperconvexity and fixed-point theorems, Abstr. Appl. Anal., 2003, 111-119, (2003) · Zbl 1057.47059
[15] 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) · Zbl 1171.47042
[16] Dhompongsa, S.; Kirk, W.A.; Sims, B., Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 65, 762-772, (2006) · Zbl 1105.47050
[17] Dhompongsa, S.; Kirk, W.A.; Panyanak, B., Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal., 8, 35-45, (2007) · Zbl 1120.47043
[18] Dress, A.; Scharlau, R., Gated sets in metric spaces, Aequationes Math., 34, 112-120, (1987) · Zbl 0696.54022
[19] Espínola, R., Darbo-sadovski’s theorem in hyperconvex metric spaces, supplemento ai rendiconti del circolo matematico di Palermo, Serie II, 40, 129-137, (1996)
[20] 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) · Zbl 1145.47036
[21] Espínola, R.; Khamsi, M.A.; Kirk, W.A. (ed.); Sims, B. (ed.), Introduction to hyperconvex spaces, 391-435, (2001), Dordrecht · Zbl 1029.47002
[22] Espínola, R.; Kirk, W.A., Fixed point theorems in \({\mathbb{R}}\)-trees with applications to graph theory, Topol. Appl., 153, 1046-1055, (2006) · Zbl 1095.54012
[23] Espínola, R.; Kirk, W.A.; López, G., Nonexpansive retracts in hyperconvex spaces, J. Math. Anal. Appl., 251, 557-570, (2000) · Zbl 0971.47036
[24] Espínola, R.; López, G., Extension of compact mappings and \({ℵ_0}\)-hyperconvexity, Nonlinear Anal., 49, 1127-1135, (2002) · Zbl 1006.54022
[25] 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) · Zbl 1232.47045
[26] Goebel K., Kirk W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990) · Zbl 0708.47031
[27] Goebel, K.; Kirk, W.A., A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math., 47, 135-140, (1973) · Zbl 0265.47044
[28] Goebel K., Reich S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Marcel Dekker, New York (1984) · Zbl 0537.46001
[29] Grünbaum, B., On some covering and intersection properties in Minkowski spaces, Pac. J. Math., 9, 487-494, (1959) · Zbl 0086.15203
[30] Heinonen J.: Lectures on Analysis on Metric Spaces. Springer, Berlin (2001) · Zbl 0985.46008
[31] Heinonen, J., Nonsmooth calculus, Bull. Am. Math. Soc. (N.S.), 44, 163-232, (2007) · Zbl 1124.28003
[32] Isbell, J.R., Injective envelopes of of Banach spaces are rigidly attached, Bull. Am. Math. Soc., 70, 727-729, (1964) · Zbl 0128.34503
[33] Khamsi, M.A., KKM and Ky Fan theorems in hyperconvex metric spaces, J. Math. Anal. Appl., 204, 298-306, (1996) · Zbl 0869.54045
[34] Khamsi, M.A., Sadovskii’s fixed point theorem without convexity, Nonlinear Anal., 53, 829-837, (2003) · Zbl 1028.47042
[35] 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) · Zbl 0767.54039
[36] Khamsi M.A., Kirk W.A.: An Introduction to Metric Spaces and Fixed Point Theory. Wiley Interscience, New York (2001) · Zbl 1318.47001
[37] 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) · Zbl 0959.47032
[38] Khan, A.R.; Hussain, N.; Thaheem, A.B., Some generalizations of Ky fan’s best approximation theorem, Anal. Theory Appl., 20, 189-198, (2004) · Zbl 1072.41021
[39] Khamsi, M.A.; Knaust, H.; Nguyen, N.T.; O’Neill, M.D., Lambda-hyperconvexity in metric spaces, Nonlinear Anal., 43, 21-31, (2001) · Zbl 0969.47036
[40] Kirk, W.A., Hyperconvexity of \({\mathbb{R}}\)-trees, Fundamenta Mathematicae, 156, 67-72, (1998) · Zbl 0913.54030
[41] Kirk, W.A.: A note on geodesically bounded \({\mathbb{R}}\)-trees. Fixed Point Theory Appl. Article ID 393470 (2010) · Zbl 1095.54012
[42] Kirk, W.A., Fixed point theorems in CAT(0) spaces and \({\mathbb{R}}\)-trees, Fixed Point Theory Appl., 4, 309-316, (2004) · Zbl 1089.54020
[43] Kirk, W.A., Krasnoselskii’s iteration process in hyperbolic space, Numer. Funct. Anal. Optimiz., 4, 371-381, (1981-1982) · Zbl 0505.47046
[44] Kirk, W.A.; Panyanak, B., Best approximation in \({\mathbb{R}}\)-trees, Numer. Funct. Anal. Optim., 28, 681-690, (2007) · Zbl 1132.54025
[45] Kirk, W.A.; Panyanak, B., Remarks on best approximation in \({\mathbb{R}}\)-trees, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 63, 133-138, (2009) · Zbl 1191.54040
[46] Kirk, W.A.; Shin, S.S., Fixed point theorems in hyperconvex spaces, Houst. J. Math., 23, 175-187, (1997) · Zbl 0957.46033
[47] Kirk, W.A.; Reich, S.; Veeramani, P., Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Opt., 24, 851-862, (2003) · Zbl 1054.47040
[48] Lacey H.E.: The Isometric Theory of Classical Banach Spaces. Springer-Verlag, New York (1974) · Zbl 0285.46024
[49] Lancien, G.; Randrianantoanina, B., On the extension of Hölder maps with values in spaces of continuous functions, Isr. J. Math., 147, 75-92, (2005) · Zbl 1277.46008
[50] Lin, M.; Sine, R., On the fixed point set of order preserving maps, Math. Zeit., 203, 227-234, (1990) · Zbl 0662.47030
[51] Lindenstrauss, J., Extension of compact operators, Mem. Am. Math. Soc. AMS, 48, 1-112, (1964) · Zbl 0141.12001
[52] Markin, J.T., A best approximation theorem for nonexpansive set-valued mappings in hyperconvex metric spaces, Rocky Mt. J. Math., 35, 2435-2441, (2009) · Zbl 1163.41005
[53] Markin, J.T., Fixed points, selections, and best approximation for multivalued mappings in \({\mathbb{R}}\)-trees, Nonlinear Anal., 67, 2712-2716, (2007) · Zbl 1128.47052
[54] Markin, J.T.; Shahzad, N., Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces, Nonlinear Anal., 70, 2435-2441, (2009) · Zbl 1163.41005
[55] Nadler, S.B., Multi-valued contraction mappings, Pac. J. Math., 30, 2059-2063, (2005) · Zbl 1097.54040
[56] Nowakowski, R.; Rival, I., Fixed-edge theorem for graphs with loops, J. Graph Theory, 3, 339-350, (1979) · Zbl 0432.05030
[57] Pia¸tek, B., Best approximation of coincidence points in metric trees, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 62, 113-121, (2008) · Zbl 1182.54055
[58] Pia¸tek, B.; Espínola, R.: Fixed points and non-convex sets in CAT(0) spaces. Topol. Meth. Nonlinear Anal (to appear) · Zbl 0694.54033
[59] Razani, A.; Salahifard, H., Invariant approximation for CAT(0) spaces, Nonlinear Anal., 72, 2421-2425, (2010) · Zbl 1188.54022
[60] Sine, R., On nonlinear contraction semigroups in sup norm spaces, Nonlinear Anal., 3, 885-890, (1979) · Zbl 0423.47035
[61] Sine, R., Hyperconvexity and approximate fixed points, Nonlinear Anal., 13, 863-869, (1989) · Zbl 0694.54033
[62] Sine, R., Hyperconvexity and nonexpansive multifunctions, Trans. Am. Math. Soc., 315, 755-767, (1989) · Zbl 0682.47029
[63] Sine, R., Hyperconvexity and approximate fixed points, Nonlinear Anal., 13, 863-869, (1989) · Zbl 0694.54033
[64] Soardi, P., Existence of fixed points of nonexpansive mappings in certain Banach lattices, Proc. Am. Math. Soc., 73, 25-29, (1979) · Zbl 0371.47048
[65] Suzuki, T., Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340, 1088-1095, (2008) · Zbl 1140.47041
[66] Young, G.S., The introduction of local connectivity by change of topology, Am. J. Math., 68, 479-494, (1946) · Zbl 0060.40204
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