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Browder’s convergence theorem for multivalued mappings without endpoint condition. (English) Zbl 1245.54042
A geodesic path joining x,y in a metric space (X,d) is an isometric mapping c:[0,1]X such that c(0)=x and c(1)=y. The space is said to be geodesic if any two points can be joined by a geodesic path. It is said to be an -tree if for any x,yX there is a unique geodesic path from x to y denoted by [x,y] such that the following condition holds: if [y,x][x,z]={x} then [y,x][x,z]=[y,z]. Let then E be a nonempty closed convex subset of a complete -tree X and let T be a multivalued nonexpansive mapping from E into the nonempty compact convex subsets of E. Choose uE and define f:EE by defining f(x) as the nearest point to u in T(x). For s(0,1) denote by t s (x) the point z[u,f(x)] that satisfies d(u,z)=s. Finally, let x s be the unique fixed point of t s . The authors prove that T has a fixed point if and only if {x s } remains bounded as s0. In this case (x s ) converges to the unique fixed point of T that is nearest to u.
MSC:
54H25Fixed-point and coincidence theorems in topological spaces
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
References:
[1]Aksoy, A. G.; Khamsi, M. A.: A selection theorem in metric trees, Proc. amer. Math. soc. 134, 2957-2966 (2006) · Zbl 1102.54022 · doi:10.1090/S0002-9939-06-08555-8
[2]Aleomraninejad, S. M. A.; Rezapour, Sh.; Shahzad, N.: Some fixed point results on a metric space with a graph, Topology appl. 159, 659-663 (2012)
[3]Amini-Harandi, A.; Farajzadeh, A. P.: Best approximation, coincidence and fixed point theorems for set-valued maps in R-trees, Nonlinear anal.: TMA 71, 1649-1653 (2009) · Zbl 1179.54050 · doi:10.1016/j.na.2009.01.001
[4]Bridson, M.; Haefliger, A.: Metric spaces of non-positive curvature, (1999)
[5]Browder, F. E.: Convergence of approximants to fixed points of nonexpansive mappings in Banach spaces, Arch. ration. Mech. anal. 24, 82-90 (1967) · Zbl 0148.13601 · doi:10.1007/BF00251595
[6]Dhompongsa, S.; Kaewkhao, A.; Panyanak, B.: On kirk’s strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces, Nonlinear anal.: TMA 75, 459-468 (2012)
[7]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
[8]Dhompongsa, S.; Kirk, W. A.; Sims, B.: Fixed points of uniformly Lipschitzian mappings, Nonlinear anal.: TMA 65, 762-772 (2006) · Zbl 1105.47050 · doi:10.1016/j.na.2005.09.044
[9]Dhompongsa, S.; Panyanak, B.: On Δ-convergence theorems in CAT(0) spaces, Comput. math. Appl. 56, 2572-2579 (2008) · Zbl 1165.65351 · doi:10.1016/j.camwa.2008.05.036
[10]Espinola, R.; Kirk, W. A.: Fixed point theorems in R-trees with applications to graph theory, Topology appl. 153, 1046-1055 (2006) · Zbl 1095.54012 · doi:10.1016/j.topol.2005.03.001
[11]Goebel, K.: On a fixed point theorem for multivalued nonexpansive mappings, Ann. univ. Mariae Curie-sklodowska sect. A math. 29, 69-72 (1975) · Zbl 0365.47032
[12]Halpern, B.: Fixed points of nonexpansive maps, Bull. amer. Math. soc. 73, 957-961 (1967) · Zbl 0177.19101 · doi:10.1090/S0002-9904-1967-11864-0
[13]Jung, J. S.: Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces, Nonlinear anal.: TMA 66, 2345-2354 (2007) · Zbl 1123.47047 · doi:10.1016/j.na.2006.03.023
[14]Kim, T. H.; Jung, J. S.: Approximating fixed points of nonlinear mappings in Banach spaces, Ann. univ. Mariae Curie-sklodowska sect. A math. 51, 149-165 (1997) · Zbl 1012.47034
[15]Kirk, W. A.: Fixed point theorems in CAT(0) spaces and R-trees, Fixed point theory appl. 2004, 309-316 (2004) · Zbl 1089.54020 · doi:10.1155/S1687182004406081
[16]Lim, T. C.: A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space, Bull. amer. Math. soc. 80, 1123-1126 (1974) · Zbl 0297.47045 · doi:10.1090/S0002-9904-1974-13640-2
[17]Lopez, G.; Xu, H. K.: Remarks on multivalued nonexpansive mappings, Soochow J. Math. 21, 107-115 (1995) · Zbl 0826.47037
[18]Markin, J. T.: Fixed points for generalized nonexpansive mappings in R-trees, Comput. math. Appl. 62, 4614-4618 (2011)
[19]Markin, J. T.: Fixed points, selections and best approximation for multivalued mappings in R-trees, Nonlinear anal.: TMA 67, 2712-2716 (2007) · Zbl 1128.47052 · doi:10.1016/j.na.2006.09.036
[20]Jr., S. B. Nadler: Multi-valued contraction mappings, Pacific J. Math. 30, 475-487 (1969) · Zbl 0187.45002
[21]Piatek, B.: Best approximation of coincidence points in metric trees, Ann. univ. Mariae Curie-sklodowska sect. A math. 62, 113-121 (2008) · Zbl 1182.54055 · doi:10.2478/v10062-008-0013-3
[22]Pietramala, P.: Convergence of approximating fixed point sets for multivalued nonexpansive mappings, Comment. math. Univ. carolin. 32, 697-701 (1991) · Zbl 0756.47039
[23]Saejung, S.: Halpern’s iteration in CAT(0) spaces, Fixed point theory appl. 2010 (2010) · Zbl 1197.54074 · doi:10.1155/2010/471781
[24]Sahu, D. R.: Strong convergence theorems for nonexpansive type and non-self multi-valued mappings, Nonlinear anal.: TMA 37, 401-407 (1999) · Zbl 0938.47039 · doi:10.1016/S0362-546X(98)00056-X
[25]Shahzad, N.: Fixed point results for multimaps in CAT(0) spaces, Topology appl. 156, 997-1001 (2009) · Zbl 1175.47049 · doi:10.1016/j.topol.2008.11.016
[26]Shahzad, N.; Zegeye, H.: Strong convergence results for nonself multimaps in Banach spaces, Proc. amer. Math. soc. 136, 539-548 (2008) · Zbl 1135.47054 · doi:10.1090/S0002-9939-07-08884-3
[27]Song, Y.; Cho, Y. J.: Iterative approximations for multivalued nonexpansive mappings in reflexive Banach spaces, Math. inequal. Appl. 12, 611-624 (2009) · Zbl 1183.47073 · doi:http://files.ele-math.com/abstracts/mia-12-47-abs.pdf
[28]Tits, J.: A theorem of Lie-kolchin for trees. Contributions to algebra: A collection of papers dedicated to Ellis kolchin, (1977) · Zbl 0373.20039
[29]Wlodarczyk, K.; Klim, D.; Plebaniak, R.: Existence and uniqueness of endpoints of closed set-valued asymptotic contractions in metric spaces, J. math. Anal. appl. 328, 46-57 (2007) · Zbl 1110.54025 · doi:10.1016/j.jmaa.2006.05.029