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Does Kirk’s theorem hold for multivalued nonexpansive mappings? (English) Zbl 1207.47054

Let \(X\) be a Banach space and \(C\) be a bounded closed convex subset of \(X\). Suppose that \(KC(C)\) is the family of nonempty compact and convex subsets of \(C\). A mapping \(T:C\to KC(C)\) is nonexpansive if \(H(Tx,Ty)\leq\|x-y\|\) for all \(x,y\in C\), where \(H(\cdot,\cdot)\) is the Hausdorff metric. W.A.Kirk and S.Massa [Houston J. Math.16, No.3, 357–364 (1990; Zbl 0729.47053)] proved that, if the asymptotic center in \(C\) of each bounded sequence of \(X\) is nonempty and compact, then \(T\) has a fixed point. The authors study and survey some connections between the properties of asymptotic centers and geometric properties of Banach spaces to obtain the same conclusion as above. They also study the same problem in modular function spaces.

MSC:

47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H04 Set-valued operators

Citations:

Zbl 0729.47053
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Full Text: DOI EuDML

References:

[1] Banach S: Sur les opérations dans les ensembles abstraits et leurs applications.Fundamenta Mathematicae 1922, 3: 133-181. · JFM 48.0201.01
[2] Kirk WA: A fixed point theorem for mappings which do not increase distances.The American Mathematical Monthly 1965, 72: 1004-1006. 10.2307/2313345 · Zbl 0141.32402 · doi:10.2307/2313345
[3] Nadler SB Jr.: Multi-valued contraction mappings.Pacific Journal of Mathematics 1969, 30: 475-488. · Zbl 0187.45002 · doi:10.2140/pjm.1969.30.475
[4] Brodskiĭ MS, Mil’man DP: On the center of a convex set.Doklady Akademii Nauk SSSR 1948, 59: 837-840.
[5] Bynum WL: Normal structure coefficients for Banach spaces.Pacific Journal of Mathematics 1980,86(2):427-436. · Zbl 0442.46018 · doi:10.2140/pjm.1980.86.427
[6] Maluta E: Uniformly normal structure and related coefficients.Pacific Journal of Mathematics 1984,111(2):357-369. · Zbl 0495.46012 · doi:10.2140/pjm.1984.111.357
[7] Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings.Bulletin of the American Mathematical Society 1967, 73: 591-597. 10.1090/S0002-9904-1967-11761-0 · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[8] Prus S: Banach spaces with the uniform Opial property.Nonlinear Analysis: Theory, Methods & Applications 1992,18(8):697-704. 10.1016/0362-546X(92)90165-B · Zbl 0786.46023 · doi:10.1016/0362-546X(92)90165-B
[9] Lin P-K, Tan K-K, Xu HK: Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings.Nonlinear Analysis: Theory, Methods & Applications 1995,24(6):929-946. 10.1016/0362-546X(94)00128-5 · Zbl 0865.47040 · doi:10.1016/0362-546X(94)00128-5
[10] Gossez J-P, Lami Dozo E: Some geometric properties related to the fixed point theory for nonexpansive mappings.Pacific Journal of Mathematics 1972, 40: 565-573. · Zbl 0223.47025 · doi:10.2140/pjm.1972.40.565
[11] Huff R: Banach spaces which are nearly uniformly convex.The Rocky Mountain Journal of Mathematics 1980,10(4):743-749. 10.1216/RMJ-1980-10-4-743 · Zbl 0505.46011 · doi:10.1216/RMJ-1980-10-4-743
[12] Goebel K, Sękowski T: The modulus of noncompact convexity.Annales Universitatis Mariae Curie-Skłodowska. Sectio A 1984, 38: 41-48. · Zbl 0607.46011
[13] Ayerbe Toledano JM, Domínguez Benavides T, López Acedo G: Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory: Advances and Applications. Volume 99. Birkhäuser, Basel, Switzerland; 1997:viii+211. · Zbl 0885.47021 · doi:10.1007/978-3-0348-8920-9
[14] Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244. · Zbl 0708.47031 · doi:10.1017/CBO9780511526152
[15] Turett, B., A dual view of a theorem of Baillon, No. 80, 279-286 (1982), New York, NY, USA
[16] Khamsi MA: On metric spaces with uniform normal structure.Proceedings of the American Mathematical Society 1989,106(3):723-726. 10.1090/S0002-9939-1989-0972234-4 · Zbl 0671.47052 · doi:10.1090/S0002-9939-1989-0972234-4
[17] Prus S: Some estimates for the normal structure coefficient in Banach spaces.Rendiconti del Circolo Matematico di Palermo 1991,40(1):128-135. 10.1007/BF02846365 · Zbl 0757.46029 · doi:10.1007/BF02846365
[18] Lami Dozo E: Multivalued nonexpansive mappings and Opial’s condition.Proceedings of the American Mathematical Society 1973, 38: 286-292. · Zbl 0268.47060 · doi:10.1090/S0002-9939-1973-0310718-0
[19] Lim TC: A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space.Bulletin of the American Mathematical Society 1974, 80: 1123-1126. 10.1090/S0002-9904-1974-13640-2 · Zbl 0297.47045 · doi:10.1090/S0002-9904-1974-13640-2
[20] Edelstein M: The construction of an asymptotic center with a fixed-point property.Bulletin of the American Mathematical Society 1972, 78: 206-208. 10.1090/S0002-9904-1972-12918-5 · Zbl 0231.47029 · doi:10.1090/S0002-9904-1972-12918-5
[21] Kirk WA, Massa S: Remarks on asymptotic and Chebyshev centers.Houston Journal of Mathematics 1990,16(3):357-364. · Zbl 0729.47053
[22] Kirk, WA, Nonexpansive mappings in product spaces, set-valued mappings and [InlineEquation not available: see fulltext.]-uniform rotundity, No. 45, 51-64 (1986), Providence, RI, USA · doi:10.1090/pspum/045.2/843594
[23] Kuczumow T, Prus S: Compact asymptotic centers and fixed points of multivalued nonexpansive mappings.Houston Journal of Mathematics 1990,16(4):465-468. · Zbl 0724.47033
[24] Xu H-K: Metric fixed point theory for multivalued mappings.Dissertationes Mathematicae 2000, 389: 39. · Zbl 0972.47041 · doi:10.4064/dm389-0-1
[25] Domínguez Benavides T, Lorenzo Ramírez P: Fixed-point theorems for multivalued non-expansive mappings without uniform convexity.Abstract and Applied Analysis 2003,2003(6):375-386. 10.1155/S1085337503203080 · Zbl 1058.47047 · doi:10.1155/S1085337503203080
[26] Domínguez Benavides T, Lorenzo Ramírez P: Asymptotic centers and fixed points for multivalued nonexpansive mappings.Annales Universitatis Mariae Curie-Skłodowska. Sectio A 2004, 58: 37-45. · Zbl 1105.47047
[27] Dhompongsa S, Kaewcharoen A, Kaewkhao A: The Domínguez-Lorenzo condition and multivalued nonexpansive mappings.Nonlinear Analysis: Theory, Methods & Applications and Methods 2006,64(5):958-970. 10.1016/j.na.2005.05.051 · Zbl 1106.47046 · doi:10.1016/j.na.2005.05.051
[28] Dhompongsa S, Domínguez Benavides T, Kaewcharoen A, Kaewkhao A, Panyanak B: The Jordan-von Neumann constants and fixed points for multivalued nonexpansive mappings.Journal of Mathematical Analysis and Applications 2006,320(2):916-927. 10.1016/j.jmaa.2005.07.063 · Zbl 1103.47043 · doi:10.1016/j.jmaa.2005.07.063
[29] Domínguez Benavides T, Gavira B: The fixed point property for multivalued nonexpansive mappings.Journal of Mathematical Analysis and Applications 2007,328(2):1471-1483. 10.1016/j.jmaa.2006.06.059 · Zbl 1112.47042 · doi:10.1016/j.jmaa.2006.06.059
[30] Saejung S: Remarks on sufficient conditions for fixed points of multivalued nonexpansive mappings.Nonlinear Analysis: Theory, Methods & Applications 2007,67(5):1649-1653. 10.1016/j.na.2006.07.037 · Zbl 1120.46009 · doi:10.1016/j.na.2006.07.037
[31] Kaewkhao A: The James constant, the Jordan-von Neumann constant, weak orthogonality, and fixed points for multivalued mappings.Journal of Mathematical Analysis and Applications 2007,333(2):950-958. 10.1016/j.jmaa.2006.12.001 · Zbl 1128.47046 · doi:10.1016/j.jmaa.2006.12.001
[32] Gavira B: Some geometric conditions which imply the fixed point property for multivalued nonexpansive mappings.Journal of Mathematical Analysis and Applications 2008,339(1):680-690. 10.1016/j.jmaa.2007.07.015 · Zbl 1132.47043 · doi:10.1016/j.jmaa.2007.07.015
[33] Jiao, H.; Guo, Y.; Wang, F., Modulus of convexity, the coefficient [InlineEquation not available: see fulltext.], and normal structure in Banach spaces, No. 2008, 5 (2008)
[34] Nakano H: Modulared Semi-Ordered Linear Spaces. Maruzen, Tokyo, Japan; 1950. · Zbl 0041.23401
[35] Musielak J, Orlicz W: On modular spaces.Studia Mathematica 1959, 18: 49-65. · Zbl 0086.08901
[36] Domínguez Benavides T, Khamsi MA, Samadi S: Asymptotically regular mappings in modular function spaces.Scientiae Mathematicae Japonicae 2001,53(2):295-304. · Zbl 0983.46028
[37] Domínguez Benavides T, Khamsi MA, Samadi S: Asymptotically nonexpansive mappings in modular function spaces.Journal of Mathematical Analysis and Applications 2002,265(2):249-263. 10.1006/jmaa.2000.7275 · Zbl 1014.47031 · doi:10.1006/jmaa.2000.7275
[38] Khamsi, MA, Fixed point theory in modular function spaces, No. 48, 31-57 (1996), Seville, Spain · Zbl 0886.46030
[39] Khamsi MA, Kozłowski WM, Reich S: Fixed point theory in modular function spaces.Nonlinear Analysis: Theory, Methods & Applications 1990,14(11):935-953. 10.1016/0362-546X(90)90111-S · Zbl 0714.47040 · doi:10.1016/0362-546X(90)90111-S
[40] Dhompongsa S, Domínguez Benavides T, Kaewcharoen A, Panyanak B: Fixed point theorems for multivalued mappings in modular function spaces.Scientiae Mathematicae Japonicae 2006,63(2):161-169. · Zbl 1107.47037
[41] Kozlowski WM: Modular Function Spaces, Monographs and Textbooks in Pure and Applied Mathematics. Volume 122. Marcel Dekker, New York, NY, USA; 1988:x+252.
[42] Musielak J: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics. Volume 1034. Springer, Berlin, Germany; 1983:iii+222.
[43] Alspach DE: A fixed point free nonexpansive map.Proceedings of the American Mathematical Society 1981,82(3):423-424. 10.1090/S0002-9939-1981-0612733-0 · Zbl 0468.47036 · doi:10.1090/S0002-9939-1981-0612733-0
[44] Domínguez Benavides T, García Falset J, Llorens-Fuster E, Lorenzo Ramírez P: Fixed point properties and proximinality in Banach spaces.Nonlinear Analysis: Theory, Methods & Applications 2009,71(5-6):1562-1571. 10.1016/j.na.2008.12.048 · Zbl 1181.47055 · doi:10.1016/j.na.2008.12.048
[45] Hu S, Papageorgiou NS: Handbook of Multivalued Analysis. Vol. I. Theory, Mathematics and Its Applications. Volume 419. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xvi+964.
[46] Goebel, K.; Kuczumov, T., Irregular convex sets with fixed point property for nonexpansive mappings, 259-264 (1979) · Zbl 0418.47031
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