Fixed points of continuous pseudocontractive mappings on convex, closed unbounded domains

Garcia-Falset, Jesús

doi:10.1090/proc/16756

Fixed points of continuous pseudocontractive mappings on convex, closed unbounded domains
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by Jesús Garcia-Falset;

Proc. Amer. Math. Soc. 152 (2024), 2863-2876

DOI: https://doi.org/10.1090/proc/16756

Published electronically: May 15, 2024

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Abstract:

The purpose in this article is to discuss under what conditions a continuous pseudocontractive mapping, in particular a nonexpansive map, has a fixed point when its domain is a closed, convex and unbounded subset of a Banach space. In fact, we will give a characterization of this property. The Halpern iterative algorithm for nonexpansive mappings is also investigated.

References

Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. MR 187120, DOI 10.1073/pnas.54.4.1041
James Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241–251. MR 394329, DOI 10.1090/S0002-9947-1976-0394329-4
Tomás Domínguez Benavides, Maria A. Japón, and Jeimer Villada Bedoya, Linearly and directionally bounded weak-star closed sets and the AFPP, Israel J. Math. 230 (2019), no. 2, 509–526. MR 3940426, DOI 10.1007/s11856-018-1810-1
Jesús García-Falset, Fixed points for mappings with the range type condition, Houston J. Math. 28 (2002), no. 1, 143–158. MR 1876945
J. Garcia-Falset and Kh. Latrach, Nonlinear functional analysis and applications, Series in Nonlinear Analysis and Applications, De Gruyter, 2023.
Jesús García-Falset and E. Llorens-Fuster, Fixed points for pseudocontractive mappings on unbounded domains, Fixed Point Theory Appl. , posted on (2010), Art. ID 769858, 17. MR 2557279, DOI 10.1155/2010/769858
J. Garcia-Falset and O. Muñiz-Pérez, Fixed point theory for 1-set contractive and pseudocontractive mappings, Appl. Math. Comput. 219 (2013), no. 12, 6843–6855. MR 3027851, DOI 10.1016/j.amc.2012.12.079
Jesús García-Falset and Omar Muñíz-Pérez, Projected dynamical systems on Hilbert spaces, J. Nonlinear Convex Anal. 15 (2014), no. 2, 325–344. MR 3184326
Jesús García-Falset and Claudio H. Morales, Existence theorems for $m$-accretive operators in Banach spaces, J. Math. Anal. Appl. 309 (2005), no. 2, 453–461. MR 2154128, DOI 10.1016/j.jmaa.2004.08.021
Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258 (German). MR 190718, DOI 10.1002/mana.19650300312
Kazimierz Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. MR 1074005, DOI 10.1017/CBO9780511526152
K. Goebel, On the minimal displacement of points under Lipschitzian mappings, Pacific J. Math. 48 (1973), 151–163.
Kazimierz Goebel and Łukasz Piasecki, Minimal displacement and related problems, revisited, Banach and function spaces IV (ISBFS 2012), Yokohama Publ., Yokohama, 2014, pp. 49–62. MR 3289763
Kazimierz Goebel and Simeon Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. MR 744194
William A. Kirk and Brailey Sims (eds.), Handbook of metric fixed point theory, Kluwer Academic Publishers, Dordrecht, 2001. MR 1904271, DOI 10.1007/978-94-017-1748-9
Benjamin Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957–961. MR 218938, DOI 10.1090/S0002-9904-1967-11864-0
Shiro Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), no. 1, 65–71. MR 412909, DOI 10.1090/S0002-9939-1976-0412909-X
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 189009, DOI 10.2307/2313345
W. A. Kirk, Fixed point theorems for nonexpansive mappings satisfying certain boundary conditions, Proc. Amer. Math. Soc. 50 (1975), 143–149. MR 380527, DOI 10.1090/S0002-9939-1975-0380527-7
W. A. Kirk and R. Schöneberg, Zeros of $m$-accretive operators in Banach spaces, Israel J. Math. 35 (1980), no. 1-2, 1–8. MR 576458, DOI 10.1007/BF02760935
Pei-Kee Lin, There is an equivalent norm on $l_1$ that has the fixed point property, Nonlinear Anal. 68 (2008), no. 8, 2303–2308. MR 2398651, DOI 10.1016/j.na.2007.01.050
R. H. Martin Jr., Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc. 179 (1973), 399–414. MR 318991, DOI 10.1090/S0002-9947-1973-0318991-4
Eva Matoušková and Simeon Reich, Reflexivity and approximate fixed points, Studia Math. 159 (2003), no. 3, 403–415. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday (Polish). MR 2052231, DOI 10.4064/sm159-3-5
Claudio Morales, Nonlinear equations involving $m$-accretive operators, J. Math. Anal. Appl. 97 (1983), no. 2, 329–336. MR 723235, DOI 10.1016/0022-247X(83)90199-3
Simeon Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), no. 1, 287–292. MR 576291, DOI 10.1016/0022-247X(80)90323-6
Simeon Reich, Approximating fixed points of nonexpansive mappings, Panamer. Math. J. 4 (1994), no. 2, 23–28. MR 1274185
Simeon Reich and Ricardo Torrejón, Zeros of accretive operators, Comment. Math. Univ. Carolin. 21 (1980), no. 3, 619–625. MR 590139
Simeon Reich, The almost fixed point property for nonexpansive mappings, Proc. Amer. Math. Soc. 88 (1983), no. 1, 44–46. MR 691276, DOI 10.1090/S0002-9939-1983-0691276-4
Simeon Reich, On fixed point theorems obtained from existence theorems for differential equations, J. Math. Anal. Appl. 54 (1976), no. 1, 26–36. MR 402554, DOI 10.1016/0022-247X(76)90232-8
Simeon Reich, Constructive techniques for accretive and monotone operators, Applied nonlinear analysis (Proc. Third Internat. Conf., Univ. Texas, Arlington, Tex., 1978) Academic Press, New York-London, 1979, pp. 335–345. MR 537545
Naoki Shioji and Wataru Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3641–3645. MR 1415370, DOI 10.1090/S0002-9939-97-04033-1
Barry Turett, A dual view of a theorem of Baillon, Nonlinear analysis and applications (St. Johns, Nfld., 1981) Lect. Notes Pure Appl. Math., vol. 80, Dekker, New York, 1982, pp. 279–286. MR 689566
Wataru Takahashi, Jen-Chih Yao, and Fumiaki Kohsaka, The fixed point property and unbounded sets in Banach spaces, Taiwanese J. Math. 14 (2010), no. 2, 733–742. MR 2655797, DOI 10.11650/twjm/1500405817
Rainer Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel) 58 (1992), no. 5, 486–491. MR 1156581, DOI 10.1007/BF01190119
Hong-Kun Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2) 66 (2002), no. 1, 240–256. MR 1911872, DOI 10.1112/S0024610702003332