Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Fixed point theorems for various classes of $ 1$-set-contractive and $ 1$-ball-contractive mappings in Banach spaces


Author: W. V. Petryshyn
Journal: Trans. Amer. Math. Soc. 182 (1973), 323-352
MSC: Primary 47H10
DOI: https://doi.org/10.1090/S0002-9947-1973-0328688-2
MathSciNet review: 0328688
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let X be a real Banach space, D a bounded open subset of X, and $ \bar D$ the closure of D. In §1 of this paper we establish a general fixed point theorem (see Theorem 1 below) for 1-set-contractions and 1-ball-contractions $ T:\bar D \to X$ under very mild conditions on T. In addition to classical fixed point theorems of Schauder, Leray and Schauder, Rothe, Kransnoselsky, Altman, and others for T compact, Theorem 1 includes as special cases the earlier theorem of Darbo as well as the more recent theorems of Sadovsky, Nussbaum, Petryshyn, and others (see §1 for further contributions and details) for T k-set-contractive with $ k < 1$, condensing, and 1-set-contractive. In §§2, 3, 4, and 5 of this paper Theorem 1 is used to deduce a number of known, as well as some new, fixed point theorems for various special classes of mappings (e.g. mappings of contractive type with compact or completely continuous perturbations, mappings of semicontractive type introduced by Browder, mappings of pseudo-contractive type, etc.) which have been recently extensively studied by a number of authors and, in particular, by Browder, Krasnoselsky, Kirk, and others (see §1 for details),


References [Enhancements On Off] (What's this?)

  • [1] M. Altman, A fixed point theorem in Banach spaces, Bull. Acad. Polon. Sci. Cl. III 5 (1957), 19-22. MR 19, 297. MR 0087064 (19:297b)
  • [2] L. P. Belluce and W. A. Kirk, Fixed point theorems for certain classes of nonexpansive mappings, Proc. Amer. Math. Soc. 20 (1969), 141-146. MR 38 #1663. MR 0233341 (38:1663)
  • [3] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965/66). 1041-1044. MR 32 #4574. MR 0187120 (32:4574)
  • [4] -, Fixed point theorems for nonlinear semicontractive mappings in Banach spaces, Arch. Rational Mech. Anal. 21 (1966), 259-269. MR 34 #641. MR 0200753 (34:641)
  • [5] -, Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 875-882. MR 38 #581. MR 0232255 (38:581)
  • [6] -, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660-665. MR 37 #5742. MR 0230179 (37:5742)
  • [7] -, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Pure Math., Vol. 18, Part II, Amer. Math. Soc., Providence, R.I. (to appear). MR 0405188 (53:8982)
  • [8] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-228. MR 36 #747. MR 0217658 (36:747)
  • [9] G. Darbo, Punti uniti in transformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84-92. MR 16, 1140. MR 0070164 (16:1140f)
  • [10] J. Daneś, Generalized concentrative mappings and their fixed points, Comment. Math. Univ. Carolinae 11 (1970), 115-136. MR 41 #7668. MR 0263063 (41:7668)
  • [11] D. E. Edmunds and J. R. L. Webb, Nonlinear operator equations in Hilbert spaces, J. Math. Anal. Appl. 34 (1971), 471-478. MR 43 #6787. MR 0281068 (43:6787)
  • [12] D. E. Edmunds, Remarks on nonlinear functional equations, Math. Ann. 174 (1967), 233-239. MR 36 #3180. MR 0220113 (36:3180)
  • [13] P. M. Fitzpatrick, A-proper mappings and their uniform limits, Ph. D. Thesis, Rutgers University, New Brunswick, N. J., 1971. MR 0303375 (46:2512)
  • [14] R. L. Frum-Ketkov, Mappings into a Banach sphere, Dokl. Akad. Nauk SSSR 175 (1967), 1229-1231 = Soviet Math. Dokl. 8 (1967), 1004-1006. MR 36 #3181. MR 0220114 (36:3181)
  • [15] S. Fucík, Fixed point theorems for a sum of nonlinear mappings, Comment. Math. Univ. Carolinae 9 (1968), 133-143. MR 38 #1567. MR 0233245 (38:1567)
  • [16] M. Furi and A. Vignoli, A fixed point theorem in complete metric spaces, Boll. Un. Mat. Ital. (4) 2 (1969), 505-509. MR 41 #1034. MR 0256378 (41:1034)
  • [17] -, On a-nonexpansive mappings and fixed points, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 48 (1970), 195-198. MR 43 #5513. MR 0279792 (43:5513)
  • [18] J. A. Gatica and W. A. Kirk, Fixed point theorems for Lipschitzian pseudo-contractive mappings, Proc. Amer. Math. Soc. 36 (1972), 111-115. MR 0306993 (46:6114)
  • [19] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251-258. MR 32 #8129. MR 0190718 (32:8129)
  • [20] H. Hanani, E. Netanyaku and M. Reichaw-Reichbach, The sphere in the image, Israel J. Math. 1 (1963), 188-195. MR 29 #453. MR 0163150 (29:453)
  • [21] V. Istrateseu and A. Istrateseu, On the theory of fixed points for some classes of mappings. IV (to appear).
  • [22] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. MR 32 #6436. MR 0189009 (32:6436)
  • [23] -, On nonlinear mappings of strongly semicontractive type, J. Math. Anal. Appl. 27 (1969), 409-412. MR 39 #6128. MR 0244814 (39:6128)
  • [24] -, Remarks on pseudo-contractive mappings, Proc. Amer. Math. Soc. 25 (1970), 820-823. MR 41 #9074. MR 0264481 (41:9074)
  • [25] -, Mappings of generalized contractive type (to appear).
  • [26] M. A. Krasnosel'skiĭ, Two remarks on the method of succesive approximations, Uspehi Mat. Nauk 10 (1955), no. 1 (63), 123-127. (Russian) MR 16, 833. MR 0068119 (16:833a)
  • [27] -, Topological methods in the theory of nonlinear integral equations, GITTL, Moscow, 1956; English transl., Macmillan, New York, 1964. MR 20 #3464; MR 28 #2414. MR 0159197 (28:2414)
  • [28] K. Kuratowski, Topologie. Vol. 1, PWN, Warsaw, 1958; English transl., Academic Press, New York; PWN, Warsaw, 1966. MR 19, 873; MR 36 #840. MR 0217751 (36:840)
  • [29] R. D. Nussbaum, The fixed point index and fixed point theorems for k-set-contractions, Ph. D. Thesis, University of Chicago, Chicago, Ill., 1969.
  • [30] -, The radius of the essential spectrum, Duke Math. J. 37 (1970), 473-478. MR 41 #9028. MR 0264434 (41:9028)
  • [31] -, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. 89 (1971), 217-258. MR 0312341 (47:903)
  • [32] W. V. Petryshyn, Fixed point theorems involving P-compact, semicontractive, and accretive operators not defined on all of Banach space, J. Math. Anal. Appl. 23 (1968), 336-354. MR 38 #588. MR 0232262 (38:588)
  • [33] W. V. Petryshyn and T. S. Tucker, On the functional equations involving nonlinear generalized P-compact operators, Trans. Amer. Math. Soc. 135 (1969), 343-373. MR 40 #804. MR 0247539 (40:804)
  • [34] W. V. Petryshyn, Structure of the fixed points sets of k-set-contractions, Arch. Rational Mech. Anal. 40 (1970/71), 312-328. MR 42 #8358. MR 0273480 (42:8358)
  • [35] -, Note on the structure of fixed point sets of 1-set-contractions, Proc. Amer. Math. Soc. 31 (1972), 189-194. MR 0285944 (44:3161)
  • [36] -, Remarks on condensing and k-set-contractive mappings, J. Math. Anal. Appl. Appl. 39 (1972), 717-741. MR 0328687 (48:7029)
  • [37] J. Reinermann, Fixpunktsätze vom Krasnoselski-Typ, Math. Z. 119 (1971), 339-344. MR 43 #3873. MR 0278142 (43:3873)
  • [38] E. Rothe, Zur Theorie der topologischen Ordnung und der Vektorfelder in Banachschen Raumen, Compositio Mat. 5 (1937), 177-197.
  • [39] B. N. Sadovskiĭ, On a fixed point principle, Funkcional. Anal. i Priložen. 1 (1967), no. 2, 74-76. (Russian) MR 35 #2184. MR 0211302 (35:2184)
  • [40] -, Measures of noncompactness and condensing maps, Problemy Mat. Anal. Slož. Sistem 2 (1968), 89-119.
  • [41] J. Schauder, Der Fixpunktsatz in Funktionalraumen, Studia Math. 2 (1930), 171-180.
  • [42] J. G. Stampfli, Adjoint abelian operators on Banach space, Canad. J. Math. 21 (1969), 505-512. MR 39 #807. MR 0239450 (39:807)
  • [43] M. M. Vaĭnberg, Variational methods for the study of non-linear operators, GITTL, Moscow, 1956; English transl., Holden-Day, San Francisco, Calif., 1964. MR 19, 567; MR 31 #638.
  • [44] G. M. Vaĭnikko and B. N. Sadovskiĭ, On the degree of (ball) condensing vector fields, Problemy Mat. Anal. Slož. Sistem. 2 (1968), 84-88. (Russian)
  • [45] J. R. L. Webb, Fixed point theorems for nonlinear semicontractive operators in Banach spaces, J. London Math. Soc. (2) 1 (1969), 683-688. MR 40 # 3392. MR 0250152 (40:3392)
  • [46] -, Mapping and fixed ponit theorems for nonlinear operators in Banach spaces, Proc. London Math. Soc. (3) 20 (1970), 451-468. MR 42 #917. MR 0266008 (42:917)
  • [47] -, Remarks on k-set-contractions, Boll. Un. Mat. Ital. 4 (1971), 614-629. MR 0293467 (45:2544)
  • [48] S. Yamamuro, Some fixed point theorems in locally convex linear spaces, Yokohoma Math. J. 11 (1963), 5-12. MR 29 #5095. MR 0167828 (29:5095)
  • [49] P. P. Zabreĭko and M. A. Krasnosel'skiĭ, A method for producing new fixed point theorems, Dokl. Akad. Nauk SSSR 176 (1967), 1233-1235 = Soviet Math. Dokl. 8 (1967), 1297-1299. MR 36 #3183.
  • [50] P. P. Zabreĭko, R. I. Kačurovskiĭ and M. A. Krasnosel'skiĭ, On a fixed point principle for operators in Hilbert space, Funkcional. Anal. i Prilozen. 1 (1967), no. 2, 93-94. (Russian) MR 35 #3505. MR 0212635 (35:3505)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47H10

Retrieve articles in all journals with MSC: 47H10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0328688-2
Keywords: Fixed point theorems, topological degree, k-set-contractions, k-ball-contractions, set-condensing and ball-condensing mappings, mappings of contractive, semicontractive, and pseudo-contractive type
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society