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 mappings satisfying inwardness conditions


Author: James Caristi
Journal: Trans. Amer. Math. Soc. 215 (1976), 241-251
MSC: Primary 47H10
DOI: https://doi.org/10.1090/S0002-9947-1976-0394329-4
MathSciNet review: 0394329
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let X be a normed linear space and let K be a convex subset of X. The inward set, $ {I_K}(x)$, of x relative to K is defined as follows: $ {I_K}(x) = \{ x + c(u - x):c \geqslant 1,u \in K\} $. A mapping $ T:K \to X$ is said to be inward if $ Tx \in {I_K}(x)$ for each $ x \in K$, and weakly inward if Tx belongs to the closure of $ {I_K}(x)$ for each $ x \in K$. In this paper a characterization of weakly inward mappings is given in terms of a condition arising in the study of ordinary differential equations. A general fixed point theorem is proved and applied to derive a generalization of the Contraction Mapping Principle in a complete metric space, and then applied together with the characterization of weakly inward mappings to obtain some fixed point theorems in Banach spaces.


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

  • [1] H. Brezis, On a characterization of flow-invariant sets, Comm. Pure Appl. Math. 23 (1970), 261-263. MR 41 #2161. MR 0257511 (41:2161)
  • [2] F. E. Browder, 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)
  • [3] -, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177(1968), 283-301. MR 37 #4679. MR 0229101 (37:4679)
  • [4] -, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660-665. MR 37 #5742. MR 0230179 (37:5742)
  • [5] M. G. Crandall, A generalization of Peano's existence theorem and flow invariance, Proc. Amer. Math. Soc. 36 (1972), 151-155. MR 46 #5708. MR 0306586 (46:5708)
  • [6] Ky Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234-240. MR 40 #4830. MR 0251603 (40:4830)
  • [7] B. R. Halpern, Fixed point theorems for outward maps, Doctoral Thesis, Univ. of California, Los Angeles, Calif., 1965.
  • [8] -, Fixed-point theorems for set-valued maps in infinite dimensional spaces, Math. Ann. 189 (1970), 87-98. MR 42 #8357. MR 0273479 (42:8357)
  • [9] B. R. Halpern and G. M. Bergman, A fixed-point theorem for inward and outward maps, Trans. Amer. Math. Soc. 130 (1968), 353-358. MR 36 #4397. MR 0221345 (36:4397)
  • [10] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508-520. MR 37 #1820. MR 0226230 (37:1820)
  • [11] 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)
  • [12] -, Fixed point theorems for nonexpansive mappings satisfying certain boundary conditions, Proc. Amer. Math. Soc. 50 (1975), 143-149. MR 0380527 (52:1427)
  • [13] R. H. Martin, Jr., Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc. 179 (1973), 399-414. MR 47 #7537. MR 0318991 (47:7537)
  • [14] W. V. Petryshyn and P. M. Fitzpatrick, Fixed point theorems for multivalued non-compact inward maps (to appear).
  • [15] R. M. Redheffer, The theorems of Bony and Brezis on flow-invariant sets, Amer. Math. Monthly 79 (1972), 740-747. MR 46 #2166. MR 0303024 (46:2166)
  • [16] S. Reich, Fixed points in locally convex spaces, Math. Z. 125 (1972), 17-31. MR 46 #6110. MR 0306989 (46:6110)
  • [17] -, Remarks on fixed points, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 52 (1972), 689-697. MR 48 #9473. MR 0331139 (48:9473)
  • [18] -, Fixed points of condensing functions, J. Math. Anal. Appl. 41 (1973), 460-467. MR 48 #971. MR 0322609 (48:971)
  • [19] -, Fixed points of non-expansive functions, J. London Math. Soc. (2) 7 (1973), 5-10. MR 48 #4855. MR 0326511 (48:4855)
  • [20] G. Vidossich, Existence comparison and asymptotic behavior of solutions of ordinary differential equations in finite and infinite dimensional Banach spaces (to appear).
  • [21] -, Nonexistence of periodic solutions of differential equations and applications to zeros of nonlinear operators (to appear).
  • [22] J. A. Yorke, Differential inequalities and non-Lipschitz scalar functions, Math. Systems Theory 4 (1970), 140-153. MR 42 #3373. MR 0268476 (42:3373)

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/S0002-9947-1976-0394329-4
Keywords: Fixed point theorems, contraction and nonexpansive mappings, inward and weakly inward conditions, complete metric space, Banach space
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society