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)

 
 

 

On the structure of the set of solutions of equations involving $ A$-proper mappings


Author: P. M. Fitzpatrick
Journal: Trans. Amer. Math. Soc. 189 (1974), 107-131
MSC: Primary 47H15
DOI: https://doi.org/10.1090/S0002-9947-1974-0336475-5
MathSciNet review: 0336475
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let X and Y be Banach spaces having complete projection schemes (say, for example, they have Schauder bases). We consider various properties of mappings $ T:D \subset X \to Y$ which are either Approximation-proper (A-proper) or the uniform limit of such mappings. In §1 general properties, including those of the generalized topological degree, of such mappings are discussed. In §2 we give sufficient conditions in order that the solutions of an equation involving a nonlinear mapping be a continuum. The conditions amount to requiring that the generalized topological degree not vanish, and that the mapping involved be the uniform limit of well structured mappings. We devote §3 to proving a result connecting the topological degree of an A-proper Fréchet differentiable mapping to the degree of its derivative. Finally, in §4, various Lipschitz-like conditions are discussed in an A-proper framework, and constructive fixed point and surjectivity results are obtained.


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

  • [1] F. E. Browder, Nonlinear elliptic boundary value problems and the generalized topological degree, Bull. Amer. Math. Soc. 76 (1970), 999-1005. MR 41 #8818. MR 0264222 (41:8818)
  • [2] -, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041-1044. MR 32 #4574. MR 0187120 (32:4574)
  • [3] -, 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)
  • [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] F. E. Browder and W. V. Petryshyn, Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces, J. Functional Analysis 3 (1969), 217-245. MR 39 #6126. MR 0244812 (39:6126)
  • [6] -, The topological degree and Galerkin approximations for noncompact operators in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 641-646. MR 37 #4678. MR 0229100 (37:4678)
  • [7] J. Cronin, Fixed points and topological degree in nonlinear analysis, Math. Surveys, no. 11, Amer. Math. Soc., Providence, R. I., 1964. MR 29 #1400. MR 0164101 (29:1400)
  • [8] K. Deimling, Fixed points of generalized P-compact operators, Math. Z. 115 (1970), 188-196. MR 41 #9073. MR 0264480 (41:9073)
  • [9] D. G. de Figueiredo, Topics in nonlinear functional analysis, Lecture Series, no. 48, University of Maryland, College Park, Md., 1967.
  • [10] D. G. de Figueiredo and L. A. Carlovitz, On the radial projection in normed spaces, Bull. Amer. Math. Soc. 73 (1967), 364-368. MR 35 #2130. MR 0211248 (35:2130)
  • [11] P. M. Fitzpatrick, A generalized degree for uniform limits of A-proper mappings, J. Math. Anal. Appl. 35 (1971), 536-552. MR 43 #6788. MR 0281069 (43:6788)
  • [12] D. Göhde, Züm Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251-258. MR 32 #8129. MR 0190718 (32:8129)
  • [13] 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)
  • [14] -, Mappings of generalized contractive type, J. Math. Anal. Appl. 32 (1970), 567-572. MR 42 #6675. MR 0271794 (42:6675)
  • [15] -, On nonlinear mappings of strongly semicontractive type, J. Math. Anal. Appl. 27 (1969), 409-412. MR 39 #6128. MR 0244814 (39:6128)
  • [16] M. A. Krasnosel'skiĭ, 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)
  • [17] M. A. Krasnosel'skiĭ and P. E. Sobolevskiĭ, Structure of the set of solutions of an equation of parabolic type, Ukrain. Mat. Ž. 16 (1964), 319-333; English transl., Amer. Math. Soc. Transl. (2) 51 (1966), 113-131. MR 29 #3763. MR 0166488 (29:3763)
  • [18] J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. No. 48 (1964). MR 31 #3828. MR 0179580 (31:3828)
  • [19] R. D. Nussbaum, The fixed point index and fixed point theorems for k-set-contractions, Ph.D. Dissertation, University of Chicago, Chicago, Ill., 1968.
  • [20] -, Degree theory for locally condensing maps, J. Math. Anal. Appl. (to appear). MR 0306986 (46:6107)
  • [21] -, Some results on the ball intersection property and the existence of non-expansive retractions, Bull. Polon. Acad. Sci. (to appear).
  • [22] W. V. Petryshyn, On the projectional solvability of nonlinear operator equations, Inform. Bull. no. 5, Internat. Congress of Math. (Moscow, 1966), ``Mir", Moscow, 1968.
  • [23] -, Projection methods in nonlinear numerical functional analysis, J. Math. Mech. 17 (1967), 353-372. MR 36 #2025. MR 0218941 (36:2025)
  • [24] -, Remarks on the approximation-solvability of nonlinear functional equations, Arch. Rational Mech. Anal. 26 (1967), 43-49. MR 36 #3186. MR 0220120 (36:3186)
  • [25] -, On the approximation-solvability of nonlinear equations, Math. Ann. 177 (1968), 156-164. MR 37 #2048. MR 0226458 (37:2048)
  • [26] -, On projectional-solvability and the Fredholm alternative for equations involving linear Aproper operators, Arch. Rational Mech. Anal. 30 (1968), 270-284. MR 37 #6776. MR 0231221 (37:6776)
  • [27] -, Invariance of domain theorem for locally A-proper mappings and its implications, J. Functional Analysis 5 (1970), 137-159. MR 42 #914. MR 0266005 (42:914)
  • [28] -, Further remarks on nonlinear P-compact operators in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 684-687. MR 33 #3148. MR 0194943 (33:3148)
  • [29] -, Iterative construction of fixed points of contractive type mappings in Banach spaces, Numerical Analysis of Partial Differential Equations (C.I.M.E. 2$ ^{o}$ Ciclo, Ispra, 1967), Edizioni Cremonese, Rome, 1968, pp. 307-339. MR 40 #3674. MR 0250435 (40:3674)
  • [30] -, 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)
  • [31] W. V. Petryshyn and T. S. Tucker, On the functional equations involving nonlinear generalized Pcompact operators, Trans. Amer. Math. Soc. 135 (1969), 343-373. MR 40 #804. MR 0247539 (40:804)
  • [32] H. Schaefer, Über die Method sukzessiver Approximationen, Jber. Deutsch. Math. Verein. 59 (1957), Abt. 1, 131-140. MR 18,811. MR 0084116 (18:811g)
  • [33] M. M. Valnberg, 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.
  • [34] G. Vidossich, On Peano phenomenon, Boll. Un. Mat. Ital. (4) 3 (1970), 33-42. MR 42 #6674. MR 0271793 (42:6674)
  • [35] J. R. L. Webb, Fixed point theorems for non-linear semi-contractive operators in Banach spaces, J. London Math. Soc. (2) 1 (1969), 683-688. MR 40 #3392. MR 0250152 (40:3392)
  • [36] Wong-Ng Ship Fah, Le degree topologique de certaines applications non-compactes, nonlinéaires, Ph.D. Dissertation, University of Montreal, 1969.
  • [37] S. Yamamuro, A note on d-ideals in some near-algebras, J. Austral. Math. Soc. 7 (1967), 129-134. MR 35 #3456. MR 0212585 (35:3456)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47H15

Retrieve articles in all journals with MSC: 47H15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0336475-5
Keywords: Complete projection scheme, A-proper, generalized topological degree, continuum, Fréchet derivative, contraction, compact, fixed point, surjective
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society