On the structure of the set of solutions of equations involving $A$-proper mappings
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- by P. M. Fitzpatrick PDF
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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
- Felix E. Browder, Nonlinear elliptic boundary value problems and the generalized topological degree, Bull. Amer. Math. Soc. 76 (1970), 999–1005. MR 264222, DOI 10.1090/S0002-9904-1970-12530-7
- 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
- Felix E. Browder, Fixed point theorems for nonlinear semicontractive mappings in Banach spaces, Arch. Rational Mech. Anal. 21 (1966), 259–269. MR 200753, DOI 10.1007/BF00282247
- Felix E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660–665. MR 230179, DOI 10.1090/S0002-9904-1968-11983-4
- 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 0244812, DOI 10.1016/0022-1236(69)90041-x
- Felix E. Browder and W. V. Petryshyn, The topological degree and Galerkin approximations for noncompact operators in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 641–646. MR 229100, DOI 10.1090/S0002-9904-1968-11973-1
- Jane Cronin, Fixed points and topological degree in nonlinear analysis, Mathematical Surveys, No. 11, American Mathematical Society, Providence, R.I., 1964. MR 0164101
- Klaus Deimling, Fixed points of generalized $P$-compact operators, Math. Z. 115 (1970), 188–196. MR 264480, DOI 10.1007/BF01109857 D. G. de Figueiredo, Topics in nonlinear functional analysis, Lecture Series, no. 48, University of Maryland, College Park, Md., 1967.
- D. G. de Figueiredo and L. A. Karlovitz, On the radial projection in normed spaces, Bull. Amer. Math. Soc. 73 (1967), 364–368. MR 211248, DOI 10.1090/S0002-9904-1967-11753-1
- P. M. Fitzpatrick, A generalized degree for uniform limits of $A$-proper mappings, J. Math. Anal. Appl. 35 (1971), 536–552. MR 281069, DOI 10.1016/0022-247X(71)90201-0
- Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258 (German). MR 190718, DOI 10.1002/mana.19650300312
- 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, Mappings of generalized contractive type, J. Math. Anal. Appl. 32 (1970), 567–572. MR 271794, DOI 10.1016/0022-247X(70)90278-7
- W. A. Kirk, On nonlinear mappings of strongly semicontractive type, J. Math. Anal. Appl. 27 (1969), 409–412. MR 244814, DOI 10.1016/0022-247X(69)90057-2
- M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, A Pergamon Press Book, The Macmillan Company, New York, 1964. Translated by A. H. Armstrong; translation edited by J. Burlak. MR 0159197
- M. A. Krasnosel′skiĭ and P. E. Sobolevskiĭ, The structure of the set of solutions of an equation of parabolic type, Ukrain. Mat. Ž. 16 (1964), 319–333 (Russian, with English summary). MR 0166488
- Joram Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 112. MR 179580 R. D. Nussbaum, The fixed point index and fixed point theorems for k-set-contractions, Ph.D. Dissertation, University of Chicago, Chicago, Ill., 1968.
- Roger D. Nussbaum, Degree theory for local condensing maps, J. Math. Anal. Appl. 37 (1972), 741–766. MR 306986, DOI 10.1016/0022-247X(72)90253-3 —, Some results on the ball intersection property and the existence of non-expansive retractions, Bull. Polon. Acad. Sci. (to appear). W. V. Petryshyn, On the projectional solvability of nonlinear operator equations, Inform. Bull. no. 5, Internat. Congress of Math. (Moscow, 1966), “Mir", Moscow, 1968.
- W. V. Petryshyn, Projection methods in nonlinear numerical functional analysis, J. Math. Mech. 17 (1967), 353–372. MR 0218941, DOI 10.1512/iumj.1968.17.17019
- W. V. Petryshyn, Remarks on the approximation-solvability of nonlinear functional equations, Arch. Rational Mech. Anal. 26 (1967), 43–49. MR 220120, DOI 10.1007/BF00283858
- W. V. Petryshyn, On the approximation-solvability of nonlinear equations, Math. Ann. 177 (1968), 156–164. MR 226458, DOI 10.1007/BF01350791
- W. V. Petryshyn, On projectional-solvability and the Fredholm alternative for equations involving linear $A$-proper operators, Arch. Rational Mech. Anal. 30 (1968), 270–284. MR 231221, DOI 10.1007/BF00281535
- W. V. Petryshyn, Invariance of domain theorem for locally $A$-proper mappings and its implications, J. Functional Analysis 5 (1970), 137–159. MR 0266005, DOI 10.1016/0022-1236(70)90041-8
- W. V. Petryshyn, Further remarks on nonlinear $P$-compact operators in Banach space, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 684–687. MR 194943, DOI 10.1073/pnas.55.4.684
- W. V. Petryshyn, Iterative construction of fixed points of contractive type mappings in Banach spaces, Numerical Analysis of Partial Differential Equations (C.I.M.E. 2 Ciclo, Ispra, 1967) Edizioni Cremonese, Rome, 1968, pp. 307–339. MR 0250435
- W. V. Petryshyn, Structure of the fixed points sets of $k$-set-contractions, Arch. Rational Mech. Anal. 40 (1970/71), 312–328. MR 273480, DOI 10.1007/BF00252680
- 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 247539, DOI 10.1090/S0002-9947-1969-0247539-7
- Helmut Schaefer, Über die Methode sukzessiver Approximationen, Jber. Deutsch. Math.-Verein. 59 (1957), no. Abt. 1, 131–140 (German). MR 84116 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.
- Giovanni Vidossich, On Peano phenomenon, Boll. Un. Mat. Ital. (4) 3 (1970), 33–42 (English, with Italian summary). MR 0271793
- J. R. L. Webb, Fixed point theorems for non-linear semicontractive operators in Banach spaces, J. London Math. Soc. (2) 1 (1969), 683–688. MR 250152, DOI 10.1112/jlms/s2-1.1.683 Wong-Ng Ship Fah, Le degree topologique de certaines applications non-compactes, nonlinéaires, Ph.D. Dissertation, University of Montreal, 1969.
- Sadayuki Yamamuro, A note on $d$-ideals in some near-algebras, J. Austral. Math. Soc. 7 (1967), 129–134. MR 0212585, DOI 10.1017/S1446788700005498
Additional Information
- © Copyright 1974 American Mathematical Society
- 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