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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Local and asymptotic approximations of nonlinear operators by $(k_{1}, \ldots k_{N})$-homogeneous operators
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by R. H. Moore and M. Z. Nashed PDF
Trans. Amer. Math. Soc. 178 (1973), 293-305 Request permission

Abstract:

Notions of local and asymptotic approximations of a nonlinear mapping F between normed linear spaces by a sum of N ${k_i}$-homogeneous operators are defined and investigated. It is shown that the approximating operators inherit from F properties related to compactness and measures of noncompactness. Nets of equi-approximable operators with collectively compact (or bounded) approximates, which arise in approximate solutions of integral and operator equations, are studied with particular reference to pointwise (or weak convergence) properties. As a by-product, the well-known result that the Fréchet (or asymptotic) derivative of a compact operator is compact is generalized in several directions and to families of operators.
References
  • Philip M. Anselone, Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. With an appendix by Joel Davis. MR 0443383
  • Josef Daneš, Generalized concentrative mappings and their fixed points, Comment. Math. Univ. Carolinae 11 (1970), 115–136. MR 263063
  • Josef Daneš, Some fixed point theorems in metric and Banach spaces, Comment. Math. Univ. Carolinae 12 (1971), 37–51. MR 287398
  • Gabriele Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92 (Italian). MR 70164
  • L. S. Gol′denšteĭn and A. S. Markus, On the measure of non-compactness of bounded sets and of linear operators, Studies in Algebra and Math. Anal. (Russian), Izdat. “Karta Moldovenjaske”, Kishinev, 1965, pp. 45–54 (Russian). MR 0209894
  • A. Granas, Über eine Klasse nichtlinearer Abbildungen in Banachschen Raumen, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 867–871 (Russian, with German summary). MR 0091432
  • 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
  • K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
  • V. B. Melamed and A. I. Perov, A generalization of a theorem of M. A. Krasnosel′skiĭon the complete continuity of the Fréchet derivative of a completely continuous operator, Sibirsk. Mat. Ž. 4 (1963), 702–704 (Russian). MR 0157240
  • Robert H. Moore, Differentiability and convergence for compact nonlinear operators, J. Math. Anal. Appl. 16 (1966), 65–72. MR 196549, DOI 10.1016/0022-247X(66)90186-7
  • M. Z. Nashed, Differentiability and related properties of nonlinear operators: Some aspects of the role of differentials in nonlinear functional analysis, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1971, pp. 103–309. MR 0276840
  • M. Z. Nashed and J. S. W. Wong, Some varaints of a fixed point theorem of Krasnoselskii and applications to nonlinear integral equations, J. Math. Mech. 18 (1969), 767–777. MR 0238140
  • Roger D. Nussbaum, Estimates for the number of solutions of operator equations, Applicable Anal. 1 (1971), no. 2, 183–200. MR 296780, DOI 10.1080/00036817108839013
  • Roger D. Nussbaum, The fixed point index and asymptotic fixed point theorems for $k$-set-contractions, Bull. Amer. Math. Soc. 75 (1969), 490–495. MR 246285, DOI 10.1090/S0002-9904-1969-12213-5
  • 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
  • B. N. Sadovskiĭ, On measures of noncompactness and concentrative operators, Problemy Mat. Anal. Slož. Sistem. 2 (1968), 89-119. P. P. Zabreĭko and A. I. Povolockiĭ, Eigenvectors for the Hammerstein operators, Dokl. Akad. Nauk SSSR 183 (1968), 758-761 = Soviet Math. Dokl. 9 (1968), 1439-1442. MR 39 #833.
  • P. P. Zabreĭko and A. I. Povolockiĭ, Theorems on the existence and uniqueness of solutions of Hammerstein equations, Dokl. Akad. Nauk SSSR 176 (1967), 759–762 (Russian). MR 0221241
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 178 (1973), 293-305
  • MSC: Primary 47H99; Secondary 46G05
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0358465-8
  • MathSciNet review: 0358465