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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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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.
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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