## 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