Local and asymptotic approximations of nonlinear operators by -homogeneous operators

Authors:
R. H. Moore and M. Z. Nashed

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

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Abstract: Notions of local and asymptotic approximations of a nonlinear mapping *F* between normed linear spaces by a sum of *N* -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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DOI:
https://doi.org/10.1090/S0002-9947-1973-0358465-8

Keywords:
Local approximations,
asymptotic approximations and derivatives,
collectively compact operators,
nonlinear operators,
Fréchet and other differentials,
measure of noncompactness,
nondifferentiable Hammerstein operators,
nonlinear integral equations,
approximations by numerical quadratures

Article copyright:
© Copyright 1973
American Mathematical Society