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Transactions of the American Mathematical Society

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



Local and asymptotic approximations of nonlinear operators by $(k_{1}, \ldots k_{N})$-homogeneous operators

Authors: R. H. Moore and M. Z. Nashed
Journal: Trans. Amer. Math. Soc. 178 (1973), 293-305
MSC: Primary 47H99; Secondary 46G05
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 ${k_i}$-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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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