Local and Asymptotic Approximations of Nonlinear Operators by (k(1), ..., k(N))-Homogeneous Operators.
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WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER
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Notions of local and asymptotic approximations of a nonlinear mapping F between normed linear spaces by a sum of N ki-homogeneous operators are defined and investigated. It is shown that the approximating operators inherit from F properties related to compactness and measures of noncompactness. As a byproduct, the well-known result that the Frechet or asymptotic derivative of a compact operator is compact is generalized in several directions and to families of operators. The notions introduced are examined within a hierarchy of other notions of local and asymptotic approximations and related differentials. 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. Author
- Theoretical Mathematics