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Operator superpositions in the spaces \(\ell _ p\). (English. Russian original) Zbl 0632.47046

Sib. Math. J. 28, No. 1-2, 63-73 (1987); translation from Sib. Mat. Zh. 28, No. 1(161), 86-98 (1987).
The superposition operator \(Fx(s)=f(s,x(s))\) is the most important operator in nonlinear analysis. There is a vast literature on the theory and applications of this operator in various function spaces; the first systematic study in the Lebesgue function spaces \(L_ p\), for instance, is contained in the book of M. A. Krasnosel’skii et al., Integral operators in spaces of summable functions (in Russian), Moscow (1966; Zbl 0145.39703). Very little attention has been given, however, to this operator between sequence space, rather than function spaces. The present paper gives a systematic account of various important properties of the superposition operator in the Lebesgue sequence spaces \(\ell_ p\) (1\(\leq p\leq \infty)\). The authors give conditions (both necessary and sufficient), under which F acts between two spaces \(\ell_ p\) and \(\ell_ q\), is locally bounded, locally continuous, bounded on balls, uniformly continuous on balls, absolutely bounded (i.e. compact), or differentiable. The last section is concerned with an application to a certain “discrete analogue” to nonlinear integral equations of Hammerstein type.
Reviewer: Jürgen Appell

MSC:

47H99 Nonlinear operators and their properties
46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables

Citations:

Zbl 0145.39703
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References:

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