Dedagich, F.; Zabrejko, P. P. 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 Cited in 4 ReviewsCited in 5 Documents MSC: 47H99 Nonlinear operators and their properties 46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables Keywords:discrete analogue to nonlinear integral equations of Hammerstein type; superposition operator; Lebesgue sequence spaces; locally bounded; locally continuous; bounded on balls; uniformly continuous on balls; absolutely bounded; compact; differentiable Citations:Zbl 0145.39703 PDFBibTeX XMLCite \textit{F. Dedagich} and \textit{P. P. Zabrejko}, Sib. Math. J. 28, No. 1--2, 63--73 (1987; Zbl 0632.47046); translation from Sib. Mat. Zh. 28, No. 1(161), 86--98 (1987) Full Text: DOI References: [1] M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations [in Russian], Gostekhizdat, Moscow (1956). [2] M. A. Krasnosel’skii, P. P. Zabreiko, E. I. Pustyl’nik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions [in Russian], Nauka, Moscow (1966). [3] Ya. B. Rutitskii, ?A generalization of Orlich coordinate spaces,? Nauch. Tr. Voronezh. Inzh.-Stroit. Inst., 271-286 (1952). [4] P. P. Zabreiko and A. I. Povolotskii, ?On the theory of Hammerstein equations,? Ukr. Mat. Zh.,22, No. 2, 271-286 (1970). · Zbl 0204.42902 · doi:10.1007/BF01125601 [5] M. A. Krasnosel’skii and Ya. B. Rutitskii, Convex Functions and Orlich Spaces [in Russian], Fizmatgiz, Moscow (1958). [6] M. A. Krasnosel’skii and Ya. B. Rutitskii, ?Orlich spaces and nonlinear integral equations,? Tr. Mosk. Mat. O-va,7, 63-120 (1958). [7] E. Hille and R. Phillips, Functional Analysis and Semigroups [Russian translation], IL, Moscow (1962). [8] J. Appell, ?Upper estimates for superposition operators, and some applications,? Ann. Acad. Sci. Fenn., Ser. Al,8, 149 (1983). · Zbl 0489.47017 [9] J. Appell and P. P. Zabreiko, ?Exact upper estimates for a superposition operator,? Dokl. Akad. Nauk BSSR,27, No. 8, 686-689 (1983). · Zbl 0525.47020 [10] J. Appell and P. P. Zabreiko, ?On a theorem of M. A. Krasnosel’skii,? Nonlinear Analysis,7, No. 7, 695-706 (1983). · Zbl 0522.47056 · doi:10.1016/0362-546X(83)90026-3 [11] P. P. Zabreiko, ?Ideal spaces of functions. I,? Vestn. Yarosl. Univ., 12-52 (1974). [12] B. N. Sadovskii, ?Limitingly compact and contracting operators,? Usp. Mat. Nauk,27, No. 2, 81-146 (1972). [13] R. Cook, Infinite Matrices and Spaces of Sequences [Russian translation], Fizmatgiz, Moscow (1960). [14] G. H. Hardy, D. E. Littlewood, and G. Polya, Inequalities, Cambridge Univ. Press (1952). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.