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.

**[1]**P. M. Anselone,*Collectively compact operator approximations*, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR**0443383 (56:1753)****[2]**J. Daneš,*Generalized concentrative mappings and their fixed points*, Comment. Math. Univ. Carolinae**11**(1970), 115-135. MR**41**#7668. MR**0263063 (41:7668)****[3]**-,*Some fixed point theorems in metric and Banach spaces*, Comment. Math. Univ. Carolinae**12**(1971), 37-51. MR**0287398 (44:4604)****[4]**G. Darbo,*Punti uniti in trasformazioni a codiminio noncompatto*, Rend. Sem. Mat. Univ. Padova**24**(1955), 84-92. MR**16**, 1140. MR**0070164 (16:1140f)****[5]**L. S. Gol' denštein and A. S. Markus,*On the measure of non-compactness of bounded sets and of linear operators*, Studies in Algebra and Math. Anal., Izdat. ``Karta Moldovenjaske", Kishinev, 1965, pp. 45-54. (Russian) MR**35**#789. MR**0209894 (35:789)****[6]**A. Granas,*Über eine Klasse nichtlinearer Abbildungen in Banachschen Raumen*, Bull. Acad. Polon. Sci. Cl. III**5**(1957), 867-871. (Russian) MR**19**, 968. MR**0091432 (19:968e)****[7]**M. A. Krasnosel' skiĭ,*Topological methods in the theory of nonlinear integral equations*, GITTL, Moscow, 1956; English transl., Macmillan, New York, 1964. MR**20**#3464; MR**28**#2414. MR**0159197 (28:2414)****[8]**K. Kuratowski,*Topologie*. Vol. 1, PWN, Warsaw, 1958; English transl., Academic Press, New York; PWN, Warsaw, 1966. MR**19**, 873; MR**36**#839. MR**0217751 (36:840)****[9]**V. B. Melamed and A. I. Perov,*A generalization of a theorem of M. A. Krasnosel' skiĭ on the complete continuity of the Fréchet derivative of a completely continuous operator*, Sibirsk. Mat. Ž.**4**(1963), 702-704. (Russian) MR**28**#476. MR**0157240 (28:476)****[10]**R. H. Moore,*Differentiability and convergence for compact nonlinear operators*, J. Math. Anal. Appl.**16**(1966), 65-72. MR**33**#4736. MR**0196549 (33:4736)****[11]**M. Z. Nashed,*Differentiability and related properties of nonlinear operators*:*Some aspects of the role of differentials in nonlinear functional analysis*, Nonlinear Functional Analysis and Applications, L. B. Rail (editor), Academic Press, New York, 1971, pp. 103-309. MR**0276840 (43:2580)****[12]**M. Z. Nashed and J. S. W. Wong,*Some variants of a fixed point theorem of Krasnoselskii and applications to nonlinear integral equations*, J. Math. Mech.**18**(1969), 767-777. MR**38**#6416. MR**0238140 (38:6416)****[13]**R. D. Nussbaum,*Estimates for the number of solutions of operator equations*, Applicable Anal.**1**(1971), 183-200. MR**0296780 (45:5839)****[14]**-,*The fixed point index and asymptotic fixed point theorems for k-set-contractions*, Bull. Amer. Math. Soc.**75**(1969), 490-495. MR**39**#7589. MR**0246285 (39:7589)****[15]**W. V. Petryshyn,*Structure of the fixed points sets of k-set-contractions*, Arch. Rational Mech. Anal.**40**(1970/71), 312-328. MR**42**#8358. MR**0273480 (42:8358)****[16]**B. N. Sadovskiĭ,*On measures of noncompactness and concentrative operators*, Problemy Mat. Anal. Slož. Sistem.**2**(1968), 89-119.**[17]**P. P. Zabreĭko and A. I. Povolockiĭ,*Eigenvectors for the Hammerstein operators*, Dokl. Akad. Nauk SSSR**183**(1968), 758-761 = Soviet Math. Dokl.**9**(1968), 1439-1442. MR**39**#833.**[18]**-,*Existence and uniqueness theorems for solutions of the Hammerstein equations*, Dokl. Akad. Nauk SSSR**176**(1967), 759-762 = Soviet Math. Dokl.**8**(1967), 1178-1182. MR**36**#4293. MR**0221241 (36:4293)**

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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