Phistable operators and inner approximationsolvability
Author:
Ram U. Verma
Journal:
Proc. Amer. Math. Soc. 117 (1993), 491499
MSC:
Primary 47H17; Secondary 47H09, 65J15
MathSciNet review:
1127144
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Abstract: We extend, by applying a theorem of Petryshyn (1970), the approximationsolvability of the nonlinear functional equations involving strongly stable Hilbert space mappings to the case of strongly stable mappingsa new and rather general class of mappings. These mappings constitute a generalization of monotone mappings. Finally, we upgrade the obtained results to the case of Banach space mappings.
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 F. Browder and W. Petryshyn, Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces, J. Funct. Anal. 3 (1969), 217245. MR 0244812 (39:6126)
 [2]
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 [3]
 A. G. Kartsatos, On the equation in Banach spaces with weakly continuous duality maps, Nonlinear Equations in Abstract Spaces, Academic Press, New York, 1978. MR 502538 (80k:47060)
 [4]
 M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii, and V. Ya. Stetsenko, Approximation solution of operator equations, English transl., WolterNoordhoff, Groningen, 1972. MR 0385655 (52:6515)
 [5]
 R. N. Mukherjee and T. Som, On phimonotone operators and variational inequalities, Bull. Calcutta Math. Soc. 80 (1988), 201205. MR 990970 (91b:47122)
 [6]
 W. Petryshyn, Nonlinear equations involving noncompact operators, Nonlinear Functional Analysis (Chicago, 1968), Proc. Sympos. Pure Math., vol. 18, part 1, Amer. Math. Soc., Providence, RI, 1970, pp. 206233. MR 0271789 (42:6670)
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 M. Théra, Existence results for the nonlinear complementarity problem and applications to nonlinear analysis, J. Math. Anal. Appl. 154 (1991), 572584. MR 1088652 (92d:47085)
 [12]
 V. B. Trushin, On the solution of certain nonlinear equations and variational inequalities, Soviet Math. Dokl. 40 (1990), 521524. MR 1036142 (91a:47087)
 [13]
 R. Verma, Role of numerical range in approximationsolvability of nonlinear functional equations, Appl. Math. Lett. 5 (1992), 2527. MR 1154606
 [14]
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 [15]
 E. Zeidler, Nonlinear functional analysis and its applications II/B: Nonlinear monotone operators, SpringerVerlag, Berlin and New York, 1990. MR 1033498 (91b:47002)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919931127144X
PII:
S 00029939(1993)1127144X
Article copyright:
© Copyright 1993
American Mathematical Society
