A note on the imbedding theorem of Browder and Ton
Author:
J. Berkovits
Journal:
Proc. Amer. Math. Soc. 131 (2003), 29632966
MSC (2000):
Primary 47H05, 78M99
Published electronically:
April 9, 2003
MathSciNet review:
1974355
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Abstract: The imbedding theorem of Browder and Ton states that for any real separable Banach space there exist a real separable Hilbert space and a compact linear injection such that is dense in We shall give a short and elementary new proof to this result. We also briefly discuss the corresponding result without the completeness assumption.
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R.A. Adams, Sobolev Spaces, Academic Press, 1975. MR 56:9247
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Y.I. Alber, The solution of nonlinear equations with monotone operators in a Banach space, Siberian Math. J. 16 (1) (1975) pp. 18. MR 51:6512
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H. Amann, An existence and unicity theorem for the Hammerstein equation in Banach spaces, Math. Z. 111 (3) (1969) pp. 175190. MR 40:7894
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J. Berkovits, On the degree theory for nonlinear mappings of monotone type, Ann. Acad. Sci. Fenn. Ser. A1, Dissertationes, 58 (1986). MR 87f:47084
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J. Berkovits and V. Mustonen, On the topological degree for mappings of monotone type, Nonlinear Anal., TMA, 10 (1986) pp. 13731383. MR 88b:47073
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J. Berkovits and M. Tienari, Topological degree for some classes of multis with applications to hyperbolic and elliptic problems involving dicontinuous nonlinearities, Dynamic Systems and Applications 5 (1996) pp. 118. MR 96m:47112
 7.
F.E. Browder and B.A. Ton, Nonlinear functional equations in Banach spaces and elliptic superregularization, Math. Z. 105 (1968) pp. 177195. MR 38:582
 8.
A.A. Khan, A regularization approach for variational inequalities, Comput. Math. Appl. 42 (12), (2001) pp. 6574. MR 2002b:49020
 9.
D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Editura Academiei, 1978. MR 80g:47056
 10.
C.G. Simader, Weak solutions of the Dirichlet problem for strongly nonlinear elliptic differential equations, Math. Z. 150 (1) (1976) pp. 126. MR 54:8018
 11.
J.R.L. Webb, On the Dirichlet problem for strongly nonlinear elliptic operators in unbounded domains, J. Lond. Math. Soc. 10 (1975) pp. 163170. MR 52:14644
 1.
 R.A. Adams, Sobolev Spaces, Academic Press, 1975. MR 56:9247
 2.
 Y.I. Alber, The solution of nonlinear equations with monotone operators in a Banach space, Siberian Math. J. 16 (1) (1975) pp. 18. MR 51:6512
 3.
 H. Amann, An existence and unicity theorem for the Hammerstein equation in Banach spaces, Math. Z. 111 (3) (1969) pp. 175190. MR 40:7894
 4.
 J. Berkovits, On the degree theory for nonlinear mappings of monotone type, Ann. Acad. Sci. Fenn. Ser. A1, Dissertationes, 58 (1986). MR 87f:47084
 5.
 J. Berkovits and V. Mustonen, On the topological degree for mappings of monotone type, Nonlinear Anal., TMA, 10 (1986) pp. 13731383. MR 88b:47073
 6.
 J. Berkovits and M. Tienari, Topological degree for some classes of multis with applications to hyperbolic and elliptic problems involving dicontinuous nonlinearities, Dynamic Systems and Applications 5 (1996) pp. 118. MR 96m:47112
 7.
 F.E. Browder and B.A. Ton, Nonlinear functional equations in Banach spaces and elliptic superregularization, Math. Z. 105 (1968) pp. 177195. MR 38:582
 8.
 A.A. Khan, A regularization approach for variational inequalities, Comput. Math. Appl. 42 (12), (2001) pp. 6574. MR 2002b:49020
 9.
 D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Editura Academiei, 1978. MR 80g:47056
 10.
 C.G. Simader, Weak solutions of the Dirichlet problem for strongly nonlinear elliptic differential equations, Math. Z. 150 (1) (1976) pp. 126. MR 54:8018
 11.
 J.R.L. Webb, On the Dirichlet problem for strongly nonlinear elliptic operators in unbounded domains, J. Lond. Math. Soc. 10 (1975) pp. 163170. MR 52:14644
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Additional Information
J. Berkovits
Affiliation:
Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, FIN90014 Oulu, Finland
Email:
juha.berkovits@oulu.fi
DOI:
http://dx.doi.org/10.1090/S0002993903070941
PII:
S 00029939(03)070941
Keywords:
Compact imbedding
Received by editor(s):
May 30, 2002
Published electronically:
April 9, 2003
Communicated by:
N. TomczakJaegermann
Article copyright:
© Copyright 2003
American Mathematical Society
