[1] Shi Ping, Ma Jipu,. Generalized regular points of a C1 1map between Banach spaces [J]. Journal of Southeast University (English Edition), 2007, 23 (1): 148-150. [doi:10.3969/j.issn.1003-7985.2007.01.030]

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Generalized regular points of a C¹ 1map between Banach spaces()

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

23

Issue:

2007 1

Page:

148-150

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2007-03-30

Info

Title:

Generalized regular points of a C¹ 1map between Banach spaces

Author(s):

Shi Ping¹;  Ma Jipu²

¹Department of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210003, China
²Department of Mathematics, Nanjing University, Nanjing 210093, China

Keywords:

Banach space;  bounded linear operator;  generalized inverse;  index;  generalized regular point;  semi-Fredholm map

PACS:

O177.91

DOI:

10.3969/j.issn.1003-7985.2007.01.030

Abstract:

Let f be a C¹ 1map between two Banach spaces E and F.It has been proved that the concept of generalized regular points of f, which is a generalization of the notion of regular points of f, has some crucial applications in nonlinearity and global analysis.We characterize the generalized regular points of f using the three integer-valued(or infinite)indices M(x₀0), M_cc(x₀0)and M_rr(x₀0)at x_0∈E 0generated by f and by analyzing generalized inverses of bounded linear operators on Banach spaces, that is, if f〓 ′(x₀0)has a generalized inverse in the Banach space B(E, F)of all bounded linear operators on E into F and at least one of the indices M(x₀0), M_cc(x₀0)and M_rr(x₀0)is finite, then x₀ 0is a generalized regular point of f if and only if the multi-index(M(x), M_cc(x), M_rr(x))is continuous at x₀0.

References:

[1] Ma Jipu.(1.2)inverses of operators between Banach spaces and local conjugacy theorem [J].Chinese Annals of Math:Series B, 1999, 20(1):57-62.
[2] Ma Jipu. Rank theorems of operators between Banach spaces [J]. Science in China: Series A, 2000, 43(1):1-5.
[3] Ma Jipu. Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds [J].Science in China, Series A, 2000, 43(12):1233-1237.
[4] Nashed M Z.Generalized inverses and applications [M].New York:Academic Press, 1976.
[5] Nashed M Z, Chen X.Convergence of Newton-like methods for singular operator equations using outer inverses [J].Numer Math , 1993, 66:235-257.
[6] Zeidler E.Nonlinear functional analysis and its applications: Ⅳ applications to mathematical physics [M].New York:Springer-Verlag, 1988.
[7] Berger M S.Nonlinearity and functional analysis [M].New York:Academic Press, 1977.
[8] Shi Ping, Ma Jipu.A note on a problem of M.S.Berger [J].Northeast Math J, 2003, 19(4):366-370.
[9] Ma Jipu.Generalized indices of operators in B(H)[J].Science in China:Series A, 1997, 40(12):1233-1238.

Memo

Memo:

Biography: Shi Ping(1963—), male, associate professor, pshi@eyou.com.

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