无界广义逆的扰动与极小不动点定理
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摘要
设E和F是Banach空间, B(E,F)表示从空间E到F的有界线性算子全体.当A∈B(E,F)具有有界的广义逆A+∈B(F,E)时, Nashed和Chen证明了一个很有用的定理:对任意满足T ? A < A+ -1的T,若使C~(-1)(A,A+,T)TN(A) - R(A),则B = A+C?1(A,A+,T)是T的一个广义逆,且N(B) = N(A+)和R(B) = R(A+),其中C(A,A+,T) = IF + (T - A)A+.在这篇文章中,我们将上述结果推广到A不必具有有界广义逆的情形.并且我们证明这里的定理包含Nashed和Chen的定理.所以我们的结果推广了上述己知的定理.另外,本文的另一个结果是利用局部凸空间中Fan-Kakutani不动点定理,将局部凸空间中集值映射的极小不动点定理进行推广,把原定理中的半范数条件减弱为次可加泛函,得到具局部凸空间中极值映射的一个极小不动点定理.最后,我们将极小不动点定理与广义逆理论相结合,得到一类不适定的半线性两点边值问题的最佳逼近解的刻画.
Let E,F be Banach spaces, B(E,F) denote all the bounded linear operatorsfrom E to F, and A+ be a generalized inverse of A. The following theorem byNashed and Chen is known well: for all T satisfying T - A < A+-1, ifC~(-1)1(A,A+,T)TN(A) ? R(A), then B = A+C?1(A,A+,T) is a generalized inverseof T, and N(B) = N(A+), R(B) = R(A+), where C(A,A+,T) = IF + (T ? A)A+,which is very useful. In this paper, the theorem is generalized to the case ofthat A+ does not need to be bounded. Let A∈B(E,F), and R(A), N(A)split F, E respectively, say F = R(A)⊕N+ and E = N(A)⊕R+. LetA+ : D(A+) = R(A)+˙N+→E be a generalized inverse of A corresponding tothe decompositions above. The following result is proved that if T∈B(E,F)satisfies N(T)∩R+ = {0}, R(T)∩N(A+) = {0} and TR(A+) = R(T), thenB = A+C?1(A,A+,T) : R(T)+˙N+→E is a generalized inverse of T with N(B) =N(A+) and R(B) = R(A+), which is a generalization of the theorem by M.Z.Nashedand Chen. Moreover, in this paper, by Fan-kakutani fixed point theorem, we gen-eralized the extreme minimum fixed point theorem for set-valued mappings in thelocal convex space in Xu’s paper. We use sub-additive function instead semi-normsto prove a extreme minimum fixed point theorem for set-valued mapping in thelocal convex space.
Let E,F be Banach spaces, B(E,F) denote all the bounded linear operatorsfrom E to F, and A+ be a generalized inverse of A. The following theorem byNashed and Chen is known well: for all T satisfying T - A < A+-1, ifC~(-1)1(A,A+,T)TN(A) ? R(A), then B = A+C?1(A,A+,T) is a generalized inverseof T, and N(B) = N(A+), R(B) = R(A+), where C(A,A+,T) = IF + (T ? A)A+,which is very useful. In this paper, the theorem is generalized to the case ofthat A+ does not need to be bounded. Let A∈B(E,F), and R(A), N(A)split F, E respectively, say F = R(A)⊕N+ and E = N(A)⊕R+. LetA+ : D(A+) = R(A)+˙N+→E be a generalized inverse of A corresponding tothe decompositions above. The following result is proved that if T∈B(E,F)satisfies N(T)∩R+ = {0}, R(T)∩N(A+) = {0} and TR(A+) = R(T), thenB = A+C?1(A,A+,T) : R(T)+˙N+→E is a generalized inverse of T with N(B) =N(A+) and R(B) = R(A+), which is a generalization of the theorem by M.Z.Nashedand Chen. Moreover, in this paper, by Fan-kakutani fixed point theorem, we gen-eralized the extreme minimum fixed point theorem for set-valued mappings in thelocal convex space in Xu’s paper. We use sub-additive function instead semi-normsto prove a extreme minimum fixed point theorem for set-valued mapping in thelocal convex space.
引文
1 M. Z. Nashed, X. Chen. Convergence of Newton-like methods for singular equa-tions using outer inverses. Numer.Math. 1996,66:235～257
2 J. P. Ma. Rank theorems of operators between Banach spaces. Sicience in China.2000,43A(1):1～5
3 J. P. Ma. Complete rank theorem of advanced calculus and singularities ofbounded linear operators. Front.Math.China. 2008,2:305～316
4王玉文. Banach空间算子广义逆理论及其应用.现代数学丛书,北京,科学出版社. 2005.
5 J. P. Ma. Three classes of smooth Banach submanifolds in B(E,F). Sciencein China. 2007,9: 1233～1239
6徐明跃,曹玉红,王玉文,局部凸空间中极值映射的极小不动点定理及其应用.系统科学与数学. 2007.12,27(6): 943～952
7张恭庆,郭懋正.泛函分析讲义下册.北京大学出版社1990
8潘吉勋,张顺明,经济均衡的数学原理,吉林大学出版社,1997
9 Holmes,R.B.,Geometric Functional Analysis and Its Applications in Banach Spaces,New York, Heidelberg, Berlin: Springer-Verlag, 1975
10王玉文,李志伟, Banach空间中Moore-Penrose广义逆与不适定问题,系统科学与数学, 1995,15(2):175～185
11 Wang Songgui, Liski E. D. Constraint Lower Ordering of Metrics and Extent-ions of the Cauchy-Schwars Inequality. Report A 304, Dept of Math. Sciences,Univ of Thampare, Finland, 1996
12 Marshall A. W. Olkin I Matrix Versions of the Cauchy and Kantorovich In-equalities. A equalities, A equation Math. 1990,40: 89～93
13 Nordstron K. Some Further Aspect of the Lower Ording Antionicty of theMoore-Penrose Inverse Commu In Statist Theory and Methods. 1989, 18: 4471～4489
14 M. P. Drazin. Pesudoinverse in associative rings and semigroups. Amer. Math.Monthly, 1958, 65: 506～514
15 M. Z. Nashed, Ed.Generralized Inverse and Applications. New York/London:Academic Press, 1976
16 A. Ben-Israel, T. N. E. Greville. Generalized Inverses: Theory and Applica-tions New York: John Wiley, 1974
17 S. L. Campbell, Jr. C. D. Meryer. Continuity properties of the Drazin inverse.Linear Algebra Appl., 1975, 10: 77～83
18§. L. Campbell, Jr. C. D. Meryer. Continuity properties of the Drazin inverse.Linear Algebra Appl., 1975, 10: 77～83
19 G. W. Groetsch. Generalized Inverse of Linear Operators. New York: MarcelDekker, 1977
20 S. R. Caradus. Generalized Inverse and Operator Theory, Queens papers in Pureand Appl. Math., 50(Queens University, Kingston, Onrario), 1978
21 M. Z. Nashed, G.F.Votruba, A unified approach to generalized inverses oflinear operator:Algebraic,topological and projectional properties, Bull. Amer.Math. Soc. 1974,80:825～830
22 Y. Y. Tseng. The charateristic value problem of Hermitam functional operatorsin a non-Hilbert spaces. Doctoral Dissertation in Math.University of Chicago,1933
23 Y. Y. Tseng. Generalized inverses of unbounded operators between two unitaryspaces. Dokl. Akad. Nauk. SSSR(N.S.), 1941,67:431～434
24 Y. Y. Tseng. Properties and classification of generalized inverses of closed op-erators. Dokl. Akad. Nauk. SSSR(N.S.)., 1949,67:607～610
25 Y. Y. Tseng. Sur les solutions des equations operatrices functionnelless entreles espaces . Unitaires, C. R. Acad. Sci. Paris, 1949.228:640～641
26 Y. Y. Tseng. Virtual solutions and general inversions. Uspehi. Mat.Nauk.(N.S.), 1956,11:213～215
27 J. P. Aubin. Applied Functional Analysis, Wiley-Inter-Science, New York,1979
28 J. Locker. The Method of Least Squares for Boundary Value Problems.Trans, Amer .Math. Soc., 1971,39:57～68
29 J. Locker. On Constructing Least Squares Solutions to Two-Point BoundryValue Problems. Trans. Amer. Math. Soc., 1975,203: 175～183
30 J. Locker. The Generalized Greens Function for an nth Order Linear Di?erentialOperator. Trans. Amer.L Math. Soc., 1997,228:243～268
31 J. Locker . Functional Analysis and Two-Point Di?erential Operators. NewYork.John Wiley& Sons, Inc. 1986
32 Erdelyi I and Ben-Israel. A Extremal Solutions of Linear Equations andGeneralized Inversion Between Hilbert Spaces . J. Math. Appl. 1972,39:298～313
33马吉溥.关于R(Ax)闭的连续算子族Ax的Moore-Pense广义逆A+x连续的充要条件.中国科学, A(1990) 561～568
34李绍宽,n Hilbert空间的算子广义逆.数学年刊, 1990,4
35 M. Z. Nashed, G. F. Votruba. A Unified Operator Theory of Generalized In-verse, Nashed Ed. Academic Press, New York, 1976. 101～109
36魏益民,匡蛟勋. Banach空间中计算线性算子Drazin逆的迭代法.复旦学报,1996, 35: 407～413
37 M. Z. Nashed. Inner, outer and generalized inverse in Banach and Hilbert spaces.Numer. Funct. Anal. Optim., 1987, 9: 261～325
38 M. Z. Nashed. A new approach to classification and regularzation of ill-posedoperator equations, Inverse and Ill-posed problems, (sankt wolfgarg,1986),53～75, Notes Rep. Math. Sci. Engrg. 4, Boston:Academic Press, MA, 1987
39 M. Z. Nashed. On nonlinear ill-posed problem II,Monotone variational inequal-ities. Theory and Applications of Nonlinear Operators of Accretive and Mono-tone Type, 230～240, Lecture Notes in Pure and Appl . Math., 178., New York:Dekker, 1986
40乔三正. Banach空间线性算子的Drazin逆.上海师范学院学报(自然科学版),1981,2:11～18
41乔三正. Banach空间线性算子的带权Drazin逆和Drazin逆的逼近.高等学校计算数学学报.1981,4:296～305
42匡蛟勋.线性算子Drazin逆的表示与逼近.高等学校计算数学学报,1982,4(2):97～106
43匡蛟勋,关于空间算子广义逆的某些逼近方法.上海师范院学报(自然科学版).1982,11:1～8
44匡蛟勋,陈学良.关于线性算子的K-条件数为极小的一些充要条件.高校计算数学学报.1987,9
45王国荣. Banach空间中线性算子的带权Drazin逆的逼近方法.高校计算数学学报. 1988,16:76～81
46 Wang Guorong. The reverse order for the Drazin inverses of multiple matrixproducts. Linear Algebra and Appl. 2002,348:265～272
47王国荣,魏益民,Banach空间线性算子的带W权Drazin逆的迭代法.上海师大学报.1999,28(1):1-7
48 Y. W. Wang. The generalized inverse operator in Banach spaces. Bull. Polish.Aca. Sci. Math. 1989,37(7-12):433～441
49王玉文,王辉,王润洁. Banach空间中线性算子的集值度量右逆的表示及应用.应用泛函分析学报. 1999,1(3):255～260
50王玉文,李志伟. Banach空间中Moore-Penrose广义逆与不适定的边值问题.系统科学与数学. 1995,20(2)
51 Wang Y W and Wang R J.Pseudoinverse and Two Objective Optimal Controlin Banach Spaces. Functions et Approx., XXI, 1992,UAM, 140～160
52李志伟,王玉文. Banach空间中非光滑指标奇异最优控制.系统科学与数学.1995,15(4):305～311
53王玉文,季大琴. Banach空间中无限维矩阵的Tseng度量广义逆.哈师大自然科学学报.1998,14(4):11～14
54王玉文,季大琴. Banach空间中线性算子的Tseng度量广义逆.系统科学与数学, 2000,20(2): 203～209
55王玉文,潘少容. Banach空间线性算子(集值)度量广义逆及其齐性选择.数学学报. 2003,46(3): 273～280
56 H. Wang , Y. W. Wang . Metric Generalized inverse of linear operator in Ba-nach spaces . Chin. Ann. Math., 2003, 24B(4): 509～520
57 Y. W. Wang, J. Liu. Metric generalized inverse of linear maniford and extremalsolution of linear inclusion in Banach Spaces. J. Math. Anal. Apll. 2005,302:360～371
2 J. P. Ma. Rank theorems of operators between Banach spaces. Sicience in China.2000,43A(1):1～5
3 J. P. Ma. Complete rank theorem of advanced calculus and singularities ofbounded linear operators. Front.Math.China. 2008,2:305～316
4王玉文. Banach空间算子广义逆理论及其应用.现代数学丛书,北京,科学出版社. 2005.
5 J. P. Ma. Three classes of smooth Banach submanifolds in B(E,F). Sciencein China. 2007,9: 1233～1239
6徐明跃,曹玉红,王玉文,局部凸空间中极值映射的极小不动点定理及其应用.系统科学与数学. 2007.12,27(6): 943～952
7张恭庆,郭懋正.泛函分析讲义下册.北京大学出版社1990
8潘吉勋,张顺明,经济均衡的数学原理,吉林大学出版社,1997
9 Holmes,R.B.,Geometric Functional Analysis and Its Applications in Banach Spaces,New York, Heidelberg, Berlin: Springer-Verlag, 1975
10王玉文,李志伟, Banach空间中Moore-Penrose广义逆与不适定问题,系统科学与数学, 1995,15(2):175～185
11 Wang Songgui, Liski E. D. Constraint Lower Ordering of Metrics and Extent-ions of the Cauchy-Schwars Inequality. Report A 304, Dept of Math. Sciences,Univ of Thampare, Finland, 1996
12 Marshall A. W. Olkin I Matrix Versions of the Cauchy and Kantorovich In-equalities. A equalities, A equation Math. 1990,40: 89～93
13 Nordstron K. Some Further Aspect of the Lower Ording Antionicty of theMoore-Penrose Inverse Commu In Statist Theory and Methods. 1989, 18: 4471～4489
14 M. P. Drazin. Pesudoinverse in associative rings and semigroups. Amer. Math.Monthly, 1958, 65: 506～514
15 M. Z. Nashed, Ed.Generralized Inverse and Applications. New York/London:Academic Press, 1976
16 A. Ben-Israel, T. N. E. Greville. Generalized Inverses: Theory and Applica-tions New York: John Wiley, 1974
17 S. L. Campbell, Jr. C. D. Meryer. Continuity properties of the Drazin inverse.Linear Algebra Appl., 1975, 10: 77～83
18§. L. Campbell, Jr. C. D. Meryer. Continuity properties of the Drazin inverse.Linear Algebra Appl., 1975, 10: 77～83
19 G. W. Groetsch. Generalized Inverse of Linear Operators. New York: MarcelDekker, 1977
20 S. R. Caradus. Generalized Inverse and Operator Theory, Queens papers in Pureand Appl. Math., 50(Queens University, Kingston, Onrario), 1978
21 M. Z. Nashed, G.F.Votruba, A unified approach to generalized inverses oflinear operator:Algebraic,topological and projectional properties, Bull. Amer.Math. Soc. 1974,80:825～830
22 Y. Y. Tseng. The charateristic value problem of Hermitam functional operatorsin a non-Hilbert spaces. Doctoral Dissertation in Math.University of Chicago,1933
23 Y. Y. Tseng. Generalized inverses of unbounded operators between two unitaryspaces. Dokl. Akad. Nauk. SSSR(N.S.), 1941,67:431～434
24 Y. Y. Tseng. Properties and classification of generalized inverses of closed op-erators. Dokl. Akad. Nauk. SSSR(N.S.)., 1949,67:607～610
25 Y. Y. Tseng. Sur les solutions des equations operatrices functionnelless entreles espaces . Unitaires, C. R. Acad. Sci. Paris, 1949.228:640～641
26 Y. Y. Tseng. Virtual solutions and general inversions. Uspehi. Mat.Nauk.(N.S.), 1956,11:213～215
27 J. P. Aubin. Applied Functional Analysis, Wiley-Inter-Science, New York,1979
28 J. Locker. The Method of Least Squares for Boundary Value Problems.Trans, Amer .Math. Soc., 1971,39:57～68
29 J. Locker. On Constructing Least Squares Solutions to Two-Point BoundryValue Problems. Trans. Amer. Math. Soc., 1975,203: 175～183
30 J. Locker. The Generalized Greens Function for an nth Order Linear Di?erentialOperator. Trans. Amer.L Math. Soc., 1997,228:243～268
31 J. Locker . Functional Analysis and Two-Point Di?erential Operators. NewYork.John Wiley& Sons, Inc. 1986
32 Erdelyi I and Ben-Israel. A Extremal Solutions of Linear Equations andGeneralized Inversion Between Hilbert Spaces . J. Math. Appl. 1972,39:298～313
33马吉溥.关于R(Ax)闭的连续算子族Ax的Moore-Pense广义逆A+x连续的充要条件.中国科学, A(1990) 561～568
34李绍宽,n Hilbert空间的算子广义逆.数学年刊, 1990,4
35 M. Z. Nashed, G. F. Votruba. A Unified Operator Theory of Generalized In-verse, Nashed Ed. Academic Press, New York, 1976. 101～109
36魏益民,匡蛟勋. Banach空间中计算线性算子Drazin逆的迭代法.复旦学报,1996, 35: 407～413
37 M. Z. Nashed. Inner, outer and generalized inverse in Banach and Hilbert spaces.Numer. Funct. Anal. Optim., 1987, 9: 261～325
38 M. Z. Nashed. A new approach to classification and regularzation of ill-posedoperator equations, Inverse and Ill-posed problems, (sankt wolfgarg,1986),53～75, Notes Rep. Math. Sci. Engrg. 4, Boston:Academic Press, MA, 1987
39 M. Z. Nashed. On nonlinear ill-posed problem II,Monotone variational inequal-ities. Theory and Applications of Nonlinear Operators of Accretive and Mono-tone Type, 230～240, Lecture Notes in Pure and Appl . Math., 178., New York:Dekker, 1986
40乔三正. Banach空间线性算子的Drazin逆.上海师范学院学报(自然科学版),1981,2:11～18
41乔三正. Banach空间线性算子的带权Drazin逆和Drazin逆的逼近.高等学校计算数学学报.1981,4:296～305
42匡蛟勋.线性算子Drazin逆的表示与逼近.高等学校计算数学学报,1982,4(2):97～106
43匡蛟勋,关于空间算子广义逆的某些逼近方法.上海师范院学报(自然科学版).1982,11:1～8
44匡蛟勋,陈学良.关于线性算子的K-条件数为极小的一些充要条件.高校计算数学学报.1987,9
45王国荣. Banach空间中线性算子的带权Drazin逆的逼近方法.高校计算数学学报. 1988,16:76～81
46 Wang Guorong. The reverse order for the Drazin inverses of multiple matrixproducts. Linear Algebra and Appl. 2002,348:265～272
47王国荣,魏益民,Banach空间线性算子的带W权Drazin逆的迭代法.上海师大学报.1999,28(1):1-7
48 Y. W. Wang. The generalized inverse operator in Banach spaces. Bull. Polish.Aca. Sci. Math. 1989,37(7-12):433～441
49王玉文,王辉,王润洁. Banach空间中线性算子的集值度量右逆的表示及应用.应用泛函分析学报. 1999,1(3):255～260
50王玉文,李志伟. Banach空间中Moore-Penrose广义逆与不适定的边值问题.系统科学与数学. 1995,20(2)
51 Wang Y W and Wang R J.Pseudoinverse and Two Objective Optimal Controlin Banach Spaces. Functions et Approx., XXI, 1992,UAM, 140～160
52李志伟,王玉文. Banach空间中非光滑指标奇异最优控制.系统科学与数学.1995,15(4):305～311
53王玉文,季大琴. Banach空间中无限维矩阵的Tseng度量广义逆.哈师大自然科学学报.1998,14(4):11～14
54王玉文,季大琴. Banach空间中线性算子的Tseng度量广义逆.系统科学与数学, 2000,20(2): 203～209
55王玉文,潘少容. Banach空间线性算子(集值)度量广义逆及其齐性选择.数学学报. 2003,46(3): 273～280
56 H. Wang , Y. W. Wang . Metric Generalized inverse of linear operator in Ba-nach spaces . Chin. Ann. Math., 2003, 24B(4): 509～520
57 Y. W. Wang, J. Liu. Metric generalized inverse of linear maniford and extremalsolution of linear inclusion in Banach Spaces. J. Math. Anal. Apll. 2005,302:360～371