无穷区间上非绝对积分及其应用
摘要
无穷区间上积分的讨论是积分理论中不可缺少的一部分。本文中，我们对无穷区间上向量值函数非绝对积分理论进行了讨论，并给出了一些初步应用。主要包括以下五部分内容：在第一部分中，我们介绍了本文所用到的基本概念和引理；在第二部分中，通过定义无穷区间上δ-精细的分法，我们给出了无穷区间上向量值函数的(H)积分的定义，并讨论其性质，还给出了原函数的刻划；在本文的第三部分中，我们着重讨论了无穷区间上向量值函数(H)积分的收敛定理；在本文的在第四部分中，我们首先应用无穷区间上向量值函数(H)积分的收敛定理给出了常微分方程整体广义解的存在性定理，其次应用强(H)积分对Banach空间常微分方程广义解进行了讨论；最后，在第五部分中，我们将模糊(H)积分推广到无穷区间上并给出了其数值计算方法。
The integral on infinite interval is an indispensable part of the theory of integral. In this paper, we discuss the problem of non-absolutely integral theory for infinite interval and give some applications. On the whole, we do as follows: Firstly, we list some conceptions and lemmas for later use. Secondly, we define δ-fine partitions for infinite interval and (H) integral of vector-valued functions on infinite interval, and discuss the properties of (H) integral, and characterize its primitives. Thirdly, we discuss convergence theorems of vector-valued (H) integral on infinite interval. Fourthly, we present some applications of convergence theorems of (H) integral of vector-valued functions on infinite interval and strongly (H) integral for differential equations. Finally, we discuss fuzzy (H) integral on infinite interval and numerical calculator.
The integral on infinite interval is an indispensable part of the theory of integral. In this paper, we discuss the problem of non-absolutely integral theory for infinite interval and give some applications. On the whole, we do as follows: Firstly, we list some conceptions and lemmas for later use. Secondly, we define δ-fine partitions for infinite interval and (H) integral of vector-valued functions on infinite interval, and discuss the properties of (H) integral, and characterize its primitives. Thirdly, we discuss convergence theorems of vector-valued (H) integral on infinite interval. Fourthly, we present some applications of convergence theorems of (H) integral of vector-valued functions on infinite interval and strongly (H) integral for differential equations. Finally, we discuss fuzzy (H) integral on infinite interval and numerical calculator.
引文
[1] John D.Depree and Charles W.Swartz, Introduction to Real Analysis, John Wiley and Sons.Inc., 1988.
[2] 丁传松，李秉彝，广义黎曼积分，北京：科学出版社，1989．
[3] Lee Peng Yee and Rukolf V(?)born(?), The Integral: An Easy Approch after Kurzweil and Henstock, Cambridge University Press, 2000.
[4] Lee Peng Yee, Lanzhou lectures on Henstock integration, World Scientific, 1989.
[5] S.Cao, The Henstock Integral for Banach-Valued Functions, SEA Bull.Math.1992, 16, 35-40.
[6] 姚小波，非绝对积分：矢值积分，空间拓拓补结构及广义微分方程φ-变差解，博士学位论文．1994．
[7] R.A.Gordon, The Mcshane integration of Banach-Valued functions, J.Illinois.Math., 1990, 34, 557-567.
[8] R.A.Gordon, Another Look at a convergence theorem for the Henstock Integral, Real Analysis Exchange, 1989/1990(15).
[9] R.A.Gordon, The Denjoy extension of the Bochner, Pettis, and Dunford integrals, Studia Mathematica, 1989, 73-91.
[10] YE Guoju and S.Schwakik, The mcshane and the pettis Integral of Banach Space-Valued functions defined on R~m, J.Illinois.Math., 2002, 46, 1125-1144.
[11] Wang Pujie, Equi-integrability and controlled convergence for the Henstock integral, Real Analysis Exchange, 1993/1994, 19(1), 236-241.
[12] Lorna I.Paredes and Chew Tuan Seng, Controlled convergence Theorem for Banach-Valued HL Integrals, Scientiae Mathematicae Japonicae Online, 2002, 6, 261-271.
[13] J.Kurzweil and J.Jamik, Equiintegrability and controlled convergence of perron-Type integrable functions, Real Analysis Exchange, 1991/1992, 17, 110-139.
[14] George Cross and Over Shisha, A New Approach to Integration, J.M.A.A., 1986, 114,289-294.
[15] James T.Lewis and Over Shisha, The Generalized Rieman, Simple, Dominated and Improper integrals, J.Approximation theory, 1983, 38, 192-199.
[16] Seymour Haber and Over Shisha, An integral related to Numerical integration, J.American Math. Society, 1973, 79, 930-932.
[17] Wu Congxin and Ye Guoju, The Mcshane integral of Banach-Valued functions defined on R~n(I),数学研究，1998, 30(2), 140-144.
[18] Ye Guoju, The Relation Between the Pettis Integral and the Mcshane Integral of Banach-Valued Functions Defined on R~n，数学进展，1998, 27(6), 558-560.
[19] Wu Congxin, Yao Xiaobo, A Riemann-Type Definition of the Bochner Integral,数学研究，1994，27(1)， 32-36．
[38] 吴从忻，姚小波，向量值函数的Mcshane积分，数学研究，1995，28(1)，41-48．
[21] 丁传松，李秉彝，一般Henstock积分的支配收敛定理，数学学报，1994，37(4)，497-506．
[22] 巩增泰，一般Henstock积分的另一个收敛定理，西北师大学报(自然科学版)，2000．36(3)，10-14．
[23] Stefan Schwabik, Generalized Differential Equations, RNDR.IVO.Vrkoc, Drsc., 1985.
[24] Stefan Schwabik, Generalized Differential Equations, World Scientific, 1992.
[25] Tuan Seng Chew and Francisco Flordeliza, On x′= f(t, x) and Henstock-Kurzweil integrals, Differential and Integral Equations, 1991, 4, 861-868.
[26] Wu Congxin,Li baolin and E.stanley Lee, Discontinuous systems and the Henstock-Kurzweil Integral, J.M.A.A, 1999, 229, 119-136.
[27] 尤秉礼，常微分方程补充教程，人民教育出版社，1982．
[28] 定光桂，巴拿赫空间引论，科学出版社，1984．
[29] 郭大钧，孙经先，抽象空间常微分方程，山东科技出版社，2002．
[30] 孙经先，增算子的不动点定理及其对Banach空间含间断项的非线性方程的应用，数学学报，1991，34：665-674．
[31] 李永祥，Banach空间含间断项微分方程的广义解，西北师范大学学报(自然科学版)，2000，36(3)：1～5．
[32] 李永祥，Banach空间微分方程广义解的正则性，甘肃科学学报，2001，13(3)：1-6．
[33] 吴从炘，马明，模糊分析学基础，国防工业出版社，1991．
[34] Hsien-Chung Wu, The improper fuzzy Riemann integral and its numerical integration, Information Sciences 1998, 111: 109-137.
[35] Barnab(?)s Bede, Sorin G.Gol, Quadrature rules for integrals of fuzzy-number-valued functions, Fuzzy Sets and systems, submitted
[36] Congxin Wu, Zengtai Gong, On Henstock integral of fuzzy-number-valued functions(I), Fuzzy Sets and Systems, 2001, 120: 523-532.
[37] Congxin Wu, Zengtai Gong, On Henstock integral of interval-valued functions and fuzzy-number-valued function, Fuzzy Sets and Systems, 2000, 115: 377-391.
[38] Wu Congxin, Ma Ming, On embedding problem of fuzzy number space:Part 2, Fuzzy Sets and Systems, 1992,45:189-202.
[39] S.Saks, Theory of Integration, 2nd, English Edition, Warsaw, 1937.
[40] 孔芳弟，无穷区间上可积函数列逐项积分的条件，西北师范大学学报(自然科学版)，2003，39(3)，31-32．
[2] 丁传松，李秉彝，广义黎曼积分，北京：科学出版社，1989．
[3] Lee Peng Yee and Rukolf V(?)born(?), The Integral: An Easy Approch after Kurzweil and Henstock, Cambridge University Press, 2000.
[4] Lee Peng Yee, Lanzhou lectures on Henstock integration, World Scientific, 1989.
[5] S.Cao, The Henstock Integral for Banach-Valued Functions, SEA Bull.Math.1992, 16, 35-40.
[6] 姚小波，非绝对积分：矢值积分，空间拓拓补结构及广义微分方程φ-变差解，博士学位论文．1994．
[7] R.A.Gordon, The Mcshane integration of Banach-Valued functions, J.Illinois.Math., 1990, 34, 557-567.
[8] R.A.Gordon, Another Look at a convergence theorem for the Henstock Integral, Real Analysis Exchange, 1989/1990(15).
[9] R.A.Gordon, The Denjoy extension of the Bochner, Pettis, and Dunford integrals, Studia Mathematica, 1989, 73-91.
[10] YE Guoju and S.Schwakik, The mcshane and the pettis Integral of Banach Space-Valued functions defined on R~m, J.Illinois.Math., 2002, 46, 1125-1144.
[11] Wang Pujie, Equi-integrability and controlled convergence for the Henstock integral, Real Analysis Exchange, 1993/1994, 19(1), 236-241.
[12] Lorna I.Paredes and Chew Tuan Seng, Controlled convergence Theorem for Banach-Valued HL Integrals, Scientiae Mathematicae Japonicae Online, 2002, 6, 261-271.
[13] J.Kurzweil and J.Jamik, Equiintegrability and controlled convergence of perron-Type integrable functions, Real Analysis Exchange, 1991/1992, 17, 110-139.
[14] George Cross and Over Shisha, A New Approach to Integration, J.M.A.A., 1986, 114,289-294.
[15] James T.Lewis and Over Shisha, The Generalized Rieman, Simple, Dominated and Improper integrals, J.Approximation theory, 1983, 38, 192-199.
[16] Seymour Haber and Over Shisha, An integral related to Numerical integration, J.American Math. Society, 1973, 79, 930-932.
[17] Wu Congxin and Ye Guoju, The Mcshane integral of Banach-Valued functions defined on R~n(I),数学研究，1998, 30(2), 140-144.
[18] Ye Guoju, The Relation Between the Pettis Integral and the Mcshane Integral of Banach-Valued Functions Defined on R~n，数学进展，1998, 27(6), 558-560.
[19] Wu Congxin, Yao Xiaobo, A Riemann-Type Definition of the Bochner Integral,数学研究，1994，27(1)， 32-36．
[38] 吴从忻，姚小波，向量值函数的Mcshane积分，数学研究，1995，28(1)，41-48．
[21] 丁传松，李秉彝，一般Henstock积分的支配收敛定理，数学学报，1994，37(4)，497-506．
[22] 巩增泰，一般Henstock积分的另一个收敛定理，西北师大学报(自然科学版)，2000．36(3)，10-14．
[23] Stefan Schwabik, Generalized Differential Equations, RNDR.IVO.Vrkoc, Drsc., 1985.
[24] Stefan Schwabik, Generalized Differential Equations, World Scientific, 1992.
[25] Tuan Seng Chew and Francisco Flordeliza, On x′= f(t, x) and Henstock-Kurzweil integrals, Differential and Integral Equations, 1991, 4, 861-868.
[26] Wu Congxin,Li baolin and E.stanley Lee, Discontinuous systems and the Henstock-Kurzweil Integral, J.M.A.A, 1999, 229, 119-136.
[27] 尤秉礼，常微分方程补充教程，人民教育出版社，1982．
[28] 定光桂，巴拿赫空间引论，科学出版社，1984．
[29] 郭大钧，孙经先，抽象空间常微分方程，山东科技出版社，2002．
[30] 孙经先，增算子的不动点定理及其对Banach空间含间断项的非线性方程的应用，数学学报，1991，34：665-674．
[31] 李永祥，Banach空间含间断项微分方程的广义解，西北师范大学学报(自然科学版)，2000，36(3)：1～5．
[32] 李永祥，Banach空间微分方程广义解的正则性，甘肃科学学报，2001，13(3)：1-6．
[33] 吴从炘，马明，模糊分析学基础，国防工业出版社，1991．
[34] Hsien-Chung Wu, The improper fuzzy Riemann integral and its numerical integration, Information Sciences 1998, 111: 109-137.
[35] Barnab(?)s Bede, Sorin G.Gol, Quadrature rules for integrals of fuzzy-number-valued functions, Fuzzy Sets and systems, submitted
[36] Congxin Wu, Zengtai Gong, On Henstock integral of fuzzy-number-valued functions(I), Fuzzy Sets and Systems, 2001, 120: 523-532.
[37] Congxin Wu, Zengtai Gong, On Henstock integral of interval-valued functions and fuzzy-number-valued function, Fuzzy Sets and Systems, 2000, 115: 377-391.
[38] Wu Congxin, Ma Ming, On embedding problem of fuzzy number space:Part 2, Fuzzy Sets and Systems, 1992,45:189-202.
[39] S.Saks, Theory of Integration, 2nd, English Edition, Warsaw, 1937.
[40] 孔芳弟，无穷区间上可积函数列逐项积分的条件，西北师范大学学报(自然科学版)，2003，39(3)，31-32．