双枝模糊值函数的Mcshane积分及其推广
|
摘要
本文共分两个部分.
第一部分：首先,在区间值函数的Mcshane积分基础上,引入了双区间值函数的Mcshane积分.其次,将模糊值函数的Mcshane积分推广为双枝模糊值函数Mcshane积分,并研究了此积分的一些基本性质.最后,结合双区间值函数和双枝模糊值函数的Mcshane积分定义,讨论了双枝模糊值函数的Mcshane积分的单调收敛定理和控制收敛定理.
第二部分：利用无穷区间上传统的δ-精细分划定义,结合模糊值函数与其截函数之间的关系,引入了无穷区间上模糊值函数的Mcshane积分.此外,针对模糊值函数给出了等度模糊Mcshane积分定义,并给出了其模糊值函数可积的等价条件.最后,定义了强模糊Mcshane积分,并在此积分意义下获得了其模糊值函数可积的充分必要条件,从而完善并丰富了模糊积分理论.
The paper mainly has two parts.
The first part:at first, based on Mcshane integral of interval-valued func-tions, Mcshane integral of both-branch-interval-valued functions is introduced. Then Mcshane integral of fuzzy-valued functions is extended to both-branch-fuzzy-valued functions and some of its basic properties are studied. At last, com-bining the definitions of both-branch-interval-valued functions and both-branch-fuzzy-valued functions, monotone convergence theorem and dominated conver-gence theorem for Mcshane integral of both-branch-fuzzy-valued functions are discussed.
The second part:using the definition of traditional fine division on infinite interval and combining the relationship between fuzzy-valued functions on in-finite interval and its cut functions, Mcshane integral of fuzzy-valued functions on infinite interval is introduced. Furthermore, aiming at fuzzy-valued functions, the definition of equivalent measure fuzzy Mcshane integral and a condition of equivalence of its integrability are given. Last but not least, strong fuzzy Mcshane integral is defined and in the sense of this integral a necessary and sufficient condi-tion for its fuzzy valued function to be integrable is obtained. Thereby the theory of fuzzy integral is improved and enriched.
第一部分：首先,在区间值函数的Mcshane积分基础上,引入了双区间值函数的Mcshane积分.其次,将模糊值函数的Mcshane积分推广为双枝模糊值函数Mcshane积分,并研究了此积分的一些基本性质.最后,结合双区间值函数和双枝模糊值函数的Mcshane积分定义,讨论了双枝模糊值函数的Mcshane积分的单调收敛定理和控制收敛定理.
第二部分：利用无穷区间上传统的δ-精细分划定义,结合模糊值函数与其截函数之间的关系,引入了无穷区间上模糊值函数的Mcshane积分.此外,针对模糊值函数给出了等度模糊Mcshane积分定义,并给出了其模糊值函数可积的等价条件.最后,定义了强模糊Mcshane积分,并在此积分意义下获得了其模糊值函数可积的充分必要条件,从而完善并丰富了模糊积分理论.
The paper mainly has two parts.
The first part:at first, based on Mcshane integral of interval-valued func-tions, Mcshane integral of both-branch-interval-valued functions is introduced. Then Mcshane integral of fuzzy-valued functions is extended to both-branch-fuzzy-valued functions and some of its basic properties are studied. At last, com-bining the definitions of both-branch-interval-valued functions and both-branch-fuzzy-valued functions, monotone convergence theorem and dominated conver-gence theorem for Mcshane integral of both-branch-fuzzy-valued functions are discussed.
The second part:using the definition of traditional fine division on infinite interval and combining the relationship between fuzzy-valued functions on in-finite interval and its cut functions, Mcshane integral of fuzzy-valued functions on infinite interval is introduced. Furthermore, aiming at fuzzy-valued functions, the definition of equivalent measure fuzzy Mcshane integral and a condition of equivalence of its integrability are given. Last but not least, strong fuzzy Mcshane integral is defined and in the sense of this integral a necessary and sufficient condi-tion for its fuzzy valued function to be integrable is obtained. Thereby the theory of fuzzy integral is improved and enriched.
引文
[1]Zadeh L A. Fuzzy sets[J]. Information and control,1965,8:338-353.
[2]罗承忠,王德森.区间值函数积分的推广与Fuzzy值函数的积分[J].模糊数学,1983,3：45-52.
[3]程其襄,张奠宙等.实变函数与泛函分析基础[M].北京：高等教育出版社,1983
[4]张文修.模糊数学基础[M].西安：西安交通大学出版社,1984.
[5]Shi Kaiquan. Both-branch fuzzy sets (Ⅰ)(Ⅱ)(Ⅲ)[J]. BUSEFAL,1998,(75):146-174.
[6]丁传松,李秉彝.广义黎曼积分[M].北京：科学出版社,1989.
[7]R.A.Gordon. The Mcshane integration of Banach-valued functions[J]. Illinois.Math,1990,34:557-567
[8]Ye Guoju and S.Schwakik. The Mcshane and the Pettis Integral of Banach Sapce-valued functions defined on [J],Illinois.Math,2002,46:1125-1144.
[9]Wu Congxin and Ye Guoju. The Mcshane integral of Banach-valued fanctions defined on Rm [J].数学研究,1998,30(2)：140-144.
[10]吴从,姚小波,向量值函数的Mcshane积分[J].数学研究,1995,28(1)：41-48.
[11]张文修,吴从.模糊数学引论[M].西安：西安交通大学出版社,1991.
[12]吴从,马明.模糊分析学基础[M].北京：国防工业出版社,1991.
[13]Ye Guoju. The Relation Between the Pettis Integral and the Mcshane Integral of Banach-valued Functions Defined on Rm [J]数学进展,1998,27(6)：558-560.
[14]吴从,马明,方锦暄.模糊分析学的机构理论[M].贵阳：贵州科技出版社,1994.
[15]刘普寅,吴孟达.模糊理论及其应用[M].北京：国防科技大学出版社,1998.
[16]Guo Caimei,Zhang Deli,Wu Congxin. Generalized fuzzy integrals of fuzzy-valued functions[J]. Fuzzy Sets and Systems,1998,97:123-128.
[17]华东师范大学数学系.数学分析(上册)[M].北京：华东高等教育出版社,2001.
[18]杨纶标,高英仪.模糊数学原理及其应用[M].广州：华南理工大学出版社,2001.
[19]Fang Jin-xuan. On the convergence theorems of generalized fuzzy integral sequence[J]. Fuzzy Sets and Systems,2001,124:117-123.
[20]Fang Jin-xuan. A note on the convergence theorems of generalized fuzzy integral sequence[J]. Fuzzy Sets and Systems,2001,127:377-381.
[21]张德利,郭彩梅,吴从.模糊积分论进展[J].模糊数学与系统,2003,4(17)：1-6.
[22]陈水利，李敬功,王向公.模糊集理论及其应用[M].北京：科学出版社,2005.
[23]Wu Congxin, Gong Zengtai. On Henstock integral of fuzzy-number-valued functions(I)(J). Fuzzy Sets and Systems,2001,120:523-533.
[24]Wu Congxin, Gong Zengtai. On Henstock of integral of interval-valued functions and fuzzy-valued func-tions[J]. Fuzzy Sets and Systems,2000,115:388-391.
[25]巩增泰,吴从.模糊数值函数Henstock积分的原函数刻画[J].模糊系统与数学,2000,14(4)：24-30.
[26]李守伟,张小平.双枝模糊值函数Henstock积分[J].山东建筑工程学院学报,2001,3(16)：75-80.
[27]Gong Zengtai,Wu Congxin. The Mcshane integral of fuzzy-valued functions[J]. Southeast Asian Bulletin of Mathematics,2000,24:365-373.
[28]Wang Ling,Gong Zengtai. The Perron integral of fuzzy-valued functions:background, definition and charac-terizations[J]. Intern J of Pure and Appl Math,2005,19(3):321-336.
[29]Gong Zengtai. On the problem of characterizing derivatives for the fuzzy-valued-functions (Ⅱ):almost every-where differentiability and strong Henstock integral[J]. Fuzzy Sets and Systems,2004,145:381-393.
[30]HENSTOCK R. Theory of Integration[M]. London:Butterworth,1963,10-30.
[31]巩增泰.Henstock引理,导函数的可积性,积分原函数的可导性[J].数学实践与认识,2005,35(9)：148-154.
[32]曾文艺,罗承忠Fuzzy值函数的收敛[J].北京师范大学学报,1995,31：164-169.
[33]曾文艺,米洪海.区间值函数和Fuzzy值函数的积分收敛(Ⅱ)[J].河北工业大学学报,1994,23：25-32.
[34]H.Roman-Flores,A.Flores-Franulic,R.C.Bassanezi,M.Rojas-Medar. On the level-continuity of fuzzy inte-grals[J]. Fuzzy sets and systems,1996,80(1996):339-344.
[35]毕淑娟,张晓东.模糊数值函数的收敛性及连续性[J].哈尔滨商业大学学报,2002,18：330-333.
[36]齐思刚.模糊值函数在模糊数积分区间上的积分[J].河北大学学报,1999,19：199-204.
[37]Luo Chengzhong, Wang Desen. Extension of the integral of interval-valued function and the integral of fuzzy valued function fuzzy mathematics[J].1983(3):30-35.
[38]程地会,吴冲.区间值函数与模糊值函数的RS积分[J].哈尔滨工业大学学报,1999,31：29-31.
[39]Wu Congxin,Wang Shuili,Ma Ming. On generalized fuzzy integrals[J]. FSS,1995,70(1):75-87.
[40]Wu Chong. Fuzzy number fuzzy measures and fuzzy-valued integrals(Ⅱ)[J]. Fuzzy Sets and Systems,1999,101(1):137-141.
[41]Wu Chong, Zhang Deli, Guo Caimei. Fuzzy number fuzzy measures and fuzzy-valued integrals(Ⅰ)[J]. Fuzzy Sets and Systems,1998,98(3):355-360.
[42]Gong Zengtai. A New Arithmetic for Fuzzy Numbers and Its Applications[J].模糊系统与数学,2008,22:101-108.
[43]孔芳弟.无穷区间上可积函数列逐项积分的条件(J).西北师范大学学报(自然科学版),2003,39(3)：31-32.
[44]Bi Shujuan,Wu Congxin. The operation property of fuzzy number and the limit and the distance of fuzzy numbers[J]. Fuzzy Systems and Mathematics,2000,14:40-45.
[45]Ma Ming. A new fuzzy arithmetic[J]. Fuzzy Sets and Systems,1999,108:83-90.
[46]Wu Congxin, Ma Ming. On embedding problem of fuzzy number spaces[J]. Fuzzy Sets and Systems,1991,44:33-38.
[47]Wu Congxin, Wu Chong. The supremum and infimum of the set of fuzzy numbers and its applications[J]. J. Math. Anal. Appl.,1997,210:499-511.
[48]汪玲,巩增泰.模糊Henstock积分[J].兰州理工大学学报,2006,32：133-137.
[49]Wu Congxin, Zhang Bokan. A note on the embedding problem for multidimensional fuzzy number spaces[J]. Fuzzy Sets and Systems,2001,121:359-362.
[50]巩增泰,郭元伟.区间值和模糊数值函数的Choquet积分[J].兰州大学学报,2009,45(4)：112-117
[2]罗承忠,王德森.区间值函数积分的推广与Fuzzy值函数的积分[J].模糊数学,1983,3：45-52.
[3]程其襄,张奠宙等.实变函数与泛函分析基础[M].北京：高等教育出版社,1983
[4]张文修.模糊数学基础[M].西安：西安交通大学出版社,1984.
[5]Shi Kaiquan. Both-branch fuzzy sets (Ⅰ)(Ⅱ)(Ⅲ)[J]. BUSEFAL,1998,(75):146-174.
[6]丁传松,李秉彝.广义黎曼积分[M].北京：科学出版社,1989.
[7]R.A.Gordon. The Mcshane integration of Banach-valued functions[J]. Illinois.Math,1990,34:557-567
[8]Ye Guoju and S.Schwakik. The Mcshane and the Pettis Integral of Banach Sapce-valued functions defined on [J],Illinois.Math,2002,46:1125-1144.
[9]Wu Congxin and Ye Guoju. The Mcshane integral of Banach-valued fanctions defined on Rm [J].数学研究,1998,30(2)：140-144.
[10]吴从,姚小波,向量值函数的Mcshane积分[J].数学研究,1995,28(1)：41-48.
[11]张文修,吴从.模糊数学引论[M].西安：西安交通大学出版社,1991.
[12]吴从,马明.模糊分析学基础[M].北京：国防工业出版社,1991.
[13]Ye Guoju. The Relation Between the Pettis Integral and the Mcshane Integral of Banach-valued Functions Defined on Rm [J]数学进展,1998,27(6)：558-560.
[14]吴从,马明,方锦暄.模糊分析学的机构理论[M].贵阳：贵州科技出版社,1994.
[15]刘普寅,吴孟达.模糊理论及其应用[M].北京：国防科技大学出版社,1998.
[16]Guo Caimei,Zhang Deli,Wu Congxin. Generalized fuzzy integrals of fuzzy-valued functions[J]. Fuzzy Sets and Systems,1998,97:123-128.
[17]华东师范大学数学系.数学分析(上册)[M].北京：华东高等教育出版社,2001.
[18]杨纶标,高英仪.模糊数学原理及其应用[M].广州：华南理工大学出版社,2001.
[19]Fang Jin-xuan. On the convergence theorems of generalized fuzzy integral sequence[J]. Fuzzy Sets and Systems,2001,124:117-123.
[20]Fang Jin-xuan. A note on the convergence theorems of generalized fuzzy integral sequence[J]. Fuzzy Sets and Systems,2001,127:377-381.
[21]张德利,郭彩梅,吴从.模糊积分论进展[J].模糊数学与系统,2003,4(17)：1-6.
[22]陈水利，李敬功,王向公.模糊集理论及其应用[M].北京：科学出版社,2005.
[23]Wu Congxin, Gong Zengtai. On Henstock integral of fuzzy-number-valued functions(I)(J). Fuzzy Sets and Systems,2001,120:523-533.
[24]Wu Congxin, Gong Zengtai. On Henstock of integral of interval-valued functions and fuzzy-valued func-tions[J]. Fuzzy Sets and Systems,2000,115:388-391.
[25]巩增泰,吴从.模糊数值函数Henstock积分的原函数刻画[J].模糊系统与数学,2000,14(4)：24-30.
[26]李守伟,张小平.双枝模糊值函数Henstock积分[J].山东建筑工程学院学报,2001,3(16)：75-80.
[27]Gong Zengtai,Wu Congxin. The Mcshane integral of fuzzy-valued functions[J]. Southeast Asian Bulletin of Mathematics,2000,24:365-373.
[28]Wang Ling,Gong Zengtai. The Perron integral of fuzzy-valued functions:background, definition and charac-terizations[J]. Intern J of Pure and Appl Math,2005,19(3):321-336.
[29]Gong Zengtai. On the problem of characterizing derivatives for the fuzzy-valued-functions (Ⅱ):almost every-where differentiability and strong Henstock integral[J]. Fuzzy Sets and Systems,2004,145:381-393.
[30]HENSTOCK R. Theory of Integration[M]. London:Butterworth,1963,10-30.
[31]巩增泰.Henstock引理,导函数的可积性,积分原函数的可导性[J].数学实践与认识,2005,35(9)：148-154.
[32]曾文艺,罗承忠Fuzzy值函数的收敛[J].北京师范大学学报,1995,31：164-169.
[33]曾文艺,米洪海.区间值函数和Fuzzy值函数的积分收敛(Ⅱ)[J].河北工业大学学报,1994,23：25-32.
[34]H.Roman-Flores,A.Flores-Franulic,R.C.Bassanezi,M.Rojas-Medar. On the level-continuity of fuzzy inte-grals[J]. Fuzzy sets and systems,1996,80(1996):339-344.
[35]毕淑娟,张晓东.模糊数值函数的收敛性及连续性[J].哈尔滨商业大学学报,2002,18：330-333.
[36]齐思刚.模糊值函数在模糊数积分区间上的积分[J].河北大学学报,1999,19：199-204.
[37]Luo Chengzhong, Wang Desen. Extension of the integral of interval-valued function and the integral of fuzzy valued function fuzzy mathematics[J].1983(3):30-35.
[38]程地会,吴冲.区间值函数与模糊值函数的RS积分[J].哈尔滨工业大学学报,1999,31：29-31.
[39]Wu Congxin,Wang Shuili,Ma Ming. On generalized fuzzy integrals[J]. FSS,1995,70(1):75-87.
[40]Wu Chong. Fuzzy number fuzzy measures and fuzzy-valued integrals(Ⅱ)[J]. Fuzzy Sets and Systems,1999,101(1):137-141.
[41]Wu Chong, Zhang Deli, Guo Caimei. Fuzzy number fuzzy measures and fuzzy-valued integrals(Ⅰ)[J]. Fuzzy Sets and Systems,1998,98(3):355-360.
[42]Gong Zengtai. A New Arithmetic for Fuzzy Numbers and Its Applications[J].模糊系统与数学,2008,22:101-108.
[43]孔芳弟.无穷区间上可积函数列逐项积分的条件(J).西北师范大学学报(自然科学版),2003,39(3)：31-32.
[44]Bi Shujuan,Wu Congxin. The operation property of fuzzy number and the limit and the distance of fuzzy numbers[J]. Fuzzy Systems and Mathematics,2000,14:40-45.
[45]Ma Ming. A new fuzzy arithmetic[J]. Fuzzy Sets and Systems,1999,108:83-90.
[46]Wu Congxin, Ma Ming. On embedding problem of fuzzy number spaces[J]. Fuzzy Sets and Systems,1991,44:33-38.
[47]Wu Congxin, Wu Chong. The supremum and infimum of the set of fuzzy numbers and its applications[J]. J. Math. Anal. Appl.,1997,210:499-511.
[48]汪玲,巩增泰.模糊Henstock积分[J].兰州理工大学学报,2006,32：133-137.
[49]Wu Congxin, Zhang Bokan. A note on the embedding problem for multidimensional fuzzy number spaces[J]. Fuzzy Sets and Systems,2001,121:359-362.
[50]巩增泰,郭元伟.区间值和模糊数值函数的Choquet积分[J].兰州大学学报,2009,45(4)：112-117