几类概周期型差分方程的解及应用
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摘要
概周期型函数理论在微分方程的应用这一研究课题备受数学工作者关注。很多数学研究者在讨论具有逐段常变量的微分方程的概周期型解的存在时,通常利用相应的差分方程的概周期型序列解。所以研究差分方程的概周期型序列解的存在性是很有必要的。
本文主要讨论三类不同的差分方程的概周期型序列解,并把其中一个方程的遥远概周期序列解的相关结果应用到一类具有逐段常变量微分方程中,考虑其遥远概周期解的唯一存在性。
主要内容如下:
第一部分讨论了一类离散的Logistic差分方程的渐近概周期序列解的存在性。在已有的结果上,对条件做了一些变动,得到了一些新的结果。
第二部分给出了一类常系数的差分方程的渐近概周期序列解存在的充要条件。这一结果的证明是从矩阵的角度来考虑,通过把N阶矩阵转化为一阶来分析问题。
第三部分讨论了一类差分方程的遥远概周期序列解的唯一存在性。并把所得到的遥远概周期序列解的相关结果应用到一类逐段常变量微分方程中,得到其遥远概周期解的唯一存在性。
本文第一部分和第二部分的结果是已有结果的进一步推广。讨论遥远概周期函数在微分方程中应用的文献不是很多,所以说第三部分的结论是较新的,这对解决其它一些问题有着一定的应用价值。
The topic that theories of almost periodic type functions are applied into differential equations has attracted more and more attention among mathematical staffs. The existence of almost periodic type solutions to a differential equation with piecewise constant argument is discussed by mathematical staffs using almost periodic type sequence solutions of the corresponding difference equation. Therefore, studying the existence of almost periodic type solutions is necessary for difference equations.
In this article, almost periodic type sequences solutions for three different types of difference equations are researched and the relevant results of remotely almost periodic sequence solution are applied into a differential equation, the existence and uniqueness of remotely almost periodic solution are gotten.
Primary contents are as follows:
In the first part of paper, existence of asymptotically almost periodic sequence solution for the discrete Logistic differential equation is discussed. In the previous results, some changes on their conditions are made and some new results are obtained.
In the second part of writings, the necessary and sufficient conditions of asymptotically almost periodic solution to the difference equation with constant coefficient are given. Results of the conclusion are proved by thinking the N-order matrix into a first-order.
In the third part of text, the existence and uniqueness of remotely almost periodic sequence solution for a class of differential equations are discussed. The relevant results of remotely almost periodic sequence solution are applied into the differential equation and the existence and uniqueness of remotely almost periodic solution are gotten.
The results of first part and second part are the further extension on the existing results. The literatures on remotely almost periodic solutions in differential equations are not many. So, the results of the third part are relatively new and it has great value to solve the practical problems.
本文主要讨论三类不同的差分方程的概周期型序列解,并把其中一个方程的遥远概周期序列解的相关结果应用到一类具有逐段常变量微分方程中,考虑其遥远概周期解的唯一存在性。
主要内容如下:
第一部分讨论了一类离散的Logistic差分方程的渐近概周期序列解的存在性。在已有的结果上,对条件做了一些变动,得到了一些新的结果。
第二部分给出了一类常系数的差分方程的渐近概周期序列解存在的充要条件。这一结果的证明是从矩阵的角度来考虑,通过把N阶矩阵转化为一阶来分析问题。
第三部分讨论了一类差分方程的遥远概周期序列解的唯一存在性。并把所得到的遥远概周期序列解的相关结果应用到一类逐段常变量微分方程中,得到其遥远概周期解的唯一存在性。
本文第一部分和第二部分的结果是已有结果的进一步推广。讨论遥远概周期函数在微分方程中应用的文献不是很多,所以说第三部分的结论是较新的,这对解决其它一些问题有着一定的应用价值。
The topic that theories of almost periodic type functions are applied into differential equations has attracted more and more attention among mathematical staffs. The existence of almost periodic type solutions to a differential equation with piecewise constant argument is discussed by mathematical staffs using almost periodic type sequence solutions of the corresponding difference equation. Therefore, studying the existence of almost periodic type solutions is necessary for difference equations.
In this article, almost periodic type sequences solutions for three different types of difference equations are researched and the relevant results of remotely almost periodic sequence solution are applied into a differential equation, the existence and uniqueness of remotely almost periodic solution are gotten.
Primary contents are as follows:
In the first part of paper, existence of asymptotically almost periodic sequence solution for the discrete Logistic differential equation is discussed. In the previous results, some changes on their conditions are made and some new results are obtained.
In the second part of writings, the necessary and sufficient conditions of asymptotically almost periodic solution to the difference equation with constant coefficient are given. Results of the conclusion are proved by thinking the N-order matrix into a first-order.
In the third part of text, the existence and uniqueness of remotely almost periodic sequence solution for a class of differential equations are discussed. The relevant results of remotely almost periodic sequence solution are applied into the differential equation and the existence and uniqueness of remotely almost periodic solution are gotten.
The results of first part and second part are the further extension on the existing results. The literatures on remotely almost periodic solutions in differential equations are not many. So, the results of the third part are relatively new and it has great value to solve the practical problems.
引文
[1] BOHR H. Zur Theorie Der Fastperiodischen[J]. Acta Math, 1925, 45: 19-127.
[2] BOCHNER S, NEUMANN V J. Almost Periodic Functions in Groups I[J]. Trans. Amer. Math. Soc, 1933, 21: 101-120.
[3] NEUMANN V J. Almost Periodic Functions in A GroupI[J]. Trans. Amer. Math. Soc, 1934, 36(3): 445-492.
[4] FRECHET M. Les Fonctions Asymptotiquement Presque-periodiques Contiues [J]. C. R. Acad. Sci. Paris, 1941, 213: 520-522.
[5] FRECHET M. Les Fonctions Asymptotiquement Presque-periodiques[J]. Revue. Sci, 1942, 79: 341-351.
[6] EBERLEIN W F. Abstract Ergodic Theorems and Weak Almost Periodic Functions[J]. Trans. Amer. Math. Soc, 1949, 67: 217-240.
[7] SARASON D. Toeplitz Operators with Semi-almost Periodic Symbols[J]. Duke. Math. J, 1997, 44(2): 357-364.
[8] SARSON D.Remotely Almost Periodic Function[J]. Contemporary Mathema- tics, 1984, 32: 237-242.
[9] ZHANG C. Pseudo Almost Periodic Functions and Their Applications[D]. Canada: University of Western Ontario, 1992: 1-50.
[10] ZHANG C. Almost Periodic Type Functions and Ergodicity[M]. Beijing: Science Press, 2003: 1-86.
[11] MEISTERS G H. On Almost Periodic Solutions of a Class of Differential Equations[J]. Proc. Amer. Math. Soc, 1959, 10: 113-119.
[12] FINK A M. Almost Periodic Differential Functions[M]. Berlin: Spring Verl- ag, 1974: 115-187.
[13] GOPALSAMY K, MHAMAD S. Canonical Solutions and Almost Periodicity in A Discrete Logistic Equation[J]. Applied Mathematics and Computation, 2000, 113: 305-323.
[14] PIAO D. Pseudo Almost Periodic Solutions of the Systems of Differential Equations with Piecewise Constant Argument[t][J]. Science in China (Series A), 2001, 44(9):1156-1161.
[15]杨喜陶.一类具有无穷时滞泛函微分方程概周期解的存在性与稳定性[J].湘潭矿业学院学报, 2001, 16(2): 78-81.
[16] YUAN R. Pseudo Almost Periodic Solutions of Second-Order Neutral Delay Differential Equations with Piecewise Constant Argument[J]. Nonlinear Anlysis, 2000, 41: 871-890.
[17]冯春华.一类时滞微分方程的概周期解[J].广西科学, 2004, 11(4): 286-288.
[18]魏凤英,王克.渐近周期函数的Logistic方程[J].生物数学学报, 2005, 20(4): 395-405.
[19]张敬信,姚慧丽.一类差分方程的渐近概周期序列解的存在性[J].黑龙江大学自然科学学报, 2007, 24(6): 81-82.
[20]朴大雄,辛娜.受迫摆方程的伪概周期解[J].中国海洋大学学报, 2007, 37(4): 573-575.
[21]郭莹.常差分方程组的概周期解[J].中国海洋大学学报, 2007, 37: 245-248.
[22]张克玉.离散Logistic方程的伪概周期解[J].井冈山学院学报, 2007, 28(2): 25-26.
[23]江利娜,张传义.带逐段常变量微分方程的伪概周期解[J].应用泛函分析学报, 2007, 9(1): 56-62.
[24] ZHANG C, JIANG L. Remotely Almost Periodic Solutions to Systems of Diffe- rential Equations with Piecewise Constant Argument[J]. Applied Mathematics Letters, 2008, 21: 761-768.
[25]姚慧丽,徐姗姗.带逐段常变量微分方程的渐近概周期解[J].哈尔滨理工大学学报, 2009, 14(4): 95-98.
[26]殷红红,姚慧丽.一类具有逐段常变量微分方程的渐近概周期解[J].哈尔滨商业大学学报, 2008, 24(1): 125-128.
[27]张克玉.二阶中立型逐段常变量微分方程的伪周期解[J].山东教育学院学报, 2009, 131(1): 65-68.
[28]姚慧丽.带逐段常变量微分方程的渐近概周期解[J].哈尔滨理工大学, 2009, 14(4): 95-98.
[29]赵丽娟,姚慧丽.二阶含逐段常变量微分方程的渐近概周期解[J].黑龙江自然科学学报, 2009, 26(6): 749-752.
[30] LI Z, CHEN F. Almost Periodic Solutions of Discrete Almost periodic Logistic Equations[J]. Mathematical and Computer Modelling, 2009, 50: 245-259.
[31]牛晶,陈晓航.一类具有延迟常变量微分方程的渐近概周期解的存在性[J].哈尔滨师范大学学报, 2009, 25(2): 50-52.
[32]王丽,张传义.带逐段常变量二阶中立型延迟微分方程的概周期解[J].数学学报, 2010, 53(2): 227-232.
[33]祝文壮,冀书关.一类二阶微分方程概周期解的存在性[J].吉林大学学报, 2007, 45(1): 5-10.
[34]郭尊光,李灿.一类积分方程的向量值遥远概周期解[J].数学实践与认识, 2009, 39(10): 253-256.
[35] ZHOU H, PIAO D. Pseudo Almost Periodic Solutions of Nonlinear Hyperbo- olic Equations with Piecewise Constant Argument[J]. Northeast Math, 2007, 26(6): 491-504.
[36]张克玉.二阶中立型逐段常变量微分方程的伪周期解[J].山东教育学院学报, 2009, 1(131): 65-68.
[37] YANG X. The Unique Existence of Almost Periodic Solutions for System of Differential Equations with Piecewise Constant Argument[J]. Acta Math Sini- ca, 2007, 50(2): 461-472.
[38]杨枫林.抛物型微分方程反问题中的概周期型和遥远概周期型解[D].哈尔滨:哈尔滨工业大学, 2006(6): 51-54.
[39]张振宇.遥远概周期函数及微分方程的解[D].黑龙江:哈尔滨工业大学, 2006(6): 14-24.
[40]江利娜.概周期型函数和概周期型微分方程的解[D].黑龙江:哈尔滨工业大学, 2007(9): 11-33.
[41]王清. Logistic方程的伪概周期解[J].中国科技信息, 2008, 8: 34-35.
[42]王寿松.单种群生长的Logistic模型[J].生物科学学报, 1990, 5(1): 21-25.
[43]杨子胥.高等代数习题集解下册[M].山东:山东科学技术出版社, 2004: 441-442.
[44] YUAN R, HONG J. Almost Periodic Solutions of Differential Equations with Piecewise Constant Argument[J]. Analysis, 1996, 16(2): 171-180.
[45]姚慧丽,牛晶.一类具有逐段常变量微分方程的渐近概周期解[J].应用泛函分析学报, 2008, 10(2): 150-154.
[46]牛晶,王似龙.逐段常变量微分方程的渐近概周期解[J].黑龙江自然科学学报, 2010, 27(3): 319-322.
[47] YUAN R, PIAO D. Pseudo Almost Periodic Solutions of Differential Equati- ons with Piecewise Constant Argument[J]. China of Math, 1999, 4: 489-494.
[2] BOCHNER S, NEUMANN V J. Almost Periodic Functions in Groups I[J]. Trans. Amer. Math. Soc, 1933, 21: 101-120.
[3] NEUMANN V J. Almost Periodic Functions in A GroupI[J]. Trans. Amer. Math. Soc, 1934, 36(3): 445-492.
[4] FRECHET M. Les Fonctions Asymptotiquement Presque-periodiques Contiues [J]. C. R. Acad. Sci. Paris, 1941, 213: 520-522.
[5] FRECHET M. Les Fonctions Asymptotiquement Presque-periodiques[J]. Revue. Sci, 1942, 79: 341-351.
[6] EBERLEIN W F. Abstract Ergodic Theorems and Weak Almost Periodic Functions[J]. Trans. Amer. Math. Soc, 1949, 67: 217-240.
[7] SARASON D. Toeplitz Operators with Semi-almost Periodic Symbols[J]. Duke. Math. J, 1997, 44(2): 357-364.
[8] SARSON D.Remotely Almost Periodic Function[J]. Contemporary Mathema- tics, 1984, 32: 237-242.
[9] ZHANG C. Pseudo Almost Periodic Functions and Their Applications[D]. Canada: University of Western Ontario, 1992: 1-50.
[10] ZHANG C. Almost Periodic Type Functions and Ergodicity[M]. Beijing: Science Press, 2003: 1-86.
[11] MEISTERS G H. On Almost Periodic Solutions of a Class of Differential Equations[J]. Proc. Amer. Math. Soc, 1959, 10: 113-119.
[12] FINK A M. Almost Periodic Differential Functions[M]. Berlin: Spring Verl- ag, 1974: 115-187.
[13] GOPALSAMY K, MHAMAD S. Canonical Solutions and Almost Periodicity in A Discrete Logistic Equation[J]. Applied Mathematics and Computation, 2000, 113: 305-323.
[14] PIAO D. Pseudo Almost Periodic Solutions of the Systems of Differential Equations with Piecewise Constant Argument[t][J]. Science in China (Series A), 2001, 44(9):1156-1161.
[15]杨喜陶.一类具有无穷时滞泛函微分方程概周期解的存在性与稳定性[J].湘潭矿业学院学报, 2001, 16(2): 78-81.
[16] YUAN R. Pseudo Almost Periodic Solutions of Second-Order Neutral Delay Differential Equations with Piecewise Constant Argument[J]. Nonlinear Anlysis, 2000, 41: 871-890.
[17]冯春华.一类时滞微分方程的概周期解[J].广西科学, 2004, 11(4): 286-288.
[18]魏凤英,王克.渐近周期函数的Logistic方程[J].生物数学学报, 2005, 20(4): 395-405.
[19]张敬信,姚慧丽.一类差分方程的渐近概周期序列解的存在性[J].黑龙江大学自然科学学报, 2007, 24(6): 81-82.
[20]朴大雄,辛娜.受迫摆方程的伪概周期解[J].中国海洋大学学报, 2007, 37(4): 573-575.
[21]郭莹.常差分方程组的概周期解[J].中国海洋大学学报, 2007, 37: 245-248.
[22]张克玉.离散Logistic方程的伪概周期解[J].井冈山学院学报, 2007, 28(2): 25-26.
[23]江利娜,张传义.带逐段常变量微分方程的伪概周期解[J].应用泛函分析学报, 2007, 9(1): 56-62.
[24] ZHANG C, JIANG L. Remotely Almost Periodic Solutions to Systems of Diffe- rential Equations with Piecewise Constant Argument[J]. Applied Mathematics Letters, 2008, 21: 761-768.
[25]姚慧丽,徐姗姗.带逐段常变量微分方程的渐近概周期解[J].哈尔滨理工大学学报, 2009, 14(4): 95-98.
[26]殷红红,姚慧丽.一类具有逐段常变量微分方程的渐近概周期解[J].哈尔滨商业大学学报, 2008, 24(1): 125-128.
[27]张克玉.二阶中立型逐段常变量微分方程的伪周期解[J].山东教育学院学报, 2009, 131(1): 65-68.
[28]姚慧丽.带逐段常变量微分方程的渐近概周期解[J].哈尔滨理工大学, 2009, 14(4): 95-98.
[29]赵丽娟,姚慧丽.二阶含逐段常变量微分方程的渐近概周期解[J].黑龙江自然科学学报, 2009, 26(6): 749-752.
[30] LI Z, CHEN F. Almost Periodic Solutions of Discrete Almost periodic Logistic Equations[J]. Mathematical and Computer Modelling, 2009, 50: 245-259.
[31]牛晶,陈晓航.一类具有延迟常变量微分方程的渐近概周期解的存在性[J].哈尔滨师范大学学报, 2009, 25(2): 50-52.
[32]王丽,张传义.带逐段常变量二阶中立型延迟微分方程的概周期解[J].数学学报, 2010, 53(2): 227-232.
[33]祝文壮,冀书关.一类二阶微分方程概周期解的存在性[J].吉林大学学报, 2007, 45(1): 5-10.
[34]郭尊光,李灿.一类积分方程的向量值遥远概周期解[J].数学实践与认识, 2009, 39(10): 253-256.
[35] ZHOU H, PIAO D. Pseudo Almost Periodic Solutions of Nonlinear Hyperbo- olic Equations with Piecewise Constant Argument[J]. Northeast Math, 2007, 26(6): 491-504.
[36]张克玉.二阶中立型逐段常变量微分方程的伪周期解[J].山东教育学院学报, 2009, 1(131): 65-68.
[37] YANG X. The Unique Existence of Almost Periodic Solutions for System of Differential Equations with Piecewise Constant Argument[J]. Acta Math Sini- ca, 2007, 50(2): 461-472.
[38]杨枫林.抛物型微分方程反问题中的概周期型和遥远概周期型解[D].哈尔滨:哈尔滨工业大学, 2006(6): 51-54.
[39]张振宇.遥远概周期函数及微分方程的解[D].黑龙江:哈尔滨工业大学, 2006(6): 14-24.
[40]江利娜.概周期型函数和概周期型微分方程的解[D].黑龙江:哈尔滨工业大学, 2007(9): 11-33.
[41]王清. Logistic方程的伪概周期解[J].中国科技信息, 2008, 8: 34-35.
[42]王寿松.单种群生长的Logistic模型[J].生物科学学报, 1990, 5(1): 21-25.
[43]杨子胥.高等代数习题集解下册[M].山东:山东科学技术出版社, 2004: 441-442.
[44] YUAN R, HONG J. Almost Periodic Solutions of Differential Equations with Piecewise Constant Argument[J]. Analysis, 1996, 16(2): 171-180.
[45]姚慧丽,牛晶.一类具有逐段常变量微分方程的渐近概周期解[J].应用泛函分析学报, 2008, 10(2): 150-154.
[46]牛晶,王似龙.逐段常变量微分方程的渐近概周期解[J].黑龙江自然科学学报, 2010, 27(3): 319-322.
[47] YUAN R, PIAO D. Pseudo Almost Periodic Solutions of Differential Equati- ons with Piecewise Constant Argument[J]. China of Math, 1999, 4: 489-494.