几类微分方程的概周期型解的存在性和唯一性
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摘要
众所周知,在讨论具有逐段常变量微分方程的概周期型解时,往往要用到相关差分方程的概周期型序列解。特别是近几年来,越来越多的数学工作者开始研究差分方程的各种解的存在性和唯一性问题,并把差分方程和微分方程结合起来,从而进一步来研究微分方程概周期型解的存在性和唯一性问题。还有一些是利用不动点理论来讨论某些方程的各种解的存在性的,并取得了很大的进展。
本文主要讨论了两类微分方程的渐近概周期解的存在性和唯一性。
具体包括以下内容:
第一部分是通过微分方程解的连续性构造一个差分方程,随后结合指数二分并利用差分方程的渐近概周期序列解,讨论了一类具有逐段常变量微分方程的渐近概周期解的存在性。
第二部分是利用指数型二分性定理和不动点定理,建立了一些保证一类具有有限时滞的微分方程有渐近概周期解的充分条件。
本文所得的结果或是对已有结果的推广,从而使得相关结论应用更加广泛,或是对已有问题所给的条件作出调整和变化。这可以为其他问题的研究提供一些方法和借鉴。
As well known, when the almost periodic type solutions of differential equations with piecewise constant argument are discussed, the almost periodic type sequence solutions of relevant difference equations are usually used. Especially in recent years, more and more mathematical workers began to study the existence and uniqueness of almost periodic type sequence solutions for difference equations, and the existence and uniqueness of almost periodic type solutions for some differential equations by joining difference equations and differential equations together. Moreover, the existence of various solutions for certain equations is discussed by using fixed point theory, and great progress is made.
The existence and uniqueness of asympototically almost periodic solutions for two types differential equations are mainly discussed in this paper.
Specific contents are as follows:
A difference equation is constructed through the continuity of solutions of differential equations in the first part after this, the existence of asympototically almost periodic solutions for a class of differential equation with piecewise constant is discussed using asympototically almost periodic sequence solutions of relevant difference equations combinating exponential dichotomy.
The exponential dichotomy theory and fixed point theory are used in the second part, and established the full conditions that assured a class of differential equations with limited delay exist the asympototically almost periodic solutions
Some of the results in the paper are promotion of the well-known results, thus the relevant conclusions will be more extensively applied. The others are obtained by changing the conditions of previous results, these methods can provide some reference for other problems.
本文主要讨论了两类微分方程的渐近概周期解的存在性和唯一性。
具体包括以下内容:
第一部分是通过微分方程解的连续性构造一个差分方程,随后结合指数二分并利用差分方程的渐近概周期序列解,讨论了一类具有逐段常变量微分方程的渐近概周期解的存在性。
第二部分是利用指数型二分性定理和不动点定理,建立了一些保证一类具有有限时滞的微分方程有渐近概周期解的充分条件。
本文所得的结果或是对已有结果的推广,从而使得相关结论应用更加广泛,或是对已有问题所给的条件作出调整和变化。这可以为其他问题的研究提供一些方法和借鉴。
As well known, when the almost periodic type solutions of differential equations with piecewise constant argument are discussed, the almost periodic type sequence solutions of relevant difference equations are usually used. Especially in recent years, more and more mathematical workers began to study the existence and uniqueness of almost periodic type sequence solutions for difference equations, and the existence and uniqueness of almost periodic type solutions for some differential equations by joining difference equations and differential equations together. Moreover, the existence of various solutions for certain equations is discussed by using fixed point theory, and great progress is made.
The existence and uniqueness of asympototically almost periodic solutions for two types differential equations are mainly discussed in this paper.
Specific contents are as follows:
A difference equation is constructed through the continuity of solutions of differential equations in the first part after this, the existence of asympototically almost periodic solutions for a class of differential equation with piecewise constant is discussed using asympototically almost periodic sequence solutions of relevant difference equations combinating exponential dichotomy.
The exponential dichotomy theory and fixed point theory are used in the second part, and established the full conditions that assured a class of differential equations with limited delay exist the asympototically almost periodic solutions
Some of the results in the paper are promotion of the well-known results, thus the relevant conclusions will be more extensively applied. The others are obtained by changing the conditions of previous results, these methods can provide some reference for other problems.
引文
[1] BOHR H. Zur Theorie Der Fastperiodischen[J]. Acta Math, 1925, 45: 19-127.
[2] BOCHNER S, NEUMANN J VON. Almost Periodic Functions in Groups I[J]. TrAns. Math. Soc, 1933, 21: 101-120.
[3] CORDUNEANU C. Almost Periodic Functions[M]. New York: Chelsea, 1968: 30-50.
[4] ZHANG C. Pseudo Almost Periodic Functions and their Applications[M]. University of Western Ontario: Ph. D. Thesis, 1992: 2-5.
[5] ZHANG C. Almost Periodic Type Functions and Ergodicity[M]. Beijing: Science Press, 2003: 1-86, 174-179.
[6] WIENER J, COOK K L. Retarted Differential equations with Piecewise Constant Delays[J]. Math. Anal. Appl., 1984, 99(2): 265-297.
[7] YUAN R, PIAO D. Pseudo Almost Periodic Solutions of Diffe- rential Equations with Piecewise Constant Argument[J]. Chin. Ann. of Math, 1999, 4: 489-494.
[8]朴大雄.带逐段常变量的微分方程的概周期解[J].数学学报, 1999, 42: 749-756.
[9]姚慧丽,张传义,吴丛炘.一类具有逐段常变量微分方程的渐近概周期解的存在性[J].哈尔滨师范大学自然科学学报, 2000: 16(6): 1-6.
[10]朴大雄.带逐段常变量[t + 1/ 2]的微分方程组的伪概周期解[J].中国科学, 2003, 33(3): 220-225.
[11]杨喜陶.逐段常变量微分方程组概周期型解存在唯一性[J].数学学报, 2007, 50(4): 461-472.
[12]江利娜.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.
[13] YUAN R, HONG J. The Existence of Almost Periodic Solution for A Class of Differential Equations with Piecewise Constant Argument[J]. Nonl- inear Analysis, 1997, 28(8): 1439-1450.
[14] HONG J, RAFAEL OBAYA, ANA SANZ. Almost Periodic Type Solutions of Some Differential Equations with Piecewise Constant Argument[J]. Nonlinear Analysis, 1997, 45: 661-688.
[15]王克,黄启昌. C h空间与具有无穷时滞的泛函微分方程的有界性和周期解[J].中国科学, 1987, 17A(3): 242-252.
[16]王全义.概周期解的存在性、唯一性与稳定性[J].数学学刊, 1997, 40(1); 80-89.
[17]杨喜陶,冯春华.一类具有无穷时滞的中立型Volterra积分微分方程的概周期解的存在唯一性[J].数学学报, 1997, 40(3): 359-402.
[18]王全义.一类中立型泛函微分方程的概周期解的存在性与稳性[J].华侨大学学报, 2002, 23(3): 222.
[19]王全义.具有无穷时滞的积分微分方程解的存在性唯一性及稳定性[J].应用数学学报, 1998, 21(2): 312-318.
[20]石磊.具有无穷时滞中立型泛函微分方程的有界性及周期解[J].科学通报, 1990, 35(6): 409-411.
[21]彭世国,朱思铭.具有无穷时滞泛函微分方程的周期解[J].数学年刊, 2002, 23A(3): 371-380.
[22]唐扬斌.中立型高维非自治系统的周期解.应用数学学报[J], 2000, 23(3): 321-328.
[23]王全义.中立型泛函微分方程的概周期解的存在唯一性[J].华侨大学学报,2005, 26(6):117-121.
[24]王全义.具时滞的中立型泛函微分方程的概周期解[J].华侨大学学报, 2004, 25(3): 247-250.
[25]杨淑芳,王鑫.二阶中立型含逐段常滞量微分方程的伪概周期解的存在性[J].国防科技大学学报, 2005, 27(3): 120-124.
[26]姚慧丽.具逐段常变量的二阶中立型微分方程渐近概周期解存在性[J].哈尔滨理工大学学报, 2007, 12(3): 103-106.
[27]王幼斌.带有逐段常变量的一阶中立型泛函微分方程的全局吸收性[J].山西大学学报, 2002, 25(1): 6-9.
[28]王鑫,贺明科.二阶中立型含逐段常变量微分方程的概周期解[J].吉首大学(自然科学版), 2004, 25(1): 7-12.
[29]袁荣.具逐段常变量中立型时滞微分方程的概周期解[J].数学年刊, 1998, 19(4): 499-506.
[30] HONG X. Almost Periodic Weak Solutions of Neutral Dealy Differential Equations with Piecewise Constant Argument[J]. Nonlinear Analysis, 2006, (4): 530-545.
[31]杨喜陶.一类具有逐段常变量神经网络系统概周期解存在性与指数稳定性[J].应用数学学报, 2006, 29(5): 789-800.
[32]杨喜陶,朱焕然.具有无穷时滞的生态竞争系统概周期解的存在性[J].生物数学学报, 2006, 20(4): 424-428.
[33] YANG X. The Unique Existence of Almost Periodic Solutions for Syst- em of Differential Equations with Piecewise Constant Argument[J]. ACTA Math Sinica, 2007, 50(2): 461-472.
[34]魏喆,张传义.遍历序列在动力系统平均方法定量研究中的应用[J].应用泛函分析学报, 2006, 8(1): 90-96.
[35] MAN N, MINH N. On the Existence of Quasi Periodic Solutions of Neutral Differential Equation[J]. Pure Appl. Anal, 2004, 3(2): 291-300.
[36]陈飞,张永红.一类两种微生物混合培养系统的概周期解[J].长春大学学报, 2006, 16(2): 10-16.
[37]郭大钧.传染病模型的非线性积分方程正解的个数[J].数学年刊, 1987, 8A(1): 79-87.
[38]马如云.传染病模型的非线性积分方程非零解的唯一性[J].生物数学学报, 1990, 5(2): 97-100.
[39]姚慧丽,张传义,一类非线性延迟积分方程渐近概周期解的存在性[J].哈尔滨理工大学学报, 2001, 6(1): 92-95.
[40]姚慧丽,张传义,一类非线性延迟积分方程正的概周期型解的存在性[J].数学学报, 2004, 47(2): 279-284.
[41]刘永建,冯春华.一类线性积分方程的概周期解[J].广西科学, 2006, 13(4): 264-266.
[42]方聪娜.中立性Volterra积分微分方程概周期解的存在唯一性[J].集美大学学报, 2009, 14(2): 47-51.
[43]姚慧丽.一类非线性延迟积分方程概周期型解的存在性[J].吉首大学学报. 2009, 30(1): 9-12.
[44]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.
[45] AIT DADS, EL HADI, CIEUTAT, PLILIPPE, LHACHIMI, LAHCEN. Positive pseudo almost periodic solutions for some nonlinear infinite delay integral equations[J]. Mathe-Matical &Computer Modelling, 2009, 49(19): 721-739.
[46]张传义,杨枫林.抛物型边值反问题中的伪概周期解[J].中国科学(A辑), 2008, 38(1): 41-52.
[47] YANG X. The Existence and Uniqueness of Positive Periodic Solutions for Certain Differential Equations with Piecewise Constant Argument[J]. Acta Math Appl Sinica, 2006, 29(6): 984-994.
[48] GUO J,YUAN R. Pseudo Almost Periodic Solutions of A Sing- ularly Perturbed Differential Equation with Piecewise Constant Argument[J]. Acta Math, 2007, 23(3): 423-438.
[49] ZHOU H, PIAO D. Pseudo Almost Periodic Solutions of Nonlinear Hyperbolic Equations with Piecewise Constant Argument[J]. North- east Math, 2007, 26(6): 491-504.
[50] XIA Y, HUANG Z, HAN M. Existence of Almost Periodic Sol- utions for Forced Perturbed Systems with Piecewise Constant Argument[J]. J.Math. Anal. Appl. , 2007, 30(6): 1-19.
[2] BOCHNER S, NEUMANN J VON. Almost Periodic Functions in Groups I[J]. TrAns. Math. Soc, 1933, 21: 101-120.
[3] CORDUNEANU C. Almost Periodic Functions[M]. New York: Chelsea, 1968: 30-50.
[4] ZHANG C. Pseudo Almost Periodic Functions and their Applications[M]. University of Western Ontario: Ph. D. Thesis, 1992: 2-5.
[5] ZHANG C. Almost Periodic Type Functions and Ergodicity[M]. Beijing: Science Press, 2003: 1-86, 174-179.
[6] WIENER J, COOK K L. Retarted Differential equations with Piecewise Constant Delays[J]. Math. Anal. Appl., 1984, 99(2): 265-297.
[7] YUAN R, PIAO D. Pseudo Almost Periodic Solutions of Diffe- rential Equations with Piecewise Constant Argument[J]. Chin. Ann. of Math, 1999, 4: 489-494.
[8]朴大雄.带逐段常变量的微分方程的概周期解[J].数学学报, 1999, 42: 749-756.
[9]姚慧丽,张传义,吴丛炘.一类具有逐段常变量微分方程的渐近概周期解的存在性[J].哈尔滨师范大学自然科学学报, 2000: 16(6): 1-6.
[10]朴大雄.带逐段常变量[t + 1/ 2]的微分方程组的伪概周期解[J].中国科学, 2003, 33(3): 220-225.
[11]杨喜陶.逐段常变量微分方程组概周期型解存在唯一性[J].数学学报, 2007, 50(4): 461-472.
[12]江利娜.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.
[13] YUAN R, HONG J. The Existence of Almost Periodic Solution for A Class of Differential Equations with Piecewise Constant Argument[J]. Nonl- inear Analysis, 1997, 28(8): 1439-1450.
[14] HONG J, RAFAEL OBAYA, ANA SANZ. Almost Periodic Type Solutions of Some Differential Equations with Piecewise Constant Argument[J]. Nonlinear Analysis, 1997, 45: 661-688.
[15]王克,黄启昌. C h空间与具有无穷时滞的泛函微分方程的有界性和周期解[J].中国科学, 1987, 17A(3): 242-252.
[16]王全义.概周期解的存在性、唯一性与稳定性[J].数学学刊, 1997, 40(1); 80-89.
[17]杨喜陶,冯春华.一类具有无穷时滞的中立型Volterra积分微分方程的概周期解的存在唯一性[J].数学学报, 1997, 40(3): 359-402.
[18]王全义.一类中立型泛函微分方程的概周期解的存在性与稳性[J].华侨大学学报, 2002, 23(3): 222.
[19]王全义.具有无穷时滞的积分微分方程解的存在性唯一性及稳定性[J].应用数学学报, 1998, 21(2): 312-318.
[20]石磊.具有无穷时滞中立型泛函微分方程的有界性及周期解[J].科学通报, 1990, 35(6): 409-411.
[21]彭世国,朱思铭.具有无穷时滞泛函微分方程的周期解[J].数学年刊, 2002, 23A(3): 371-380.
[22]唐扬斌.中立型高维非自治系统的周期解.应用数学学报[J], 2000, 23(3): 321-328.
[23]王全义.中立型泛函微分方程的概周期解的存在唯一性[J].华侨大学学报,2005, 26(6):117-121.
[24]王全义.具时滞的中立型泛函微分方程的概周期解[J].华侨大学学报, 2004, 25(3): 247-250.
[25]杨淑芳,王鑫.二阶中立型含逐段常滞量微分方程的伪概周期解的存在性[J].国防科技大学学报, 2005, 27(3): 120-124.
[26]姚慧丽.具逐段常变量的二阶中立型微分方程渐近概周期解存在性[J].哈尔滨理工大学学报, 2007, 12(3): 103-106.
[27]王幼斌.带有逐段常变量的一阶中立型泛函微分方程的全局吸收性[J].山西大学学报, 2002, 25(1): 6-9.
[28]王鑫,贺明科.二阶中立型含逐段常变量微分方程的概周期解[J].吉首大学(自然科学版), 2004, 25(1): 7-12.
[29]袁荣.具逐段常变量中立型时滞微分方程的概周期解[J].数学年刊, 1998, 19(4): 499-506.
[30] HONG X. Almost Periodic Weak Solutions of Neutral Dealy Differential Equations with Piecewise Constant Argument[J]. Nonlinear Analysis, 2006, (4): 530-545.
[31]杨喜陶.一类具有逐段常变量神经网络系统概周期解存在性与指数稳定性[J].应用数学学报, 2006, 29(5): 789-800.
[32]杨喜陶,朱焕然.具有无穷时滞的生态竞争系统概周期解的存在性[J].生物数学学报, 2006, 20(4): 424-428.
[33] YANG X. The Unique Existence of Almost Periodic Solutions for Syst- em of Differential Equations with Piecewise Constant Argument[J]. ACTA Math Sinica, 2007, 50(2): 461-472.
[34]魏喆,张传义.遍历序列在动力系统平均方法定量研究中的应用[J].应用泛函分析学报, 2006, 8(1): 90-96.
[35] MAN N, MINH N. On the Existence of Quasi Periodic Solutions of Neutral Differential Equation[J]. Pure Appl. Anal, 2004, 3(2): 291-300.
[36]陈飞,张永红.一类两种微生物混合培养系统的概周期解[J].长春大学学报, 2006, 16(2): 10-16.
[37]郭大钧.传染病模型的非线性积分方程正解的个数[J].数学年刊, 1987, 8A(1): 79-87.
[38]马如云.传染病模型的非线性积分方程非零解的唯一性[J].生物数学学报, 1990, 5(2): 97-100.
[39]姚慧丽,张传义,一类非线性延迟积分方程渐近概周期解的存在性[J].哈尔滨理工大学学报, 2001, 6(1): 92-95.
[40]姚慧丽,张传义,一类非线性延迟积分方程正的概周期型解的存在性[J].数学学报, 2004, 47(2): 279-284.
[41]刘永建,冯春华.一类线性积分方程的概周期解[J].广西科学, 2006, 13(4): 264-266.
[42]方聪娜.中立性Volterra积分微分方程概周期解的存在唯一性[J].集美大学学报, 2009, 14(2): 47-51.
[43]姚慧丽.一类非线性延迟积分方程概周期型解的存在性[J].吉首大学学报. 2009, 30(1): 9-12.
[44]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.
[45] AIT DADS, EL HADI, CIEUTAT, PLILIPPE, LHACHIMI, LAHCEN. Positive pseudo almost periodic solutions for some nonlinear infinite delay integral equations[J]. Mathe-Matical &Computer Modelling, 2009, 49(19): 721-739.
[46]张传义,杨枫林.抛物型边值反问题中的伪概周期解[J].中国科学(A辑), 2008, 38(1): 41-52.
[47] YANG X. The Existence and Uniqueness of Positive Periodic Solutions for Certain Differential Equations with Piecewise Constant Argument[J]. Acta Math Appl Sinica, 2006, 29(6): 984-994.
[48] GUO J,YUAN R. Pseudo Almost Periodic Solutions of A Sing- ularly Perturbed Differential Equation with Piecewise Constant Argument[J]. Acta Math, 2007, 23(3): 423-438.
[49] ZHOU H, PIAO D. Pseudo Almost Periodic Solutions of Nonlinear Hyperbolic Equations with Piecewise Constant Argument[J]. North- east Math, 2007, 26(6): 491-504.
[50] XIA Y, HUANG Z, HAN M. Existence of Almost Periodic Sol- utions for Forced Perturbed Systems with Piecewise Constant Argument[J]. J.Math. Anal. Appl. , 2007, 30(6): 1-19.