分数阶微分方程的概周期型解
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摘要
本文主要包括三部分内容:第一部分介绍概周期型函数空间的逐步扩张及相关性质;第二部分介绍了整数阶微分方程的概周期型解的相关理论;最后一部分是关于分数阶微分方程的概周期型解的相关理论。
概周期函数理论是由Danish的数学家H.Bohr于1925—1926首先提出来的,后来由S.Bochner,H.Weyl, A.Besicovitch, V.V.Stepanov等人发展了他的理论。分数阶微积分运算包括分数阶微分运算和分数阶积分运算,它的含义就是将普通意义下的微积分运算的运算阶次从整数阶推广到分数和复数的情况。在相关的文章中,作者Daniela Araya等引入了? -预解族的概念,证明了Riemann-Liouville分数阶微分方程
在适当条件下概自守适度解的存在性;之后,作者Hui-Sheng等对该文中的条件进行放宽,得到了相似的结论。
在本文中,我们利用Banach不动点定理得到了分数阶微分方程在适当条件下伪概周期解的存在唯一性,并在某种程度上推广了前人的结果。
This paper consists of three parts: the first part introduces the space of almost periodic type functions and the related nature of the gradual expansion; the second part describes the interger-order differential equations of the almost periodic solutions of relevant theories; the last part takes on the existence and uniqueness of almost periodic type solutions of abstract fractional differential equations.
Theory of almost periodic function was first proposed by Danish mathematician H. Bohr in 1925-1926, and later was developed by S. Bochner, H. Weyl, A. Besicovitch, and V.V.Stepanov. Fractional calculus includes fractional derivatives and fractional integrals. It means to generalize the differentiation and integration into fractional and complex order. In a related article, the authors Daniela Araya introduced the concept of ? -resolvent families to prove the existence of almost automorphic mild solutions to the following fractional differential equation in the sense of Riemann-Liouville. Later, the author Hui-Sheng relaxed the conditions in literature and obtained the same results.
In this paper, we utilize Banach fix-point theorem to get pseudo almost periodic mild solutions and pseudo almost automorphic mild solutions which extends the results in the paper introduced before to a certain extent under appropriate assumptions.
概周期函数理论是由Danish的数学家H.Bohr于1925—1926首先提出来的,后来由S.Bochner,H.Weyl, A.Besicovitch, V.V.Stepanov等人发展了他的理论。分数阶微积分运算包括分数阶微分运算和分数阶积分运算,它的含义就是将普通意义下的微积分运算的运算阶次从整数阶推广到分数和复数的情况。在相关的文章中,作者Daniela Araya等引入了? -预解族的概念,证明了Riemann-Liouville分数阶微分方程
在适当条件下概自守适度解的存在性;之后,作者Hui-Sheng等对该文中的条件进行放宽,得到了相似的结论。
在本文中,我们利用Banach不动点定理得到了分数阶微分方程在适当条件下伪概周期解的存在唯一性,并在某种程度上推广了前人的结果。
This paper consists of three parts: the first part introduces the space of almost periodic type functions and the related nature of the gradual expansion; the second part describes the interger-order differential equations of the almost periodic solutions of relevant theories; the last part takes on the existence and uniqueness of almost periodic type solutions of abstract fractional differential equations.
Theory of almost periodic function was first proposed by Danish mathematician H. Bohr in 1925-1926, and later was developed by S. Bochner, H. Weyl, A. Besicovitch, and V.V.Stepanov. Fractional calculus includes fractional derivatives and fractional integrals. It means to generalize the differentiation and integration into fractional and complex order. In a related article, the authors Daniela Araya introduced the concept of ? -resolvent families to prove the existence of almost automorphic mild solutions to the following fractional differential equation in the sense of Riemann-Liouville. Later, the author Hui-Sheng relaxed the conditions in literature and obtained the same results.
In this paper, we utilize Banach fix-point theorem to get pseudo almost periodic mild solutions and pseudo almost automorphic mild solutions which extends the results in the paper introduced before to a certain extent under appropriate assumptions.
引文
1 E. Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin.Dyn. Sys. 2007:277~285
2 A. M. Fink. Almost Periodic Differential Equations. Springer-Verlag, 1974:158~183
3 T. Diagana, G.M. N'Guérékata. Almost automorphic solutions to semilinear evolution equations, Funct. Differ. Equ. 2006, 13 (2) 195~206
4 J.A. Goldstein, G.M. N’Guerekata. Almost automorphic solutions of semilinear evolution equations, Proc. Amer. Math. Soc. 2005,133(8) 2401~2408
5 G.M. N'Guérékata. Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Acad. Plenum, New York, Boston, Moscow,London, 2001:68~105
6 J.Hale. Theory of Functional Differential Equations. Springer-Verlag, 1977:166~188
7 Daniela Araya, Carlos Lizama. Almost automorphic mild solutions to fractional differential equations,Nonlinear Analysis. 2008, 69:3692~3705
8 G.M. N'Guérékata. Existence and Uniqueness of Almost Automorphic Mild Solutions of Some Semilinear Abstract Differential Equations, Semigroup Forum Vol.2004, 69:80~86
9 T. Diagana, G.M. N'Guérékata, N. Van Minh. Almost automorphic solutions of evolution equations. Proc. Amer. Math. Soc. 2004, 132:3289~3298
10 Toka Diagana. Pseudo almost periodic solutions to some differential equations, Nonlinear Analysis. 2005, 60:1277~1286
11 Toka Diagana. Existence of pseudo-almost automorphic solutions to some abstract differential equations with S p-pseudo-almost automorphic coefficients. Nonlinear Analysis. 2009, 70:3781~3790
12 Claudio Cuevas, Carlos Lizama. Almost Automorphic Solutions To A Class Of Semilinear Fractional Differential Equations,2000:2038~2043
13 J. Liang, J. Zhang, T.-J. Xiao. Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 2008,340:1493~1499
14 T-J Xiao, J. Liang, J. Zhang. Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum. 2008, 76 (3):518~524
15 J. Liang, G.M. N'Guérékata, T.-J. Xiao, J. Zhang. Some properties of pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Anal. 2009, 70:2731~2735
16 E. Bazhlekova. Fractional evolution equations in Banach spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001:5~12
17 D. Bugajewski, T. Diagana. Almost automorphy of the convolution operator and applications to differential and functional differential equations, Nonlinear Stud. 2006, 13 (2):129~140
18 Hui-Sheng, Ding, Jin Liang, Ti-Jun Xiao. Almost Automorphic Solutions to Abstract Fractional Differential Equations,Hindawi Publishing Corporation, Advances in Difference Equations, Volume 2010, article ID 508374, 9 pages
19 H. S. Ding, J. Liang, and T. J. Xiao. Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces. preprint.
20 El Hadi Ait Dadsa, Philippe Cieutatb, Khalil Ezzinbia. The existence of pseudo-almost periodic solutions for some nonlinear differential equations in Banach spaces Nonlinear Analysis. 2008, 69:1325~1342
21 M. Bahaj, O. Sidki. Almost periodic solutions of semilinear equations with analytic semigroups in Banach spaces, Electron. J. Differential Equations 2002 (98) :1~11
22 Toka Diagana. Existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations, J. Math. Anal. Appl. 2009, 350:18~28
23 V. Lakshmikantham, A.S. Vatsala. Basic theory of fractional differential equations, Nonlinear Analysis 2008, 69:2677~2682
24 K. Diethelm, N.J. Ford. Analysis of fractional differential equations, J. Math. Anal. Appl. 2002, 265:229~248
25 Toka Diagana, Weighted pseudo-almost periodic solutions to some differential equations, Nonlinear Analysis. 2008, 68:2250~2260
26 H.X. Li, F.L. Huang, J.Y. Li. Composition of pseudo almost-periodic functions and semilinear differential equations, J. Math. Anal. Appl. 2001,255 (2) :436~446
27 E. Ait Dads, O. Arino. Exponential dichotomy and existence of pseudo almost periodic solutions of some differential equations, Nonlinear Anal. 1996, 27 (4) :369~386
28 Philippe Cieutat, Samir Fatajou, G.M. N'Guérékata. Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations, Applicable Analysis. 2010, Vol 89:11~27
29 S. Boulite, L. Maniar, G.M. N'Guérékata. Almost Automorphic Solutions for Hyperbolic Semilinear Evolution Equations, Semigroup Forum. 2005, Vol. 71:231~240
30 Haewon Lee, Hadi Alkahby, Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear Analysis. 2008, 69:2158~2166
31 T. Diagana, Weighted pseudo almost periodic functions and applications, 2006, 343 (10) :643~646
32 A. A. Pankov. Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations. Kluwer Academic Publishers, 1990:5~43
33 K.J.Engel, R.Nagel. One Parameter Semigroup of Linear Evolution Equations:20~89
34 A.Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983:1~239
35 C.Y. Zhang. Pseudo almost periodic solutions of some differential equations, J. Math. Anal. Appl. 1994, 181 (1): 62~76
36 C.Y. Zhang. Integration of vector-valued pseudo almost periodic functions, Proc. Amer. Math. Soc. 1994, 121 (1) :167~174
37 C.Y. Zhang. Pseudo almost periodic solutions of some differential equations. II, J. Math. Anal. Appl. 1995, 192 (2) :543~561
38 C.Y. Zhang. Almost Periodic Type Functions and Ergodicity, Science Press/Kluwer Academic Publishers, 2003:226~239
39张恭庆,郭懋正.泛函分析讲义.北京大学出版社, 1987:151~163
40何崇佑.概周期微分方程.高等教育出版社, 1992:1~53
41林振声.概周期微分方程与积分流形.科学技术出版社, 1986:48~79
42列维坦.概周期函数.高等教育出版社, 1956:11~118
2 A. M. Fink. Almost Periodic Differential Equations. Springer-Verlag, 1974:158~183
3 T. Diagana, G.M. N'Guérékata. Almost automorphic solutions to semilinear evolution equations, Funct. Differ. Equ. 2006, 13 (2) 195~206
4 J.A. Goldstein, G.M. N’Guerekata. Almost automorphic solutions of semilinear evolution equations, Proc. Amer. Math. Soc. 2005,133(8) 2401~2408
5 G.M. N'Guérékata. Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Acad. Plenum, New York, Boston, Moscow,London, 2001:68~105
6 J.Hale. Theory of Functional Differential Equations. Springer-Verlag, 1977:166~188
7 Daniela Araya, Carlos Lizama. Almost automorphic mild solutions to fractional differential equations,Nonlinear Analysis. 2008, 69:3692~3705
8 G.M. N'Guérékata. Existence and Uniqueness of Almost Automorphic Mild Solutions of Some Semilinear Abstract Differential Equations, Semigroup Forum Vol.2004, 69:80~86
9 T. Diagana, G.M. N'Guérékata, N. Van Minh. Almost automorphic solutions of evolution equations. Proc. Amer. Math. Soc. 2004, 132:3289~3298
10 Toka Diagana. Pseudo almost periodic solutions to some differential equations, Nonlinear Analysis. 2005, 60:1277~1286
11 Toka Diagana. Existence of pseudo-almost automorphic solutions to some abstract differential equations with S p-pseudo-almost automorphic coefficients. Nonlinear Analysis. 2009, 70:3781~3790
12 Claudio Cuevas, Carlos Lizama. Almost Automorphic Solutions To A Class Of Semilinear Fractional Differential Equations,2000:2038~2043
13 J. Liang, J. Zhang, T.-J. Xiao. Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 2008,340:1493~1499
14 T-J Xiao, J. Liang, J. Zhang. Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum. 2008, 76 (3):518~524
15 J. Liang, G.M. N'Guérékata, T.-J. Xiao, J. Zhang. Some properties of pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Anal. 2009, 70:2731~2735
16 E. Bazhlekova. Fractional evolution equations in Banach spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001:5~12
17 D. Bugajewski, T. Diagana. Almost automorphy of the convolution operator and applications to differential and functional differential equations, Nonlinear Stud. 2006, 13 (2):129~140
18 Hui-Sheng, Ding, Jin Liang, Ti-Jun Xiao. Almost Automorphic Solutions to Abstract Fractional Differential Equations,Hindawi Publishing Corporation, Advances in Difference Equations, Volume 2010, article ID 508374, 9 pages
19 H. S. Ding, J. Liang, and T. J. Xiao. Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces. preprint.
20 El Hadi Ait Dadsa, Philippe Cieutatb, Khalil Ezzinbia. The existence of pseudo-almost periodic solutions for some nonlinear differential equations in Banach spaces Nonlinear Analysis. 2008, 69:1325~1342
21 M. Bahaj, O. Sidki. Almost periodic solutions of semilinear equations with analytic semigroups in Banach spaces, Electron. J. Differential Equations 2002 (98) :1~11
22 Toka Diagana. Existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations, J. Math. Anal. Appl. 2009, 350:18~28
23 V. Lakshmikantham, A.S. Vatsala. Basic theory of fractional differential equations, Nonlinear Analysis 2008, 69:2677~2682
24 K. Diethelm, N.J. Ford. Analysis of fractional differential equations, J. Math. Anal. Appl. 2002, 265:229~248
25 Toka Diagana, Weighted pseudo-almost periodic solutions to some differential equations, Nonlinear Analysis. 2008, 68:2250~2260
26 H.X. Li, F.L. Huang, J.Y. Li. Composition of pseudo almost-periodic functions and semilinear differential equations, J. Math. Anal. Appl. 2001,255 (2) :436~446
27 E. Ait Dads, O. Arino. Exponential dichotomy and existence of pseudo almost periodic solutions of some differential equations, Nonlinear Anal. 1996, 27 (4) :369~386
28 Philippe Cieutat, Samir Fatajou, G.M. N'Guérékata. Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations, Applicable Analysis. 2010, Vol 89:11~27
29 S. Boulite, L. Maniar, G.M. N'Guérékata. Almost Automorphic Solutions for Hyperbolic Semilinear Evolution Equations, Semigroup Forum. 2005, Vol. 71:231~240
30 Haewon Lee, Hadi Alkahby, Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear Analysis. 2008, 69:2158~2166
31 T. Diagana, Weighted pseudo almost periodic functions and applications, 2006, 343 (10) :643~646
32 A. A. Pankov. Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations. Kluwer Academic Publishers, 1990:5~43
33 K.J.Engel, R.Nagel. One Parameter Semigroup of Linear Evolution Equations:20~89
34 A.Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983:1~239
35 C.Y. Zhang. Pseudo almost periodic solutions of some differential equations, J. Math. Anal. Appl. 1994, 181 (1): 62~76
36 C.Y. Zhang. Integration of vector-valued pseudo almost periodic functions, Proc. Amer. Math. Soc. 1994, 121 (1) :167~174
37 C.Y. Zhang. Pseudo almost periodic solutions of some differential equations. II, J. Math. Anal. Appl. 1995, 192 (2) :543~561
38 C.Y. Zhang. Almost Periodic Type Functions and Ergodicity, Science Press/Kluwer Academic Publishers, 2003:226~239
39张恭庆,郭懋正.泛函分析讲义.北京大学出版社, 1987:151~163
40何崇佑.概周期微分方程.高等教育出版社, 1992:1~53
41林振声.概周期微分方程与积分流形.科学技术出版社, 1986:48~79
42列维坦.概周期函数.高等教育出版社, 1956:11~118