半线性中立型分数阶微分方程S -渐近ω周期解

数学物理学报 ›› 2014, Vol. 34 ›› Issue (1): 16-26.

半线性中立型分数阶微分方程S -渐近ω周期解

舒小保|戴斌祥

湖南大学数学与计量经济学院\quad 长沙 |410082；中南大学数学与计算科学学院\quad 长沙 |410075

收稿日期:2012-03-17 修回日期:2013-08-25 出版日期:2014-02-25 发布日期:2014-02-25
基金资助:
国家自然基金(11271115) 和教育部博士点基金(200805321017)资助.

S-asymptotically ω-periodic Solutions of Semi-linear Neutral Fractional Differential Equations

SHU Xiao-Bao, DAI Bin-Xiang

College of Mathematics and Econometrics, Hunan University, Changsha |410082； School of Mathematical Science and Computing Technology, Central South University, Changsha |410075

Received:2012-03-17 Revised:2013-08-25 Online:2014-02-25 Published:2014-02-25
Supported by:
国家自然基金(11271115) 和教育部博士点基金(200805321017)资助.

摘要/Abstract

摘要：

在Banach空间X中, 研究了如下半线性Caputo -分数阶中立型微分方程S -渐近ω周期解的存在性
D_t^α(x(t)+F(t, x_t))+A x(t) =G(t, x_t), t≥0,
x(0)=φ ∈B.

其中0<α<1, -A是解析半群{T(t)}_t≥0 的无穷小生成元.

关键词: 分数阶微分方程, 柯西问题, mild解, 解析半群, 中立型分数阶微分方程

Abstract:

This paper is concerned with the existence of the S-asymptotically ω-periodic solutions of the semilinear neutral Caputo
fractional equation
D_t^α(x(t)+F(t, x_t))+A x(t) =G(t, x_t), t≥0,
x(0)=φ ∈B.

considered in a Banach space X, where 0<α<1, -A is the infinitesimal generator of an analytic semigroup {T(t)}_t≥0.

Key words: Fractional abstract differential equation, Cauchy problem, Mild solutions, Analytic semigroups, Neutral fractional differential equations

中图分类号:

35R11

舒小保, 戴斌祥. 半线性中立型分数阶微分方程S -渐近ω周期解[J]. 数学物理学报, 2014, 34(1): 16-26.

SHU Xiao-Bao, DAI Bin-Xiang. S-asymptotically ω-periodic Solutions of Semi-linear Neutral Fractional Differential Equations[J]. Acta mathematica scientia,Series A, 2014, 34(1): 16-26.

参考文献

[1] Pavi P. Agarwal, Claudio Cuevas, Herme Soto, et al. Asymptotic periodicity for some evolution equations in Banach spaces. Nonlinear Anal, 2011, 74: 1769--1798

[2] Cuevas C, Hern\'{a}ndez E. Pseudo-almost periodic solutions for abstract partial functional differential equations. Appl Math Lett, 2009, 22: 534--538

[3] Santos J P C, Cuevas C. Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral
equations. Appl Math Lett, 2010, 23: 960--965

[4] Araya D, Lizama C. Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal, 2008, 69: 3692--3705

[5] Cuesta E. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Discrete Contin Dyn Sys (Suppl), 2007: 277--285

[6] El-Borai M M. Some probability densities and fundamental solutions of fractional evolution equations. Chaos, Solitons and Fractals, 2002, 14: 433--440

[7] Cuevas C, Souza J C. S-asymptotically ω-periodic solutions of semilinear fractional integro-differential equations. Appl Math Lett, 2009, 22: 865--870

[8] Agarwal R P, Benchohra M, Slimani B A. Existence results for differential equations with fractional order and impulses. Mem Differential Equations Math Phys, 2008, 44: 1--21

[9] Friedman A. Partial Differential Equations. New York: Holt Rinehart and Winston, 1969

[10] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983

[11] El-Borai M M. On some fractional evolution equations with nonlocal conditions. Pure and Appl Math, 2005, 24(3): 405--413

[12] Li F B, Li M, Zheng Q. Fractional Evolution Equations Governed by Coercive Differential Operators. Abstract and Applied Analysis, 2009, Art. ID 438690, 14pp

[13] Mainardi F, Paradisi P, Gorenflo R. Probability Distributions Generated by Fractional Diffusion Equations. Dordrecht: Kertesz, I. Kondor (Eds.), Econophysics: An Emerging Science, Kluwer, 2000

[14] Hino Y, Murakami S, Naito T. Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473, Berlin, Heidelberg: Springer-Verlag, 1991

[15] Pierri M. S-asymptotically ω-periodic functions on Banach spaces and applications to differential equations
[D]. Universidade de s\~{a}o Paulo, 2009

[16] Lunardi A. Analytic Semi-groups and Optimal Regularity in Parabolic Problems. Basel Boston Berlin: Birkh\"{a}ser, 1995

[17] Butzer P L, Westphal U. An Access to Fractional Differentiation via Fractional Difference Quotients, 116-145, Lecture Notes in Math 457, Berlin: Springer, 1975

[18] Agarwal R P, Cuevas, Claudio, Soto H, El-Gebeily M. Asymptotic periodicity for some evolution equations in Banach spaces. Nonlinear Anal, 2011, 74(5): 1769--1798

[19] Dos Santos J P C, Cuevas C, De Andrade B. Existence results for a fractional equation with state-dependent delay.
Adv. Difference Equ 2011, Art. ID 642013, 15pp

[20] Cuevas C, Pierri M, Sepulveda A. Weighted S-asymptotically ω-periodic solutions of a class of fractional differential equations. Adv Difference Equ 2011, Art. ID 584874, 13pp

[21] Caicedo A, Cuevas C. S-asymptotically ω-periodic solutions of abstract partial neutral integro-differential equations.
Funct Differ Equ, 2010, 17(1/2): 59--77

[22] Agarwal R P, de Andrade B, Cuevas C. On type of periodicity and ergodicity to a class of integral equations with infinite delay. J Nonlinear Convex Anal, 2010, 11(2): 309--333

[23] de Andrade B, Cuevas C. S-asymptotically ω-periodic and asymptotically ω-periodic solutions to semilinear
Cauchy problems with non dense domain. Nonlinear Anal, 2010, 72: 3190--3208

[24] Cuevas C, Lizama C. S-asymptotically ω-periodic solutions for semilinear Volterra equations. Math. Methods Appl Sci, 2010, 33(13): 1628--1636

[25] Agarwal R P, de Andrade B, Cuevas C. On type of periodicity and ergodicity to a class of fractional order differential equations. Adv Difference Equ 2010, Art. ID 179750, 25pp

[26] Cuevas, Claudio, de Souza J C. Existence of S-asymptotically ω-periodic solutions for fractional order functional integro-differential equations with infinite delay. Nonlinear Anal, 2010, 72(3/4): 1683--1689