N维非线性时滞分数阶微分方程初值问题的全局解

数学物理学报 ›› 2018, Vol. 38 ›› Issue (5): 903-910.

N维非线性时滞分数阶微分方程初值问题的全局解

王俊¹,王天璐²,温艳华¹,周先锋^1,*()

¹ 安徽大学数学科学学院合肥 230601
² 西北工业大学数学系西安 710072

收稿日期:2017-06-22 出版日期:2018-10-26 发布日期:2018-11-09
通讯作者: 周先锋 E-mail:zhouxf@ahu.edu.cn
基金资助:
国家自然科学基金(11471015);国家自然科学基金(11371027);国家自然科学基金(11601003);安徽省自然科学基金(1508085MA01);安徽省自然科学基金(1608085MA12);安徽省自然科学基金(1708085MA15);安徽省高等学校自然科学基金(KJ2016A023)

Global Solutions of IVP for N-Dimensional Nonlinear Fractional Differential Equations with Delay

Jun Wang¹,Tianlu Wang²,Yanhua Wen¹,Xianfeng Zhou^1,*()

¹ School of Mathematical Sciences, Anhui University, Hefei 230601
² Department of Mathematics, Northwestern Polytechnical University, Xi'an 710072

Received:2017-06-22 Online:2018-10-26 Published:2018-11-09
Contact: Xianfeng Zhou E-mail:zhouxf@ahu.edu.cn
Supported by:
the NSFC(11471015);the NSFC(11371027);the NSFC(11601003);the Natural Science Foundation of Anhui Province(1508085MA01);the Natural Science Foundation of Anhui Province(1608085MA12);the Natural Science Foundation of Anhui Province(1708085MA15);the Program of Natural Science Research for Universities of Anhui Province(KJ2016A023)

摘要/Abstract

摘要：

微分方程初值问题全局解的存在性是研究李雅普诺夫意义下稳定性的先决条件。该文旨在研究初值问题（1.10）-（1.11）全局解的存在性.首先得到了初值问题（1.10）-（1.11）局部解的存在性，推广了文献[14]的结果；然后基于得到的延拓定理，证明了初值问题（1.10）-（1.11）全局解的存在性和唯一性.

关键词: 非线性, 分数阶, 存在性, 时滞, 全局解

Abstract:

The existence of global solutions of initial value problems (IVP) for differential equations is a precondition to study their stability in Lyapunov sense. This paper aims to investigate the existence of the global solutions of the IVP (1.10)-(1.11). The existence of a local solution of the IVP (1.10)-(1.11) is obtained first, which is an extension of the paper[14]. Then based on our extension theorem, we prove that the existence and uniqueness of the global solution of the IVP (1.10)-(1.11).

Key words: Nonlinear, Fractional, Existence, Delay, Global solution

中图分类号:

O175.14

王俊,王天璐,温艳华,周先锋. N维非线性时滞分数阶微分方程初值问题的全局解[J]. 数学物理学报, 2018, 38(5): 903-910.

Jun Wang,Tianlu Wang,Yanhua Wen,Xianfeng Zhou. Global Solutions of IVP for N-Dimensional Nonlinear Fractional Differential Equations with Delay[J]. Acta mathematica scientia,Series A, 2018, 38(5): 903-910.

参考文献

1	Podlubny I . Fractional Differential Equations. New York: Academic Press, 1993
2	Lakshmikantham V , Leela S , Vasundhara Devi J . Theory and Applications of Fractional Differential Equations. Amsterdan: Elsevier Science, 2006
3	Diethelm K . The Analysis of Fractional Differential Equations. Heideberg: Springer, 2010
4	Kilbas A A , Srivastava Hari M , Trujillo J J . Theory and Applications of Fractional Differential Equations. Amsterdan: Elsevier Science, 2006
5	Zhou Y . Basic Theorem of Fractional Differential Equations. Singapore: World Scientific, 2014
6	Glockle W G , Nonnenmacher T F . A fractional calculus approach of self-similar protein dynamics. Biophys J, 1995, 68: 46- 53
7	Hilfer R . Applications of Fractional Calculus in Physics. Singapore: World Scientific, 2000
8	Metzler F , Schick W , Kilian H G , Nonnenmacher T F . Relaxation in filled polymers:A fractional calculus approach. J Chen Phys, 1995, 103: 7180- 7186
9	Tarasov V E , Aifantis E C . Non-standard extensions of gradient elasticity:Fractional non-locality, memory and fractality. Commun Nonlinear Sci Numer Simulat, 2015, 22: 197- 227
10	Benchohra M , Henderson J , Ntouyas S K , Ouahab A . Existence results for fractional order functional differential equations with infinite delay. J Math Anal Appl, 2008, 338: 1340- 1350
11	Zhou Y , Jiao F , Li J . Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal, 2009, 71: 3249- 3256
12	Zhou Y , Jiao F , Li J . Existence and uniqueness for p-type fractional neutral differential equations. Nonlinear Anal, 2009, 71: 2724- 2733
13	Agarwal R P , Zhou Y , He Y . Existence of fractional neutral functional differential equations. Comput Math Appl, 2010, 59: 1095- 1100
14	Lakshmikantham V . Theory of fractional functional differential equations. Nonlinear Anal, 2008, 69: 3337- 3343
15	Jankowski T . Initial value problems for neutral fractional differential equations involving a RiemannLiouvill derivative. Appl Math Comput, 2013, 219 (14): 7772- 7774
16	Wang J , Zhou Y . Existence of mild solutions for fractional delay evolution systems. Appl Math Comput, 2011, 218: 357- 367
17	Agarwal R , Zhou Y , et al. Fractional functional differential equations with causal operators in Banach spaces Math. Comput Model, 2011, 54: 1440- 1452
18	Liu Y . Piecewise continuous solutions of initial value problems of singular fractional differential equations with impulse effects. Acta Math Sci, 2016, 3636B (5): 1492- 1508
19	Vinodkumar A , Malar K , et al. Existence, uniqueness and stability of random impulsive fractional differential equations. Acta Math Sci, 2016, 36B (2): 428- 442
20	Wen Y , Zhou X F , Zhang Z . Lyapunov method for nonlinear fractional differential systems with delay. Nonlinear Dyn, 2015, 82: 1015- 1025
21	Liu S , Zhou X F , et al. Stability of fractional nonlinear singular systems and its applications in synchronization of complex dynamical networks. Nonlinear Dyn, 2016, 84: 2377- 2385
22	Liu S , Wu X , et al. Asymptotical stability of Riemann-Liouville fractional nonlinear systems. Nonlinear Dyn, 2016, 86: 65- 71
23	Li M , Wang J . Finite time stability of fractional delay differential equations. Appl Math Lett, 2017, 64: 170- 176
24	Agarwal R , O'Regan D , et al. Practical stability with respect to initial time difference for Caputo fractional differential equations. Commun Nonlinear Sci Numer Simulat, 2017, 42: 106- 120
25	Balachandran K , Govindaraj V , et al. Controllability of fractional damped dynamical systems. Appl Math Comput, 2015, 257: 66- 73
26	Ge F D , Zhou H C , Kou C H . Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique. Appl Math Comput, 2016, 275: 107- 120
27	Zhou X F , Hu L G , et al. Controllability of a fractional linear time-invariant neutral dynamical system. Appl Math Lett, 2013, 26: 418- 424
28	Sakthivel R , Ganesh R , et al. Approximate controllability of nonlinear fractional dynamical systems. Commun Nonlinear Sci Numer Simulat, 2013, 18: 3498- 3508
29	Ding X . Controllability and optimality of linear time-invariant neutral control systems with different fractional orders. Acta Math Sci, 2015, 35B (5): 1003- 1013
30	Bhrawy A H , Zaky M A . Numerical algorithm for the variable-order Caputo fractional functional differential equation. Nonlinear Dyn, 2016, 85: 1815- 1823
31	Diethelm K , Ford N J , Freed A . A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn, 2002, 29: 3- 22
32	Hale J K , Verduyn Lunel Sjocrd M . Introduction to Functional Differential Equations. New York: SpringVerlag, 1993

[1]	刘鹏杰, 邵虎, 李佳宜, 董端端. 一个基于 BFGS 更新的惯性三项共轭梯度投影法[J]. 数学物理学报, 2025, 45(5): 1698-1710.
[2]	朱伟鹏, 李金禄, 吴星. 带阻尼Boussinesq方程的一类大初值整体光滑解[J]. 数学物理学报, 2025, 45(4): 1077-1085.
[3]	马晶晶, 魏娜. 上半空间分数阶 $p$-Laplace 算子相关的抛物方程解的单调性[J]. 数学物理学报, 2025, 45(4): 1086-1099.
[4]	倪思妍, 邹天芳, 赵才地. 格点时滞反应扩散方程的拉回吸引子与不变测度[J]. 数学物理学报, 2025, 45(4): 1128-1143.
[5]	祝鹏, 陈艳萍, 徐先宇. 非均匀网格上时间分数阶扩散—波动方程的 BDF2 型有限元方法[J]. 数学物理学报, 2025, 45(4): 1268-1290.
[6]	鹿高杰, 韩众, 刘露. 非局域反时空高阶非线性薛定谔方程的达布变换及其精确解[J]. 数学物理学报, 2025, 45(3): 767-775.
[7]	刘羽, 陈光淦, 李树勇. 随机 Kuramoto-Sivashinsky 方程行波解的非线性稳定[J]. 数学物理学报, 2025, 45(3): 790-806.
[8]	姚旺进, 张慧萍. 一类具有瞬时和非瞬时脉冲的 $\psi$-Caputo 型分数阶微分方程的多解性[J]. 数学物理学报, 2025, 45(3): 807-823.
[9]	任红越, 周立群. 一类比例时滞随机神经网络的均方指数同步及应用[J]. 数学物理学报, 2025, 45(3): 888-901.
[10]	刘安淇,余婷,向长林. 一类双调和映照型偏微分方程组正则性研究[J]. 数学物理学报, 2025, 45(2): 408-417.
[11]	孟笑莹,陆璐. 含分数阶 $p$-Laplacian 算子基尔霍夫方程解的存在性及其渐近行为[J]. 数学物理学报, 2025, 45(2): 434-449.
[12]	李建军,李阳晨. 一类具有对数非线性源项的分数阶 $p$-Laplace 扩散方程解的存在性和爆破[J]. 数学物理学报, 2025, 45(2): 465-478.
[13]	王国正,史振霞. 具有移动环境的 Lotka-Volterra 合作系统周期强迫波的存在性[J]. 数学物理学报, 2025, 45(2): 479-492.
[14]	李彬,谢莉. 具奇异敏感的趋向性犯罪模型中的警察威慑效应[J]. 数学物理学报, 2025, 45(2): 512-533.
[15]	张倩. 拟线性薛定谔方程组在有界区域上的正规化解[J]. 数学物理学报, 2025, 45(1): 1-30.