带有奇异非线性项的分数微分方程周期解的存在性与唯一性

Abstract

Abstract:

The purpose of this paper is to give some sufficient conditions for the existence and uniqueness of solutions to the fractional differential equation as follows
D^αx(t)=f(t,x(t)),t∈J:=(0,1], 0<α<1,
t^1-αx(t)=x(1),
where D^α denotes the Riemann-Liouville fractional derivative, f may be singular at t=0. Lower and upper solutions method, maximum principle together with iterative technique are employed. An example is presented to illustrate the application of results obtained.

Key words: Fractional differential equation, Periodic boundary value problem, Existence, Uniqueness, Singularity

CLC Number:

O175.08

Feng Yuqiang, Wang Weimin, Li Shougui. Existence and Uniqueness Results for the Periodic Boundary Value Problems of Fractional Differential Equations with Singular Nonlinearities[J].Acta mathematica scientia,Series A, 2015, 35(6): 1059-1070.

Trendmd

References

[1] Podlubny I. Fractional Differential Equations. San Diego:Academic Press, 1999
[2] 胡雷, 张淑琴, 侍爱玲.分数阶微分方程耦合系统共振边值问题解的存在性. 数学物理学报, 2014, 34(5):1313-1326
[3] Chen P, Li Y, Li Q. Existence of mild solutions for fractional evolution equations with nonlocal initial conditions. Ann Polon Math, 2014, 110(1):13-24
[4] Lakshmikantham V. Theory of fractional functional differential equations. Nonlinear Anal, 2008, 69(10):3337-3343
[5] Bai Z, Qiu T. Existence of positive solution for singular fractional differential equation. Appl Math Comput, 2009, 215(1):2761-2767
[6] Zhang S. Monotone iterative nethod for initial value problem involving Riemann-Liouville fractional derivatives. Nonlinear Anal, 2009, 71(5/6):2087-2093
[7] Benchohra M, Hellar M. Global uniqueness results for fractional partial hyperbolic differential equations with state-dependent delay. Ann Polon Math, 2014, 110(3):259-281
[8] Belmekki M, Nieto Juan J, Rodriguez-Lopez R. Existence of periodic solution for a nonlinear fractional differential equations. Boundary Value Problem, 2009, Article ID 324561, 18 pages
[9] Nieto Juan J. Maximum principles for fractional differential equations derived from Mittag-Leffler functions. Appl Math Lett, 2010, 23(10):1248-1251
[10] Wei Z, Dong W, Che J. Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative. Nonlinear Anal, 2010, 73(10):3232-3238
[11] Nieto Juan J. Comparision results for periodic boundary value problem of fractional differential equations. Fractional Differential Calculus, 2011, 1(1):99-104
[12] Karami H, Babakhani A, Baleanu D. Existence results for a class of fractional differential equations with periodic boundary conditions and with delay. Abstract and Appllied Analysis, 2013, Article ID 176180, 8 pages
[13] 李福义, 冯锦峰, 沈沛龙. 一类减算子的不动点定理及其应用. 数学学报, 1999, 42(2):193-196

[1]	Wang Jun, Wang Tianlu, Wen Yanhua, Zhou Xianfeng. Global Solutions of IVP for N-Dimensional Nonlinear Fractional Differential Equations with Delay [J]. Acta mathematica scientia,Series A, 2018, 38(5): 903-910.
[2]	Wang Huiwen, Zeng Hongjuan, Li Fang. An Existence Result for Impulsive Boundary Value Problem for Fractional Differential Equations with Multiple Base Points [J]. Acta mathematica scientia,Series A, 2018, 38(4): 697-715.
[3]	Yao Shaowen, Cheng Zhibo. Positive Periodic Solution for Damped Duffing Equation with Singularity [J]. Acta mathematica scientia,Series A, 2018, 38(3): 543-548.
[4]	Xing Chao, Ren Mengzhang, Luo Hong. The Existence of Global Weak Solutions to Thermohaline Circulation Equations [J]. Acta mathematica scientia,Series A, 2018, 38(2): 284-290.
[5]	Zhang Shuqin, Hu Lei. Initial Value Problem for Nonlinear Fractional Differential Equations with Variable-order Derivative and a Parameter [J]. Acta mathematica scientia,Series A, 2018, 38(2): 291-303.
[6]	Zhu Xincai. Existence of Constrained Minimizers for Schrödinger-Poisson Equations in R^N [J]. Acta mathematica scientia,Series A, 2018, 38(1): 61-70.
[7]	Su Xiao, Wang Shubin. Finite Time Blow-Up for the Damped Semilinear Wave Equations with Arbitrary Positive Initial Energy [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1085-1093.
[8]	Li Haixia. Stability and Uniqueness of Positive Solutions for a Food-Chain Model [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1094-1104.
[9]	Li Qin, Yang Zuodong. Infinitely Many High Energy Solutions of p-Kirchhoff-Type System with Sign-Changing Weight [J]. Acta mathematica scientia,Series A, 2017, 37(5): 869-876.
[10]	Li Qingwei, Gao Wenjie, Han Yuzhu. Existence of Solutions to Some Singular Elliptic Problems with Degenerate Coercivity [J]. Acta mathematica scientia,Series A, 2017, 37(5): 877-894.
[11]	Huang Decheng, Chen Shouxin. New Proof for the Existence of Minimizing Energy Solutions for the Ginzburg-Landau Equations [J]. Acta mathematica scientia,Series A, 2017, 37(2): 299-306.
[12]	Qiu Hua, Zhang Yingshu, Fang Shaomei. The Global Existence Result of a Leray-α-Oldroyd Model to the Incompressible Viscoelastic Flow [J]. Acta mathematica scientia,Series A, 2017, 37(1): 102-112.
[13]	Yuan Hailong, Li Yanling. Existence and Stability of Coexistence States for a Lotka-Volterra Competition Model [J]. Acta mathematica scientia,Series A, 2017, 37(1): 173-184.
[14]	Zhang Hongwei, Liu Gongwei, Hu Qingying. Initial Boundary Value Problem for a Class Wave Equation with Logarithmic Source Term [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1124-1136.
[15]	Liu Yang. Potential Well and Application to Non-Newtonian Filtration Equations at Critical Initial Energy Level [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1211-1220.

Existence and Uniqueness Results for the Periodic Boundary Value Problems of Fractional Differential Equations with Singular Nonlinearities

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10