Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations

Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations

We discuss the initial value problem for the nonlinear fractional differential equation L(D)u=f(t,u),  t∈(0,1],  u(0)=0, where L(D)=Dsn-an-1Dsn-1-⋯-a1Ds1, 0<s1<s2<⋯<sn<1, and aj<0, j=1,2,…,n-1, Dsj is the standard Riemann-Liouville fractional derivative and f:[0,1]×ℝ→ℝ is a given c...

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Main Authors: Qiuping Li, Shurong Sun, Ping Zhao, Zhenlai Han
Format: Article
Language:English
Published: Hindawi Limited 2012-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2012/615230
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spelling doaj-c646bbb1b46544d28fbd7fa244f3ce232020-11-24T22:20:04ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/615230615230Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential EquationsQiuping Li0Shurong Sun1Ping Zhao2Zhenlai Han3School of Mathematical Sciences, University of Jinan, Shandong, Jinan 250022, ChinaSchool of Mathematical Sciences, University of Jinan, Shandong, Jinan 250022, ChinaSchool of Control Science and Engineering, University of Jinan, Shandong, Jinan 250022, ChinaSchool of Mathematical Sciences, University of Jinan, Shandong, Jinan 250022, ChinaWe discuss the initial value problem for the nonlinear fractional differential equation L(D)u=f(t,u),  t∈(0,1],  u(0)=0, where L(D)=Dsn-an-1Dsn-1-⋯-a1Ds1, 0<s1<s2<⋯<sn<1, and aj<0, j=1,2,…,n-1, Dsj is the standard Riemann-Liouville fractional derivative and f:[0,1]×ℝ→ℝ is a given continuous function. We extend the basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique to the initial value problem. Some existence and uniqueness results are established.http://dx.doi.org/10.1155/2012/615230
collection DOAJ
language English
format Article
sources DOAJ
author Qiuping Li
Shurong Sun
Ping Zhao
Zhenlai Han
spellingShingle Qiuping Li
Shurong Sun
Ping Zhao
Zhenlai Han
Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations
Abstract and Applied Analysis
author_facet Qiuping Li
Shurong Sun
Ping Zhao
Zhenlai Han
author_sort Qiuping Li
title Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations
title_short Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations
title_full Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations
title_fullStr Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations
title_full_unstemmed Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations
title_sort existence and uniqueness of solutions for initial value problem of nonlinear fractional differential equations
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2012-01-01
description We discuss the initial value problem for the nonlinear fractional differential equation L(D)u=f(t,u),  t∈(0,1],  u(0)=0, where L(D)=Dsn-an-1Dsn-1-⋯-a1Ds1, 0<s1<s2<⋯<sn<1, and aj<0, j=1,2,…,n-1, Dsj is the standard Riemann-Liouville fractional derivative and f:[0,1]×ℝ→ℝ is a given continuous function. We extend the basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique to the initial value problem. Some existence and uniqueness results are established.
url http://dx.doi.org/10.1155/2012/615230
work_keys_str_mv AT qiupingli existenceanduniquenessofsolutionsforinitialvalueproblemofnonlinearfractionaldifferentialequations
AT shurongsun existenceanduniquenessofsolutionsforinitialvalueproblemofnonlinearfractionaldifferentialequations
AT pingzhao existenceanduniquenessofsolutionsforinitialvalueproblemofnonlinearfractionaldifferentialequations
AT zhenlaihan existenceanduniquenessofsolutionsforinitialvalueproblemofnonlinearfractionaldifferentialequations
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