数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (4): 1337-1346.doi: 10.1016/S0252-9602(11)60320-2

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THREE POINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

 Mujeeb ur Rehman1, Rahmat Ali Khan2, Naseer Ahmad Asif2   

  1. 1. National University of Sciences and Technology (NUST), Centre for Advanced Mathematics and Physics, Sector H-12 Islamabad, Pakistan;
    2. University of Malakand, Chakdara Dir(L), Khyber Pakhtunkhawa, Pakistan
  • 收稿日期:2009-09-27 修回日期:2010-11-20 出版日期:2011-07-20 发布日期:2011-07-20

THREE POINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

 Mujeeb ur Rehman1, Rahmat Ali Khan2, Naseer Ahmad Asif2   

  1. 1. National University of Sciences and Technology (NUST), Centre for Advanced Mathematics and Physics, Sector H-12 Islamabad, Pakistan;
    2. University of Malakand, Chakdara Dir(L), Khyber Pakhtunkhawa, Pakistan
  • Received:2009-09-27 Revised:2010-11-20 Online:2011-07-20 Published:2011-07-20

摘要:

In this paper, we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type
cDδ0+u(t) = f(t, u(t), cDσ0+u(t)), t ∈ [0, T],
u(0) = ∂u(η), u(T) = βu(η),
where 1 < δ < 2, 0 < σ < 1, α, β∈ R, η ∈ (0, T), αη(1 −β ) + (1 − α)(T − βη) ≠ 0 and cDδ0+, cDσ0+ are the Caputo fractional derivatives. We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results. Examples
are also included to show the applicability of our results.

关键词: fractional differential equations, three point boundary conditions, existence and uniqueness results

Abstract:

In this paper, we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type
cDδ0+u(t) = f(t, u(t), cDσ0+u(t)), t ∈ [0, T],
u(0) = ∂u(η), u(T) = βu(η),
where 1 < δ < 2, 0 < σ < 1, α, β∈ R, η ∈ (0, T), αη(1 −β ) + (1 − α)(T − βη) ≠ 0 and cDδ0+, cDσ0+ are the Caputo fractional derivatives. We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results. Examples
are also included to show the applicability of our results.

Key words: fractional differential equations, three point boundary conditions, existence and uniqueness results

中图分类号: 

  • 34A08