Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations. (English) Zbl 1214.34007

Rehman, Mujeeb Ur; Khan, Rahmat Ali

Appl. Math. Lett. 23, No. 9, 1038-1044 (2010).

The authors investigate the existence and uniqueness of solutions for the multi-point boundary value problem for fractional differential equations of the form
\[ D_t^\alpha y(t)= f(t,y(t),D_t^\beta y(t)),\,\,t\in (0,1),\tag{1} \]
\[ y(0)=0, \,\,D_t^\beta y(1)-\sum_{i=1}^{m-2}\zeta_iD_t^\beta y(\xi_i)=y_0,\tag{2} \]
where \(1<\alpha\leq 2\), \(0<\beta<1\), \(0<\xi_i<1,\) \(i=1,2,\dots,m-2\), \(\xi_i\geq 0\) with \(\gamma=\sum_{i=1}^{m-2}\zeta_i\xi_i^{\alpha-\beta-1}<1\) and \(D_t^\alpha\) represents the Riemann-Liouville fractional derivative.
The main tool used by the authors is based on fixed point theory. Specifically, they use the contraction mapping principle and the Schauder fixed point theorem.

Reviewer: Claudio Cuevas (Pernambuco)

MSC:

34A08	Fractional ordinary differential equations
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

fractional differential equations; multi-point boundary conditions; existence and uniqueness

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