Existence results for some fractional differential equations with nonlocal conditions. (English) Zbl 1204.34010

Mophou, G. Massengo; Nakoulima, O.; N’Guérékata, G. M.

Nonlinear Stud. 17, No. 1, 15-21 (2010).

The authors consider the Cauchy problem for the fractional semi linear differential equation with nonlocal conditions in \({\mathbb R}^n\) \[ D^qx(t)=Ax(t)+f(t,x(t)),~~~t \in I=[0,T], \] \[ x(0)=x_0+g(x), \] where \(A\) is a \(n\times n\) matrix, \(f\) is a nonlinear function, \(g \in C[I, {\mathbb R}^n]\) is a continuous function and \(0<q<1\).
They obtain an existence result using an approach similar to the approach in a paper by G. M. N’Guérékata [Nonlinear Anal., Theory Methods Appl. 70, No. 5, A, 1873–1876 (2009; Zbl 1166.34320)].

Reviewer: J. Vasundhara Devi (Visakhapatnam)

MSC:

34A08	Fractional ordinary differential equations and fractional differential inclusions
34G20	Nonlinear differential equations in abstract spaces
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations

Keywords:

Cauchy problem; abstract fractional differential equation; nonlocal condition