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

##### 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