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Existence of the mild solution for some fractional differential equations with nonlocal conditions. (English) Zbl 1180.34006

Summary: We are concerned in this paper with the existence of mild solutions to the Cauchy Problem for the fractional differential equation with nonlocal conditions:
\[ D^q x(t)= Ax(t)+ t^n f(t,x(t),Bx(t)), \quad t\in[0,T],\;n\in\mathbb Z^+,\;x(0)+g(x)=x_0, \]
where \(0<q<1\), \(A\) is the infinitesimal generator of a \(C_0\)-semigroup of bounded linear operators on a Banach space \(X\).

MSC:

34A08 Fractional ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
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