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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:

${D}^{q}x\left(t\right)=Ax\left(t\right)+{t}^{n}f\left(t,x\left(t\right),Bx\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right],\phantom{\rule{4pt}{0ex}}n\in {ℤ}^{+},\phantom{\rule{4pt}{0ex}}x\left(0\right)+g\left(x\right)={x}_{0},$

where $0, $A$ is the infinitesimal generator of a ${C}_{0}$-semigroup of bounded linear operators on a Banach space $X$.

MSC:
 34A08 Fractional differential equations 34G20 Nonlinear ODE in abstract spaces
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