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Mild solutions for semilinear fractional differential equations. (English) Zbl 1179.34002
The authors study the fractional semilinear differential equation with nonlocal conditions $$D^{q}x(t)=-Ax(t)+f(t,x(t),Bx(t)),\qquad t\in [0,T],$$ $$x(0)+g(x)=x_0,$$ where $T>0,$ $0<q<1,$ $-A$ generates an analytic compact semigroup $(S(t))_{t\ge0} $ of uniformly bounded linear operators on a Banach space $X$. By using the Krasnoselkii and the contraction mapping principle, the existence and uniqueness of a mild solution for a fractional semilinear differential equation with non local conditions are given.

34A08Fractional differential equations
26A33Fractional derivatives and integrals (real functions)
34G20Nonlinear ODE in abstract spaces
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