We study the existence of mild solutions for the two-term fractional order abstract differential equation $${D}^{lpha+1}_t u(t) + mu {D}_t^{eta} u(t) - Au(t) = D^{lpha-1}_t f(t,u(t)), quad tin [0,1], quad 0 < lpha leq eta leq 1, mu geq 0,$$ with nonlocal initial conditions and where $A$ is a linear operator of sectorial type. To achieve our goal, we use a new mixed method, which combines a generalization of the theory of $C_0$-semigroups, Hausdorff measure of noncompactness and a fixed point argument.
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机译：我们研究了二阶分数阶抽象微分方程$$ {D} ^ { alpha + 1} _t u（t）+ mu {D} _t ^ { beta} u（t）的温和解的存在性-Au（t）= D ^ { alpha-1} _t f（t，u（t））， quad t in [0,1]， quad 0 < alpha leq beta leq 1， mu geq 0，$$具有非局部初始条件，其中$ A $是部门类型的线性运算符。为了实现我们的目标，我们使用了一种新的混合方法，该方法结合了$ C_0 $-半群理论的一般化，非紧致性的Hausdorff测度和不动点参数。
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