Mild solutions for semilinear fractional differential equations.

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Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays (2009, methodology): “where T> 0, 0 13 ] established the existence and uniqueness of mild solutions in α-norm to the following semilinear fractional differential equations with nonlocal conditions by means of the Banach contraction principle and Schauder’s fixed point theorem in a Banach space X: ⎧ ⎨”
Existence Results for Fractional Order Semilinear Integro-Differential Evolution Equations with Infinite Delay (2010, background): “[5] extended the results of [18] to the following semilinear fractional order integro-differential equations with nonlocal conditions in a Banach space X: { Dx(t) +Ax(t) = f ( t, x(t), ∫ t 0 e (t, s, x(s)) ds ) , t ∈ J := [0, T ], x(0) = g(x) + x0.”