Existence of the mild solution for fractional semilinear initial value problems

By O. Jaradat, A. Al-Omari, Shaher Momani. 2008

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Fractional differential equations: a novel study of local and global solutions in Banach spaces (2013, unknown): “u(t) = T(t)u0 + ∫ t 0 (t − s)T(t − s) f (s, u(s)) ds, t ≥ 0, as can been seen in [38, 49, 50].”
A singular boundary value problem for nonlinear differential equations of fractional order (2009, background): “For only several among numerous advances in the theory and applications of differential equations of fractional order, we refer the reader to [3–7, 11, 15, 16, 20] and the references therein.”
On the new concept of solutions and existence results for impulsive fractional evolution equations (2011, unknown): “[38].”
Mild solutions for semilinear fractional differential equations. (2009, background): “Lemma 2.3. (S (t))t 0 is an immediately compact semigroup in X , and hence it is immediately norm-continuous. Definition 2.4 ([ 14 ]).”
Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations (2017, background): “It is well known that the contraction mapping principle and strict contraction mapping principle require the contraction constant k to be such that 0 < k < 1, for example in the papers ([1, 19, 23, 24, 32, 37, 38]) and the references therein.”