Existence of Mild Solutions for a Class of Fractional Evolution Equations with Compact Analytic Semigroup

and Applied Analysis 3 iii For every t > 0, AS t is bounded in X and there existsMα > 0 such that ‖AS t ‖ ≤ Mαt−α. 2.2 iv A−α is a bounded linear operator for 0 ≤ α ≤ 1 in X. In the following, we denote by C J,Xα the Banach space of all continuous functions from J into Xα with supnorm given by ‖u‖C supt∈J‖u t ‖α for u ∈ C J,Xα . From Lemma 2.1 iv , since A−α is a bounded linear operator for 0 ≤ α ≤ 1, there exists a constant Cα such that ‖A−α‖ ≤ Cα for 0 ≤ α ≤ 1. For any t ≥ 0, denote by Sα t the restriction of S t to Xα. From Lemma 2.1 i and ii , for any x ∈ Xα, we have ‖S t x‖α ‖A · S t x‖ ‖S t ·Aαx‖ ≤ ‖S t ‖ · ‖Ax‖ ‖S t ‖ · ‖x‖α, ‖S t x − xα‖ ‖A · S t x −Aαx‖ ‖S t ·Aαx −Aαx‖ −→ 0 2.3 as t → 0. Therefore, S t t ≥ 0 is a strongly continuous semigroup in Xα, and ‖Sα t ‖α ≤ ‖S t ‖ for all t ≥ 0. To prove our main results, the following lemma is also needed. Lemma 2.2 see 27 . Sα t t ≥ 0 is an immediately compact semigroup in Xα, and hence it is immediately norm-continuous. Let us recall the following known definitions in fractional calculus. For more details, see 16–20, 23 . Definition 2.3. The fractional integral of order σ > 0 with the lower limits zero for a function f is defined by