On the Aronszajn Property for an Implicit Differential Equation of Fractional Order

Abstract

In this paper we investigate some topological properties of solution sets of an implicit differential equation of fractional order in Banach spaces.

On the Convergence of Successive Approximations for a Fractional Differential Equation in Banach Spaces

“…Since the sequence ( u n (·)−u n−1 (·) ) is equicontinuous and uniformly bounded, from the definition of φ(·) and Arzela's Lemma we deduce that for fixed t ∈ J there exists a subsequence (n k ) such that lim k→∞ u n k +1 (t) − u n k (t) = φ(t) and u n k (s) − u n k −1 (s) → φ 1 (s) uniformly in s ∈ J. Replacing n by n k in (7) and passing to the limit as k → ∞, we obtain the inequality…”

“…[1,5]). Recently, the theory of fractional differential equations has gained considerable popularity and importance, and as a result, several research papers and monographs have been published in this field (see, for example, [7,12,13] and the references therein).…”