Differ Equ Dyn Syst https://doi.org/10.1007/s12591-018-0426-6 ORIGINAL RESEARCH Semilinear Conformable Fractional Differential Equations in Banach Spaces 1 1 Anjali Jaiswal · D. Bahuguna © Foundation for Scientiﬁc Research and Technological Innovation 2018 Abstract We introduce the concept of a mild solution of conformable fractional abstract initial value problem. We establish the existence and uniqueness theorem using the contraction principle. As a regularity result for a linear problem, we show that the mild solution is in fact a strong solution. We give an example to demonstrate the applicability of the established theoretical results. Keywords Conformable fractional derivative · Fractional-order differential equation · Banach ﬁxed point theorem Mathematics Subject Classiﬁcation MSC 34G10 · MSC 34G20 Introduction Here we study the following problem: (T y)(t ) = Ay(t ) + f (t, y(t )), α ∈ (0, 1], (1) y(a) = y , a < t ≤ T < ∞, where A is inﬁnitesimal generator of a C semigroup T (t ), t ≥ 0 on a Banach space X and f :[a, T ]× X → X is continuous in t and Lipschitz continuous in y and (T y)(t ) is (left) conformable fractional derivative of y at t where (T y)(t
Differential Equations and Dynamical Systems – Springer Journals
Published: May 28, 2018
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