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In this paper we consider an initial value problem for a fractional differential equation formulated in a Banach space X where the fractional derivative is Riemann-Liouville type of order 0 < α < 1. We establish the existence and uniqueness of a strong solution of the problem by applying the method of semi-discretization in time, also known as the method of lines or more popularly as Rothe's method. The dual space X * of X is assumed to be uniformly convex. In the final section, we illustrate the applicability of the theoretical results with the help of an example.

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