Electronic Journal of Differential Equations, Vol. 2009(2009), No. 18, pp. 1-13. Title: Existence of convex and non convex local solutions for fractional differential inclusions Author: Rabha W. Ibrahim (Univ. Kebangsaan, Malaysia) Abstract: In this paper, we establish the existence theorems for a class of fractional differential inclusion of order $n-1 < \alpha \leq n $. The study holds in two cases, when the set-valued function has convex and non-convex values. Submitted February 20, 2008. Published January 20, 2009. Math Subject Classifications: 34G10, 26A33, 34A12, 42B05. Key Words: Fractional calculus; set-valued function; extremal solution; maximal solution; minimal solution; differential inclusions.