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Abstract. In this paper we are concerned with nonlinear implicit fractional differential equations with initial conditions. We prove the existence and uniqueness results by using modified version of contraction principle. Further, our prime aim is to present various Ulam-Hyers stability and E α -Ulam-Hyers stability results via successive approximation method.

We establish existence and uniqueness results for fractional order delay differential equations. It is proved that successive approximation method can also be successfully applied to study Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability, generalized Ulam-Hyers-Rassias stability, Eα-Ulam-Hyers stability and generalized Eα-Ulam-Hyers stability of fractional order delay differential equations.

Abstract. The aim of this paper is to establish the existence result for implicit differential equation of fractional (arbitrary) order via topological transversality theorem known as Leray-Schauder alternative. Further we prove the uniqueness results. The Grownwall's lemma for singular kernels play an important role to prove our results. We verify our results by providing an example. Mathematics subject classification (2010): 26A33, 34A08.