Integro-differential equations of fractional order (Q691362)

For a Cauchy type problem for a two-dimensional integro-differential equation of fractional order the global unique existence of a solution is proved if the nonlinearity satisfies a global Lipschitz condition with a sufficiently small Lipschitz constant. The existence is proved also under a sublinear growth condition of the nonlinearity and also under a more restrictive condition. The proofs are straightforward applications of the fixed point theorems of Banach and Leray-Schauder, using a known Gronwall lemma for the considered problems.