In this work we study several questions concerning to abstract fractional Cauchy problems of order α ∈ (0, 1). Concretely, we analyze the existence of local mild solutions for the problem, and its possible continuation to a maximal interval of existence. The case of critical nonlinearities and corresponding regular mild solutions is also studied. Finally, by establishing some general comparison results, we apply them to conclude the global well-posedness of a fractional partial differential equation coming from heat conduction theory.

In this paper we study the Riemann-Liouville fractional integral of order α > 0 as a linear operator from L p (I, X) into itself, when 1 ≤ p ≤ ∞, I = [t 0 , t 1 ] (orand X is a Banach space. In particular, when I = [t 0 , t 1 ], besides proving that this linear operator is bounded, we obtain necessary and sufficient conditions to ensure its compactness. We also prove that Riemann-Liouville fractional integral defines a C 0 −semigroup but does not defines a uniformly continuous semigroup. We close this study by presenting lower and higher bounds to the norm of this operator.

In the present work, by taking advantage of a so-called practical limitation of fractional derivatives, namely, the absence of a simple chain and Leibniz's rules, we proposed a generalized fractional calculus of variation where the Lagrangian function depends on fractional derivatives of differentiable functions. The Euler-Lagrange equation we obtained generalizes previously results and enables us to construct simple Lagrangians for nonlinear systems. Furthermore, in our main result, we formulate a Noether-type theorem for these problems that provides us with a means to obtain conservative quantities for nonlinear systems. In order to illustrate the potential of the applications of our results, we obtain Lagrangians for some nonlinear chaotic dynamical systems, and we analyze the conservation laws related to time translations and internal symmetries.

In this work we study the Riemann-Liouville fractional integral of order α ∈ (0, 1/p) as an operator from L p (I; X) into L q (I; X), withwe prove an important property on the uniform continuity of this operator, regarding the order of integration, that allowed us to deduce necessary and sufficient conditions to ensure the compactness of the Riemann-Liouville fractional integral, which was an open problem.

In this work I consider the abstract Cauchy problems with Caputo fractional time derivative of order α ∈ (0, 1], and discuss the continuity of the respective solutions regarding the parameter α. I also present a study about the continuity of the Mittag-Leffler families of operators (for α ∈ (0, 1]), induced by sectorial operators.Definition 1. Consider α ∈ (0, 1] and T a positive real value.