When addressing ordinary differential equations in infinite dimensional Banach spaces, an interesting question that arises concerns the existence (or non existence) of blowing up solutions in finite time. In this manuscript we discuss this question for the fractional differential equation cD α t u = f (t, u) proving that when f is locally Lipschitz, however does not maps bounded sets into bounded sets, we can construct a maximal local solution that not "blows up" in finite time.2010 Mathematics Subject Classification. 34A08, 26A33, 45J05.

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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.

This manuscript is dedicated to prove a new inequality that involves an important case of Leibniz rule regarding Riemann–Liouville and Caputo fractional derivatives of order α∈(0,1). In the context of partial differential equations, the aforesaid inequality allows us to address the Faedo–Galerkin method to study several kinds of partial differential equations with fractional derivative in the time variable; particularly, we apply these ideas to prove the existence and uniqueness of solution to the fractional version of the 2D unsteady Stokes equations in bounded domains.

We describe the Kantor square (and Kantor product) of multiplications, extending the classification proposed in [J. Algebra Appl. 2017, 16 (9), 1750167]. Besides, we explicitly describe the Kantor square of some low dimensional algebras and give constructive methods for obtaining new transposed Poisson algebras and Poisson-Novikov algebras; and for classifying Poisson structures and commutative post-Lie structures on a given algebra.