Rodrigo Ponce

A connection between almost periodic functions defined on timescales and ℝ

In this paper, we prove a strong connection between almost periodic functions on timescales and almost periodic functions on . An application to difference equations on is given.

Maximal regularity for degenerate differential equations with infinite delay in periodic vector-valued function spaces

Let A and M be closed linear operators defined on a complex Banach space X and a ∈ L(R+) be an scalar kernel. We use operator-valued Fourier multipliers techniques to obtain necessary and sufficient conditions to guarantee the existence and uniqueness of periodic solutions to the equation d dt (Mu(t)) = αAu(t) + ∫ t −∞ a(t− s)Au(s)ds+ f(t), t > 0, with initial condition Mu(0) = Mu(2π), solely in terms of spectral properties of the data. Our results are obtained in the scales of periodic Besov, Triebel-Lizorkin and Lebesgue vector-valued function spaces.

Existence of Mild Solutions to Nonlocal Fractional Cauchy Problems via Compactness

We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases and

Approximate controllability for fractional differential equations of sobolev type via properties on resolvent operators

Abstract This paper treats the approximate controllability of fractional differential systems of Sobolev type in Banach spaces. We first characterize the properties on the norm continuity and compactness of some resolvent operators (also called solution operators). And then via the obtained properties on resolvent operators and fixed point technique, we give some approximate controllability results for Sobolev type fractional differential systems in the Caputo and Riemann-Liouville fractional derivatives with order 1 < α < 2, respectively. Particularly, the existence or compactness of an operator E−1 is not necessarily needed in our results.

Weighted pseudo almost automorphic solutions to a semilinear fractional differential equation with Stepanov-like weighted pseudo almost automorphic nonlinear term

In this paper, we investigate some existence results of weighted pseudo almost automorphic solutions to a semilinear fractional differential equation in Banach spaces with Stepanov-like weighted pseudo almost automorphic nonlinear term. Our main results are based upon ergodicity and composition theorems of Stepanov-like weighted pseudo almost automorphic functions combined with fixed point techniques.

Weighted pseudo antiperiodic solutions for fractional integro-differential equations in Banach spaces

In this paper we prove the existence of weighted pseudo antiperiodic mild solutions for fractional integro-differential equations in the form D α u ( t ) = Au ( t ) + ? - ∞ t a ( t - s ) Au ( s ) ds + f ( t , u ( t ) ) , t ? R , where f ( ? , u ( ? ) ) is Stepanov-like weighted pseudo antiperiodic and A generates a resolvent family of bounded and linear operators on a Banach space X , a ? L loc 1 ( R + ) and α 0 . Here the fractional derivative is considered in the sense of Weyl. Also, we give a short proof to show that the vector-valued space of Stepanov-like weighted pseudo antiperiodic functions is a Banach space.