Publicaciones

We study existence and uniqueness of almost automorphic solutions for nonlinear partial difference-differential equations modeled in abstract form as (Formula presented.) for (Formula presented.) where (Formula presented.) is the generator of a (Formula presented.) -semigroup defined on a Banach space (Formula presented.) , (Formula presented.) denote fractional difference in Weyl-like sense and (Formula presented.) satisfies Lipchitz conditions of global and local type. We introduce the notion of (Formula presented.) -resolvent sequence (Formula presented.) and we prove that a mild solution of (Formula presented.) corresponds to a fixed point of (Formula presented.) We show that such mild solution is strong in case of the forcing term belongs to an appropriate weighted Lebesgue space of sequences. Application to a model of population of cells is given. © 2015 Taylor & Francis.

Let X be a complex Banach space. The connection between algebra homomorphisms defined on subalgebras of the Banach algebra l1(N0) and fractional versions of Cesàro sums of a linear operator T ? B(X) is established. In particular, we show that every (C, ?)-bounded operator T induces an algebra homomorphism — and it is in fact characterized by such an algebra homomorphism. Our method is based on some sequence kernels, Weyl fractional difference calculus and convolution Banach algebras that are introduced and deeply examined. To illustrate our results, improvements to bounds for Abel means, new insights on the (C, ?)-boundedness of the resolvent operator for temperated a-times integrated semigroups, and examples of bounded homomorphisms are given in the last section. © 2016, Hebrew University of Jerusalem.

We characterize the well-posedness of a third order in time equation with infinite delay in Holder spaces, solely in terms of spectral properties concerning the data of the problem. Our analysis includes the case of the linearized Kuznetzov and Westerwelt equations. We show in case of the Laplacian operator the new and surprising fact that for the standard memory kernel g(t) = tv-1/?(?)e - at the third order problem is ill-posed whenever 0 < ? < 1 and a is inversely proportional to one of the terms of the given model.

The paper is devoted to understand the large time behaviour and decay of the solution of the discrete heat equation in the one dimensional mesh Z on ℓp spaces, and its analogies with the continuous-space case. We do a deep study of the moments of the discrete gaussian kernel (which is given in terms of Bessel functions), in particular the mass conservation principle; that is reflected on the large time behaviour of solutions. We prove asymptotic pointwise and ℓp decay results for the fundamental solution. We use that estimates to get rates on the ℓp decay and large time behaviour of solutions. For the ℓ2 case, we get optimal decay by use of Fourier techniques. © 2021 Elsevier Inc.

We develop an operator-theoretical method for the analysis on well posedness of partial differential-difference equations that can be modeled in the form (*) {Delta(alpha) u(n) = Au(n + 2) + f(n, u(n)), n is an element of N-0, 1 < alpha <= 2; u(0) = u(0); u(1) = u(1); where A is a closed linear operator defined on a Banach space X. Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences. Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on sequences of strongly continuous families of bounded operators generated by A, and natural restrictions on the nonlinearity f. Finally we present some original examples to illustrate our results.

We characterize the solutions of the Poisson equation and the domain of its associated one-sided Hilbert transform for (C, alpha)-bounded operators, alpha > 0. This extends known results for power bounded operators to the setting of Cesaro bounded operators of fractional order, thus generalizing the results substantially. In passing, we obtain a generalization of the mean ergodic theorem in our framework. Examples are given to illustrate the theory.

We study a class of abstract nonlinear integral equations of convolution type defined on a Banach space. We prove the existence of a unique, locally mild solution and an extension property when the nonlinear term satisfies a local Lipschitz condition. Moreover, we guarantee the existence of the global mild solution and blow up profiles for a large class of kernels and nonlinearities. If the nonlinearity has critical growth, we prove the existence of the local ?-mild solution. Our results improve and extend recent results for special classes of kernels corresponding to nonlocal in time equations. We give an example to illustrate the application of the theorems so obtained. © 2018 Rocky Mountain Mathematics Consortium.

In this paper we show the unexpected property that extension from local to global without loss of regularity holds for the solutions of a wide class of vector-valued differential equations, in particular for the class of fractional abstract Cauchy problems in the subdiffusive case. The main technique is the use of the algebraic structure of these solutions, which are defined by new versions of functional equations defining solution families of bounded operators. The convolution product and the double Laplace transform for functions of two variables are useful tools which we apply also to extend these solutions. Finally we illustrate our results with different concrete examples. © 2016 by the Tusi Mathematical Research Group.

We show the existence of uniformly locally attractive solutions for a nonlinear Volterra integral equation of convolution type with a general kernel. We use methods and techniques of fixed point theorems and properties of measure of noncompactness. We extend earlier results obtained in the context of integral equations of fractional order. We give new insights about a new and striking relation between the size of data and the fractional order alpha > 0 of the kernel k(t) = t(alpha-1)/Gamma(alpha).