Maximal Regularity for Perturbed Integral Equations on the Line

We characterize the existence and uniqueness of solutions for a perturbed linear integral equation with infinite delay in H?lder spaces. The method is based on the theory of operator-valued Fourier multipliers.     

S‐asymptotically ω‐periodic solutions for semilinear Volterra equations

We study S‐asymptotically ω‐periodic mild solutions of the semilinear Volterra equation u′(t)=(a* Au)(t)+f(t, u(t)), considered in a Banach space X, where A is the generator of an (exponentially) stable resolvent family. In particular, we extend the recent results for semilinear fractional integro‐differential equations considered in (Appl. Math. Lett. 2009; 22:865–870) and for semilinear Cauchy problems of first order given in (J. Math. Anal. Appl. 2008; 343(2): 1119–1130). Applications to integral equations arising in viscoelasticity theory are shown. Copyright © 2010 John Wiley & Sons, Ltd.     

Hydrodynamics of bioreactor systems for liquid-liquid contacting

Applied Biochemistry and Biotechnology - Two liquid-liquid bioreactors, a stirred-tank and a novel electrostatic-dispersion system, are being used to investigate biodesulfurization of oil by...     

Solutions to stochastic fractional oscillation equations

We formulate a fractional stochastic oscillation equation as a generalization of Bagley’s fractional differential equation. We do this in analogy with the case for Basset’s equation, which gives rise to fractional stochastic relaxation equations. We analyze solutions under some conditions of spatial regularity of the operators considered.     

Controllability results for the Moore–Gibson–Thompson equation arising in nonlinear acoustics

We show that the Moore–Gibson–Thomson equation

$\tau {?}_{ttt}y+\alpha {?}_{tt}y?c^2\mathrm{\Delta }y?b\mathrm{\Delta }{?}_{t}y=k{?}_{tt}(y^2)+{\chi }_{\omega (t)}u,$

is controlled by a force that is supported on an moving subset $\omega (t)$ of the domain, satisfying a geometrical condition. Using the concept of approximately outer invertible map, a generalized implicit function theorem and assuming that $\gamma :=\alpha ?\frac{\tau c^2}{b}>0$, the local null controllability in the nonlinear case is established. Moreover, the analysis of the critical value $\gamma =0$ for the linear equation is included.     

On a connection between powers of operators and fractional Cauchy problems

Many phenomena in mathematical physics and in the theory of stochastic processes are recently described through fractional evolution equations. We investigate a general framework for connections between ordinary non-homogeneous equations in Banach spaces and fractional Cauchy problems. When the underlying operator generates a strongly continuous semigroup, it is known, using a subordination argument, that the fractional evolution equation is well posed. In this case, we provide an explicit form of the solution involving special functions, one example being the Airy function.     

On approximation and representation of K-regularized resolvent families

We study the problem of approximation and representation for a family of strongly continuous operators defined in a Banach space. It allows us to extend, and in some cases to improve results from the theory ofC ₀-semigroups of operators to, among others, the theories of cosine families, n-times integrated semigroups, resolvent families and k-generalized solutions by means of an unified method.The author was supported by FONDECYT grants 1980812; 1970722 and DICYT (USACH).     

Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

We prove representation results for solutions of a time‐fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. Our focus is in the problem , where 0<β ≤ 2, 0<α ≤ 1, , (?Δ_d)^α is the discrete fractional Laplacian, and is the Caputo fractional derivative of order β. We discuss important special cases as consequences of the representations obtained.     

Uniform continuity and compactness for resolvent families of operators

We characterize the uniform continuity and the compactness of a resolvent family of operators {R(t)_t0 for a Volterra equation of convolution type denned in a Banach spaceX. In particular, we extend similar results to those for semigroups of operators and cosine families of operators studied in other works.Work partially supported by DICYT 91-33 and FONDECYT 91-0471.