Linear Volterra Integral Equations

The Kurzweil-Henstock integral formalism is applied to establish the existence of solutions to the linear integral equations of Volterra-typewhere the functions are Banach-space valued. Special theorems on existence of solutions concerning the Lebesgu3 integral setting are obtained. These sharpen earlier results.     

On the Norms of Singular Integral Operators on Contours with Intersections

The exact bounds are obtained for the norm of the singular integral operator S on the family of rays originating at the same point. These bounds, with the use of the localization technique, are then extended to the essential norm of S on piecewise smooth curves with finitely many points of self intersection. Submitted: April 13, 2007. Accepted: September 19, 2007.     

On Limit Cycles Appearing by Polynomial Perturbation of Darbouxian Integrable Systems

We prove an existential finiteness result for integrals of rational 1-forms over the level curves of Darbouxian integrals. Received: May 2007, Revision: March 2008, Accepted: March 2008     

Multiple Solutions for Asymptotically Linear Elliptic Systems

This paper is concerned with the following periodic Hamiltonian elliptic system

Elements of the Theory of Linear Volterra Operators in Banach Spaces

The new definition of Volterra operator introduced in [5] allows specification of the classical theory of linear equations in Banach spaces to equations with such operators. Here we specially address relations between properties of the given linear equation with Volterra operator and properties of its conjugate. As well we treat the theory of Noetherian and Fredholm equations.     

Sharp Two Weight Inequalities for Commutators of Riemann-Liouville and Weyl Fractional Integral Operators

Let b be a BMO function, and the commutator of order k for the Weyl fractional integral. In this paper we prove weighted strong type (p, p) inequalities (p > 1) and weighted endpoint estimates (p = 1) for the operator and for the pairs of weights of the type (w, ), where w is any weight and is a suitable one-sided maximal operator. We also prove that, for weights, the operator is controlled in the L ^p (w) norm by a composition of the one-sided fractional maximal operator and the one-sided Hardy-Littlewood maximal operator iterated k times. These results improve those obtained by an immediate application of the corresponding two-sided results and provide a different way to obtain known results about the operators . The same results can be obtained for the commutator of order k for the Riemann-Liouville fractional integral This research has been partially supported by Spanish goverment Grant MTM2005-8350-C03-02. The first author was also supported by CONICET, ANPCyT and CAI+D-UNL. The second author was also supported by Junta de Andalucía Grant FQM 354.     

Optimal State Feedback Input-Output Stabilization of Infinite-Dimensional Discrete Time-Invariant Linear Systems

We study the optimal input-output stabilization of discrete time-invariant linear systems in Hilbert spaces by state feedback. We show that a necessary and sufficient condition for this problem to be solvable is that the transfer function has a right factorization over H-infinity. A necessary and sufficient condition in terms of an (arbitrary) realization is that each state which can be reached in a finite time from the zero initial state has a finite cost. Another equivalent condition is that the control Riccati equation has a solution (in general unbounded and even non densely defined). The optimal state feedback input-output stabilization problem can then be solved explicitly in terms of the smallest solution of this control Riccati equation. We further show that after renorming the state space in terms of the solution of the control Riccati equation, the closed-loop system is not only input-output stable, but also strongly internally stable. Received: July 4, 2007. Revised: October 17, 2007.     

On Integer Solutions to Linear Equations

A magic square is an n × n matrix with non-negative integer entries, such that the sum of the entries in each row and column is the same. We study the enumeration and P-recursivity of these in the case in which the sum along each row and column is fixed, with the size n of the matrix as the variable. A method is developed that nicely proves some known results about the case when the row and column sum is 2, and we prove new results for the case when the sum is 3. Received December 23, 2005     

Non-supercyclicity of Volterra Convolution and Related Operators

We show that if V ^α (α > 0) is the Riemann-Liouville fractional integration operator and T is an invertible operator on L ²(0, 1) which commutes with V , then TV ^α is not supercyclic on L ²(0, 1); in particular, many Volterra convolution operators are not supercyclic. The technique is based on an argument used by Gallardo-Gutiérrez and Montes-Rodríguez to show that V is not supercyclic.     

Boundedness of Commutators of Marcinkiewicz Integral with Rough Variable Kernel

In this paper the authors give the boundedness of the commutator of Marcinkiewicz integral with rough variable kernel, which is an extension of the result in [5] . Research supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

Existence of Solutions for Second Order Partial Neutral Functional Differential Equations

We establish existence of mild solutions for a class of abstract second-order partial neutral functional differential equations with unbounded delay in a Banach space. The work of the second author was supported by FONDECYT-CONICYT, Grants 1050314 and 7050034.     

Integral representation of velocity in a flow with separation simulated by a vortex sheet

The goal of this paper is to present a new integral operator that defines the velocity of a streamline flow in terms of the intensity of a free vortex sheet. This operator is important for numerical simulation of flows.     

Integral Equations on Function Spaces and Dichotomy on the Real Line

The purpose of this paper is to give new and general characterizations for uniform dichotomy and uniform exponential dichotomy of evolution families on the real line. We consider two general classes denoted and and we prove that if V,W are Banach function spaces with and , then the admissibility of the pair for an evolution family implies the uniform dichotomy of . In addition, we consider a subclass and we prove that if , then the admissibility of the pair implies the uniform exponential dichotomy of the family . This condition becomes necessary if . Finally, we present some applications of the main results.     

The Aumann-Gould Integral

We introduce and study a multi-valued integral of Aumann-type where the selectors are Gould integrals of real functions with respect to finitely additive vector-valued measures. We also discuss the relationship between this kind of integral and the M-Gould set-valued integral with respect to a finitely additive multimeasure M. Received: November 17, 2006, Revised: September 17, 2007 and January, 8, 2008, Accepted: March 13, 2008.     

Inverse-Definiteness of the Fourth-Order Symmetric Differential Operator （Ⅰ）

Abstract   We give a linear symmetric differential operator L defined by

Nonclassical Solutions of Fully Nonlinear Elliptic Equations

We prove the existence of non-smooth solutions to fully non-linear elliptic equations. Received: June 2006, Accepted: January 2007     

Blow-up Solutions for Mixed Nonlinear Schrodinger Equations

This paper is concerned with the initial boundary-value problem for the following nonlinear evolution equation: Фt=iαФxx βФ^2Ф^-x γ|Ф|^2Фx ig(|Ф|^2)Ф Under certain conditions on the initial data and the function g(s),we study the existence and nonexistence of global solution for this equation.The blow-up solution and the blow-up time are also investigated.     

On a Bernoulli-integrable magnetohydrodynamic system

A geometric procedure is used to isolate four Bernoulli integrals for a nonlinear, two-dimensional magnetohydrodynamic system. The general solution of the latter may thereby be constructed.     

Riesz Systems and Controllability of Heat Equations with Memory

The main result we derive is the proof that a particular set of functions related to the controllability of the heat equation with memory and finite signal speed, with suitable kernel, is a Riesz system. Riesz systems are important tools in applied mathematics, for example for the solution of inverse problems. In this paper we shows that the Riesz system we identify can be used to give a constructive method for the computation of the control steering a given initial condition to a prescribed target.

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This paper fits into the research program of the GNAMPA-INDAM .