Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay

In this paper, we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay. We shall rely on a nonlinear alternative of Leray–Schauder. For the global existence and uniqueness we apply a recent nonlinear alternative of Leray–Schauder type in Fréchet spaces, due to M. Frigon and A. Granas [Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (2) (1998) 161–168]. The goal of this paper is to extend the problems considered by A. Ouahab [Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006) 456–472].     

Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces

In this article, a recent nonlinear alternative for contraction maps in Fréchet spaces due to Frigon and Granas [1998, Résultats de type Leray-Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22, 161–168] is used to investigate the existence and uniqueness of solutions for fractional order functional differential equations with infinite delay.     

Some Uniqueness Results for Functional Damped Semilinear Differential Equations in Frechet Spaces

A recent nonlinear alternative for contraction maps in Frechet spaces due to Frigon and Granas （Resultats de type Leray-Schauder pour des contractions sur des espaces de Frechet, Ann. Sci. Math. Quebec 22, （2）, 161-168 （1998））, combined with semigroup theory, is used to investigate the existence and uniqueness of mild solutions for first- and second-order functional semi linear and neutral damped differential equations in Frechet space.     

A Stirling-like method with Hölder continuous first derivative in Banach spaces

In this paper, the convergence of a Stirling-like method used for finding a solution for a nonlinear operator in a Banach space is examined under the relaxed assumption that the first Fréchet derivative of the involved operator satisfies the Hölder continuity condition. Many results exist already in the literature to cover the stronger case when the second Fréchet derivative of the involved operator satisfies the Lipschitz/Hölder continuity condition. Our convergence analysis is done by using recurrence relations. The error bounds and the existence and uniqueness regions for the solution are obtained. Finally, two numerical examples are worked out to show that our convergence analysis leads to better error bounds and existence and uniqueness regions for the fixed points.     

The Newton Method for Operators with Hölder Continuous First Derivative

We analyze the convergence of the Newton method when the first Fréchet derivative of the operator involved is Hölder continuous. We calculate also the R-order of convergence and provide some a priori error bounds. Based on this study, we give some results on the existence and uniqueness of the solution for a nonlinear Hammerstein integral equation of the second kind.     

On Lipschitz conditions for ordinary differential equations in Fréchet spaces

We will give an existence and uniqueness theorem for ordinary differential equations in Fréchet spaces using Lipschitz conditions formulated with a generalized distance and row-finite matrices.     

Maurey's Extension Theorem and Grothendieck's Probleme Des Topologies

Relating geometric notions like type and cotype from local Banachspace theory with some important topological invariants forFréchet spaces (for example the density condition), weprove a version of Maurey's celebrated extension theorem forthe category of Fréchet spaces, and apply it to Grothendieck'sproblème des topologies.     

Existence of positive solutions of nonlinear fractional delay differential equations

This paper investigates the existence and uniqueness of positive solutions for a class of nonlinear fractional delay differential equations. Using a nonlinear alternative of Leray-Schauder type, we show the existence of positive solutions for the equations in question.     

Sur les dérivées des opérateurs du type Navier-Stokes

Résumé Dans cette note on caractérise les opérateurs linéaires qui représentent la dérivée de Fréchet en un point d'un opérateur nonlinéaire du type envisagé (dans [2]) comme modèle fonctionnel de l'opérateur dans l'espace des vitesses qui gouverne le problème aux limites homogène pour les équations de Navier-Stokes stationnaires.     

Inward contractions on metric spaces

We study the existence of constrained fixed points of contractions in arbitrary complete metric spaces from a global and local point of view. In particular, we provide generalizations of results due to Lim, Downing and Kirk and others. Some aspects of the topological transversality in the spirit of Frigon and Granas of contractions under constraints are also considered.     

Identification problem for damped sine-Gordon equation with point sources

We establish the existence and uniqueness of solutions for sine-Gordon equations in a multidimensional setting. The equations contain a point-like source. Furthermore, the continuity and the Gâteaux differentiability of the solution map is established. An identification problem for parameters governing the equations is set, and is shown to have a solution. The objective function is proved to be Fréchet differentiable with respect to the parameters. An expression for the Fréchet derivative in terms of the solutions of the direct and the adjoint systems is presented. A criterion for optimal parameters is formulated as a bang-bang control principle. An application of these results to the one-dimensional sine-Gordon equation is considered.     

Almost periodicity in Fréchet spaces

In this paper we deal with almost periodic functions with values in a Fréchet space. We apply obtained results to prove the existence of solutions of the initial value problem as well as the Volterra integral equation in this class of functions. We also introduce and investigate asymptotically almost periodic functions with values in a Fréchet space.     

Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise

We prove existence, uniqueness and Lipschitz dependence on the initial datum for mild solutions of stochastic partial differential equations with Lipschitz coefficients driven by Wiener and Poisson noise. Under additional assumptions, we prove Gâteaux and Fréchet differentiability of solutions with respect to the initial datum. As an application, we obtain gradient estimates for the resolvent associated to the mild solution. Finally, we prove the strong Feller property of the associated semigroup.     

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

The initial value problem of a nonlinear fractional differential equation is discussed in this paper. Using the nonlinear alternative of Leray-Schauder type and the contraction mapping principle,we obtain the existence and uniqueness of solutions to the fractional differential equation,which extend some results of the previous papers.     

Partial subdifferentials, derivates and Rademacher's Theorem

In this paper, we present new partial subdifferentiation formulas in nonsmooth analysis, based upon the study of two directional derivatives. Simple applications of these formulas include a new elementary proof of Rademacher's Theorem in , as well as some results on Gâteaux and Fréchet differentiability for locally Lipschitz functions in a separable Hilbert space.

RÉSUMÉ. Dans cet article, nous présentons de nouvelles formules de sousdifférentiation partielle en analyse nonlisse, basées sur l'étude de deux dérivées directionnelles. Une simple application de ces formules nous permet d'obtenir une nouvelle preuve élémentaire du théorème de Rademacher dans , ainsi que certains résultats sur la différentiabilité Gâteaux ou Fréchet des fonctions localement Lipschitz sur un espace de Hilbert séparable.

     

Existence of invariant manifolds for stochastic equations in infinite dimension

We provide a Frobenius type existence result for finite-dimensional invariant submanifolds for stochastic equations in infinite dimension, in the spirit of Da Prato and Zabczyk (Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992). We recapture and make use of the convenient calculus on Fréchet spaces, as developed by Kriegl and Michor (The Convenient Setting for Global Analysis, Surveys and Monographs, Vol. 53, Amer. Math. Soc., Providence, RI, 1997). Our main result is a weak version of the Frobenius theorem on Fréchet spaces. As an application, we characterize all finite-dimensional realizations for a stochastic equation which describes the evolution of the term structure of interest rates.     

Existence results for fractional order semilinear functional differential equations with nondense domain

In this paper, we establish sufficient conditions for existence and uniqueness of solutions for some nondensely defined semilinear functional differential equations involving the Riemann-Liouville derivative. Our approach is based on integrated semigroup theory, the Banach contraction principle, and the nonlinear alternative of Leray-Schauder type.     

Controllability results for functional semilinear differential inclusions in Fréchet spaces

In this paper, a recent nonlinear alternative for multivalued admissible contractions in Fréchet spaces due to Frigon combined with semigroups theory is used to investigate the controllability of some classes of semilinear functional and neutral functional differential inclusions in Fréchet spaces.     

非线性一阶常微分方程解的存在惟一性

讨论了一类非线性一阶常微分方程边值问题解的存在惟一性.得到了当参数在一定的范围取值时解存在惟一的充分条件,并包含了一些已知结果.主要结果基于Leray-Schauder非线性抉择理论和Banach不动点定理.     

Projectively bounded Fréchet measures

A scalar valued set function on a Cartesian product of -algebras is a Fréchet measure if it is a scalar measure independently in each coordinate. A basic question is considered: is it possible to construct products of Fréchet measures that are analogous to product measures in the classical theory? A Fréchet measure is said to be projectively bounded if it satisfies a Grothendieck type inequality. It is shown that feasibility of products of Fréchet measures is linked to the projective boundedness property. All Fréchet measures in a two dimensional framework are projectively bounded, while there exist Fréchet measures in dimensions greater than two that are projectively unbounded. A basic problem is considered: when is a Fréchet measure projectively bounded? Some characterizations are stated. Applications to harmonic and stochastic analysis are given.