Abstract differential inequalities and the Cauchy problem for infinite-dimensional linear functional differential equations

Andrei Rontó,Jiří Šremr

doi:10.1155/jia.2005.235

Abstract

We establish optimal, in a sense, unique solvability conditions of the Cauchy problem for a wide class of linear functional differential equations in a Banach space with a solid wedge. The conditions are formulated in terms of certain abstract functional differential inequalities.

Highlights

It is well known that, in the theory of functional differential equations, the study of the Cauchy problem requires much more effort than in the case of an ordinary differential equation
We suggest a new approach to the Cauchy problem, which is based on the use of order-theoretical methods, and establish considerably more general versions of
Due to the use of rather general preorderings, which may not be, and often are not orderings, the theorems that we prove here allow one to establish the unique solvability of the Cauchy problem for linear functional differential equations in the cases where the operator determining the equation may not be positive in any natural sense

Summary

Introduction

It is well known that, in the theory of functional differential equations, the study of the Cauchy problem requires much more effort than in the case of an ordinary differential equation. Due to the use of rather general preorderings, which may not be, and often are not orderings, the theorems that we prove here allow one to establish the unique solvability of the Cauchy problem for (finite- or infinite-dimensional) linear functional differential equations in the cases where the operator determining the equation may not be positive in any natural sense.

Results

Conclusion