Abstract differential inequalities and the Cauchy problem for infinite-dimensional linear functional differential equati
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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. 1. 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. One can easily show that even the simplest scalar initial value problem u (t) = p(t)u(θ),
t ∈ [a,b],
u(τ) = 0,
(1.1) (1.2)
where the function p : [a,b] → R is integrable and θ ∈ [a,b] is a fixed number, may have infinitely many solutions. From the theoretical viewpoint, the Cauchy problem for functional differential equations, therefore, should be put amongst the other boundary value problems because the question on its solvability is almost as far from being obvious as is that of any other problem for this extremely general kind of equations. At present, unfortunately, there are not but a few fruitful, leading one to sharp and easy-to-verify conditions, approaches to the Cauchy problem for general functional differential equations, the most powerful and efficient one being based on the use of differential inequalities and developed most extensively for ordinary differential equations (see, e.g., [3, 5, 6, 7]). It should be noted, however, that the techniques used in the works cited are essentially finite-dimensional, often even one-dimensional, in which circumstance excludes any opportunity to study, for example, countable systems of differential equations. Moreover, the majority of significant results on the solvability of the general Cauchy problem are currently available for the scalar equations only [2, 3]. In this paper, 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 Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:3 (2005) 235–250 DOI: 10.1155/JIA.2005.235
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Cauchy problem for infinite-dimensional linear equations
the related results of [2, 3]. The solvability conditions obtained here involve abstract functional differential inequalities understood in a rather broad sense; they are constructed on the base of a certain preordering of the given Banach space. The approach based on the study of operators preserving a certain preordering in the given Banach space, firstly, is equally applicable in finite- and infinite-dimensional cases, without any loss in the sharpness of estimates, and, secondly, provides a unified way to obtain solvability conditions for various equations with apparently different properties. 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 also in th
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