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

ABSTRACT DIFFERENTIAL INEQUALITIES AND THE CAUCHY PROBLEM FOR INFINITE-DIMENSIONAL LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS ANDREI RONTÓ AND JIŘÍ ŠREMR Received 27 August 2003 and in revised form 18 November 2003 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. 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 236 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 the cases where the operator determining the equation may not be positive in any natural sense. In the “positive” cases, that is, if the preorderings are generated by cones, we obtain a statement (namely Theorem 4.4) containing the corresponding results of [2, 3]. In the proofs of the main Theorems 4.1 and 4.4, we use our previous results on the estimates of spectra of certain classes of linear operators [11] which may not necessarily be isotone with respect to any proper cone. We do not give applications of our general theorems to any concrete classes of equations here. To demonstrate the practical realisation of the ideas on an example, we only obtain a generalised version of a theorem from [2] concerning a scalar linear equation with a single transformation of argument. 2. Notation and definitions The following notation is used in the sequel. (i) R = (−∞, ∞), R+ = [0, ∞), R− = (−∞,0]. (ii) X,  ·  is a Banach space. (iii) C([a,b],X) is the Banach space of continuous functions u : [a,b] → X endowed by the norm
   C [a,b],X  u −→ max u(t). t ∈[a,b] (2.1) (iv) L([a,b],X) is the Banach space of Bochner integrable functions u : [a,b] → X endowed by the norm
 L [a,b],X  u −→ b a   u(t)dt. (v) mesΩ is the Lebesgue measure of a set Ω. (vi) r(A) is the spectral radius of a linear operator A. (vii) A(H) := {Ax | x ∈ H } is the image of a set H ⊂ X under the mapping A. (viii) IntB is the set of interior points of a set B. (ix)
K and K : see Definitions 2.4 and 2.6. (x) blade K: see Definition 2.2 and formula (2.3). (xi) ᏮK (τ,Ω;[a,b],X): see Definition 2.9. (xii) CK,Ω ([a,b],X): see formula (5.5). (2.2) A. Rontó and J. Šremr 237 The two subsections below contain a number of definitions used in the sequel. 2.1. Wedges. We recall some definitions from the theory of linear semigroups in Banach spaces (see, e.g., [8, 9]). Definition 2.1. A nonempty closed set K in a Banach space X is called a wedge (see, e.g., [8]) if the following conditions are satisfied: (i) K + K ⊂ K, (ii) λK ⊂ K for an arbitrary λ ∈ [0, ∞). Here, by definition, we set K + K := {x1 + x2 | {x1 ,x2 } ⊂ K } and, similarly, λK := {λx | x ⊂ K }. Definition 2.2. The set K ∩ (−K) is referred to as the blade [8] of the wedge K. We use the following notation for the blade: K ∩ (−K) =: bladeK. (2.3) Remark 2.3. In the original terminology introduced by Kreı̆n and Rutman [9], a set K satisfying conditions (i) and (ii) of Definition 2.1 is called a linear semigroup. The presence of a wedge in a Banach space X allows one to introduce a natural preordering there. More precisely, we introduce the following standard. Definition 2.4. Two elements {x1 ,x2 } ⊂ X are said to be in relation x1 K x2 if and only if they satisfy the relation x2 − x1 ∈ K. In a similar way, the relation
K is introduced: x1
K x2 if and only if x2 K x1 . Thus, we have   K = x ∈ X | x
K 0 ,   (2.4) blade K = x ∈ X | 0 K x K 0 . Definition 2.5. A wedge K ⊂ X will be called proper if it does not coincide with the entire X and is different from the zero-dimensional subspace {0}. Definition 2.6. A wedge K ⊂ X is said to be solid [9] if its interior is nonempty. In the case of a solid wedge K, following [9], we write x K 0 if and only if x ∈ Int K. Definition 2.7. The wedge K is called a cone [8, 9] if it has trivial blade, that is, when blade K = {0}. (2.5) 2.2. Definition of a (τ,Ω)-positive operator. The set ᏮK (τ,Ω;[a,b],X). Here, we introduce the classes of operators frequently used in the sequel. Let τ be a point in [a,b], Ω a subset of [a,b], and K ⊂ X a wedge. 238 Cauchy problem for infinite-dimensional linear equations Definition 2.8. An operator l : C([a,b],X) (...truncated)