Linear functional differential equations possessing solutions with a given growth rate
- PDF / 701,537 Bytes
- 17 Pages / 468 x 680 pts Page_size
- 29 Downloads / 165 Views
We establish optimal, in a sense, conditions under which, for arbitrary forcing terms from a suitable class, a linear inhomogeneous functional differential equation in a preordered Banach space possesses solutions satisfying a certain growth restriction. 1. Introduction When studying mathematical models of various dynamic phenomena, it is often desirable not only to prove the existence of a solution satisfying the given initial or boundary conditions but also to ensure that the solution in question possesses certain qualitative properties (e.g., has at least a prescribed number of zeroes on the given interval). It should be noted that, in most cases, the techniques commonly used to describe the qualitative behaviour of solutions do not provide any algorithm to find the solution itself. In this paper, we prove two theorems which provide conditions under which a linear functional differential equation has a unique absolutely continuous solution passing through a given point, having a given growth rate, representable as a uniformly convergent functional series with the coefficients defined recursively, and positive in a certain sense. More precisely, we are interested in the solutions of the abstract functional differential equation u (t) = (u)(t) + f (t),
t ∈ [a,b],
(1.1)
satisfying the condition sup t ∈[a,b]
u(t) − c
φ(t)
< +∞,
(1.2)
where φ : [a,b] → R is a given nonnegative continuous function possessing at least one zero on the interval [a,b]. Throughout the paper, we exclude from consideration the trivial case where φ(t) = 0 for all t ∈ [a,b] and do not mention this explicitly in the statements. Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:1 (2005) 49–65 DOI: 10.1155/JIA.2005.49
50
Linear functional differential equations
Here, : C([a,b],X) → L([a,b],X) is a continuous linear operator, X is a Banach space, c is a given vector from X, and f : [a,b] → X is an arbitrary Bochner integrable function satisfying a certain growth restriction. By a solution of problem (1.1), (1.2), we mean an absolutely continuous abstract function u : [a,b] → X possessing property (1.2) and satisfying (1.1) almost everywhere on [a,b]. The main purpose of this paper is to give optimal, in a sense, conditions under which problem (1.1), (1.2) is solvable for a sufficiently wide class of pairs (c, f ) from X × L([a,b],X), and to provide a series expansion of a solution. 2. Notation The following notation is used in the sequel. (1) R = (−∞,+∞), N = {1,2,... }. (2) X, · is a Banach space. (3) C([a,b],X) is the Banach space of continuous functions u : [a,b] → X endowed with the norm
C [a,b],X u −→ max u(t). t ∈[a,b]
(2.1)
(4) B([a,b],X) is the Banach space of bounded functions u : [a,b] → X endowed with the norm B [a,b],X u −→ sup u(t).
(2.2)
t ∈[a,b]
(5) Cφ ([a,b],X) is the Banach space of continuous functions u : [a,b] → X satisfying the condition sup t ∈[a,b]
u(t)
φ(t)
< +∞.
(2.3)
The norm in Cφ ([a,b],X) is denoted by the symbol
Data Loading...
