ON UNIQUE SOLVABILITY OF THE INITIAL VALUE PROBLEM FOR ...

Unique Solvability of the Non-Linear Cauchy Problem 47
The main goal of this paper is to establish new conditions sufficient
for unique solvability of the Cauchy problem for certain classes of manydimensional
systems of nonlinear functional differential equations. Similar
topics for linear and nonlinear problems with the Cauchy and more general
conditions were addressed, in particular, in [2], [3, 4], [5], [6], [9], [10], [11].
In this work, for nonlinear functional differential systems determined by
operators that may be defined on the space of the absolutely continuous
functions only, we prove several new theorems close to some results of [5],
[11] concerning existence and uniqueness of the Cauchy problem. The proof
of the main results is based on application of Theorem 1 from [4] which, in
its turn, was established using Theorem 3 from [7].
1. Problem Formulation
Here we consider the system of functional differential equations of the
general form
u ′ k (t) = (f ku)(t), t ∈ [a, b], k = 1, 2, . . . , n, (1)
subjected to the initial condition
u k (a) = c k , k = 1, 2, . . . , n, (2)
where −∞ < a < b < +∞, n ∈ N, f k : D([a, b], R n ) → L 1 ([a, b], R),
k = 1, 2, . . . , n, are generally speaking nonlinear continuous operators, and
{c k | k = 1, 2, . . . , n} ⊂ R (see Section 2 for the notation). It should be
noted that, in contrast to the case considered in [5], [11], setting (1) covers,
in particular, neutral differential equations because the expressions for f k u,
k = 1, 2, . . . , n, in (1) may contain various terms with derivatives.
2. Notation and Definition
The following notation is used throughout the paper.
(1) R := (−∞, ∞), N := {1, 2, 3, . . . }.
(2) ‖x‖ := max 1≤k≤n |x k | for x = (x k ) n k=1 ∈ Rn .
(3) C([a, b], R n ) is the Banach space of continuous functions [a, b] → R n
equipped with the norm
C([a, b], R n ) ∋ u ↦−→ max
s∈[a,b] ‖u(s)‖.
(4) D([a, b], R n ) is the Banach space of absolutely continuous functions
[a, b] → R n equipped with the norm
∫ b
D([a, b], R n ) ∋ u ↦−→ ‖u(a)‖ + ‖u ′ (s)‖ ds.
a