ABSTRACT
Let X be a Banach space of norm || • || and le t / : [a,b] x X —• Xbe a continuous function. A solution of class C1 ([«,£]) to the Cauchy problem (CP) in X
(1.1)
satisfies the Volterra integral equation
(1.2)
and vice versa (i.e., (1.1) and (1.2) are equivalent in X). In other words, the (ODE) x' =f(t,x) and the Volterra equation (1.2) have the same set of solutions in X. This is not true in the case of a general Volterra integral equation
differential equation in C([a,b],X) (Larrieu [3]) (1.4)
Io) If x is a solution of the (VIE) (1.3) on [to,c] then the function y : [to,c] —> C([a,b];X) given by
(2.1)
is a solution of (I A). 2°) If y is a solution 0/(1.4), i.e., if
Io) The key fact here is the property
(2.3)
Therefore y given by (2.1) satisfies
(2.4)
for all s e [to,c] so (y(t))(t) = x(t) as x is a solution of (1.3). Thus ^(¿)(i) = x(t) so (2.4) can be rewritten as
(2.5)
This means that y satisfies (2.2), i.e., (1.4). 2o) Viceversa, if y satisfies (2.2), i.e., if (2.5) holds, then
(2.6)
3 Flow-invariance Problems
In spite of the fact that a (VIE) in X can be reduced to an (ODE) in C([a, b],X), the flow-invariance of a set D with respect to a (VIE) is not a natural problem. Let us prove why. Consider the particular case of (1.3) with/(¿) = xo e D. In view of Definition 1.1, it would be very legitimate to say that D is flow-invariant with respect to (VIE)
(3.1)
is for every to e [a, b) and XQ e D, the corresponding solution x of (3.1) remains in D for all t > to in the domain of x. This leads to the following necessary condition for D to be a (FIS) of (3.1).
