The proof of the following theorem is based on the retract method, which is well known for ordinary differential equations and goes back to Ważewski . Below we will assume that the function f, except for the indicated conditions, satisfies all the assumptions given in Section 2.
Theorem 2 Let . Let be delta differentiable functions such that for each . If, moreover, every point is the point of strict egress for the set Ω with respect to equation (1), then there exists an rd-continuous initial function satisfying
such that the initial problem
defines a solution y of (1) on the interval satisfying
Proof The idea of the proof is simple. We suppose that the statement of the theorem is not valid. Then it is possible to prove that there exists a retraction of a segment with onto a two-point set . But it is well known that the boundary of a nonempty (closed) interval cannot be its retract (see ). So, in our case, such a retractive mapping cannot exist because it is incompatible with continuity.
Without any special comment, throughout the proof, we use the property that the initial value problem in question has a unique solution and the property of continuous dependence of solutions on their initial data.
Suppose now that satisfying the inequality
and generating the solution which satisfies (7) for any does not exist. This means that for any rd-continuous initial function satisfying the inequality
there exists a , such that for a corresponding solution of the initial problem
Let us define auxiliary mappings , and .
First, define the mapping , where
for , , , we define
for satisfying , and , we put and define
for , , and , we put and define
Second, we define the mapping , where
Third, we define the mapping , where
We will show that the composite mapping
is continuous with respect to the second coordinate of the point . The definition of the mapping P implies that only two resulting points are possible, namely either or .
We consider the first possibility, i.e., . Let for all . Then
Let . Then and the continuity of the mapping is obvious. Indeed, if
is the initial problem defining the solution , ε is a sufficiently small number and
then due to the property of continuous dependence of solutions on their initial data, and
Let . By the assumption of the theorem, every boundary point of ∂ Ω is the point of strict egress for the set Ω with respect to equation (1). Then for the solution defined by (9), we have
either with or with , . (We do not describe all the possibilities for the occurrence of the first or of the second alternative.) In both alternatives we get again. Hence, the mapping P is continuous in the considered case.
We proceed analogously with the case .
The continuity of the mapping P was proved for initial functions satisfying for all and . The desired retraction can be defined as a mapping of the second coordinates realized by P. Then the mapping
is continuous and
i.e., the points , are stationary.
In this situation we proved that there exists a retraction of the set onto the two-point set (see Definition 3). In regard to the above mentioned fact, this is impossible. Our supposition is false, and there exists the initial problem (6) such that the corresponding solution satisfies the inequalities (7) for every . The theorem is proved. □