3. UNIQUENESS AND EXISTENCE OF SOLUTION To prove the uniqueness of the solution, we use the ideas of [19], [18].
We derive the integral representation equivalent to (3). To this end, we rewrite (3) as
(21)
where .
Integrating equation (21) over the domain we find
We get
(22)
Theorem 4. Under assumption of Theorems 1 and 2, problem (3)-(7) has a unique solution.
Proof. Let and are solutions of the problem (3)-(7) and, moreover,
Then, taking into account (22), we have
(23)
where is the solution defined between and .
From Theorem 1 we have that
(24)
Let . Then for we get an equation with bounded coefficients and the problem
where , , .
Hence by the maximum principle, we have
Due to the boundedness of the functions , , , we estimate the components of (23):
Let . Then
where .
Analogously we get
(25)
Next, we dividing (25) by we end up with
(26)
Now we estimate the integral term
Consider the auxiliary problem
We introduce the function
We have
Hence, by the principle of maximum
As
that
Consequently,
Since the right-hand side of (26) tends to zero when , then for sufficiently small we come to a contradiction. Consequently, , and further for .
The uniqueness of the solution of the problem for any is established as follows.
Let . If , then the issue will be resolved. Otherwise, assuming that the parameter is bounded and, repeating the calculations performed above in the interval , we again arrive at a contradiction.