In this section, we will prove the continuous dependence of the solution using an iteration method. The continuous dependence upon the data for linear problems by different methods is shown in [4, 5].
Theorem 1 Under the following assumptions, the solution depends continuously upon the data.
) Let the function
be continuous with respect to all arguments in
and satisfy the following condition:
where , ,
(A2) , ,
Proof Let and be two sets of data which satisfy the conditions (A1)-(A3).
be the solutions of problem (1)-(4) corresponding to the data ϕ
The solutions of (1)-(4),
, are presented in the following form, respectively:
From the condition of the theorem, we have
. We will prove that the other sequential approximations satisfy this condition.
where , , and , , .
First of all, we write
in (6)-(7). We consider
to both sides and applying the Cauchy inequality, Hölder inequality, Lipschitz condition and Bessel inequality to the right-hand side of (8) respectively, we obtain
In the same way, for a general value of N
, we have
(The sequence is convergent, then we can write , ∀N.)
It follows from the estimation ([, pp.76-77]) that .
for the last equation
If and , then . □