In this section we show that if for any initial point, there exists a trajectory of the dynamical system induced by a nonexpansive set-valued mapping , which converges to an invariant set , then a convergent trajectory also exists for a nonstationary dynamical system induced by approximations of .
Let be such that is a closed set for each and
Let , , a nonempty closed subset of ,
and for each integer , let satisfy
Assume that for each , there exists a sequence such that
Then for each , there is a sequence such that
We begin the proof of Theorem 3.1 with two lemmata.
Let , a natural number, ,
and let . Then there is a natural number and a sequence such that
Choose a natural number such that
and a sequence such that
There is a sequence such that
We are now going to define by induction a sequence .
To this end, assume that is an integer and that we have already defined , , such that
(Clearly, this assumption holds for .)
By (3.11) and (3.1),
By (3.15), there is such that
Together with (3.3), this implies that
and there is
When combined with (3.16) and (3.13), this implies that
Thus, by (3.18) and (3.20), the assumption we have made concerning also holds for . Therefore, we have indeed defined by induction a sequence such that
and (3.13) holds for all integers . By (3.11), there is an integer such that
Together with (3.8) and (3.13), this inequality implies that
Lemma 3.2 is proved.
, a natural number,
Then for all integers .
We intend to show by induction that for all integers ,
Clearly, for inequality (3.27) does hold. Assume now that is an integer and (3.27) holds. Then there is
By (3.24) and (3.3), there is
By (3.29) and (3.1),
and, in view of (3.30), there is
By (3.33), (3.28), and (3.2),
By (3.35), (3.31), (3.34), and (3.27),
Thus, the assumption we have made concerning also holds for . Therefore, we may conclude that inequality (3.27) indeed holds for all integers . Together with (3.26), this implies that for all integers ,
Lemma 3.3 is proved.
Completion of the Proof of Theorem 3.1
Let . Since , it follows from Lemma 3.2 that there exist a sequence and a strictly increasing sequence of natural numbers , constructed by induction, such that
and for each integer ,
It now follows from (3.39) and Lemma 3.3 that
Theorem 3.1 is proved.