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

Theorem 3.1.

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.

Lemma 3.2.

Let , a natural number, ,

and let . Then there is a natural number and a sequence such that

Proof.

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

such that

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.

Lemma 3.3.

Let ,

, a natural number,

Then for all integers .

Proof.

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

such that

By (3.24) and (3.3), there is

such that

By (3.29) and (3.1),

and, in view of (3.30), there is

such that

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.