It is a standard way to use contractions to approximate nonexpansive mappings. We follow this idea and use contractions to approximate the nonexpansive mapping in order to modify Byrne's algorithm. More precisely, we introduce the following algorithm which is viewed as a modification of Byrne's algorithm. The purpose for such a modification lies in the hope of strong convergence.
For an arbitrary guess , the sequence is generated by the iterative algorithm
where is a sequence in such that
Note that a prototype of is for all .
To prove the convergence of algorithm (4.1) (see Theorem 4.3 below), we need a lemma below.
Set , where with being the spectral radius of the self-adjoint operator
(i) is an averaged mapping; namely, , where is a constant and is a nonexpansive mapping from into itself.
(ii); consequently, .
That which is averaged is actually proved in .
To see (ii), we first observe that the inclusion holds trivially. It remains to prove the implication: . To see this, we notice that the relation is equivalent to the relation . It turns out that
Since the solution set , we can take . Now since , we have by (2.5),
It follows from (4.2) and (4.3) that
This shows that ; that is, .
Finally, since , and both and are averaged, we have .
The sequence generated by algorithm (4.1) converges strongly to the minimum-norm solution of SFP (1.1).
Define operators and on by
where is averaged by Lemma 4.2.
It is readily seen that is a contraction with contractive constant . Namely,
Also we may rewrite algorithm (4.1) as
We first prove that is a bounded sequence. Indeed, since , we can take any (thus by Lemma 4.2) to deduce that
Substituting (4.9) into (4.8), we get
By induction, we can easily show that, for all ,
In particular, is bounded.
We now claim that
To see this, we compute
Letting be a constant such that for all , we find
Substituting (4.14) into (4.13), we arrive at
By virtue of the assumptions (a)–(c), we can apply Lemma 2.3 to (4.15) to obtain (4.12). Consequently we also have
This follows from the following computations:
Therefore, the demiclosedness principle (Lemma 2.1) ensures that each weak limit point of is a fixed point of the nonexpansive mapping , that is, a point of the solution set of SFP (1.1).
One of the key ingredients of the proof is the following conclusion:
where is the minimum-norm element of (i.e., the projection ). Since
to prove (4.18), it suffices to prove that
To prove (4.20), we use Lemma 4.2 to get and is averaged. Write for some and nonexpansive mapping . Then we derive, by taking a point , that
It turns out that (for some constant for all )
Now since , (4.23) implies (4.20).
To prove (4.21), we take a subsequence of so that
Since is bounded, we may further assume with no loss of generality that converges weakly to a point . Noticing that and that is the projection of the origin onto , and applying (2.5), we arrive at
This is (4.21).
Finally we prove in norm. To see this, we compute
satisfies the property (due to (4.18))
We therefore can apply Lemma 2.3 to (4.26) to conclude that . This completes the proof.