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Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces
© Yao et al.; licensee Springer. 2014
- Received: 15 February 2014
- Accepted: 9 May 2014
- Published: 23 May 2014
An iterative algorithm is introduced for the construction of the minimum-norm fixed point of a pseudocontraction on a Hilbert space. The algorithm is proved to be strongly convergent.
MSC:47H05, 47H10, 47H17.
- fixed point
- nonexpansive mapping
where is a sequence in the unit interval , T is a self-mapping of a closed convex subset C of a Hilbert space H, and the initial guess is an arbitrary (but fixed) point of C.
This algorithm, however, does not converge in the strong topology in general (see [, Corollary 5.2]).
Browder and Petryshyn  studied weak convergence of Mann’s algorithm (1.1) for the class of strict pseudocontractions (in the case of constant stepsizes for all n; see  for the general case of variable stepsizes). However, Mann’s algorithm fails to converge for Lipschitzian pseudocontractions (see the counterexample of Chidume and Mutangadura ). It is therefore an interesting question of inventing iterative algorithms which generate a sequence converging in the norm topology to a fixed point of a Lipschitzian pseudocontraction (if any). The interest of pseudocontractions lies in their connection with monotone operators; namely, T is a pseudocontraction if and only if the complement is a monotone operator.
where is the metric projection from onto C, is the adjoint of A, is a constant, and is such that .
It is therefore an interesting problem to invent iterative algorithms that can generate sequences which converge strongly to the minimum-norm solution of a given fixed point problem. The purpose of this paper is to solve such a problem for pseudocontractions. More precisely, we shall introduce an iterative algorithm for the construction of fixed points of Lipschitzian pseudocontractions and prove that our algorithm (see (3.1) in Section 3) converges in the strong topology to the minimum-norm fixed point of the mapping.
For the existing literature on iterative methods for pseudocontractions, the reader can consult [10, 12–26]; for finding minimum-norm solutions of nonlinear fixed point and variational inequality problems, see [27–29]; and for related iterative methods for nonexpansive mappings, see [2, 3, 30, 31] and the references therein.
for all ; or
is monotone on C: for all .
It is immediately clear that nonexpansive mappings are pseudocontractions.
Consequently, is nonexpansive.
In the sequel we shall use the following notations:
stands for the set of fixed points of S;
stands for the weak convergence of to x;
stands for the strong convergence of to x.
Below is the so-called demiclosedness principle for nonexpansive mappings.
Lemma 2.1 (cf. )
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a nonexpansive mapping with fixed points. If is a sequence in C such that and , then .
We also need the following lemma whose proof can be found in literature (cf. ).
Finally, we state the following elementary result on convergence of real sequences.
Lemma 2.3 ()
either or .
Then converges to 0.
We shall prove that this sequence strongly converges to the minimum-norm fixed point of T provided and satisfy certain conditions. To this end, we need the following lemma.
converges in norm, as , to the minimum-norm fixed point of S.
which implies that is a self-contraction of C. Hence has a unique fixed point which is the unique solution of fixed point equation (3.3).
Hence, is bounded and so is .
However, . This together with (3.9) guarantees that . The net is therefore relatively compact, as , in the norm topology.
Therefore, . That is, is the unique fixed point in of the contraction . Clearly this is sufficient to conclude that the entire net converges in norm to as .
Therefore, is the minimum-norm fixed point of S. This completes the proof. □
We are now in a position to prove the strong convergence of algorithm (3.2).
Then the sequence generated by algorithm (3.2) converges strongly to the minimum-norm fixed point of T.
Next, we show that .
[Here for .]
Set (i.e., S is a resolvent of the monotone operator ). We then have that S is a nonexpansive self-mapping of C and (cf. Theorem 6 of ).
converges strongly to the minimum-norm fixed point of S (and of T as ). Without loss of generality, we may assume that for all n.
We can therefore apply Lemma 2.3 to (3.25) and conclude that as . This completes the proof. □
Therefore, and satisfy all three conditions (i)-(iii) in Theorem 3.2.
To show an application of our results, we deal with the following problem.
At which value does approach as n goes to infinity?
We claim that and it can be easily derived by applying Theorem 3.2.
Yonghong Yao was supported in part by NSFC 71161001-G0105. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.
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