- Open Access
Ergodicity of the implicit midpoint rule for nonexpansive mappings
© Xu et al.; licensee Springer. 2015
- Received: 24 October 2014
- Accepted: 8 December 2014
- Published: 6 January 2015
We prove a mean ergodic theorem for the implicit midpoint rule for nonexpansivemappings in a Hilbert space. We obtain weak convergence for the general case andstrong convergence for certain special cases.
MSC: 47J25, 47N20, 34G20, 65J15.
- implicit midpoint rule
- nonexpansive mapping
- Hilbert space
converge weakly to a fixed point of T. This mean ergodic theorem wasextended by Bruck  to the setting of Banach spaces that are uniformly convex and have aFréchet differentiable norm. Baillon and Clement  also investigated ergodicity of the nonlinear Volterra integral equationsin Hilbert spaces.
converge weakly to a solution to the inclusion .
where the initial guess is arbitrarily chosen, for all n, and is a nonexpansive mapping with fixed points.
The IMR (1.3) is proved to converge weakly  in the Hilbert space setting provided the sequence satisfies the two conditions:
(C1) for all and some , and
However, this algorithm may fail to converge weakly without the assumption (C2). Wetherefore turn our attention to the ergodic convergence of the algorithm. We willshow that for any sequence in the interval , the mean averages as defined by (1.2) will always converge weakly to afixed point of T as long as is an approximate fixed point of T(i.e., ). We will also show that under certain additionalconditions the means can converge in norm to a fixed point of T.This paper is organized as follows. In the next section we introduce the concept ofnearest point projections and properties of nonexpansive mappings. The main resultsof this paper (i.e., weak and strong ergodicity of the IMR (1.3)) arepresented in Section 3.
We need the following characterization of projections.
for all ;
for all ;
for all .
In the rest of this paper we always assume .
We need the demiclosedness principle of nonexpansive mappings as described below.
Next we need the following lemma (not hard to prove).
where for all n, and is a nonexpansive mapping.
The following properties of the IMR (2.3) are proved in .
The convergence of the IMR (2.3) is proved in .
Theorem 2.5LetCbe a nonempty closed convex subset of a Hilbert spaceHand be a nonexpansive mapping with . Assume is generated by the IMR (2.3) where the sequence of parameters satisfies the conditions (C1) and (C2) in theIntroduction. Then converges weakly to a fixed point ofT.
Set and let be the nearest point projection from H toF.
Lemma 3.1The sequence is convergent in norm.
That is, is decreasing and (3.3) is proven.
The strong convergence of follows immediately from the fact(3.3). □
Hence, (3.5) holds.
Theorem 3.3LetCbe a closed convex subset of a Hilbert spaceHand let be a nonexpansive mapping such that . Assume is any sequence of positive numbers in the unit interval and let be the sequence generated by the IMR (2.3). Define the means by (3.1), where the weights are all positive and satisfy the condition (3.2). Assume, inaddition, . Then converges weakly to a pointz, where (in norm).
(Here M is a constant such that for all k.)
Consequently, by Lemma 2.2, each weak cluster point of falls in F.
for all n.
It turns out that (3.8) with .
This together with the fact that implies that . That is, z is the only weak cluster pointof the sequence and therefore, we must have weakly. □
Next we show that in some circumstances, the sequence can converge strongly.
The fixed point setFofThas nonempty interior.
Tis compact, namely, Tmaps bounded sets to relatively norm-compact sets.
Proof (i) By assumption, we have and such that
for all such that .
for all such that .
Since T is compact and since is weakly convergent, is relatively norm-compact. This together with (3.7) evidently implies that is relatively norm-compact. Therefore, must converge in norm to . □
The authors are grateful to the anonymous referees for their helpful comments andsuggestions, which improved the presentation of this manuscript. This projectwas funded by the Deanship of Scientific Research (DSR), King AbdulazizUniversity, under grant No. (49-130-35-HiCi). The authors, therefore,acknowledge technical and financial support of KAU.
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