1 Introduction
The first mean ergodic theorem for nonlinear noncompact operators was proved byBaillon [1]. Let C be a closed convex subset of a Hilbert space Hand let
be a nonexpansive mapping (i.e.,
for all
) with fixed points. Then, for each
, the Cesàro means
be a nonexpansive mapping (i.e.,
for all
) with fixed points. Then, for each
, the Cesàro means
converge weakly to a fixed point of T. This mean ergodic theorem wasextended by Bruck [2] to the setting of Banach spaces that are uniformly convex and have aFréchet differentiable norm. Baillon and Clement [3] also investigated ergodicity of the nonlinear Volterra integral equationsin Hilbert spaces.
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It is quite natural to consider ergodic convergence of iterative algorithms in thecase where the sequences generated by the algorithms either are not guaranteed toconverge or not convergent at all. For instance, the double-backward method ofPassty [4] generates a sequence
in the recursive manner:
in the recursive manner:
(1.1)
where A and B are maximal monotone operators in a Hilbert spacesuch that
is also maximal monotone and the inclusion
is solvable, and
and
are the resolvents of A and B,respectively, that is,
and
. It is well known [5] that the sequence
generated by the double-backward method (1.1) failsto converge weakly, in general. However, Passty [4] showed that if the sequence of parameters,
, is in
, then the averages
is also maximal monotone and the inclusion
is solvable, and
and
are the resolvents of A and B,respectively, that is,
and
. It is well known [5] that the sequence
generated by the double-backward method (1.1) failsto converge weakly, in general. However, Passty [4] showed that if the sequence of parameters,
, is in
, then the averages
(1.2)
converge weakly to a solution to the inclusion
.
.The implicit midpoint rule (IMR) for nonexpansive mappings in a Hilbert spaceH, inspired by the IMR for ordinary differential equations [6‐12], was introduced in [13]. This rule generates a sequence
via the semi-implicit procedure:
via the semi-implicit procedure:
(1.3)
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where the initial guess
is arbitrarily chosen,
for all n, and
is a nonexpansive mapping with fixed points.
is arbitrarily chosen,
for all n, and
is a nonexpansive mapping with fixed points.The IMR (1.3) is proved to converge weakly [13] in the Hilbert space setting provided the sequence
satisfies the two conditions:
satisfies the two conditions:(C1)
for all
and some
, and
for all
and some
, and(C2)
.
.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.
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 3.2 Preliminaries
Let C be a nonempty closed convex subset of a Hilbert space H.Recall that the nearest point projection from H to C,
, is defined by
, is defined by
(2.1)
We need the following characterization of projections.
Lemma 2.1LetCbe a nonempty closed convex subset of a Hilbert spaceH. Given
and
, then
if and only if any one of the following properties is satisfied:
and
, then
if and only if any one of the following properties is satisfied: (i)
for all
;
for all
;(ii)
for all
;
for all
;(iii)
for all
.
for all
.Recall that a mapping
is said to be nonexpansive if
is said to be nonexpansive if
A point
such that
is said to be a fixed point of T. The set ofall fixed points of T is denoted by
, namely,
such that
is said to be a fixed point of T. The set ofall fixed points of T is denoted by
, namely,
In the rest of this paper we always assume
.
.We need the demiclosedness principle of nonexpansive mappings as described below.
Lemma 2.2[14]
LetCbe a closed convex subset of a Hilbert spaceHand let
be a nonexpansive mapping. Then the mapping
is demiclosed in the sense that, for any sequence
ofC, the following implication holds:
be a nonexpansive mapping. Then the mapping
is demiclosed in the sense that, for any sequence
ofC, the following implication holds:
Next we need the following lemma (not hard to prove).
Lemma 2.3[15]
For each integer
,
such that
, points
, and any nonexpansive mapping
, we have
,
such that
, points
, and any nonexpansive mapping
, we have
(2.2)
Recall also that the implicit midpoint rule (IMR) [13] generates a sequence
by the recursion process
by the recursion process
(2.3)
where
for all n, and
is a nonexpansive mapping.
for all n, and
is a nonexpansive mapping.The following properties of the IMR (2.3) are proved in [13].
Lemma 2.4Let
be any sequence in
and let
be the sequence generated by the IMR (2.3). Then
be any sequence in
and let
be the sequence generated by the IMR (2.3). Then(i)
for all
and
. In particular,
is bounded, and moreover, we have
for all
and
. In particular,
is bounded, and moreover, we have
(2.4)
(ii)
.
.(iii)
.
.The convergence of the IMR (2.3) is proved in [13].
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.
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.3 Ergodicity
In this section we discuss the ergodic convergence of the sequence
generated by the IMR (2.3), that is, the convergenceof the means
generated by the IMR (2.3), that is, the convergenceof the means
(3.1)
where
is a sequence of positive numbers such that
is a sequence of positive numbers such that
(3.2)
Set
and let
be the nearest point projection from H toF.
and let
be the nearest point projection from H toF.Lemma 3.1The sequence
is convergent in norm.
is convergent in norm.Proof First observe that
(3.3)
As a matter of fact, we get for
, by Lemma 2.1(i) and Lemma 2.4(i),
, by Lemma 2.1(i) and Lemma 2.4(i),
That is,
is decreasing and (3.3) is proven.
is decreasing and (3.3) is proven.Applying the inequality (Lemma 2.1(iii))
(3.4)
to the case where
and
(with
) together with Lemma 2.4(i), we get
and
(with
) together with Lemma 2.4(i), we get
The strong convergence of
follows immediately from the fact(3.3). □
follows immediately from the fact(3.3). □Remark 3.2 The limit of
, which we denote by
, can also be identified as the asymptotic center ofthe sequence
with respect to the fixed point set F ofT. In other words,
, which we denote by
, can also be identified as the asymptotic center ofthe sequence
with respect to the fixed point set F ofT. In other words,
(3.5)
As a matter of fact, by (3.4) we get, for any
,
,
Upon taking limsup we immediately obtain
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).
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).Proof Let
which is well defined by Lemma 3.1. ByLemma 2.1(ii), we have, for each k,
which is well defined by Lemma 3.1. ByLemma 2.1(ii), we have, for each k,
It turns out that, for
,
,
(Here M is a constant such that
for all k.)
for all k.)By multiplying by
and then summing up from
to n, we conclude
and then summing up from
to n, we conclude
(3.6)
We now claim that
(3.7)
Consequently, by Lemma 2.2, each weak cluster point of
falls in F.
falls in F.To see (3.7), we will prove that
(3.8)
for all n big enough, where
as
. For the sake of simplicity, we may, due to theassumption
, assume that
as
. For the sake of simplicity, we may, due to theassumption
, assume that
(3.9)
for all n.
Let
for
and let M be a constant such that
. For each n, we put
for
and apply (2.2) to get
for
and let M be a constant such that
. For each n, we put
for
and apply (2.2) to get
(3.10)
Combining (3.9) and (3.10), we derive that
(3.11)
It turns out that (3.8) with
.
.Now since
in norm, we see that the means
in norm, as well. Consequently, if
is a subsequence weakly converging to some point
, it follows from (3.6) that
in norm, we see that the means
in norm, as well. Consequently, if
is a subsequence weakly converging to some point
, it follows from (3.6) that
(3.12)
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. □
implies that
. That is, z is the only weak cluster pointof the sequence
and therefore, we must have
weakly. □Remark 3.4 In Theorem 3.3 we assumed that
. This assumption is guaranteed if the sequence
satisfies the condition (C2) in the Introduction,that is,
. Indeed, by (C2) and Lemma 2.4(ii), we find
. This assumption is guaranteed if the sequence
satisfies the condition (C2) in the Introduction,that is,
. Indeed, by (C2) and Lemma 2.4(ii), we find
(3.13)
Since the definition of IMR (2.3) yields
we also have
(3.14)
Combining (3.13) and (3.14), we infer that
Remark 3.5 If we assume (3.9) holds for all
, then we need some more delicate technicalitiesdealing with (3.10). We may proceed as follows. Decompose
(for
) as
, then we need some more delicate technicalitiesdealing with (3.10). We may proceed as follows. Decompose
(for
) as
where
As
, we may assume
. Repeating the argument for (3.10) and (3.11), we get
, we may assume
. Repeating the argument for (3.10) and (3.11), we get
Let
. We finally obtain, for
,
. We finally obtain, for
,
Next we show that in some circumstances, the sequence
can converge strongly.
can converge strongly.Theorem 3.6Let the assumptions of Theorem 3.3 holds. Then thesequence
converges in norm to the point
if, in addition, any one of the following conditions issatisfied:
converges in norm to the point
if, in addition, any one of the following conditions issatisfied: (i)
The fixed point setFofThas nonempty interior.
(ii)
Tis a contraction, that is,
where
is a constant. In this case, the sequence
generated by the IMR (2.3) converges in norm to the unique fixed pointofT.
is a constant. In this case, the sequence
generated by the IMR (2.3) converges in norm to the unique fixed pointofT.(iii)
Tis compact, namely, Tmaps bounded sets to relatively norm-compact sets.
Proof (i) By assumption, we have
and
such that
and
such that-
for all
such that
.
Therefore, upon substituting
for u in (3.6) we obtain
for u in (3.6) we obtain
(3.15)
for all
such that
.
such that
.Taking the supremum in (3.15) over
such that
immediately yields
such that
immediately yields
This verifies that
in norm.
in norm.(ii)
Since T is a contraction, T has a unique fixed point which is denoted by p. By (2.3) we deduce that (noticing
)
)
It turns out that
and hence
Since
, we must have
. However, since the sequence
is decreasing, we must have
. Namely,
in norm, and so
in norm.
, we must have
. However, since the sequence
is decreasing, we must have
. Namely,
in norm, and so
in norm.(iii)
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
. □
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
. □Acknowledgements
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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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read andapproved the final manuscript.