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 Open Access
Ergodicity of the implicit midpoint rule for nonexpansive mappings
 HongKun Xu^{7, 8}Email author,
 Maryam A Alghamdi^{9}Email author and
 Naseer Shahzad^{8}Email author
https://doi.org/10.1186/1029242X20154
© Xu et al.; licensee Springer. 2015
 Received: 24 October 2014
 Accepted: 8 December 2014
 Published: 6 January 2015
Abstract
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.
Keywords
 ergodic
 implicit midpoint rule
 nonexpansive mapping
 projection
 Hilbert space
1 Introduction
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.
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 [13] in the Hilbert space setting provided the sequence satisfies the two conditions:
(C1) 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.
2 Preliminaries
We need the following characterization of projections.
 (i)
for all ;
 (ii)
for all ;
 (iii)
for all .
In the rest of this paper we always assume .
We need the demiclosedness principle of nonexpansive mappings as described below.
Lemma 2.2[14]
Next we need the following lemma (not hard to prove).
Lemma 2.3[15]
where for all n, and is a nonexpansive mapping.
The following properties of the IMR (2.3) are proved in [13].
 (i)
 (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.
3 Ergodicity
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.
 (i)
The fixed point setFofThas nonempty interior.
 (ii)
 (iii)
Tis compact, namely, Tmaps bounded sets to relatively normcompact sets.
Proof (i) By assumption, we have and such that

for all such that .
for all such that .
 (ii)
 (iii)
Since T is compact and since is weakly convergent, is relatively normcompact. This together with (3.7) evidently implies that is relatively normcompact. Therefore, must converge in norm to . □
Declarations
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. (4913035HiCi). The authors, therefore,acknowledge technical and financial support of KAU.
Authors’ Affiliations
References
 Baillon JB: Un théorème de type ergodique pour les contractions nonlinéaires dans un espace de Hilbert.C. R. Acad. Sci. Paris Sér. AB 1975,280(22):A1511A1514. (in French)MATHGoogle Scholar
 Bruck RE: A simple proof of the mean ergodic theorem for nonlinear contractions inBanach spaces.Isr. J. Math. 1979,32(2–3):107–116. 10.1007/BF02764907View ArticleMATHGoogle Scholar
 Baillon JB, Clement P: Ergodic theorems for nonlinear Volterra equations in Hilbert space.Nonlinear Anal. 1981,5(7):789–801. 10.1016/0362546X(81)900535MathSciNetView ArticleMATHGoogle Scholar
 Passty GB: Ergodic convergence to a zero of the sum of monotone operators in Hilbertspaces.J. Math. Anal. Appl. 1979, 72:383–390. 10.1016/0022247X(79)902348MathSciNetView ArticleMATHGoogle Scholar
 Lions PL, Mercier B: Splitting algorithms for the sum of two nonlinear operators.SIAM J. Numer. Anal. 1979, 16:964–979. 10.1137/0716071MathSciNetView ArticleMATHGoogle Scholar
 Auzinger W, Frank R: Asymptotic error expansions for stiff equations: an analysis for the implicitmidpoint and trapezoidal rules in the strongly stiff case.Numer. Math. 1989, 56:469–499. 10.1007/BF01396649MathSciNetView ArticleMATHGoogle Scholar
 Bader G, Deuflhard P: A semiimplicit midpoint rule for stiff systems of ordinary differentialequations.Numer. Math. 1983, 41:373–398. 10.1007/BF01418331MathSciNetView ArticleMATHGoogle Scholar
 Deuflhard P: Recent progress in extrapolation methods for ordinary differentialequations.SIAM Rev. 1985,27(4):505–535. 10.1137/1027140MathSciNetView ArticleMATHGoogle Scholar
 Edith, E: Numerical and approximative methods in some mathematical models.Ph.D. thesis, BabesBolyai University of ClujNapoca (2006)Google Scholar
 Schneider C: Analysis of the linearly implicit midpoint rule for differentialalgebraequations.Electron. Trans. Numer. Anal. 1993, 1:1–10.MathSciNetMATHGoogle Scholar
 Somalia S: Implicit midpoint rule to the nonlinear degenerate boundary valueproblems.Int. J. Comput. Math. 2002,79(3):327–332. 10.1080/00207160211930MathSciNetView ArticleGoogle Scholar
 Somalia S, Davulcua S: Implicit midpoint rule and extrapolation to singularly perturbed boundaryvalue problems.Int. J. Comput. Math. 2000,75(1):117–127. 10.1080/00207160008804969MathSciNetView ArticleGoogle Scholar
 Alghamdi MA, Alghamdi MA, Shahzad N, Xu HK: The implicit midpoint rule for nonexpansive mappings.Fixed Point Theory Appl. 2014., 2014: Article ID 96Google Scholar
 Goebel K, Kirk WA Cambridge Studies in Advanced Mathematics 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.View ArticleGoogle Scholar
 Zarantonello EH: Projections on convex sets in Hilbert space and spectral theory. In Contributions to Nonlinear Functional Analysis. Edited by: Zarantonello EH. Academic Press, New York; 1971:237–424.View ArticleGoogle Scholar
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