Nonlinear Analysis 66 (2007) 2230–2236www.elsevier.com/locate/na

On Mann iteration in Hilbert spaces

Arif Rafiq

COMSATS Institute of Information Technology, Islamabad, Pakistan

Received 1 February 2006; accepted 1 March 2006

Abstract

Let K be a compact convex subset of a real Hilbert space H ; T : K → K a hemicontractive map. Let{αn} be a real sequence in [0,1] satisfying appropriate conditions; then for arbitrary x0 ∈ K , the sequence{xn} defined iteratively by xn = αnxn−1 + (1 − αn)T xn , n ≥ 1 converges strongly to a fixed point of T .c© 2006 Elsevier Ltd. All rights reserved.

MSC: primary 05C38; 15A15; secondary 05A15; 15A18

Keywords: Hilbert space; Mann iteration; Pseudocontractive maps

1. Introduction

Let H be a Hilbert space. A mapping T : H → H is said to be pseudocontractive (see, e.g.,[1,2]) if

‖T x − T y‖2 ≤ ‖x − y‖2 + ‖(I − T )x − (I − T )y‖2, ∀x, y ∈ H (1.1)

and is said to be strongly pseudocontractive if there exists k ∈ (0, 1) such that

‖T x − T y‖2 ≤ ‖x − y‖2 + k‖(I − T )x − (I − T )y‖2, ∀x, y ∈ H. (1.2)

Let F(T ) := {x ∈ H : T x = x} and let K be a nonempty subset of H . A map T : K → K iscalled hemicontractive if F(T ) = ∅ and

‖T x − x∗‖2 ≤ ‖x − x∗‖2 + ‖x − T x‖2 ∀x ∈ H, x∗ ∈ F(T ).

It is easy to see that the class of pseudocontractive maps with fixed points is a subclass of theclass of hemicontractions. The following example, due to Rhoades [27], shows that the inclusion

E-mail address: arafiq@comsats.edu.pk.

0362-546X/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2006.03.012

A. Rafiq / Nonlinear Analysis 66 (2007) 2230–2236 2231

is proper. For x ∈ [0, 1], define T : [0, 1] → [0, 1] by T x = (1−x23 )

32 . It is shown in [27] that T

is not Lipschitz and so cannot be nonexpansive. A straightforward computation (see, e.g., [30])shows that T is pseudocontractive. For the importance of fixed points of pseudocontractions thereader may consult [1].

In the last ten years or so, numerous papers have been published on the iterative approximationof fixed points of Lipschitz strongly pseudocontractive (and correspondingly Lipschitz stronglyaccretive) maps using the Mann iteration process (see, e.g., [18]). Results which had been knownonly for Hilbert spaces and only for Lipschitz maps have been extended to more general Banachspaces (see, e.g., [3–14,17,22–34] and the references cited therein) and to more general classes ofmaps (see, e.g., [4–15,17,19–26,28–34] and the references cited therein). This success, however,has not carried over to arbitrary Lipschitz pseudocontraction T even when the domain of theoperator T is a a compact convex subset of a Hilbert space. In fact, it is still an open questionwhether or not the Mann iteration process converges under this setting. In 1974, Ishikawaintroduced an iteration process which, in some sense, is more general than that of Mann andwhich converges, under this setting, to a fixed point of T . He proved the following theorem.

Theorem 1. If K is a compact convex subset of a Hilbert space H , T : K �→ K is a Lipschitzianpseudocontractive map and x0 is any point in K , then the sequence {xn} converges strongly to afixed point of T , where xn is defined iteratively for each positive integer n ≥ 0 by

xn+1 = (1 − αn)xn + αn T yn, (1.3)

yn = (1 − βn)xn + βn T xn,

where {αn}, {βn} are sequences of positive numbers satisfying the conditions

(i) 0 ≤ αn ≤ βn < 1; (ii) limn→∞ βn = 0; (iii)

∑n≥0

αnβn = ∞.

Since its publication in 1974, Theorem 1, as far as we know, has never been extended tomore general Banach spaces. In [19], Qihou extended the theorem to the slightly more generalclass of Lipschitz hemicontractions and in [20] he proved, under the setting of Theorem 1, thatthe convergence of the recursion formula (1.3) to a fixed point of T when T is a continuoushemicontractive map, under the additional hypothesis that the number of fixed points of T isfinite. The iteration process (1.3) is generally referred to as the Ishikawa iteration process in thelight of [16]. Another iteration process which has been studied extensively in connection withfixed points of pseudocontractive maps is the following: For K a convex subset of a Banach spaceE , and T : K → K , the sequence {xn} is defined iteratively by x1 ∈ K ,

xn+1 = (1 − cn)xn + cn T xn, n ≥ 1, (1.4)

where {cn} is a real sequence satisfying the following conditions:

(iv) 0 ≤ cn < 1; (v) lim cn = 0; (vi)∞∑

n=1

cn = ∞.

The iteration process (1.4) is generally referred to as the Mann iteration process in the light of[18].

In 1995, Liu [17] introduced what he called Ishikawa and Mann iteration processes witherrors as follows:

2232 A. Rafiq / Nonlinear Analysis 66 (2007) 2230–2236

(1-a) For K a nonempty subset of E and T : K → E , the sequence {xn} defined by

x1 ∈ K , (1.5)

xn+1 = (1 − αn)xn + αn T yn + un,

yn = (1 − βn)xn + βnT xn + vn, n ≥ 1,

where {αn}, {βn} are sequences in [0, 1] satisfying appropriate conditions and∑ ‖un‖ <

∞,∑ ‖vn‖ < ∞, is called the Ishikawa iteration process with errors.

(1-b) With K , E and T as in part (a), the sequence {xn} defined by

x1 ∈ K , (1.6)

xn+1 = (1 − αn)xn + αn T xn + un, n ≥ 1,

where {αn} is a sequence in [0,1] satisfying appropriate conditions and∑ ‖un‖ < ∞, is called

the Mann iteration process with errors.While it is known that consideration of error terms in iterative processes is an important part of

the theory, it is also clear that the iteration processes with errors introduced by Liu in (a) and (b)are unsatisfactory. The occurrence of errors is random and the conditions imposed on the errorterms in (a) and (b) which imply, in particular, that they tend to zero as n tends to infinity are,therefore, unreasonable. In 1997, Xu [33] introduced the following more satisfactory definitions.(1-c) Let K be a nonempty convex subset of E and T : K → K a mapping. For any givenx1 ∈ K , the sequence {xn} defined iteratively by

xn+1 = anxn + bnT yn + cnun, (1.7)

yn = a′nxn + b′

nT xn + c′nvn, n ≥ 1,

where {un}, {vn} are bounded sequences in K and {an}, {bn}, {cn}, {a′n}, {b′

n} and {c′n} are

sequences in [0, 1] such that an + bn + cn = a′n + b′

n + c′n = 1 ∀ n ≥ 1, is called the Ishikawa

iteration sequence with errors in the sense of Xu.(1-d) If, with the same notation and definitions as in (1-c), b′

n = c′n = 0, for all integers n ≥ 1,

then the sequence {xn}, now defined by

x1 ∈ K (1.8)

xn+1 = anxn + bnT xn + cnun, n ≥ 1,

is called the Mann iteration sequence with errors in the sense of Xu. We remark that if K isbounded (as is generally the case), the error terms un, vn are arbitrary in K .

In [7], Chidume and Chika Moore proved the following theorem.

Theorem 2. Let K be a compact convex subset of a real Hilbert space H ; T : K → K acontinuous hemicontractive map. Let {an}, {bn}, {cn}, {a′

n}, {b′n} and {c′

n} be real sequences in[0, 1] satisfying the following conditions:

(vii) an + bn + cn = 1 = a′n + b′

n + c′n ∀ n ≥ 1;

(viii) lim bn = lim b′n = 0;

(ix)∑

cn < ∞; ∑c′

n < ∞;(x)

∑αnβn = ∞; ∑

αnβnδn < ∞, where δn := ‖T xn − T yn‖2;(xi) 0 ≤ αn ≤ βn < 1 ∀ n ≥ 1, where αn := bn + cn; βn := b′

n + c′n.

A. Rafiq / Nonlinear Analysis 66 (2007) 2230–2236 2233

For arbitrary x1 ∈ K , define the sequence {xn} iteratively by

xn+1 = anxn + bnT yn + cnun,

yn = a′nxn + b′

nT xn + c′nvn, n ≥ 1,

where {un}, {vn} are arbitrary sequences in K . Then, {xn} converges strongly to a fixed pointof T .

They also gave the following remark in [7].

Remark 1.1. In connection with the iterative approximation of fixed points of pseudocontractions, the

following question is still open. Does the Mann iteration process always converge forcontinuous pseudocontractions, or for even Lipschitz pseudocontractions?

2. Let E be a Banach space and K be a nonempty compact convex subset of E . Let T : K → Kbe a Lipschitz pseudocontractive map. Under this setting, even for E = H , a Hilbert space,the answer to the above question is not known. There is, however, an example [15] of adiscontinuous pseudocontractive map T with a unique fixed point for which the Mann iterationprocess does not always converge to the fixed point of T . Let H be the complex plane andK := {z ∈ H : |z| ≤ 1}. Define T : K → K by

T (reiθ ) =

⎧⎪⎨⎪⎩

2rei(θ+ π3 ), for 0 ≤ r ≤ 1

2,

ei(θ+ 2π3 ), for

1

2< r ≤ 1.

Then, zero is the only fixed point of T . It is shown in [13] that T is pseudocontractive and thatwith cn = 1

n+1 , the sequence {zn} defined by zn+1 = (1 − cn)zn + cnT zn, z0 ∈ K , n ≥ 1,does not converge to zero. Since the T in this example is not continuous, the above questionremains open.

In [10], Chidume and Mutangadura, provide an example of a Lipschitz pseudocontractivemap with a unique fixed point for which the Mann iteration sequence failed to converge and theystated there “This resolves a long standing open problem”.

However the purpose of this paper is to introduce the following Mann type implicit iterationprocess associated with hemicontractive mappings to have a strong convergence in the setting ofHilbert spaces.

Let K be a closed convex subset of a real normed space X and T : K → K be a mapping.Define {xn} in the following way:

x0 ∈ K ,

xn = αn xn−1 + (1 − αn)T xn, (1.9)

where {αn} be a real sequence in [0, 1] satisfying some appropriate conditions.

2. Preliminaries

We shall make use of the following results.

Lemma 1 ([31]). Suppose that {ρn}, {σn} are two sequences of nonnegative numbers such thatfor some real number N0 ≥ 1,

ρn+1 ≤ ρn + σn ∀n ≥ N0.

2234 A. Rafiq / Nonlinear Analysis 66 (2007) 2230–2236

(a) If∑

σn < ∞, then, lim ρn exists.(b) If

∑σn < ∞ and {ρn} has a subsequence converging to zero, then lim ρn = 0.

Lemma 2. For all x , y ∈ H and λ ∈ [0, 1], the following well-known identity [16] holds:

‖(1 − λ)x + λy‖2 = (1 − λ)‖x‖2 + λ‖y‖2 − λ(1 − λ)‖x − y‖2.

3. Main results

Now we prove our main results.

Theorem 3. Let K be a compact convex subset of a real Hilbert space H ; T : K → K ahemicontractive map. Let {αn} be a real sequence in [0, 1] satisfying {αn} ⊂ [δ, 1 − δ] for someδ ∈ (0, 1). For arbitrary x0 ∈ K , define the sequence {xn} by (1.9). Then {xn} converges stronglyto a fixed point of T .

Proof. Let x∗ ∈ K be a fixed point of T . Using the fact that T is hemicontractive we obtain

‖T xn − x∗‖2 ≤ ‖xn − x∗‖2 + ‖xn − T xn‖2. (3.1)

With the help of (1.9), Lemma 2 and (3.1), we obtain the following estimates:

‖xn − x∗‖2 = ‖αn xn−1 + (1 − αn)T xn − x∗‖2

= ‖αn(xn−1 − x∗) + (1 − αn)(T xn − x∗)‖2

= αn‖xn−1 − x∗‖2 + (1 − αn)‖T xn − x∗‖2

− αn(1 − αn)‖xn−1 − T xn‖2. (3.2)

Substituting (3.1) in (3.2), we get

‖xn − x∗‖2 ≤ αn‖xn−1 − x∗‖2 + (1 − αn)‖xn − x∗‖2

+ (1 − αn)‖xn − T xn‖2 − αn(1 − αn)‖xn−1 − T xn‖2. (3.3)

Also

‖xn − T xn‖2 = ‖αn xn−1 + (1 − αn)T xn − T xn‖2

= α2n‖xn−1 − T xn‖2. (3.4)

Substituting (3.4) in (3.3), we get

‖xn − x∗‖2 ≤ αn‖xn−1 − x∗‖2 + (1 − αn)‖xn − x∗‖2 − αn(1 − αn)2‖xn−1 − T xn‖2,

which implies

‖xn − x∗‖2 ≤ ‖xn−1 − x∗‖2 − (1 − αn)2‖xn−1 − T xn‖2,

and from the condition {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1), we conclude that the inequality

‖xn − x∗‖2 ≤ ‖xn−1 − x∗‖2 − δ2‖xn−1 − T xn‖2, (3.5)

holds for all fixed points x∗ of T . Moreover

δ2‖xn−1 − T xn‖2 ≤ ‖xn−1 − x∗‖2 − ‖xn − x∗‖2

A. Rafiq / Nonlinear Analysis 66 (2007) 2230–2236 2235

and thus

δ2∞∑j=0

‖x j−1 − T x j‖2 ≤∞∑j=0

(‖x j−1 − x∗‖2 − ‖x j − x∗‖2

)

= ‖x0 − x∗‖2.

Hence∞∑

j=0

‖x j−1 − T x j‖2 < ∞. (3.6)

This implies that

limn→∞ ‖xn−1 − T xn‖ = 0.

From (3.4) it further implies that

limn→∞ ‖xn − T xn‖ = 0.

By compactness of K this immediately implies that there is a subsequence {xn j } of {xn} whichconverges to a fixed point of T , say y∗. Since (3.5) holds for all fixed points of T we have

‖xn − y∗‖2 ≤ ‖xn−1 − y∗‖2 − δ2‖xn−1 − T xn‖2,

and in view of (3.6) and Lemma 1 we conclude that ‖xn − y∗‖ → 0 as n → ∞, i.e., xn → y∗as n → ∞. The proof is complete. �

Corollary 1. Let H , K , T , be as in Theorem 3 and {αn} be a real sequence in [0, 1] satisfying{αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1). Let PK : H → K be the projection operator of H ontoK . Then the sequence {xn} defined iteratively by

xn = PK (αn xn−1 + (1 − αn)T xn), n ≥ 1,

converges strongly to a fixed point of T .

Proof. The operator PK is nonexpansive (see, e.g., [2]). K is a Chebyshev subset of H , so PK isa single-valued map. Hence, we have the following estimate:

‖xn − x∗‖2 = ‖PK (αn xn−1 + (1 − αn)T xn) − PK x∗‖2

≤ ‖αn xn−1 + (1 − αn)T xn − x∗‖2

= ‖αn(xn−1 − x∗) + (1 − αn)(T xn − x∗)‖2

≤ αn‖xn−1 − x∗‖2 + (1 − αn)‖xn − x∗‖2 − αn(1 − αn)2‖xn−1 − T xn‖2,

which implies

‖xn − x∗‖2 ≤ ‖xn−1 − x∗‖2 − (1 − αn)2‖xn−1 − T xn‖2.

The set K ∪ T (K ) is compact and so the sequence {‖xn − T xn‖} is bounded. The rest of theargument follows exactly as in the proof of Theorem 3 and the proof is complete. �

References

[1] F.E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. PureMath. XVIII (2) (1976).

2236 A. Rafiq / Nonlinear Analysis 66 (2007) 2230–2236

[2] F.E. Browder, W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal.Appl. 20 (1967) 197–228.

[3] C.E. Chidume, Iterative approximation of Lipschitz strictly pseudocontractive mappings, Proc. Amer. Math. Soc.99 (2) (1987) 283–288.

[4] C.E. Chidume, Approximation of fixed points of strongly pseudocontractive mappings, Proc. Amer. Math. Soc. 120(2) (1994) 545–551.

[5] C.E. Chidume, Global iteration schemes for strongly pseudocontractive maps, Proc. Amer. Math. Soc. 126 (9)(1998) 2641–2649.

[6] C.E. Chidume, Iterative solution of nonlinear equations of strongly accretive type, Math. Nachr. 189 (1998) 49–60.[7] C.E. Chidume, C. Moore, Fixed point iteration for pseudocontractive maps, Proc. Amer. Math. Soc. 127 (4) (1999)

1163–1170.[8] C.E. Chidume, M.O. Osilike, Ishikawa iteration process for nonlinear Lipschitz strongly accretive mappings,

J. Math. Anal. Appl. 192 (1995) 727–741.[9] C.E. Chidume, M.O. Osilike, Nonlinear accretive and pseudocontractive operator equations in Banach spaces,

Nonlinear Anal. 31 (7) (1998) 779–789.[10] C.E. Chidume, S.A. Mutangadura, An example on the Mann iteration method for Lipschitz pseudocontractions,

Proc. Amer. Math. Soc. 129 (8) (2001) 2359–2363.[11] M.G. Crandall, A. Pazy, On the range of accretive operators, Israel J. Math. 27 (1977) 235–246.[12] L. Deng, On Chidume’s open problems, J. Math. Anal. Appl. 174 (2) (1993) 441–449.[13] L. Deng, Iteration process for nonlinear Lipschitzian strongly accretive mappings in L p spaces, J. Math. Anal.

Appl. 188 (1994) 128–140.[14] L. Deng, X.P. Ding, Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth

Banach spaces, Nonlinear Anal. 24 (7) (1995) 981–987.[15] T.L. Hicks, J.R. Kubicek, On the Mann iteration process in Hilbert space, J. Math. Anal. Appl. 59 (1977) 498–504.[16] S. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 4 (1) (1974) 147–150.[17] L.S. Liu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach

spaces, J. Math. Anal. Appl. 194 (1995) 114–125.[18] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506–610.[19] L. Qihou, On Naimpally and Singh’s open questions, J. Math. Anal. Appl. 124 (1987) 157–164.[20] L. Qihou, The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings, J. Math.

Anal. Appl. 148 (1990) 55–62.[21] L. Qihou, Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive

mappings, Nonlinear Anal. 26 (11) (1996) 1835–1842.[22] S. Reich, An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal. 2 (1978)

85–92.[23] S. Reich, Constructive techniques for accretive and monotone operators, in: Applied Nonlinear Analysis, Academic

Press, New York, 1979, pp. 335–345.[24] S. Reich, Constructing zeros of accretive operators, I II, Appl. Anal. 9 (1979) 159–163.[25] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979)

274–276.[26] S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl.

75 (1980) 287–292.[27] B.E. Rhoades, Comments on two fixed point iteration procedures, J. Math. Anal. Appl. 56 (1976) 741–750.[28] J. Schu, On a theorem of C.E. Chidume concerning the iterative approximation of fixed points, Math. Nachr. 153

(1991) 313–319.[29] J. Schu, Iterative construction of fixed points of strictly pseudocontractive mappings, Appl. Anal. 40 (1991) 67–72.[30] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158

(1991) 407–413.[31] K.K. Tan, H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,

J. Math. Anal. Appl. 178 (1993) 301–308.[32] X.L. Weng, Fixed point iteration for local strictly pseudocontractive mappings, Proc. Amer. Math. Soc. 113 (1991)

727–731.[33] Y. Xu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations,

J. Math. Anal. Appl. 224 (1998) 91–101.[34] Z.B. Xu, G.F. Roach, A necessary and sufficient condition for convergence of steepest descent approximation to

accretive operator equations, J. Math. Anal. Appl. 167 (1992) 340–354.