Asymptotically strict pseudocontractive mappings in the intermediate sense

Nonlinear Analysis 70 (2009) 3502–3511

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Asymptotically strict pseudocontractive mappings in the intermediate sense D.R. Sahu a , Hong-Kun Xu b,∗ , Jen-Chih Yao b a

Department of Mathematics, Banaras Hindu University, Varanasi-221005, India

b

Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan

article

info

Article history: Received 14 April 2008 Accepted 15 July 2008 MSC: primary 47H09 secondary 46B20 47H10

a b s t r a c t It is proved that the modified Mann iteration process: xn+1 = (1 − αn )xn + αn T n xn , n ∈ N, where {αn } is a sequence in (0, 1) with δ ≤ αn ≤ 1 − κ − δ for some δ ∈ (0, 1), converges weakly to a fixed point of an asymptotically κ -strict pseudocontractive mapping T in the intermediate sense which is not necessarily Lipschitzian. We also develop CQ method for this modified Mann iteration process which generates a strongly convergent sequence. © 2008 Elsevier Ltd. All rights reserved.

Keywords: Demiclosedness principle Asymptotically nonexpansive mapping Asymptotically strict pseudocontractive mapping Metric projection

1. Introduction Let C be a nonempty subset of a normed space X and T : C → C a mapping. Recall the following concepts. (i) T is nonexpansive if

kTx − Tyk ≤ kx − yk for all x, y ∈ C . (ii) T is asymptotically nonexpansive (cf. [7]) if there exists a sequence {kn } of positive numbers satisfying the property limn→∞ kn = 1 and

kT n x − T n yk ≤ kn kx − yk for all integers n ≥ 1 and x, y ∈ C . (iii) T is uniformly Lipschitzian if there exists a constant L > 0 such that

kT n x − T n yk ≤ Lkx − yk for all integers n ≥ 1 and all x, y ∈ C .

∗

Corresponding author. Tel.: +886 7 5252000; fax: +886 7 525 3809. E-mail addresses: [email protected] (D.R. Sahu), [email protected] (H.-K. Xu), [email protected] (J.-C. Yao).

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.07.007

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(iv) T is asymptotically nonexpansive in the intermediate sense [2] provided T is uniformly continuous and lim sup sup (kT n x − T n yk − kx − yk) ≤ 0. n→∞

x,y∈C

It is clear that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is uniformly Lipschitzian. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [7] as an important generalization of the class of nonexpansive mappings. The existence of fixed points of asymptotically nonexpansive mappings was proved by Goebel and Kirk [7] as below: Theorem 1.1 (Theorem 1, Goebel and Kirk [7]). If C is a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive mapping T : C → C has a fixed point in C . An iterative method for the approximation of fixed points of asymptotically nonexpansive mappings was developed by Schu [20] in the following interesting result: Theorem 1.2 (Schu [20]). Let C be a nonempty closed convex bounded P∞ subset of a Hilbert space H and T : C → C an asymptotically nonexpansive mapping with sequence {kn } such that n=1 (kn − 1) < ∞. Let {αn } be a sequence in [0, 1] satisfying the condition δ ≤ αn ≤ 1 − δ for all n ∈ N and for some δ > 0. Then the sequence {xn } generated from arbitrary x1 ∈ C by xn+1 = (1 − αn )xn + αn T n xn

for all ∈ N

(1.1)

converges weakly to a fixed point of T . Iterative methods for approximation of fixed points of asymptotically nonexpansive mappings have been further studied by authors (see e.g. [3–5,10,14,17–19,21,23,24,27] and references therein). The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck, Kuczumow and Reich [2] and iterative methods for the approximation of fixed points of such types of non-Lipschitzian mappings have been studied by Agarwal, O’Regan and Sahu [1], Bruck, Kuczumow and Reich [2], Chidume, Shahzad and Zegeye [6], Kim and Kim [11] and many others. Recently, Kim and Xu [13] introduced the concept of asymptotically κ -strict pseudocontractive mappings in Hilbert space as below: Definition 1.3. Let C be a nonempty subset of a Hilbert space H. A mapping T : C → C is said to be an asymptotically κ -strict pseudocontractive mapping with sequence {γn } if there exist a constant κ ∈ [0, 1) and a sequence {γn } in [0, ∞) with limn→∞ γn = 0 such that

kT n x − T n yk2 ≤ (1 + γn )kx − yk2 + κkx − T n x − (y − T n y)k2

(1.2)

for all x, y ∈ C and n ∈ N. They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically κ -strict pseudocontractive mapping with sequence {γn } is a uniformly L-Lipschitzian mapping with L = n o sup

√ κ+ 1+(1−κ)γn 1+κ

:n∈N .

In this paper we will study some properties and convergence of some iteration processes for the class of asymptotically

κ -strict pseudocontractive mappings in the intermediate sense which are not necessarily Lipschitzian.

Definition 1.4. Let C be a nonempty subset of a Hilbert space H. A mapping T : C → C will be called an asymptotically κ -strict pseudocontractive mapping in the intermediate sense with sequence {γn } if there exist a constant κ ∈ [0, 1) and a sequence {γn } in [0, ∞) with limn→∞ γn = 0 such that lim sup sup (kT n x − T n yk2 − (1 + γn )kx − yk2 − κkx − T n x − (y − T n y)k2 ) ≤ 0. n→∞

x,y∈C

(1.3)

Throughout this paper we assume that cn := max{0, sup (kT n x − T n yk2 − (1 + γn )kx − yk2 − κkx − T n x − (y − T n y)k2 )}. x,y∈C

Then cn ≥ 0 for all n ∈ N, cn → 0 as n → ∞ and (1.3) reduces to the relation

kT n x − T n yk2 ≤ (1 + γn )kx − yk2 + κkx − T n x − (y − T n y)k2 + cn for all x, y ∈ C and n ∈ N.

(1.4)

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Remark 1.5. (1) T is not necessarily uniformly L-Lipschitzian (see Lemma 2.6). (2) When cn = 0 for all n ∈ N in (1.4) then T is an asymptotically κ -strict pseudocontractive mapping with sequence {γn }. Example 1.6. Let X = R and C = [0, 1]. For each x ∈ C , we define kx, 0,

 Tx =

if x ∈ [0, 1/2], if x ∈ (1/2, 1],

where 0 < k < 1. Then T : C → C is discontinuous at x = 1/2 and hence T is not Lipschitzian. Set C1 := [1, 1/2] and C2 := (1/2, 1]. Hence

|T n x − T n y| = kn |x − y| ≤ |x − y| for all x, y ∈ C1 and n ∈ N and

|T n x − T n y| = 0 ≤ |x − y| for all x, y ∈ C2 and n ∈ N. For x ∈ C1 and y ∈ C2 , we have

|T n x − T n y| = |kn x − 0| = |kn (x − y) + kn y| ≤ kn |x − y| + kn |y| ≤ |x − y| + kn for all n ∈ N. Thus,

|T n x − T n y|2 ≤ (|x − y| + kn )2 ≤ |x − y|2 + k|x − T n x − (y − T n y)|2 + kn K for all x, y ∈ C , n ∈ N and for some K > 0. Therefore, T is an asymptotically κ -strict pseudocontractive mapping in the intermediate sense. The paper is organized as follows: In Section 2, we will recall the useful definitions and lemmas. In Section 3, we study the demiclosedness principle and weak convergence of the modified Mann iteration process (1.1) for the class of asymptotically κ -strict pseudocontractive mappings in the intermediate sense. In Section 4, we give a further modification of the modified Mann iteration process (1.1) for asymptotically κ -strict pseudocontractive mappings in the intermediate sense which generates a strongly convergent sequence. The results presented in this paper are more general than the known results in the context of the class of asymptotically κ -strict pseudocontractive mappings in the intermediate sense. 2. Preliminaries Let H be a real Hilbert space with inner product h·, ·i and norm k · k, respectively and let C be a closed convex subset of H. For every point x ∈ H, there exists a unique nearest point in C , denoted by PC x, such that

kx − PC xk ≤ kx − yk for all y ∈ C . PC is called the metric projection of H onto C . We know that PC is a nonexpansive mapping of H onto C . It is also known that PC satisfies

kPC x − PC yk2 ≤ hPC x − PC y, x − yi for every x, y ∈ H . We will adopt the following notations: 1. * for weak convergence and → for strong convergence. 2. wω ({xn }) = {x : ∃ xnj * x} denotes the weak w -limit set of {xn }. 3. F (T ) = {x ∈ C : Tx = x} denotes the set of fixed points of a self-mapping T on a set C . We need some facts and tools which are listed as lemmas below: Lemma 2.1 ([18,22]). Let {δn }, {βn } and {γn } be three sequences of nonnegative numbers satisfying the recursive inequality:

δn+1 ≤ βn δn + γn for all n ∈ N. P∞ P∞ If βn ≥ 1, n=1 (βn − 1) < ∞ and n=1 γn < ∞, then limn→∞ δn exists. Lemma 2.2 (Proposition 2.4, Agarwal, O’Regan and Sahu [1]). Let {xn } be a bounded sequence in a reflexive Banach space X . If

ωw ({xn }) = {x}, then xn * x.

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Lemma 2.3. Let C be closed convex subset of a real Hilbert space H and let PC be the metric projection mapping from H onto C . Given x ∈ H and z ∈ C , then z = PC x

hx − z , y − z i ≤ 0 for all y ∈ C .

if and only if

Lemma 2.4. Let H be a real Hilbert space. Then the following hold: (a) kx − yk2 = kxk2 − kyk2 − 2hx − y, yi for all x, y ∈ H . (b) k(1 − t )x + tyk2 = (1 − t )kxk2 + t kyk2 − t (1 − t )kx − yk2 for all t ∈ [0, 1] and for all x, y ∈ H. (c) If {xn } is a sequence in H such that xn * x, it follows that lim sup kxn − yk2 = lim sup kxn − xk2 + kx − yk2 n→∞

n→∞

for all y ∈ H .

(2.1)

Lemma 2.5 (cf. [25]). Let H be a real Hilbert space. Given a closed convex subset of H and points x, y, z ∈ H and given also a real number a ∈ R, the set

{v ∈ C : ky − vk2 ≤ kx − vk2 + hz , vi + a} is convex (and closed). Lemma 2.6. Let C be a nonempty subset of a Hilbert space H and T : C → C an asymptotically κ -strict pseudocontractive mapping in the intermediate sense with sequence {γn }. Then

kT n x − T n yk ≤

1



1−κ

κkx − yk +

p

(1 + (1 − κ)γn )kx − yk2 + (1 − κ)cn



for all x, y ∈ C and n ∈ N. Proof. For x, y ∈ C , we have

kT n x − T n yk2 ≤ (1 + γn )kx − yk2 + κkx − T n x − (y − T n y)k2 + cn ≤ (1 + γn )kx − yk2 + κ(kx − yk + kT n x − T n yk)2 + cn ≤ (1 + κ + γn )kx − yk2 + κ(2kx − yk kT n x − T n yk + kT n x − T n yk2 ) + cn . It gives us that

(1 − κ)kT n x − T n yk2 − 2κkx − yk kT n x − T n yk − (1 + κ + γn )kx − yk2 − cn ≤ 0, which is a quadratic inequality in kT n x − T n yk. Hence the result follows.



Lemma 2.7. Let C be a nonempty subset of a Hilbert space H and T : C → C a uniformly continuous asymptotically κ -strict pseudocontractive mapping in the intermediate sense with sequence {γn }. Let {xn } be a sequence in C such that kxn − xn+1 k → 0 and kxn − T n xn k → 0 as n → ∞. Then kxn − Txn k → 0 as n → ∞. Proof. Since T is an asymptotically κ -strict pseudocontractive mapping in the intermediate sense, we obtain from Lemma 2.6 that

k T n +1 x n − T n +1 x n +1 k ≤

1 1−κ

  p κkxn − xn+1 k + (1 + (1 − κ)γn+1 )kxn − xn+1 k2 + (1 − κ)cn+1 .

Note that kxn − xn+1 k → 0 which implies that kT n+1 xn − T n+1 xn+1 k → 0. Observe that

kxn − Txn k ≤ kxn − xn+1 k + kxn+1 − T n+1 xn+1 k + kT n+1 xn+1 − T n+1 xn k + kT n+1 xn − Txn k.

(2.2)

By the uniform continuity of T , we have

kTxn − T n+1 xn k → 0 as n → ∞. Since xn − T n xn → 0 and xn − xn+1 → 0, it follows from (2.2) and (2.3) that limn→∞ kxn − Txn k = 0.

(2.3)

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3. Weak convergence of the modified Mann iteration process First, we give some basic properties of asymptotically κ -strict pseudocontractive mappings in the intermediate sense. Proposition 3.1 (Demiclosedness principle). Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C a continuous asymptotically κ -strict pseudocontractive mapping in the intermediate sense. Then I − T is demiclosed at zero in the sense that if {xn } is a sequence in C such that xn * x ∈ C and lim supm→∞ lim supn→∞ kxn − T m xn k = 0, then (I − T )x = 0. Proof. Assume that T is a continuous asymptotically κ -strict pseudocontractive mapping in the intermediate sense with sequence {γn }. Let {xn } be a sequence in C such that xn * x ∈ C and lim sup lim sup kxn − T m xn k = 0. m→∞

(3.1)

n→∞

By Lemma 2.6, we have

kT m x n − T m x k ≤

1 1−κ

  p κkxn − xk + (1 + (1 − κ)γm )kxn − xk2 + (1 − κ)cm

≤ K0 for all m, n ∈ N and for some constant K 0 > 0. Define

ϕ(x) := lim sup kxn − xk2 for all x ∈ H . n→∞

Since xn * x, it follows form (2.1) that

ϕ(y) = ϕ(x) + kx − yk2 for all y ∈ H .

(3.2)

Since T is an asymptotically κ -strict pseudocontractive mapping in the intermediate sense, by relation (1.4), we have

ϕ(T m x) = lim sup kxn − T m xk2 n→∞

≤ lim sup(kxn − T m xn k + kT m xn − T m xk)2 n→∞

≤ lim sup(kxn − T m xn k2 + kT m xn − T m xk2 + 2kxn − T m xn k kT m xn − T m xk) n→∞

≤ lim sup(kxn − T m xn k2 + kT m xn − T m xk2 + 2kxn − T m xn kK 0 ) n→∞

≤ lim sup kT m xn − T m xk2 + lim sup(kxn − T m xn k2 + 2kxn − T m xn kK 0 ) n→∞

n→∞

≤ lim sup((1 + γm )kxn − xk2 + κkxn − T m xn − (x − T m x)k2 + cm ) n→∞

+ lim sup(kxn − T m xn k2 + 2kxn − T m xn kK 0 ) n→∞

≤ ϕ(x) + κ lim sup kxn − T m xn − (x − T m x)k2 + ϕ(x)γm + cm n→∞

+ lim sup(kxn − T m xn k2 + 2kxn − T m xn kK 0 ) for all m ∈ N. n→∞

By (3.2), we have

ϕ(x) + kx − T m xk2 = ϕ(T m x) ≤ ϕ(x) + κ lim sup kxn − T m xn − (x − T m x)k2 + ϕ(x)γm + cm n→∞

+ lim sup(kxn − T m xn k2 + 2kxn − T m xn kK 0 ), n→∞

which implies that

kx − T m xk2 ≤ κ lim sup kxn − T m xn − (x − T m x)k2 + ϕ(x)γm + cm n→∞

+ lim sup(kxn − T m xn k2 + 2kxn − T m xn kK 0 ).

(3.3)

n→∞

Since lim supm→∞ lim supn→∞ kxn − T m xn k = 0, it follows from (3.3) that lim sup kx − T m xk2 ≤ κ lim sup kx − T m xk2 . m→∞

m→∞

It means that T x → x as m → ∞. Therefore, the continuity of T implies that (I − T )x = 0. m



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Remark 3.2. Proposition 3.1 extends the demiclosed principles studied for certain classes of nonlinear mappings in Gornicki [8], Kim and Xu [13], Marino and Xu [15] and Xu [26]. Proposition 3.3. Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C a continuous asymptotically κ -strict pseudocontractive mapping in the intermediate sense. Then F (T ) is closed and convex. Proof. Since T is continuous, F (T ) is closed. To see convexity of F (T ), consider x, y ∈ F (T ). Let z = (1 − t )x + ty for t ∈ (0, 1). Note that

kx − z k = t kx − yk and ky − z k = (1 − t )kx − yk. By Lemma 2.4(b), we have

kz − T n z k2 = k(1 − t )(x − T n z ) + t (y − T n z )k2 = (1 − t )kx − T n z k2 + t ky − T n z k2 − t (1 − t )kx − yk2 ≤ (1 − t )[(1 + γn )kx − z k2 + κkz − T n z k2 + cn ] + t [(1 + γn )ky − z k2 + κkz − T n z k2 + cn ] − t (1 − t )kx − yk2 ≤ (1 − t )[(1 + γn )t 2 kx − yk2 + κkz − T n z k2 + cn ] + t [(1 + γn )(1 − t )2 kx − yk2 + κkz − T n z k2 + cn ] − t (1 − t )kx − yk2 = κkz − T n z k2 + t (1 − t )γn kx − yk2 + cn . It means that T n z → z as n → ∞. The continuity of T implies that z ∈ F (T ).



We now prove the weak convergence of (1.1) for asymptotically κ -strict pseudocontractive mappings in the intermediate sense. Theorem 3.4. Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C a uniformly continuous asymptotically κ -strict pseudocontractive mapping in the intermediate sense with sequence {γn } such thatPF (T ) 6= ∅ and P∞ ∞ n=1 γn < ∞. Assume that {αn } is a sequence in (0, 1) such that 0 < δ ≤ αn ≤ 1 − κ − δ < 1 and n=1 αn cn < ∞. Let {xn }∞ be a sequence in C generated by the modified Mann iteration process: n=1 xn+1 = (1 − αn )xn + αn T n xn

for all n ∈ N.

(3.4)

Then {xn } converges weakly to an element of F (T ). Proof. Let p be an element in F (T ). Using Lemma 2.4(b), we obtain

kxn+1 − pk2 = = ≤ ≤

k(1 − αn )(xn − p) + αn (T n xn − p)k2 (1 − αn )kxn − pk2 + αn kT n xn − pk2 − αn (1 − αn )kxn − T n xn k2 (1 − αn )kxn − pk2 + αn [(1 + γn )kxn − pk2 + κkxn − T n xn k2 + cn ] − αn (1 − αn )kxn − T n xn k2 (1 + γn )kxn − pk2 − αn (1 − αn − κ)kxn − T n xn k2 + αn cn

≤ (1 + γn )kxn − pk2 − δ 2 kxn − T n xn k2 + αn cn ≤ (1 + γn )kxn − pk + αn cn . P∞ By Lemma 2.1, (3.6) and the assumption n=1 αn cn < ∞, we obtain that 2

lim kxn − pk exists.

n→∞

(3.5) (3.6)

(3.7)

Suppose limn→∞ kxn − pk = r for some r > 0. It is easy to see from (3.5) that

δ 2 kxn − T n xn k2 ≤ kxn − pk2 − kxn+1 − pk2 + αn cn , which implies that limn→∞ kxn − T n xn k = 0. Observe that

kxn+1 − xn k = αn kxn − T n xn k ≤ (1 − κ − δ)kxn − T n xn k → 0 as n → ∞. Since kxn+1 − xn k → 0, kxn − T n xn k → 0 as n → ∞ and T is uniformly continuous, we obtain from Lemma 2.7 that kxn − Txn k → 0 as n → ∞. By the boundedness of {xn }, there exists a subsequence {xnk } of {xn } such that xnk * x. Note that T is uniformly continuous and kxn − Txn k → 0, we see that kxn − T m xn k → 0 for all m ∈ N. By Proposition 3.1, we obtain x ∈ F (T ). To complete the proof, it suffices to show that ωw ({xn }) consists of exactly one point, namely, x. Suppose there exists another subsequence

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{xnk } of {xn } which converges weakly to some z 6= x. As in the case of x, we must have z ∈ F (T ). It follows from (3.7) that limn→∞ kxn − xk and limn→∞ kxn − z k exist. Since X satisfies the Opial condition, we have lim kxn − xk = lim kxnj − xk < lim kxnj − z k = lim kxn − z k,

n→∞

j→∞

j→∞

n→∞

lim kxn − z k = lim kxnk − z k < lim kxnk − z k = lim kxn − z k,

n→∞

k→∞

k→∞

n→∞

which is a contradiction. Hence x = z so ωw ({xn }) is a singleton. Thus, {xn } converges weakly to x by Lemma 2.2.



We remark that Theorem 3.4 is more general than the results studied in Huang and Lan [9], Kim and Xu [13], Marino and Xu [15] and Schu [20]. As a consequence of Theorem 3.4, we may derive the following result. Corollary 3.5. Let C be a nonempty closed convex subset of a Hilbert space H and T : C → C a uniformly continuous asymptotically κ -strict pseudocontractive mapping inPthe intermediate sense with F (T ) 6= ∅. Assume that {αn } is a sequence ∞ in (0, 1) such that 0 < δ ≤ αn ≤ 1 − κ − δ < 1 and n=1 αn cn < ∞. Let {xn }∞ n=1 be a sequence in C generated by the modified Mann iteration process defined by (3.4). Then {xn } converges weakly to an element of F (T ). Corollary 3.6 (Theorem 3.1, Kim and Xu [13]). Let C be a nonempty closed convex subset of a Hilbert P∞space H and T : C → C an asymptotically κ -strict pseudocontractive mapping with sequence {γn } such that F (T ) 6= ∅ and n=1 γn < ∞. Assume that {αn } is a sequence in (0, 1) such that 0 < δ ≤ αn ≤ 1 − κ − δ < 1. Let {xn }∞ n=1 be a sequence in C generated by the modified Mann iteration process defined by (3.4). Then {xn } converges weakly to an element of F (T ). 4. The CQ method for the Mann iteration process It is proved in Theorems 1.2 and 3.4 that the modified Mann iteration method (1.1) is in general not strongly convergent for either asymptotically nonexpansive mappings or uniformly continuous asymptotically κ -strict pseudocontractive mappings in the intermediate sense. So to get strong convergence, one has to modify the iteration method (1.1). The main result of this section is the following which is more general than Theorem 4.1 of Kim and Xu [13]. Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C a uniformly continuous asymptotically κ -strict pseudocontractive mapping in the intermediate sense with sequence {γn } such that F (T ) is nonempty and bounded. Let {αn } be a sequence in [0, 1] such that 0 < δ ≤ αn ≤ 1 − κ for all n ∈ N. Let {xn } be the sequence in C generated by the following (CQ) algorithm:

 u = x1 ∈ C chosen arbitrary,    yn = (1 − αn )xn + αn T n xn , Cn = {z ∈ C : kyn − z k2 ≤ kxn − z k2 + θn },    Qn = {z ∈ C : hxn − z , u − xn i ≥ 0}, xn+1 = PCn ∩Qn (u) for all n ∈ N, where θn = cn + γn ∆n and ∆n = sup{kxn − z k : z ∈ F (T )} < ∞. Then {xn } converges strongly to PF (T ) (u). Proof. We break the proof into the following six steps: Step 1. Cn is convex. Indeed, the defining inequality in Cn is equivalent to the inequality

h2(xn − yn ), vi ≤ kxn k2 − kyn k2 + θn , it follows from Lemma 2.5 that Cn is convex. Step 2. F (T ) ⊂ Cn . Let p ∈ F (T ). From (4.1), we have

kyn − pk2 = ≤ ≤ ≤ ≤

k(1 − αn )(xn − p) + αn (T n xn − p)k2 (1 − αn )kxn − pk2 + αn kT n xn − pk2 − αn (1 − αn )kxn − T n xn k2 (1 − αn )kxn − pk2 + αn ((1 + γn )kxn − pk2 + κkxn − T n xn k2 + cn ) − αn (1 − αn )kxn − T n xn k2 kxn − pk2 + αn (κ − (1 − αn ))kxn − T n xn k2 + cn + γn ∆n kxn − pk2 + cn + γn ∆n .

Hence p ∈ Cn . Step 3. F (T ) ⊂ Cn ∩ Qn for all n ∈ N. It is suffices to show that F (T ) ⊂ Qn . We prove this by induction.

(4.1)

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For n = 1, we have F (T ) ⊂ C = Q1 . Assume that F (T ) ⊂ Qn . Since xn+1 is the projection of u onto Cn ∩ Qn , it follows that

hxn+1 − z , u − xn+1 i ≥ 0 for all z ∈ Cn ∩ Qn . As F (T ) ⊂ Cn ∩ Qn , the last inequality holds, in particular for all z ∈ F (T ). By the definition Qn+1 , Qn+1 = {z ∈ C : hxn+1 − z , u − xn+1 i ≥ 0}, it follows that F (T ) ⊂ Qn+1 . By the principle of mathematical induction, we have F (T ) ⊂ Qn

for all n ∈ N.

Step 4. kxn − xn+1 k → 0. By the definition of Qn , we have xn = PQn (u) and

ku − xn k ≤ ku − yk for all y ∈ F (T ) ⊂ Qn . Note that boundedness of F (T ) implies that {kxn − uk} is bounded. Since xn = PQn (u) which together with the fact that xn+1 ∈ Cn ∩ Qn ⊆ Qn implies that

ku − xn k ≤ ku − xn+1 k. Thus, {kxn − uk} is increasing. Since {kxn − uk} is bounded, we obtain that limn→∞ kxn − uk exists. Observe that xn = PQn (u) and xn+1 ∈ Qn which imply that

hxn+1 − xn , xn − ui ≥ 0. Using Lemma 2.4(a), we obtain

kxn+1 − xn k2 = kxn+1 − u − (xn − u)k2 = kxn+1 − uk2 − kxn − uk2 − 2hxn+1 − xn , xn − ui ≤ kxn+1 − uk2 − kxn − uk2 → 0 as n → ∞. Step 5. kxn − Txn k → 0. By the definition of yn , we have

kxn − T n xn k = αn−1 kxn − yn k ≤ αn−1 (kxn − xn+1 k + kxn+1 − yn k) ≤ δ −1 (kxn − xn+1 k + kxn+1 − yn k).

(4.2)

Since xn+1 ∈ Cn , we have

kyn − xn+1 k2 ≤ kxn − xn+1 k2 + cn + γn ∆n → 0. It follows from (4.2) that

kxn − T n xn k → 0 as n → ∞.

(4.3)

By Step 4 and (4.3), we obtain from Lemma 2.7 that xn − Txn → 0 as n → ∞. Step 6. xn → v ∈ F (T ). Since H is reflexive and {xn } is bounded, we get that ωw ({xn }) is nonempty. First, we show that ωw ({xn }) is a singleton. Assume that {xni } is subsequence of {xn } such that xni * v ∈ C . Since xn − Txn → 0 by Step 4, it follows from the uniform continuity of T that xn − T m xn → 0 for all m ∈ N. By Proposition 3.1, v ∈ wω ({xn }) ⊂ F (T ). Since xn+1 = PCn ∩Qn (u), we obtain that

ku − xn+1 k ≤ ku − PF (T ) (u)k for all n ∈ N. Observe that u − xni * u − v. By the weak lower semicontinuity of norm,

ku − PF (T ) (u)k ≤ ku − vk ≤ lim inf ku − xni k i→∞

≤ lim sup ku − xni k i→∞

≤ ku − PF (T ) (u)k,

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which yields

ku − PF (T ) (u)k = ku − vk and lim ku − xni k = ku − PF (T ) (u)k.

i→∞

(4.4)

Hence v = PF (T ) (u) by the uniqueness of the nearest point projection of u onto F (T ). Thus, kxni − uk → kv − uk. It shows that xni − u → v − u, i.e., xni → v . Since {xni } is an arbitrary weakly convergent subsequence, it follows that ωw ({xn }) = {v} and hence from Proposition 2.2 we have xn * v . It is easy to see as (4.4) that kxn − uk → kv − uk. Therefore, xn → v . Corollary 4.2 (Theorem 2.2, Kim and Xu [12]). Let C be a nonempty closed convex bounded subset of a real Hilbert space H and T : C → C an asymptotically nonexpansive mapping with sequence {kn } in [1, ∞). Let {αn } be a sequence in [0, 1] such that 0 < δ ≤ αn ≤ 1. Define a sequence {xn }∞ n=1 in C by the following algorithm:

 u = x1 ∈ C chosen arbitrary,    yn = (1 − αn )xn + αn T n xn , Cn = {z ∈ C : kyn − z k2 ≤ kxn − z k2 + θn },    Qn = {z ∈ C : hxn − z , u − xn i ≥ 0}, xn+1 = PCn ∩Qn (u),

(4.5)

where θn = (k2n − 1) diam(C )2 for all n ∈ N. Then {xn } converges strongly to PF (T ) (u). Corollary 4.3 (Theorem 3.4, Nakajo and Takahashi [16]). Let C be a nonempty closed convex bounded subset of a real Hilbert space H and T : C → C a nonexpansive mapping with F (T ) 6= ∅. Let {αn } be a sequence in [0, 1] such that 0 < δ ≤ αn ≤ 1. Define a sequence {xn }∞ n=1 in C by the following algorithm:

 u = x1 ∈ C chosen arbitrary,    yn = (1 − αn )xn + αn T n xn , Cn = {z ∈ C : kyn − z k ≤ kxn − z k},   Qn = {z ∈ C : hxn − z , u − xn i ≥ 0},  xn+1 = PCn ∩Qn (u) for all n ∈ N.

(4.6)

Then {xn } converges strongly to PF (T ) (u). Acknowledgement The first author wishes to acknowledge the support of the Department of Science and Technology, India, given in the program year 2007-2008, Project No. SR/FTP/MS-04. References [1] R.P. Agarwal, Donal O’Regan, D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (1) (2007) 61–79. [2] R.E. Bruck, T. Kuczumow, S. Reich, Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloq. Math. 65 (1993) 169–179. [3] S.S. Chang, Y.J. Cho, H. Zhou, Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings, J. Korean Math. Soc. 38 (2001) 1245–1260. [4] C.E. Chidume, B. Ali, Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 326 (2) (2007) 960–973. [5] C.E. Chidume, E.U. Ofoedu, H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 280 (2003) 364–374. [6] C.E. Chidume, N. Shahzad, H. Zegeye, Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense, Numer. Funct. Anal. Optim. 25 (2004) 239–257. [7] K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1) (1972) 171–174. [8] J. Gornicki, Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces, Comment Math. Univ. Carolin. 30 (2) (1989) 249–252. [9] N.J. Huang, H.Y. Lan, A new iterative approximation of fixed points for asymptotically contractive type mappings in Banach spaces, Indian J. Pure Appl. Math. 35 (4) (2004) 441–453. [10] S.H. Khan, H. Fukharuddin, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal. 61 (2005) 1295–1301. [11] G.E. Kim, T.H. Kim, Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces, Comp. Math. Appl. 42 (2001) 1565–1570. [12] T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (5) (2006) 1140–1152. [13] T.H. Kim, H.K. Xu, Convergence of the modified Mann’s iteration method for asymptotically strict pseudocontractions, Nonlinear Anal. 68 (2008) 2828–2836. [14] Z.Q. Liu, S.M. Kang, Weak and strong convergence for fixed points of asymptotically nonexpansive mappings, Acta Math. Sinica 20 (2004) 1009–1018. [15] G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–346.

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