Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

Volume 2010, Article ID 281070,13pages doi:10.1155/2010/281070

Research Article

Weak and Strong Convergence Theorems for

Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

Jing Zhao

and Songnian He

1College of Science, Civil Aviation University of China, Tianjin 300300, China

2Tianjin Key Laboratory For Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China

Correspondence should be addressed to Jing Zhao,[email protected] Received 23 June 2010; Accepted 19 October 2010

Academic Editor: W. A. Kirk

Copyrightq2010 J. Zhao and S. He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the convergence of Ishikawa iteration process for the class of asymptoticallyκ-strict pseudocontractive mappings in the intermediate sense which is not necessarily Lipschitzian. Weak convergence theorem is established. We also obtain a strong convergence theorem by using hybrid projection for this iteration process. Our results improve and extend the corresponding results announced by many others.

1. Introduction and Preliminaries

Throughout this paper, we always assume thatHis a real Hilbert space with inner product

·,·and norm · . and → denote weak and strong convergence, respectively.ωwxn denotes the weakω-limit set of{xn}, that is, ωwxn {x ∈ H : ∃xnj x}. Let Cbe a nonempty closed convex subset ofH. It is well known that for every pointx∈H, there exists a unique nearest point inC, denoted byPCx, such that

x−P_Cx ≤x−y, 1.1 for ally∈C.P_Cis called the metric projection ofHontoC.P_Cis a nonexpansive mapping of HontoCand satisfies

x−y, PCx−PCy

≥PCx−PCy², ∀x, y∈H. 1.2

LetT:C → Cbe a mapping. In this paper, we denote the fixed point set ofTbyFT. Recall thatT is said to be uniformlyL-Lipschitzian if there exists a constantL >0, such that

Tⁿx−Tⁿy≤Lx−y, ∀x, y∈C, ∀n≥1. 1.3

Tis said to be nonexpansive if

Tx−Ty≤x−y, ∀x, y∈C. 1.4 T is said to be asymptotically nonexpansive if there exists a sequence {kn}in 1,∞with lim_{n→ ∞}kn1, such that

Tⁿx−Tⁿy≤knx−y, ∀x, y∈C, ∀n≥1. 1.5 The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk1 as a generalization of the class of nonexpansive mappings.T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

lim sup

n→ ∞ sup

x,y∈C

Tⁿx−Tⁿy−x−y≤0. 1.6

Observe that if we define

τnmax

0,sup

x,y∈C

Tⁿx−Tⁿy−x−y

, 1.7

thenτn → 0 asn → ∞. It follows that1.6is reduced to

Tⁿx−Tⁿy≤x−yτn, ∀x, y∈C, ∀n≥1. 1.8 The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al.2. It is known3that ifCis a nonempty closed convex bounded subset of a uniformly convex Banach spaceEandT is asymptotically nonexpansive in the intermediate sense, thenThas a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings.

Recall thatT is said to be aκ-strict pseudocontraction if there exists a constantκ ∈ 0,1, such that

Tx−Ty²≤x−y²κI−Tx−I−Ty², ∀x, y∈C. 1.9

T is said to be an asymptoticallyκ-strict pseudocontraction with sequence{γn}if there exist a constantκ∈0,1and a sequence{γ_n} ⊂0,∞withγ_n → 0 asn → ∞, such that

Tⁿx−Tⁿy² ≤

1γ_nx−y²κI−Tⁿx−I−Tⁿy², ∀x, y∈C, n≥1.

1.10 The class of asymptoticallyκ-strict pseudocontractions was introduced by Qihou4in 1996 see also5. Kim and Xu6studied 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 sup{κ

1 1−κγn/1κ:n∈N}.

Recently, Sahu et al. 7 introduced a class of new mappings: asymptotically κ- strict pseudocontractive mappings in the intermediate sense. Recall thatT is said to be an asymptotically κ-strict pseudocontraction in the intermediate sense with sequence {γn} if there exist a constant κ ∈ 0,1and a sequence {γ_n} ⊂ 0,∞ with γ_n → 0 as n → ∞, such that

lim sup

n→ ∞ sup

x,y∈C

Tⁿx−Tⁿy²−

1γ_nx−y²−κI−Tⁿx−I−Tⁿy²

≤0. 1.11

Throughout this paper, we assume that

c_nmax

0,sup

x,y∈C

Tⁿx−Tⁿy²−

1γ_nx−y²−κI−Tⁿx−I−Tⁿy² .

1.12

It follows thatc_n → 0 asn → ∞and1.11is reduced to the relation Tⁿx−Tⁿy²≤

1γnx−y²κI−Tⁿx−I−Tⁿy²cn, ∀x, y∈C. 1.13 They obtained a weak convergence theorem of modified Mann iterative processes for the class of mappings which is not necessarily Lipschitzian. Moreover, a strong convergence theorem was also established in a real Hilbert space by hybrid projection methods; see7for more details.

In this paper, we consider the problem of convergence of Ishikawa iterative processes for the class of asymptoticallyκ-strict pseudocontractive mappings in the intermediate sense.

In order to prove our main results, we also need the following lemmas.

Lemma 1.1 see 8, 9. Let {δn}, {βn}, and {γn} be three sequences of nonnegative numbers satisfying the recursive inequality

δn1≤βnδnγn, ∀n≥1. 1.14

Ifβ_n≥1,_∞

n1β_n−1<∞and_∞

n1γ_n<∞, then lim_{n→ ∞}δ_nexists.

Lemma 1.2see10. Let{xn}be a bounded sequence in a reflexive Banach spaceX. Ifωwxn

{x}, thenx_n x.

Lemma 1.3see11. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Given x∈Handz∈C, thenzPCxif and only ifx−z, y−z ≤0, for ally∈C.

Lemma 1.4see11. For a real Hilbert spaceH, the following identities hold:

ix−y²x²− y²−2x−y, y, for allx, y∈H,

iitx 1−ty²tx²1−ty²−t1−tx−y²,for allt∈0,1, for allx, y∈H;

iii(Opial condition) If{x_n}is a sequence inHweakly convergent toz, then lim sup

n→ ∞

x_n−y²lim sup

n→ ∞ x_n−z²z−y², ∀y∈H. 1.15 Lemma 1.5 see 7. LetC be a nonempty subset of a Hilbert space H and T : C → C an asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense with sequence{γ_n}. Then

Tⁿx−Tⁿy≤ 1 1−κ

κx−y

1 1−κγ_nx−y² 1−κc_n

,

∀x, y∈C, ∀n∈N.

1.16

Lemma 1.6. LetCbe a nonempty subset of a Hilbert spaceH andT : C → Can asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence{γ_n}. Letn∈N. Ifγ_n<1, then

Tⁿx−Tⁿy≤ 1 1−κ

κ√

2−κx−y√ cn

, ∀x, y∈C. 1.17

Proof. Ifγn<1, forx, y∈C, we obtain fromLemma 1.5that Tⁿx−Tⁿy≤ 1

1−κ

κx−y

1 1−κγ_nx−y² 1−κc_n

≤ 1 1−κ

κx−y

2−κx−y²c_n

≤ 1 1−κ

κx−y√

2−κx−y√ cn

₂

1 1−κ

κ√

2−κx−y√c_n .

1.18

Lemma 1.7see7. LetCbe a nonempty subset of a Hilbert spaceHandT :C → Ca uniformly continuous asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense with sequence {γn}. Let{xn}be a sequence inCsuch thatxn−xn1 → 0 andxn−Tⁿxn → 0 asn → ∞, thenx_n−Tx_n → 0 asn → ∞.

Lemma 1.8 see7, Proposition 3.1. Let C be a nonempty closed convex subset of a Hilbert spaceH and T : C → Ca continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense. ThenI−T is demiclosed at zero in the sense that if{xn}is a sequence inCsuch thatxn x∈Cand lim sup_{m→ ∞}lim sup_{n→ ∞}xn−T^mxn0, thenI−Tx0.

Lemma 1.9see7. LetCbe a nonempty closed convex subset of a Hilbert spaceHandT :C → C a continuous asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense. ThenFT is closed and convex.

2. Main Results

Theorem 2.1. LetC be a nonempty closed convex subset of a Hilbert spaceH and T : C → C a uniformly continuous asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense with sequence {γn} such thatFT/∅. Let{xn}^∞_n1 be a sequence inCgenerated by the following Ishikawa iterative process:

x1∈C, ynβnTⁿxn

1−βn xn, x_n1α_nTⁿy_n 1−α_nx_n, ∀n≥1,

2.1

where{α_n}and{β_n}are sequences in0,1. Assume that the following restrictions are satisfied:

i_∞

n1α_nc_n<∞and_∞

n11γ_n²−1<∞,

ii0 < a ≤ αn ≤ βn ≤ b for some a > 0 and b ∈ 0,−1 − κ² 1−κ⁴2κ√

2−κ²1−κ²/2κ√

2−κ².

Then the sequence{xn}given by2.1converges weakly to an element ofFT. Proof. Letp∈FT. From1.13andLemma 1.4, we see that

y_n−p²β_nTⁿx_n−p 1−β_nx_n−p² βnTⁿxn−p²

1−βnxn−p²−βn 1−βn

xn−Tⁿxn²

≤β_n

1γ_nx_n−p²κx_n−Tⁿx_n²c_n

1−β_nx_n−p²−β_n 1−β_n

x_n−Tⁿx_n²

≤

1γnxn−p²−βn

1−βn−κ

xn−Tⁿxn²βncn.

2.2

Without loss of generality, we may assume thatγn<1 for alln∈N. Since

xn−yn²xn−βnTⁿxn−1−βnxn²β_n²xn−Tⁿxn², 2.3

it follows fromLemma 1.6that

yn−Tⁿyn²βnTⁿxn−Tⁿyn 1−βnxn−Tⁿyn² βnTⁿxn−Tⁿyn²

1−βnxn−Tⁿyn²−βn 1−βn

xn−Tⁿxn²

≤ β_n 1−κ²

κ√

2−κx_n−y_n√c_n2

1−βnxn−Tⁿyn²−βn 1−βn

xn−Tⁿxn²

≤2β³_n

κ√ 2−κ 1−κ

₂

xn−Tⁿxn² 2βncn

1−κ²

1−β_nx_n−Tⁿy_n²−β_n 1−β_n

x_n−Tⁿx_n².

2.4

By2.2and2.4, we obtain that Tⁿy_n−p²

≤

1γnyn−p²κyn−Tⁿyn²cn

≤

1γn₂xn−p²−βn

1γn

1−βn−κ

xn−Tⁿxn²

β_n 1γ_n

c_n2κβ_n³

κ√ 2−κ 1−κ

₂

x_n−Tⁿx_n² 2κβncn

1−κ² κ

1−β_nx_n−Tⁿy_n²−κβ_n 1−β_n

x_n−Tⁿx_n²c_n

1γ_n2x_n−p²−β_n

⎡

⎣1γ_n

1−β_n−κ

−2κβ²_n

κ√ 2−κ 1−κ

₂ κ

1−β_n⎤

⎦

× xn−Tⁿxn²κ

1−βnxn−Tⁿyn²cnM1,

2.5

whereM1sup_n≥1{βn1γn 2κβn/1−κ²1}. It follows from2.5andαn≤βnthat xn1−p²

α_nTⁿy_n−p 1−α_nx_n−p²

αnTⁿyn−p² 1−αnxn−p²−αn1−αnTⁿyn−xn²

≤αn

1γn₂xn−p²−αnβn

⎡

⎣1γn

1−βn−κ

−2κβ²_n

κ√ 2−κ 1−κ

₂ κ

1−βn⎤

⎦

× xn−Tⁿxn²αnκ

1−βnxn−Tⁿyn²

α_nc_nM1 1−α_nx_n−p²−α_n1−α_nTⁿy_n−x_n²

≤

1γ_n2x_n−p²−α_nβ_n

⎡

⎣1γ_n 1−β_n

−κγ_n−2κβ²_n

κ√ 2−κ 1−κ

₂

−κβ_n

⎤

⎦

× xn−Tⁿxn²−αn

1−αn−κ

1−βnxn−Tⁿyn²αncnM1

≤

1γn₂xn−p²−αnβn

⎡

⎣1γn 1−βn

−κγn−2κβ²_n

κ√ 2−κ 1−κ

₂

−κβn

⎤

⎦

× xn−Tⁿx_n²α_nc_nM1.

2.6

From the conditioniiandγ_n → 0, we see that there existsn0such that 1γn

1−βn

−κγn−2κβ²_n

κ√ 2−κ 1−κ

₂

−κβn

≥1−β_n−κγ_n−2β²_n

κ√ 2−κ 1−κ

₂

−κβ_n

≥1−2βn−κγn−2β²_n

κ√ 2−κ 1−κ

2

≥1−2b−2b²

κ√ 2−κ 1−κ

₂

−κγ_n

≥ 1 2

⎛

⎝1−2b−2b²

κ√ 2−κ 1−κ

₂⎞

⎠>0, ∀n≥n0.

2.7

By2.6, we have

xn1−p² ≤

1γn₂xn−p²αncnM1, ∀n≥n0. 2.8 In view ofLemma 1.1and the conditioni, we obtain that limn→ ∞xn−pexists. For any n≥n0, it is easy to see from2.6and2.7that

a² 2

⎛

⎝1−2b−2b²

κ√ 2−κ 1−κ

₂⎞

⎠x_n−Tⁿx_n²

≤

1γ_n2x_n−p²−x_n1−p²α_nc_nM1,

2.9

which implies that

nlim→ ∞x_n−Tⁿx_n0. 2.10

Note that

x_n1−x_nα_nTⁿy_n−x_n

≤α_nTⁿy_n−Tⁿx_nα_nTⁿx_n−x_n

≤ αn

1−κ

κ√

2−κxn−yn√ cn

αnTⁿxn−xn

α_nβ_n 1−κ

κ√ 2−κ

x_n−Tⁿx_nα_n√c_n

1−κ α_nTⁿx_n−x_n.

2.11

From2.10, we have

nlim→ ∞xn1−xn0. 2.12

SinceT is uniformly continuous, we obtain from2.10,2.12andLemma 1.7that

nlim→ ∞xn−Txn0. 2.13

By the boundedness of {xn}, there exist a subsequence {xnk} of {xn} such that xnk x.

Observe thatT is uniformly continuous andx_n−Tx_n → 0 asn → ∞, for anym∈Nwe havexn−T^mxn → 0 asn → ∞. FromLemma 1.8, we see thatx∈FT.

To complete the proof, it suffices to show thatωw{xn}consists of exactly one point, namely,x. Suppose there exists another subsequence{x_nj}of{x_n}such that{x_nj}converges

weakly to somez∈Candz /x. As in the case ofx, we can also see thatz∈FT. It follows that lim_{n→ ∞}x_n−xand lim_{n→ ∞}x_n−zexist. SinceHsatisfies the Opial condition, we have

nlim→ ∞xn−x lim

k→ ∞xnk−x< lim

k→ ∞xnk−z lim

n→ ∞xn−z,

n→ ∞limx_n−z lim

j→ ∞

x_nj−z< lim

j→ ∞

x_nj−x lim

n→ ∞x_n−x, 2.14

which is a contradiction. We see x z and hence ω_w{x_n} is a singleton. Thus, {x_n} converges weakly toxbyLemma 1.2.

Corollary 2.2. LetC be a nonempty closed convex subset of a Hilbert spaceH and T : C → C a uniformly continuous asymptoticallyκ-strict pseudocontractive mapping with sequence{γ_n}such thatFT/∅. Let{xn}^∞_n1be a sequence inCgenerated by the following Ishikawa iterative process:

x1∈C, ynβnTⁿxn

1−βn xn, x_n1α_nTⁿy_n 1−α_nx_n, ∀n≥1,

2.15

where{αn}and{βn}are sequences in0,1. Assume that the following restrictions are satisfied:

i_∞

n11γn²−1<∞,

ii0 < a ≤ α_n ≤ β_n ≤ b for some a > 0 and b ∈ 0,−1 − κ² 1−κ⁴2κ√

2−κ²1−κ²/2κ√

2−κ².

Then the sequence{xn}given by2.15converges weakly to an element ofFT.

Next, we modify Ishikawa iterative process to get a strong convergence theorem.

Theorem 2.3. LetC be a nonempty closed convex subset of a Hilbert spaceH and T : C → C a uniformly continuous asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense with sequence{γ_n}such thatFT/∅and bounded. Let{α_n}and{β_n}are sequences in0,1. Let {x_n}^∞_n1be a sequence inCgenerated by the modified Ishikawa iterative process:

x1∈C, y_nβ_nTⁿx_n

1−β_n x_n, znαnTⁿyn 1−αnxn, C_n

z∈C:z_n−z²≤ xn−z²θ_n−ρ_nx_n−Tⁿx_n² , Qn{z∈C:xn−z, x1−xn ≥0},

xn1PCn∩Qnx1,

2.16

where θn αncnM1 2γn γ_n²Δn, M1 sup_n≥1{βn1 γn 2κβn/1− κ² 1}, Δn sup{x_n−z²:z∈FT}<∞andρ_nα_nβ_n1−2β_n−κγ_n−2β²_nκ√

2−κ/1−κ²for each

n≥1. Assume that the control sequences{αn}and{βn}are chosen such that 0< a≤αn≤βn≤bfor somea > 0 andb∈0,−1−κ²

1−κ⁴2κ√

2−κ²1−κ²/2κ√

2−κ². Then the sequence{xn}given by2.16converges strongly to an element ofFT.

Proof. We break the proof into six steps.

Step 1Cn∩Qnis closed and convex for eachn≥1. It is obvious thatQnis closed and convex andCnis closed for eachn ≥1. Note that the defining inequality inCn is equivalent to the inequality

2x_n−z_n, z ≤ x_n²− zn²θ_n−ρ_nx_n−Tⁿx_n², 2.17

it is easy to see thatCnis convex for eachn≥1. Hence,Cn∩Qnis closed and convex for each n≥1.

Step 2FT⊂C_n∩Q_nfor eachn≥1. Letp∈FT. Following2.6,2.7and the algorithm 2.16, we have

zn−p² ≤

1γn₂xn−p²

−α_nβ_n

⎡

⎣1γ_n 1−β_n

−κγ_n−2κβ²_n

κ√ 2−κ 1−κ

₂

−κβ_n

⎤

⎦

× xn−Tⁿxn²αncnM1

≤

1γ_n2x_n−p²−α_nβ_n

⎡

⎣1−2β_n−2β²_n

κ√ 2−κ 1−κ

₂

−κγ_n

⎤

⎦

× xn−Tⁿx_n²α_nc_nM1

xn−p²−ρnxn−Tⁿxn²αncnM1

2γnγ_n²xn−p²

≤xn−p²−ρnxn−Tⁿxn²θn,

2.18

where θ_n α_nc_nM1 2γ_n γ_n²Δ_n, M1 sup_n≥1{β_n1 γ_n 2κβ_n/1 −κ² 1}, Δ_n sup{xn−z² :z∈FT}<∞andρnαnβn1−2βn−κγn−2β²_nκ√

2−κ/1−κ²for eachn≥1. Hencep∈C_nfor eachn≥1.

Next, we show thatFT⊂Qnfor eachn≥1. We prove this by induction. Forn1, we haveFT⊂CQ1. Assume thatFT⊂Qnfor somen >1. Sincexn1is the projection ofx1ontoC_n∩Q_n, we have

xn1−z, x1−xn1 ≥0, ∀z∈Cn∩Qn. 2.19

By the induction consumption, we know thatFT⊂Cn∩Qn. In particular, for anyp∈FT we have

xn1−p, x1−xn1

≥0. 2.20

This implies thatp∈Qn1. That is,FT⊂Qn1. By the principle of mathematical induction, we get FT ⊂ Q_n and henceFT ⊂ C_n ∩Q_n for alln ≥ 1. This means that the iteration algorithm2.16is well defined.

Step 3limn→ ∞xn−x1exists and{xn}is bounded. In view of2.16, we see thatxnPQnx1

andx_n1P_Cn∩Qnx1 ∈Q_n. It follows that

xn−x1 ≤ xn1−x1 2.21 for eachn≥1. We, therefore, obtain that the sequence{x_n−x1}is nondecreasing. Noticing thatFT⊂Q_nandx_nP_Qnx1, we have

x1−x_n ≤x1−p, ∀p∈FT. 2.22 This shows that the sequence{x_n−x1}is bounded. Therefore, the limit of{x_n−x1}exists and{x_n}is bounded.

Step 4xn1−xn → 0. Observe thatxnPQnx1andxn1 ∈Qnwhich imply

x_n1−x_n, x1−x_n ≤0. 2.23

UsingLemma 1.4, we obtain

x_n1−x_n² x_n1−x1−x_n−x1²

x_n1−x1²− xn−x1²−2x_n1−x_n, x_n−x1

≤ xn1−x1²− xn−x1².

2.24

Hence, we obtain thatx_n1−x_n → 0 asn → ∞.

Step 5x_n−Tx_n → 0 asn → ∞. In view ofx_n1∈C_n, we have

z_n−x_n1² ≤ xn−x_n1²θ_n−ρ_nx_n−Tⁿx_n². 2.25

On the other hand, we see that

zn−xn1² zn−xnxn−xn1²

zn−xn²xn−xn1²2zn−xn, xn−xn1. 2.26

Combing2.25and2.26and notingznαnTⁿyn 1−αnxn, we obtain that α²_nTⁿy_n−x_n²2

α_n

Tⁿy_n−x_n

, x_n−x_n1

≤θ_n−ρ_nx_n−Tⁿx_n². 2.27

From the assumption and2.7, we see that there existsn0∈Nsuch that

1−2β_n−κγ_n−2β²_n

κ√ 2−κ 1−κ

₂

≥ 1 2

⎛

⎝1−2b−2b²

κ√ 2−κ 1−κ

2⎞

⎠>0, ∀n≥n0.

2.28

For anyn≥n0, it follows from the definition ofρ_nand2.27that

a² 2

⎛

⎝1−2b−2b²

κ√ 2−κ 1−κ

₂⎞

⎠x_n−Tⁿx_n²≤θ_n2α_nTⁿy_n−x_n· xn−x_n1. 2.29

Noting thatθ_n → 0 asn → ∞andStep 4, we obtain that

nlim→ ∞xn−Tⁿxn0. 2.30

It follows fromStep 4,2.30andLemma 1.7thatxn−Txn → 0 asn → ∞.

Step 6 x_n → x ∈ FT asn → ∞, where x P_FTx1. Since H is reflexive and {x_n} is bounded, we get that ω_w{x_n}is nonempty. First, we show that ω_w{x_n}is a singleton.

Assume that{xni}is subsequence of{xn}such thatxni x∈C. Observe thatTis uniformly continuous andx_n−Tx_n → 0 asn → ∞, for anym ∈ Nwe havex_n−T^mx_n → 0 as n → ∞. FromLemma 1.8, we see thatx∈ω_w{x_n}⊂FT.

Sincexn1 PCn∩Qnx1, we obtain that

x1−x_n1 ≤x1−P_FTx1, 2.31 for eachn≥1. Observe thatx1−x_ni x1−xasn → ∞. By the weak lower semicontinuity of norm, we have

x1−P_FTx1≤ x1−x ≤lim inf

n→ ∞ x1−x_ni ≤lim sup

n→ ∞ x1−x_ni ≤x1−P_FTx1.

2.32

This implies that

x1−PFTx1x1−x, 2.33

nlim→ ∞x1−xnix1−PFTx1. 2.34 Hencex PFTx1 by the uniqueness of the nearest point projection ofx1 ontoFT. Since {xni} is an arbitrary weakly convergent subsequence, it follows thatωw{xn} {x}and hencex_n x. It is easy to see as2.34thatx1−x_n → x1−x. SinceHhas the Kadec-Klee property, we obtain thatx1−xn → x1−x, that is,xn → xPFTx1asn → ∞. This completes the proof.

Acknowledgments

This research is supported by Fundamental Research Funds for the Central Universities ZXH2009D021 and supported by the Science Research Foundation Program in Civil Aviation University of Chinano. 09CAUC-S05as well.

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