Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces

Journal of Computational and Applied Mathematics 228 (2009) 458–465

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Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces Yeol Je Cho a , Xiaolong Qin b,∗ a

Department of Mathematics Educational and RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea

b

Department of Mathematics, Gyeongsang National University, Chinju 660-701, Republic of Korea

article

a b s t r a c t

info

Article history: Received 24 August 2007 Received in revised form 14 March 2008

In this paper, we introduce a modified Ishikawa iterative process for approximating a fixed point of nonexpansive mappings in Hilbert spaces. We establish some strong convergence theorems of the general iteration scheme under some mild conditions. The results improve and extend the corresponding results of many others. © 2008 Elsevier B.V. All rights reserved.

MSC: 47H09 47H10 Keywords: Nonexpansive mapping Strong convergence Fixed point Hilbert space Modified Ishikawa iterative process

1. Introduction and preliminaries Let H be a real Hilbert space and T a nonlinear mapping on H. Recall that T is nonexpansive if

kTx − Tyk ≤ kx − yk,

∀x, y ∈ H .

A point x ∈ H is a fixed point of T provided Tx = x. Denote by F (T ) the set of fixed points of T , that is, F (T ) = {x ∈ H : Tx = x}. Recall that a self-mapping f is a contraction on H if there exists a constant α ∈ (0, 1) such that

kf (x) − f (y)k ≤ αkx − yk,

∀x, y ∈ H .

An operator A is said to be strongly positive if there exists a constant γ¯ > 0 such that

hAx, xi ≥ γ¯ kxk2 ,

∀x ∈ H .

(1.1)

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems (see [1, 5,17–19] and the references therein). A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping on a real Hilbert space H: min x∈C

∗

1 2

hAx, xi − hx, bi,

Corresponding author. E-mail addresses: [email protected] (Y.J. Cho), [email protected] (X. Qin).

0377-0427/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2008.10.004

(1.2)

Y.J. Cho, X. Qin / Journal of Computational and Applied Mathematics 228 (2009) 458–465

459

where C is the fixed point set of a nonexpansive mapping S, A is a strongly positive linear bounded self-adjoint operator and b is a given point in H. In Xu [18], it is proved that the sequence {xn } defined by



x0 ∈ H arbitrary chosen, xn+1 = (I − αn A)Sxn + αn b,

∀n ≥ 0,

converges strongly to the unique solution of the minimization problem (1.2) provided the sequence {αn } satisfies certain conditions. Recently, Marino and Xu [5] introduced a new iterative scheme by the viscosity approximation method which was first introduced by Modafi [7] as follows:



x0 ∈ H arbitrary chosen, xn+1 = (I − αn A)Sxn + αn γ f (xn ),

∀n ≥ 0.

They proved the sequence {xn } generated by the above iterative scheme converges strongly to the unique solution of the variational inequality

h(A − γ f )x∗ , x − x∗ i ≥ 0,

∀x ∈ C ,

which is the optimality condition for the minimization problem min x∈C

1 2

hAx, xi − h(x),

(1.3)

where C is the fixed point set of a nonexpansive mapping S and h is a potential function for γ f (i.e., h0 (x) = γ f (x) for all x ∈ H .) Two classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one is introduced by Mann [6] and is defined as follows:



x0 ∈ H arbitrary chosen, xn+1 = (1 − αn )xn + αn Txn ,

(1.4)

∀n ≥ 0,

where the sequence {αn } is in the interval (0, 1). The second iteration process is referred to as Ishikawa’s iteration process [3] which is defined recursively by x0 ∈ H arbitrary chosen, yn = βn xn + (1 − βn )Txn , xn+1 = αn xn + (1 − αn )Tyn ,

(

(1.5)

∀n ≥ 0,

where {αn } and {βn } are sequences in the interval (0, 1). But both (1.4) and (1.5) have only weak convergence, in general (see [2] for an example). For example, P Reich [11] shows that, if E is a uniformly convex and has a Fréchet differentiable norm and the sequence {αn } is such that ∞ n=0 αn (1 − αn ) = ∞, then the sequence {xn } generated by the process (1.4) converges weakly to a point in F (T ) (an extension of this result to the process (1.5) can be found in [15]). Therefore, many authors attempt to modify (1.4) and (1.5) to have strong convergence (see [4,8–10,13,14,16,20,21] and the references therein). In this paper, motivated by Kim and Xu [4], Marino and Xu [5] and Tan and Xu [15], we introduce a composite iterative algorithm {xn } defined as follows:

 x ∈ H arbitrary chosen,   0 zn = γn xn + (1 − γn )Txn ,  yn = βn xn + (1 − βn )Tzn , xn+1 = αn γ f (xn ) + δn xn + ((1 − δn )I − αn A)yn ,

(1.6)

∀n ≥ 0 ,

where f is a contraction, T is a nonexpansive mapping and A is a strongly positive linear bounded self-adjoint operator, and prove, under certain appropriate assumptions on the sequences {αn }, {βn }, {γn } and {δn }, that {xn } defined by (1.6) converges strongly to a fixed point of T , which solves some variational inequality and is also the optimality condition for the minimization problem (1.3). Next, we consider some special cases of the iterative scheme. (1) If {γn } = 1 for all n ≥ 0 in (1.6), then (1.6) reduces to x0 ∈ H arbitrary chosen, yn = βn xn + (1 − βn )Txn , xn+1 = αn γ f (xn ) + δn xn + ((1 − δn )I − αn A)yn ,

(

∀n ≥ 0.

(1.7)

(2) If {βn } = 0 and {γn } = 1 for all n ≥ 0 in (1.6), then (1.6) collapses to



x0 ∈ H arbitrary chosen, xn+1 = αn γ f (xn ) + δn xn + ((1 − δn )I − αn A)Txn ,

∀n ≥ 0 .

(1.8)

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(3) If {γn } = γ = 1 and {βn } = 0 for all n ≥ 0 and A = I (: the identity mapping) in (1.6), then (1.6) reduces to



x0 ∈ H arbitrary chosen, xn+1 = αn f (xn ) + δn xn + (1 − δn − αn )Txn ,

∀n ≥ 0.

(1.9)

Our purpose in this paper is to introduce the composite iterative algorithm for approximating a fixed point of nonexpansive mappings, which solves some variational inequality. We establish the strong convergence of the composite iteration algorithm {xn } defined by (1.6). The main results improve and extend the corresponding results announced by many others. In order to prove our main results, we also need the following lemmas. Lemma 1.1. In a Hilbert space H, the following inequality holds

kx + yk2 ≤ kxk2 + 2hy, (x + y)i,

∀ x, y ∈ H .

Lemma 1.2 ([12]). Let {xn } and {yn } be bounded sequences in a Banach space X and {βn } be a sequence in [0, 1] with 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1. Suppose xn+1 = (1 − βn )yn + βn xn for all integers n ≥ 0 and lim sup(kyn+1 − yn k − kxn+1 − xn k) ≤ 0. n→∞

Then limn→∞ kyn − xn k = 0. Lemma 1.3 ([17]). Assume that {αn } is a sequence of nonnegative real numbers such that

αn+1 ≤ (1 − γn )αn + δn , where {γn } is a sequence in (0, 1) and {δn } is a sequence such that (i)

P∞

n=1

γn = ∞;

P∞ (ii) lim supn→∞ γδn ≤ 0 or n=1 |δn | < ∞. n Then limn→∞ αn = 0. Lemma 1.4 ([5]). Assume that A is a strongly positive linear bounded self-adjoint operator on a Hilbert space H with coefficient γ¯ > 0 and 0 < ρ ≤ kAk−1 . Then kI − ρ Ak ≤ 1 − ρ γ¯ . Lemma 1.5 ([5]). Let H be a Hilbert space. Let A be a strongly positive linear bounded self-adjoint operator with coefficient γ¯ > 0. γ¯ Assume that 0 < γ < α . Let T be a nonexpansive mapping with a fixed point xt of the contraction x 7→ t γ f (x) + (1 − tA)Tx. Then {xt } converges strongly as t → 0 to a fixed point x¯ of T , which solves the variational inequality

h(A − γ f )¯x, x¯ − z i ≤ 0,

∀z ∈ F (T ).

2. Main results Now, we are ready to give our main results in this paper. Theorem 2.1. Let H be a real Hilbert space H, f be a contraction on H and T be a nonexpansive mapping such that F (T ) 6= ∅. Let A be a strongly positive linear bounded self-adjoint operator with the coefficient γ¯ > 0 such that kAk ≤ 1. Assume that γ¯ 0 < γ < α . Let {αn }, {βn }, {γn } and {δn } be sequences in [0, 1] satisfying the following conditions: (C1) n=0 αn = ∞ and limn→∞ αn = 0; (C2) 0 < lim infn→∞ δn ≤ lim supn→∞ δn < 1; (C3) (1 + βn )γn − 2βn > a for some a ∈ (0, 1), as n → ∞; (C4) limn→∞ |βn+1 − βn | = limn→∞ |γn+1 − γn | = 0.

P∞

Then the sequence {xn } defined by (1.6) converges strongly to q ∈ F (T ), where q = PF (T ) (γ f + (I − A))q and which also solves the following variational inequality

hγ f (q) − Aq, p − qi ≤ 0,

∀p ∈ F (T ).

Y.J. Cho, X. Qin / Journal of Computational and Applied Mathematics 228 (2009) 458–465

461

Proof. Since αn → 0 as n → ∞ by condition (C1), we may assume, without loss of generality, that αn < (1 − δn )kAk−1 for all n ≥ 0. Noticing that A is a linear bounded self-adjoint operator, one has

kAk = sup{|hAx, xi| : x ∈ H , kxk = 1}. Observing that

h((I − δn )I − αn A)x, xi = 1 − δn − αn hAx, xi ≥ 1 − δn − αn kAk ≥ 0, we obtain (1 − δn )I − αn A is positive. It follows that

k(1 − δn )I − αn Ak = sup{h((1 − δn )I − αn A)x, xi : x ∈ H , kxk = 1} = sup{1 − δn − αn hAx, xi : x ∈ H , kxk = 1} ≤ 1 − δn − αn γ¯ . Next, we observe that {xn } is bounded. Indeed, take a fixed point p of T and notice that

kzn − pk ≤ γn kxn − pk + (1 − γn )kTxn − pk ≤ kxn − pk and

kyn − pk ≤ βn kxn − pk + (1 − βn )kTzn − pk ≤ βn kxn − pk + (1 − βn )kzn − pk ≤ kxn − pk. It follows that

kxn+1 − pk = ≤ ≤ =

kαn (γ f (xn ) − Ap) + δn (xn − p) + ((1 − δn )I − αn A)(yn − p)k αn kγ f (xn ) − Apk + δn kxn − pk + (1 − δn − αn γ¯ )kyn − pk αn γ kf (xn ) − f (p)k + αn kγ f (p) − Apk + (1 − αn γ¯ )kxn − pk [1 − αn (γ¯ − γ α)]kxn − pk + αn kγ f (p) − Apk.

By simple inductions, we have

  kAp − γ f (p)k kxn − pk ≤ max kx0 − pk, , γ¯ − γ α which gives that the sequence {xn } is bounded, so are {yn } and {zn }. Next, we claim that lim kxn+1 − xn k = 0.

n→∞

(2.1)

Observing that



zn = γn xn + (1 − γn )Txn , zn−1 = γn−1 xn−1 + (1 − γn−1 )Txn−1 ,

we obtain zn − zn−1 = (1 − γn )(Txn − Txn−1 ) + γn (xn − xn−1 ) + (γn−1 − γn )(Txn−1 − xn−1 ). It follows that

kzn − zn−1 k ≤ kxn − xn−1 k + |γn−1 − γn |kTxn−1 − xn−1 k.

(2.2)

Noticing that



yn = βn xn + (1 − βn )Tzn , yn−1 = βn−1 xn−1 + (1 − βn−1 )Tzn−1 ,

we obtain yn − yn−1 = (1 − βn )(Tzn − Tzn−1 ) + βn (xn − xn−1 ) + (Tzn−1 − xn−1 )(βn−1 − βn ). It follows that

kyn − yn−1 k ≤ (1 − βn )kTzn − Tzn−1 k + βn kxn − xn−1 k + kTzn−1 − xn−1 k|βn−1 − βn | ≤ (1 − βn )kzn − zn−1 k + βn kxn − xn−1 k + kTzn−1 − xn−1 k|βn−1 − βn |.

(2.3)

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Y.J. Cho, X. Qin / Journal of Computational and Applied Mathematics 228 (2009) 458–465

Substituting (2.2) into (2.3), we get

kyn − yn−1 k ≤ kxn − xn−1 k + M1 (|γn−1 − γn | + |βn−1 − βn |),

(2.4)

where M1 is an appropriate constant such that





M1 ≥ max sup{kxn−1 − Txn−1 k, } sup{kxn−1 − Tzn−1 k} n≥1

n ≥1 x

−δ x

for all n ≥ 0. Put ln = n+11−δ n n . That is, xn+1 = (1 − δn )ln + δn xn . n Now, we compute ln+1 − ln . Observing that

αn γ f (xn ) + ((1 − δn )I − αn A)yn αn+1 γ f (xn+1 ) + ((1 − δn+1 )I − αn+1 A)yn+1 − 1 − δn+1 1 − δn αn αn+1 (γ f (xn+1 ) − Ayn+1 ) + (Ayn − γ f (xn )) + yn+1 − yn , = 1 − δn+1 1 − δn

ln+1 − ln =

one has

kln+1 − ln k ≤

αn+1 1 − δn+1

kγ f (xn+1 ) − Ayn+1 k +

αn 1 − δn

kAyn − γ f (xn )k + kyn+1 − yn k,

which combines with (2.4) yields that

kln+1 − ln k ≤

αn+1 αn kγ f (xn+1 ) − Ayn+1 k + kAyn − γ f (xn )k 1 − δn+1 1 − δn + kxn − xn+1 k + M1 (|γn−1 − γn | + |βn−1 − βn |).

It follows that

kln+1 − ln k − kxn − xn+1 k ≤

αn+1 1 − δn+1

kγ f (xn+1 ) − Ayn+1 k +

αn 1 − δn

kAyn − γ f (xn )k

+ M1 (|γn+1 − γn | + |βn+1 − βn |). Observing conditions (C1) and (C4) and taking the superior limit as n → ∞ get lim sup(kln+1 − ln k − kxn+1 − xn k) ≤ 0. n→∞

We can obtain limn→∞ kln − xn k = 0 easily by Lemma 1.2. Since xn+1 − xn = (1 − δn )(ln − xn ), one obtains that (2.1) holds. Observing that xn − yn = xn − xn+1 + xn+1 − yn

= xn − xn+1 + αn (γ f (xn ) − Ayn ) + δn (xn − yn ), we arrive at

(1 − δn )(xn − yn ) = xn − xn+1 + αn (γ f (xn ) − Ayn ). That is,

(1 − δn )kxn − yn k ≤ kxn − xn+1 k + αn kγ f (xn ) − Ayn k. From conditions (C1), (C2) and (2.1), it follows that lim kyn − xn k = 0.

n→∞

On the other hand, one has

kTxn − xn k ≤ ≤ ≤ ≤ ≤ ≤

kxn − yn k + kyn − Txn k kxn − yn k + kyn − Tzn k + kTzn − Txn k kxn − yn k + βn kxn − Tzn k + kzn − xn k kxn − yn k + βn kxn − Txn k + βn kTxn − Tzn k + kzn − xn k kxn − yn k + βn kxn − Txn k + (1 + βn )kzn − xn k kxn − yn k + βn kxn − Txn k + (1 − γn )(1 + βn )kTxn − xn k,

(2.5)

Y.J. Cho, X. Qin / Journal of Computational and Applied Mathematics 228 (2009) 458–465

463

which implies

[(1 + βn )γn − 2βn ]kTxn − xn k ≤ kxn − yn k. From condition (C3) and (2.5), it follows that lim kTxn − xn k = 0.

(2.6)

n→∞

Observe that PF (T ) (γ f + (I − A)) is a contraction. Indeed, for all x, y ∈ H, we have

kPF (S ) (γ f + (I − A))(x) − PF (S ) (γ f + (I − A))(y)k ≤ ≤ ≤ =

k(γ f + (I − A))(x) − (γ f + (I − A))(y)k γ kf (x) − f (y)k + kI − Akkx − yk γ αkx − yk + (1 − γ¯ )kx − yk [1 − (¯x − γ α)]kx − yk.

Banach’s Contraction Mapping Principle guarantees that PF (S ) (γ f + (I − A)) has a unique fixed point, say q ∈ H. That is, q = PF (S ) (γ f + (I − A))(q). Next, we claim that lim suphγ f (q) − Aq, xn − qi ≤ 0,

(2.7)

n→∞

where q = limt →0 xt with xt being the fixed point of the contraction x 7→ t γ f (x) + (I − tA)Tx. Noticing that xt solves the fixed point equation xt = t γ f (xt ) + (I − tA)Txt . It follows that

kxt − xn k = k(I − tA)(Txt − xn ) + t (γ f (xt ) − Axn )k. It follows from Lemma 1.1 that

kxt − xn k2 = k(I − tA)(Txt − xn ) + t (γ f (xt ) − Axn )k2 ≤ (1 − γ¯ t )2 kTxt − xn k2 + 2t hγ f (xt ) − Axn , xt − xn i ≤ (1 − 2γ¯ t + (γ¯ t )2 )kxt − xn k2 + fn (t ) + 2t hγ f (xt ) − Axt , xt − xn i + 2t hAxt − Axn , xt − xn i,

(2.8)

where fn (t ) = (2kxt − xn k + kxn − Txn k)kxn − Txn k → 0 (n → 0).

(2.9)

It follows from (1.1) and (2.8) that

hAxt − γ f (xt ), xt − xn i ≤

γ¯ t 2

hAxt − Axn , xt − xn i +

1 2t

fn (t ).

(2.10)

Letting n → ∞ in (2.10) and noting (2.9) yield lim suphAxt − γ f (xt ), xt − xn i ≤ n→∞

t 2

M2 ,

(2.11)

where M2 > 0 is an appropriate constant such that M2 ≥ γ¯ hAxt − Axn , xt − xn i for all t ∈ (0, 1) and n ≥ 0. Taking t → 0 in (2.11), we have lim sup lim suphAxt − γ f (xt ), xt − xn i ≤ 0. t →0

(2.12)

n→∞

On the other hand, one has

hγ f (q) − Aq, xn − qi = hγ f (q) − Aq, xn − qi − hγ f (q) − Aq, xn − xt i + hγ f (q) − Aq, xn − xt i − hγ f (q) − Axt , xn − xt i + hγ f (q) − Axt , xn − xt i − hγ f (xt ) − Axt , xn − xt i + hγ f (xt ) − Axt , xn − xt i. It follows that lim suphγ f (q) − Aq, xn − qi ≤ kγ f (q) − Aqkkxt − qk + (kAk + γ α)kxt − qk lim kxn − xt k n→∞

n→∞

+ lim suphγ f (xt ) − Axt , xn − xt i. n→∞

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Therefore, from (2.12), we have lim suphγ f (q) − Aq, xn − qi = lim sup lim suphγ f (q) − Aq, xn − qi n→∞

t →0

n→∞

≤ lim sup lim suphγ f (xt ) − Axt , xn − xt i t →0

n→∞

≤ 0. Hence (2.7) holds. Now, from Lemma 1.1, it follows that

kxn+1 − qk2 = ≤ ≤ ≤ ≤

k((1 − δn )I − αn A)(yn − q) + δn (xn − q) + αn (γ f (xn ) − Aq)k2 k((1 − δn )I − αn A)(yn − q) + δn (xn − q)k2 + 2αn hγ f (xn ) − Aq, xn+1 − qi [(1 − δn − αn γ¯ )kyn − qk + δn kxn − qk]2 + 2αn hγ f (xn ) − Aq, xn+1 − qi (1 − αn γ¯ )2 kxn − qk2 + 2αn γ αkxn − qkkxn+1 − qk + 2αn hγ f (q) − Aq, xn+1 − qi (1 − αn γ¯ )2 kxn − qk2 + αn γ α(kxn − qk2 + kxn+1 − qk2 ) + 2αn hγ f (q) − Aq, xn+1 − qi,

which implies that

(1 − αn γ¯ )2 + αn γ α 2αn kxn − qk2 + hγ f (q) − Aq, xn+1 − qi 1 − αn γ α 1 − αn γ α (1 − 2αn γ¯ + αn αγ ) αn2 γ¯ 2 2αn = kxn − qk2 + kxn − qk2 + hγ f (q) − Aq, xn+1 − qi 1 − αn γ α 1 − αn γ α 1 − αn γ α   2αn (γ¯ − αγ ) ≤ 1− kxn − qk2 1 − αn γ α   2αn (γ¯ − αγ ) 1 αn γ¯ 2 + hγ f (q) − Aq, xn+1 − qi + M3 , 1 − αn γ α γ¯ − αγ 2(γ¯ − αγ )

kxn+1 − qk2 ≤

where M3 is an appropriate constant such that M3 = supn→∞ {kxn − qk} for all n ≥ 0. Put ln = tn = γ¯ −αγ hγ f (q) − Aq, xn+1 − qi + 1

γ¯ 2

αn M . 2(γ¯ −αγ ) 3

2αn (γ¯ −αγ ) 1−αn αγ

and

Then we have

kxn+1 − qk2 ≤ (1 − ln )kxn − qk + ln tn .

(2.13)

It follows from condition (C1) and (2.7) that lim ln = 0,

n→∞

∞ X

ln = ∞,

n=1

lim sup tn ≤ 0. n→∞

Applying Lemma 1.3 to (2.13), it follows that xn → q as n → ∞. This completes the proof.



As corollaries of Theorem 2.1, we have the following results immediately: Corollary 2.2. Let H be a real Hilbert space H, f be a contraction on H and T be a nonexpansive mapping such that F (T ) 6= ∅. Let γ¯ A be a strongly positive linear bounded self-adjoint operator with coefficient γ¯ > 0 such that kAk ≤ 1. Assume that 0 < γ < α . Let {αn }, {βn } and {δn } be sequences in [0, 1] satisfying the following conditions: (C1) n=0 αn = ∞ and limn→∞ αn = 0; (C2) 0 < lim infn→∞ δn ≤ lim supn→∞ δn < 1; (C3) 1 − βn > a for some a ∈ (0, 1); (C4) limn→∞ |βn+1 − βn | = 0.

P∞

Then the sequence {xn } defined by (1.7) converges strongly to q ∈ F (T ), where q = PF (T ) (γ f + (I − A))q and which also solves some variational inequality

hγ f (q) − Aq, p − qi ≤ 0,

∀p ∈ F (T ).

Corollary 2.3. Let H be a real Hilbert space H, f be a contraction on H and T be a nonexpansive mapping such that F (T ) 6= ∅. Let γ¯ A be a strongly positive linear bounded self-adjoint operator with coefficient γ¯ > 0 such that kAk ≤ 1. Assume that 0 < γ < α . Let {αn } and {δn } be sequences in [0, 1] satisfying the following conditions: (C1) n=0 αn = ∞ and limn→∞ αn = 0; (C2) 0 < lim infn→∞ δn ≤ lim supn→∞ δn < 1.

P∞

Y.J. Cho, X. Qin / Journal of Computational and Applied Mathematics 228 (2009) 458–465

465

Then the sequence {xn } defined by (1.8) converges strongly to q ∈ F (T ), where q = PF (T ) (γ f + (I − A))q and which also solves the following variational inequality

hγ f (q) − Aq, p − qi ≤ 0,

∀p ∈ F (T ).

Corollary 2.4. Let H be a real Hilbert space H, f be a contraction on H and T be a nonexpansive mapping such that F (T ) 6= ∅. Let {αn } and {δn } be sequences in [0, 1] satisfying the following conditions: (C1) n=0 αn = ∞ and limn→∞ αn = 0; (C2) 0 < lim infn→∞ δn ≤ lim supn→∞ δn < 1.

P∞

Then the sequence {xn } defined by (1.9) converges strongly to q ∈ F (T ), where q = PF (T ) f (q) and which also solves the following variational inequality

hf (q) − q, p − qi ≤ 0,

∀p ∈ F (T ).

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