Iterative methods for strict pseudo-contractions in Hilbert spaces

Nonlinear Analysis 67 (2007) 2258–2271 www.elsevier.com/locate/na

Iterative methods for strict pseudo-contractions in Hilbert spaces Genaro Lopez Acedo a , Hong-Kun Xu b,∗ a Departmento de An´alisis, Facultad de Matem´aticas, Universidad de Sevilla, Apdo. 1160, 41080-Sevilla, Spain b School of Mathematical Sciences, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South Africa

Received 4 July 2006; accepted 31 August 2006

Abstract N be N strict pseudo-contractions defined on a closed convex subset C of a real Hilbert space H . Consider the problem Let {Ti }i=1 of finding a common fixed point of these mappings and consider the parallel and cyclic algorithms for solving this problem. We will prove the weak convergence of these algorithms. Moreover, by applying additional projections, we further prove that these algorithms can be modified to have strong convergence. c 2006 Elsevier Ltd. All rights reserved.

MSC: primary 47H09; secondary 65J15 Keywords: Strict pseudo-contraction; Iterative method; Parallel algorithm; Cyclic algorithm; Fixed point; Projection

1. Introduction Strict pseudo-contractions in Hilbert spaces were introduced by Browder and Petryshyn [2]. Given a closed convex subset C of a Hilbert space H , a mapping T : C → C is said to be a strict pseudo-contraction [2] if there exists a constant 0 ≤ κ < 1 such that kT x − T yk2 ≤ kx − yk2 + κk(I − T )x − (I − T )yk2

(1.1)

for all x, y ∈ C. (If (1.1) holds, we also say that T is a κ-strict pseudo-contraction.) These mappings are extensions of nonexpansive mappings which satisfy the inequality (1.1) with κ = 0. That is, T : C → C is nonexpansive if kT x − T yk ≤ kx − yk for all x, y ∈ C. Iterative methods for nonexpansive mappings have been extensively investigated; see [1,3,5,6,8,11–13,17–21,24, 26–31] and the references therein. Related work can also be found in [9,10,15,22,25,32]. However iterative methods for strict pseudo-contractions are far less developed than those for nonexpansive mappings though Browder and Petryshyn [2] initiated their work in 1967; the reason is probably that the second term appearing in the right-hand side of (1.1) impedes the convergence analysis for iterative algorithms used to find ∗ Corresponding author. Tel.: +27 31 260 7418; fax: +27 31 260 7806.

E-mail addresses: [email protected] (G.L. Acedo), [email protected] (H.-K. Xu). c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.08.036

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

2259

a fixed point of the strict pseudo-contraction T . However, on the other hand, strict pseudo-contractions have more powerful applications than nonexpansive mappings do in solving inverse problems (see Scherzer [23]). Therefore it is interesting to develop the theory of iterative methods for strict pseudo-contractions. As a matter of fact, Browder and Petryshyn [2] show that if a κ-strict pseudo-contraction T has a fixed point in C, then starting with an initial x0 ∈ C, the sequence {xn } generated by the recursive formula: xn+1 = αxn + (1 − α)T xn ,

(1.2)

where α is a constant such that κ < α < 1, converges weakly to a fixed point of T . Quite recently Marino and Xu [16] have extended Browder and Petryshyn’s above-mentioned result by proving that the sequence {xn } generated by the following Mann’s algorithm [14]: xn+1 = αn xn + (1 − αn )T xn ,

(1.3)

converges weakly to a fixed point of T , provided the control sequence {αn }∞ n=0 satisfies the conditions that κ < αn < 1 for all n and ∞ X

(αn − κ)(1 − αn ) = ∞.

(1.4)

n=0

This result can also be viewed as the Hilbert space version for strict pseudo-contractions of Reich’s Banach space result for nonexpansive mappings which states that if T is a nonexpansive self-mapping, with a fixed point, of a closed convex subset C of a uniformly convex Banach space with a Frechet differentiable norm, then the sequence {xn } generated by the Mann’s algorithm (1.3) converges weakly to a fixed point of T provided the sequence {αn } of parameters satisfies the conditions that 0 < αn < 1 for all n and that ∞ X

αn (1 − αn ) = ∞.

(1.5)

n=0

(Note that if T is nonexpansive, then T is κ-strict pseudo-contraction with κ = 0; hence condition (1.4) reduces to condition (1.5).) In this paper we are concerned with the problem of finding a point x such that x∈

N \

Fix(Ti ),

(1.6)

i=1 N are N strict pseudo-contractions defined on a closed convex subset C where N ≥ 1 is a positive integer and {Ti }i=1 of a Hilbert space H . Here Fix(Ti ) = {z ∈ C : Ti z = z} is the set of fixed points of Ti , 1 ≤ i ≤ N . Let T be defined by

T =

N X

λi Ti

i=1

PN where λi > 0 for all i such that i=1 λi = 1. We will see that T is a strict pseudo-contraction on C and N Fix(T ). Hence Marino and Xu’s result [16] applies to T . That is, we will show that the sequence Fix(T ) = ∩i=1 i {xn } generated by the algorithm: xn+1 = αn xn + (1 − αn )

N X

λi Ti xn

(1.7)

i=1

will converge weakly to a solution to the problem (1.6). N in (1.7) to depend on n, the Moreover, we shall consider a more general situation by allowing the weights {λi }i=1 number of steps of the iteration. That is, we consider the algorithm which generates a sequence {xn } in the following

2260

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

way: xn+1 = αn xn + (1 − αn )

N X

(n)

λi Ti xn .

(1.8)

i=1 (n)

N we shall also prove the weak convergence, Under appropriate assumptions on the sequences of the weights {λi }i=1 to a solution of the problem (1.6), of the algorithm (1.8). N Another approach to the problem (1.6) is the cyclic algorithm. (For convenience, we relabel the mappings {Ti }i=1 N −1 as {Ti }i=0 .) This means that beginning with an x0 in C, we define the sequence {xn } cyclically by

x1 = α0 x0 + (1 − α0 )T0 x0 , x2 = α1 x1 + (1 − α1 )T1 x1 , .. . x N = α N −1 x N −1 + (1 − α N −1 )TN −1 x N −1 , x N +1 = α N x N + (1 − α N )T0 x N , .. . In a more compact form, xn+1 can be written as xn+1 = αn xn + (1 − αn )T[n] xn ,

(1.9)

where T[n] = Ti , with i = n (mod N ), 0 ≤ i ≤ N − 1. We will show that this cyclic algorithm (1.9) is also weakly convergent if the sequence {αn } of parameters is appropriately chosen. However, the convergence of both algorithms (1.8) and (1.9) can only be weak in an infinite-dimensional space (see [4]; see also [7]). So in order to have strong convergence, one must modify these algorithms. Some such modifications for the Mann’s algorithm (1.3) can be found in [10,18,11,12,15–17,25]. We will propose two modifications for the algorithm (1.8) and (1.9). These modified algorithms are obtained by applying additional projections onto the intersections of two half-spaces and are guaranteed to have strong convergence. Our modification for the algorithm (1.8) produces a sequence {xn } as follows: xn+1 = PCn ∩Q n x0 ,

(1.10)

where Cn and Q n are given by Cn = {z ∈ C : kyn − zk2 ≤ kxn − zk2 − (1 − αn )(αn − κ)kxn − An xn k2 } P N (n) where An = i=1 λi Ti and yn = αn xn + (1 − αn )An xn , and Q n = {z ∈ C : hxn − z, x0 − xn i ≥ 0}.

(1.11)

As for the algorithm (1.9), we propose the following modification that produces the sequence {xn } given by the same formula (1.10) with Cn given by Cn = {z ∈ C : kyn − zk2 ≤ kxn − zk2 − (1 − αn )(αn − κ)kxn − T[n] xn k2 } where yn = αn xn + (1 − αn )T[n] xn , and with Q n given by the same formula (1.11). The organization of this paper is as follows. In Section 2, we include some useful properties of Hilbert spaces (identities and projections). We also discuss useful properties of strict pseudo-contractions. These properties will be utilized in the proofs to the main results of this paper. In Section 3 we prove the weak convergence of the parallel algorithm (1.8) and in Section 4 that of the cyclic algorithm (1.9) for strict pseudo-contractions. The final section, Section 5, is devoted to the strong convergence of the algorithm (1.10) for strict pseudo-contractions. We will use the notation:

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

2261

1. * for weak convergence and → for strong convergence. 2. ωw (xn ) = {x : ∃xn j * x} denotes the weak ω-limit set of {xn }. 2. Preliminaries We need some facts and tools in a real Hilbert space H which are listed as lemmas below (see [17] for necessary proofs of Lemmas 2.2 and 2.4). Lemma 2.1. Let H be a real Hilbert space. There hold the following identities. (i) kx − yk2 = kxk2 − kyk2 − 2hx − y, yi ∀x, y ∈ H . (ii) kt x + (1 − t)yk2 = tkxk2 + (1 − t)kyk2 − t (1 − t)kx − yk2 ∀t ∈ [0, 1], ∀x, y ∈ H . (iii) If {xn } is a sequence in H weakly convergent to z, then lim sup kxn − yk2 = lim sup kxn − zk2 + kz − yk2 n→∞

∀y ∈ H.

n→∞

Lemma 2.2. Let H be a real Hilbert space. Given a closed convex subset C ⊂ 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). Recall that given a closed convex subset K of a real Hilbert space H , the nearest point projection PK from H onto K assigns to each x ∈ H its nearest point denoted as PK x in K from x to K ; that is, PK x is the unique point in K with the property for all y ∈ K .

kx − PK xk ≤ kx − yk

Lemma 2.3. Let K be a closed convex subset of real Hilbert space H . Given x ∈ H and z ∈ K , then z = PK x if and only if there holds the relation: hx − z, y − zi ≤ 0 for all y ∈ K . Lemma 2.4. Let K be a closed convex subset of H . Let {xn } be a sequence in H and u ∈ H . Let q = PK u. Suppose {xn } is such that ωw (xn ) ⊂ K and satisfies the condition kxn − uk ≤ ku − qk

for all n.

(2.1)

Then xn → q. Lemma 2.5. Let K be a closed convex subset of H . Let {xn } be a bounded sequence in H . Assume • The weak ω-limit set ωw (xn ) ⊂ K . • For each z ∈ K , limn→∞ kxn − zk exists. Then {xn } is weakly convergent to a point in K . Proof. To see that {xn } is weakly convergent, we take p, q ∈ ωw (xn ) and let {xn i } and {xm j } be subsequences of {xn } such that xni * p and xm j * q, respectively. By assumption, p, q ∈ K and limn→∞ kxn − vk exists, where v ∈ { p, q}. By Lemma 2.1(iii), we obtain lim kxn − pk2 = lim kxm j − pk2

n→∞

j→∞

= lim kxm j − qk2 + kq − pk2 j→∞

= lim kxn i − qk2 + kq − pk2 i→∞

2262

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

= lim kxn i − pk2 + 2kq − pk2 i→∞

= lim kxn − pk2 + 2kq − pk2 . n→∞

Hence p = q and {xn } is weakly convergent.



The following proposition lists some useful properties for strict pseudo-contractions. (See also [2,23].) Proposition 2.6. Assume C is a closed convex subset of a Hilbert space H . (i) If T : C → C is a κ-strict pseudo-contraction, then T satisfies the Lipschitz condition 1+κ kx − yk ∀x, y ∈ C. (2.2) 1−κ If T : C → C is a κ-strict pseudo-contraction, then the mapping I − T is demiclosed (at 0). That is, if {xn } is a sequence in C such that xn * x˜ and (I − T )xn → 0, then (I − T )x˜ = 0. If T : C → C is a κ-strict pseudo-contraction, then the fixed point set Fix(T ) of T is closed and convex so that the projection PFix(T ) is well defined. Given an integer N ≥ 1, assume, for each 1 ≤ i ≤ N , Ti : C P → C is a κi -strict pseudo-contraction for PN N N is a positive sequence such that some 0 ≤ κi < 1. Assume {λi }i=1 i=1 λi = 1. Then i=1 λi Ti is a κ-strict pseudo-contraction, with κ = max{κi : 1 ≤ i ≤ N }. N and {λ } N be given as in (iv) above. Suppose that {T } N has a common fixed point. Then Let {Ti }i=1 i i=1 i i=1 ! N N X \ Fix λi Ti = Fix(Ti ). kT x − T yk ≤

(ii) (iii) (iv)

(v)

i=1

i=1

Proof. The proofs of (i)–(iii) can be found in Marino and Xu [16]. To prove (iv), we only need to consider the case of N = 2 (the general case can be proved by induction). Set G = (1 − λ)T1 + λT2 , where λ ∈ (0, 1) and for i = 1, 2, Ti is a κi -strict pseudo-contraction. Set κ = max{κ1 , κ2 }. We have two ways to prove that G is a κ-strict pseudo-contraction. The first one is a direct way. That is, we verify directly the following inequality: kGx − Gyk2 ≤ kx − yk2 + κk(I − G)x − (I − G)yk2

(2.3)

∀x, y ∈ C.

Indeed, we have k(I − G)x − (I − G)yk2 = k(1 − λ)[(I − T1 )x − (I − T1 )y] + λ[(I − T2 )x − (I − T2 )y]k2 = (1 − λ)k(I − T1 )x − (I − T1 )yk2 + λk(I − T2 )x − (I − T2 )yk2 − λ(1 − λ)k[(I − T1 )x − (I − T1 )y] − [(I − T2 )x − (I − T2 )y]k2 . We also compute kGx − Gyk2 = k(1 − λ)(T1 x − T1 y) + λ(T2 x − T2 y)k2 = (1 − λ)kT1 x − T1 yk2 + λkT2 x − T2 yk2 − λ(1 − λ)k(T1 x − T1 y) − (T2 x − T2 y)k2 ≤ (1 − λ)(kx − yk2 + κ1 k(I − T1 )x − (I − T1 )yk2 ) + λ(kx − yk2 + κ2 k(I − T2 )x − (I − T2 )yk2 ) − λ(1 − λ)k(T1 x − T1 y) − (T2 x − T2 y)k2 ≤ kx − yk2 + κ[(1 − λ)k(I − T1 )x − (I − T1 )yk2 + λk(I − T2 )x − (I − T2 )yk2 ] − λ(1 − λ)k(T1 x − T1 y) − (T2 x − T2 y)k2 = kx − yk2 + κk(I − G)x − (I − G)yk2 − (1 − κ)λ(1 − λ)k(T1 x − T1 y) − (T2 x − T2 y)k2 ≤ kx − yk2 + κk(I − G)x − (I − G)yk2 . Hence G is a κ-strict pseudo-contraction.

by (2.4)

(2.4)

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

2263

The second way is somewhat indirect. Observe that T : C → C is a κ-strict pseudo-contraction if and only if there holds the following 1−κ k(I − T )x − (I − T )yk2 , 2 Indeed, putting V = I − T , we see that (1.1) holds if and only if hx − y, (I − T )x − (I − T )yi ≥

k(I − V )x − (I − V )yk2 ≤ kx − yk2 + κkV x − V yk2 ,

x, y ∈ C.

x, y ∈ C.

(2.5)

(2.6)

But k(I − V )x − (I − V )yk2 = kx − yk2 − 2hx − y, V x − V yi + kV x − V yk2 .

(2.7)

Substituting (2.7) into (2.6), we obtain (2.5). Now if, for i = 1, 2, Ti is a κi -strict pseudo-contraction and G = (1 − λ)T1 + λT2 , with λ ∈ (0, 1), noting κ = max{κ1 , κ2 } and also noticing (2.4), we get hx − y, (I − G)x − (I − G)yi = (1 − λ)hx − y, (I − T1 )x − (I − T2 )yi + λhx − y, (I − T2 )x − (I − T2 )yi 1−κ [(1 − λ)k(I − T1 )x − (I − T1 )yk2 + λk(I − T2 )x − (I − T2 )yk2 ] ≥ 2 1−κ ≥ k(I − G)x − (I − G)yk2 . 2 Hence G is a κ-strict pseudo-contraction. To prove (v), again we can assume N = 2. It suffices to prove that Fix(G) ⊂ Fix(T1 ) ∩ Fix(T2 ), where G = (1 − λ)T1 + λT2 with 0 < λ < 1. Let x ∈ Fix(G) and write V1 = I − T1 and V2 = I − T2 . Take z ∈ Fix(T1 ) ∩ Fix(T2 ) to deduce that kz − xk2 = k(1 − λ)(z − T1 x) + λ(z − T2 x)k2 = (1 − λ)kz − T1 xk2 + λkz − T2 xk2 − λ(1 − λ)kT1 x − T2 xk2 ≤ (1 − λ)(kz − xk2 + κkx − T1 xk2 ) + λ(kz − xk2 + κkx − T2 xk2 ) − λ(1 − λ)kT1 x − T2 xk2 = kz − xk2 + κ[(1 − λ)kV1 xk2 + λkV2 xk2 ] − λ(1 − λ)kV1 x − V2 xk2 . It follows that λ(1 − λ)kV1 x − V2 xk2 ≤ κ[(1 − λ)kV1 xk2 + λkV2 xk2 ].

(2.8)

Since (1 − λ)V1 x + λV2 x = 0, we have (1 − λ)kV1 xk2 + λkV2 xk2 = λ(1 − λ)kV1 x − V2 xk2 . This together with (2.8) implies that (1 − κ)λ(1 − λ)kV1 x − V2 xk2 ≤ 0. Since 0 < λ < 1 and κ < 1, we get kV1 x − V2 xk = 0 which implies T1 x = T2 x which in turns implies that T1 x = T2 x = x since (1 − λ)T1 x + λT2 x = x. Thus, x ∈ Fix(T1 ) ∩ Fix(T2 ).  3. Parallel algorithm Recall that, given a self-mapping T of a closed convex subset C of a real Hilbert space H , Mann’s algorithm [14] generates a sequence {xn } in C by the recursive formula xn+1 = αn xn + (1 − αn )T xn ,

n ≥ 0,

(3.1)

where the initial guess x0 ∈ C us arbitrary, and where {αn }∞ n=0 is a real control sequence in the interval (0, 1). Mann’s algorithm has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results is proved by Reich [21], which confirms the weak convergence of the sequence {xn } generated by

2264

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

(3.1) in a uniformly convex Banach space with a Frechet P∞differentiable norm if the mapping T is nonexpansive and if the control sequence {αn }∞ n=0 αn (1 − αn ) = ∞. Recently Marino and Xu [16] extended n=0 satisfies the assumption Reich’s result to strict pseudo-contractions in the Hilbert space setting. Theorem 3.1 ([16]). Let C be a closed convex subset of a Hilbert space H . Let T : C → C be a κ-strict pseudocontraction for some 0 ≤ κ < 1 and assume that T admits a fixed point in C. Let {xn }∞ n=0 be the sequence generated by Mann’s algorithm (3.1). Assume that the control sequence {αn }∞ is chosen so that κ < αn < 1 for all n and n=0 ∞ X

(αn − κ)(1 − αn ) = ∞.

(3.2)

n=0

Then {xn } converges weakly to a fixed point of T . Our first result is the following Theorem 3.2. Let C be a closed convex subset of a Hilbert space H . Let N ≥ 1 be an integer. Let, for each 1 ≤ i ≤ N , Ti : C → C be a κi -strict pseudo-contraction for some 0 ≤ κi < 1. Let κ = max{κi : 1 ≤ i ≤ N }. Assume the N Fix(T ) is nonempty. Assume also {λ } N is a finite sequence of positive numbers such common fixed point set ∩i=1 i i i=1 PN ∞ that i=1 λi = 1. Given x0 ∈ C, let {xn }n=0 be the sequence generated by Mann’s algorithm: xn+1 = αn xn + (1 − αn )

N X

λi Ti xn .

(3.3)

i=1

Assume the control sequence {αn }∞ n=0 is chosen so that κ < αn < 1 for all n and ∞ X

(αn − κ)(1 − αn ) = ∞.

(3.4)

n=0 N . Then {xn } converges weakly to a common fixed point of {Ti }i=1

Proof. Put A=

N X

λi Ti .

i=1 N Fix(T ). Then by Proposition 2.6, A is a κ-strict pseudo-contraction and Fix(A) = ∩i=1 i We can rewrite the algorithm (3.3) as

xn+1 = αn xn + (1 − αn )Axn . Now apply Theorem 3.1 to conclude that the sequence {xn } converges weakly to a fixed point of A.



N are constant in the sense that they are independent of n, the number In the algorithm (3.3), the weights {λi }i=1 N to be step of steps of the iteration process. Below we consider a more general case by allowing the weights {λi }i=1 dependent. That is, initializing with x0 , we define {xn } by the algorithm:

xn+1 = αn xn + (1 − αn )

N X

(n)

λi Ti xn .

(3.5)

i=1

Theorem 3.3. Let C be a closed convex subset of a Hilbert space H . Let N ≥ 1 be an integer. Let, for each 1 ≤ i ≤ N , Ti : C → C be a κi -strict pseudo-contraction for some 0 ≤ κi < 1. Let κ = max{κi : 1 ≤ i ≤ N }. Assume the N Fix(T ) is nonempty. Assume also for each n, {λ(n) } N is a finite sequence of positive common fixed point set ∩i=1 i i i=1 P N (n) (n) numbers such that i=1 λi = 1 for all n and infn≥1 λi > 0 for all 1 ≤ i ≤ N . Given x0 ∈ C, let {xn }∞ n=0 be the

2265

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

sequence generated by the algorithm (3.5). Assume that the control sequence {αn }∞ n=0 is chosen so that κ < αn < 1 for all n and ∞ X

(αn − κ)(1 − αn ) = ∞.

(3.6)

n=0

Assume also that v u N ∞ uX X (n+1) (n) t |λi − λi | < ∞. n=0

(3.7)

i=1

N . Then {xn } converges weakly to a common fixed point of {Ti }i=1

Proof. Write, for each n ≥ 1, An =

N X

(n)

λi Ti .

i=1

By Proposition 2.6(iv), each An is a κ-strict pseudo-contraction on C, and the algorithm (3.5) can be rewritten as xn+1 = αn xn + (1 − αn )An xn .

(3.8)

N (i.e., F = ∩ N Fix(T )) and take a p ∈ F to deduce Denote by F the common fixed point set of the mappings {Ti }i=1 i i=1 that

kxn+1 − pk2 = kαn (xn − p) + (1 − αn )(An xn − p)k2 = αn kxn − pk2 + (1 − αn )kAn xn − pk2 − αn (1 − αn )kxn − An xn k2 ≤ αn kxn − pk2 + (1 − αn )(kxn − pk2 + κkxn − An xn k2 ) − αn (1 − αn )kxn − An xn k2 . Hence (αn − κ)(1 − αn )kxn − An xn k2 ≤ kxn − pk2 − kxn+1 − pk2 .

(3.9)

Since κ < αn < 1 for all n, we get kxn+1 − pk ≤ kxn − pk; that is, the sequence {kxn − pk} is decreasing. Also (3.9) implies that ∞ X

(αn − κ)(1 − αn )kxn − An xn k2 ≤ kx0 − pk2 < ∞.

(3.10)

n=0

Using the condition (3.6), we see that (3.10) implies that lim inf kxn − An xn k = 0. n→∞

(3.11)

We next prove that the limn→∞ kxn − An xn k actually exists. To see this, we compute kxn+1 − An+1 xn+1 k2 = αn kxn − An+1 xn+1 k2 + (1 − αn )kAn xn − An+1 xn+1 k2 − αn (1 − αn )kxn − An xn k2 .

(3.12)

Furthermore we compute kxn − An+1 xn+1 k2 = k(xn − xn+1 ) + (xn+1 − An+1 xn+1 )k2 = kxn − xn+1 k2 + kxn+1 − An+1 xn+1 k2 + 2hxn − xn+1 , xn+1 − An+1 xn+1 i. Since we can write An+1 xn+1 = An xn+1 + yn ,

(3.13)

2266

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

where yn =

N X

(n+1)

(λi

(n)

− λi )Ti xn+1 ,

i=1

kAn xn − An+1 xn+1 k2 = kAn xn − An xn+1 − yn k2 = kAn xn − An xn+1 k2 − 2hAn xn − An xn+1 , yn i + kyn k2 ≤ kxn − xn+1 k2 + κk(xn − An xn ) − (xn+1 − An xn+1 )k2 − 2hAn xn − An xn+1 , yn i + kyn k2 = kxn − xn+1 k2 + κ[kxn − An xn k2 + kxn+1 − An xn+1 k2 − 2hxn − An xn , xn+1 − An xn+1 i] − 2hAn xn − An xn+1 , yn i + kyn k2 ,

(3.14)

since xn − xn+1 = (1 − αn )(xn − An xn ) and since kxn+1 − An xn+1 k2 = k(xn+1 − An+1 xn+1 ) + (An+1 xn+1 − An xn+1 )k2 = kxn+1 − An+1 xn+1 k2 + 2hxn+1 − An+1 xn+1 , yn i + kyn k2 it follows from (3.14) that kAn xn − An+1 xn+1 k2 ≤ (κ + (1 − αn )2 )kxn − An xn k2 + κkxn+1 − An+1 xn+1 k2 − 2κhxn − An xn , xn+1 − An+1 xn+1 i + 2κhxn+1 − An+1 xn+1 , yn i + κkyn k2 − 2κhxn − An xn , yn i − 2hAn xn − An xn+1 , yn i + kyn k2 .

(3.15)

Substituting (3.13) and (3.15) into (3.12), we obtain kxn+1 − An+1 xn+1 k2 ≤ (1 − αn )[αn (1 − αn ) + κ + (1 − αn )2 − αn ]kxn − An xn k2 + [αn + κ(1 − αn )]kxn+1 − An+1 xn+1 k2 + 2(1 − αn )(αn − κ)hxn − Axn , xn+1 − An+1 xn+1 i + (1 − αn )(2κhxn+1 − An+1 xn+1 , yn i + κkyn k2 ) − 2κ(1 − αn )hxn − An xn , yn i + (1 − αn )(−2hAn xn − An xn+1 , yn i + kyn k2 ).

(3.16)

Since {xn } is bounded, we can find a constant M > 0 so that 2κ(|hxn+1 − An+1 xn+1 , yn i| + |hxn − An xn , yn i|) + 2|hAn xn − An xn+1 , yn i| + (1 + κ)kyn k2 ≤ Mkyn k. (3.17) Defining βn = kxn − An xn k for n ≥ 1 and using (3.17), we have from (3.16) 2 (1 − αn )(1 − κ)βn+1 ≤ (1 − αn )[(1 + κ − 2αn )βn2 + 2(αn − κ)βn βn+1 + Mkyn k].

Since 1 − αn > 0, we can divide by 1 − αn to get the quadratic inequality in βn+1 , 2 (1 − κ)βn+1 − 2(αn − κ)βn βn+1 − (1 + κ − 2αn )βn2 − Mkyn k ≤ 0.

Solving this inequality, we obtain 1 [(αn − κ)βn + 1−κ r Mkyn k ≤ βn + . 1−κ

βn+1 ≤

Assumption (3.7) implies that ∞ p X n=0

kyn k < ∞.

q

(1 − αn )2 βn2 + (1 − κ)Mkyn k] (3.18)

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

2267

Hence by (3.18) we conclude that limn→∞ kxn − An xn k exists. Thus (3.11) implies that lim kxn − An xn k = 0.

(3.19)

n→∞

Next we show that ωw (xn ) ⊂ F.

(3.20)

To see this, we take z ∈ ωw (xn ) and assume that xnl * z as l → ∞ for some subsequence {xnl } of {xn }. With no loss of generality, we may assume that (nl )

λi

→ λi

(as l → ∞), 1 ≤ i ≤ N . PN It is easily seen that each λi > 0 and i=1 λi = 1. We also have Anl x → Ax

(3.21)

(as l → ∞) for all x ∈ C,

where A=

N X

λi Ti .

i=1 N Fix(T ) = F. Since Note that by Proposition 2.6, A is a κ-strict pseudo-contraction and Fix(A) = ∩i=1 i

kxnl − Axnl k ≤ kxnl − Anl xnl k + kAnl xnl − Axnl k ≤ kxnl − Anl xnl k +

N X

(nl )

|λi

− λi |kTi xnl k,

i=1

we obtain by virtue of (3.19) and (3.21) kxnl − Axnl k → 0. So by the demiclosedness principle (Proposition 2.6(ii)), it follows that z ∈ Fix(A) = F and hence (3.20) holds. Since we have shown that limn→∞ kxn − pk exists for all p ∈ F, an application of Lemma 2.5 ensures that {xn } converges weakly to some point in F.  4. Cyclic algorithm N −1 Let C be a closed convex subset of a Hilbert space H and let {Ti }i=0 be N κ-strict pseudo-contractions on C such that the common fixed point set

F :=

N\ −1

Fix(Ti ) 6= ∅.

i=0 ∞ Let x0 ∈ C and let {αn }∞ n=0 be a sequence in (0,1). The cyclic algorithm generates a sequence {x n }n=1 in the following way:

x1 = α0 x0 + (1 − α0 )T0 x0 , x2 = α1 x1 + (1 − α1 )T1 x1 , .. . x N = α N −1 x N −1 + (1 − α N −1 )TN −1 x N −1 , x N +1 = α N x N + (1 − α N )T0 x N , .. . In general, xn+1 is defined by xn+1 = αn xn + (1 − αn )T[n] xn , where T[n] = Ti , with i = n (mod N ), 0 ≤ i ≤ N − 1.

(4.1)

2268

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

Theorem 4.1. Let C be a closed convex subset of a Hilbert space H . Let N ≥ 1 be an integer. Let, for each 0 ≤ i ≤ N − 1, Ti : C → C be a κi -strict pseudo-contraction for some 0 ≤ κi < 1. Let κ = max{κi : 1 ≤ i ≤ N }. N −1 N −1 Assume the common fixed point set ∩i=0 F(Ti ) of {Ti }i=0 is nonempty. Given x0 ∈ C, let {xn }∞ n=0 be the sequence is chosen so that κ +ε ≤ αn ≤ 1−ε generated by the cyclic algorithm (4.1). Assume that the control sequence {αn }∞ n=0 N −1 . for all n and some ε ∈ (0, 1). Then {xn } converges weakly to a common fixed point of the family {Ti }i=0 N −1 Proof. Pick a p ∈ F = ∩i=0 Fix(Ti ). We first show that the real sequence {kxn − pk}∞ n=0 is decreasing; hence limn→∞ kxn − pk exists. To see this, using Lemma 2.1(ii), we obtain

kxn+1 − pk2 = kαn (xn − p) + (1 − αn )(T[n] xn − p)k2 = αn kxn − pk2 + (1 − αn )kT[n] xn − pk2 − αn (1 − αn )kxn − T[n] xn k2 ≤ αn kxn − pk2 + (1 − αn )(kxn − pk2 + κkxn − T[n] xn k2 ) − αn (1 − αn )kxn − T[n] xn k2 = kxn − pk2 − (αn − κ)(1 − αn )kxn − T[n] xn k2 .

(4.2)

Since κ + ε ≤ αn ≤ 1 − ε, we get by (4.2) ε 2 kxn − T[n] xn k2 ≤ kxn − pk2 − kxn+1 − pk2 .

(4.3)

It follows that the sequence {kxn − pk} is decreasing (and hence limn→∞ kxn − pk exists) and that lim kxn − T[n] xn k = 0.

n→∞

(4.4)

This implies that lim kxn+1 − xn k = lim (1 − αn )kxn − T[n] xn k = 0.

n→∞

n→∞

(4.5)

Claim: ωw (xn ) ⊂ F. Indeed, assume x¯ ∈ ωw (xn ) and xn i * x¯ for some subsequence {xn i } of {xn }. We may further assume n i = l (mod N ) for all i. Since by (4.5), we also have xn i + j * x¯ for all j ≥ 0, we deduce that kxn i + j − T[l+ j] xn i + j k = kxn i + j − T[ni + j] xn i + j k → 0. Then the demiclosedness principle (Proposition 2.6(ii)) implies that x¯ ∈ Fix(T[l+ j] ) for all j. This ensures that x¯ ∈ F. Since we also have the existence of limn→∞ kxn − zk for every z ∈ F, an application of Lemma 2.5 ensures that {xn } is weakly convergent to a point in F.  5. Strong convergence In an infinite-dimensional Hilbert space, Mann’s algorithm has only weak convergence, in general, even for nonexpansive mappings (see the example in [4]; see also [7]). Hence in order to have strong convergence, one has to modify Mann’s algorithm. Some modifications have recently been obtained (see [18,11,12,17,29]). These modifications are for either nonexpansive or asymptotically nonexpansive mappings. A modification of Mann’s algorithm for a single strict pseudo-contraction which has strong convergence was recently obtained in [16]. Below we consider a modification of Mann’s algorithm for a finite family of strict pseudo-contractions. Theorem 5.1. Let C be a closed convex subset of a Hilbert space H . Given an integer N ≥ 1, let, for each 1 ≤ i ≤ N , Ti : C → C be a κi -strict pseudo-contraction for some 0 ≤ κi < 1. Let κ = max{κi : 1 ≤ i ≤ N }. Assume the N F(T ) of {T } N is nonempty. Assume also for each n, {λ(n) } N is a finite sequence common fixed point set F := ∩i=1 i i i=1 i i=1 P N (n) (n) of positive numbers such that i=1 λi = 1 and infn≥1 λi > 0 for all 1 ≤ i ≤ N . Let the mapping An be defined by An x =

N X i=1

(n)

λi Ti x,

x ∈ C.

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

Given x0 ∈ C, let {xn }∞ n=1 be the sequence generated by the following algorithm:  x0 ∈ C chosen arbitrarily,    y = α x + (1 − α )A x ,   n n n n n n Cn = {z ∈ C : kyn − zk2 ≤ kxn − zk2 − (1 − αn )(αn − κ)kxn − An xn k2 },   Q = {z ∈ C : hxn − z, x0 − xn i ≥ 0},    n xn+1 = PCn ∩Q n x0 .

2269

(5.1)

Assume that the control sequence {αn }∞ n=0 is chosen so that 0 ≤ αn < 1 for all n. Then {x n } converges strongly to PF x0 . Proof. First observe that Cn is convex by Lemma 2.2. Next we show that Fix(T ) ⊂ Cn for all n. Indeed, like in the proof of (3.9), we have, for all p ∈ F, kyn − pk2 ≤ kxn − pk2 − (1 − αn )(αn − κ)kxn − An xn k2 .

(5.2)

So p ∈ Cn for all n. Next we show that F ⊂ Qn

for all n ≥ 0.

(5.3)

We prove this by induction. For n = 0, we have F ⊂ C = Q 0 . Assume that F ⊂ Q n for some n > 0. Since xn+1 is the projection of x0 onto Cn ∩ Q n , by Lemma 2.3 we have hxn+1 − z, x0 − xn+1 i ≥ 0

∀z ∈ Cn ∩ Q n .

As F ⊂ Cn ∩ Q n by the induction assumption, the last inequality holds, in particular, for all z ∈ F. This together with the definition of Q n+1 implies that F ⊂ Q n+1 . Hence (5.3) holds for all n ≥ 0. Notice that the definition of Q n actually implies xn = PQ n x0 . This together with that fact F ⊂ Q n further implies kxn − x0 k ≤ k p − x0 k

for all p ∈ F.

In particular, {xn } is bounded and kxn − x0 k ≤ kq − x0 k,

where q = PF x0 .

(5.4)

The fact that xn+1 ∈ Q n asserts that hxn+1 − xn , xn − x0 i ≥ 0. This together with Lemma 2.1(i) implies kxn+1 − xn k2 = k(xn+1 − x0 ) − (xn − x0 )k2 = kxn+1 − x0 k2 − kxn − x0 k2 − 2hxn+1 − xn , xn − x0 i ≤ kxn+1 − x0 k2 − kxn − x0 k2 . It turns out that kxn+1 − xn k → 0.

(5.5)

By the fact xn+1 ∈ Cn we get kxn+1 − yn k2 ≤ kxn+1 − xn k2 − (1 − αn )(αn − κ)kxn − An xn k2 .

(5.6)

Moreover, since yn = αn xn + (1 − αn )An xn , we deduce that kxn+1 − yn k2 = αn kxn+1 − xn k2 + (1 − αn )kxn+1 − An xn k2 − αn (1 − αn )kxn − An xn k2 .

(5.7)

Substitute (5.7) into (5.6) to get (1 − αn )kxn+1 − An xn k2 ≤ (1 − αn )kxn+1 − xn k2 + (1 − αn )κkxn − An xn k2 . Since αn < 1 for all n, the last inequality becomes kxn+1 − An xn k2 ≤ kxn+1 − xn k2 + κkxn − An xn k2 .

(5.8)

But, on the other hand, we compute kxn+1 − An xn k2 = kxn+1 − xn k2 + 2hxn+1 − xn , xn − An xn i + kxn − An xn k2 .

(5.9)

2270

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

Combining (5.9) and (5.8) we obtain (1 − κ)kxn − An xn k2 ≤ −2hxn+1 − xn , xn − An xn i. Therefore, 2 kxn+1 − xn k → 0. (5.10) 1−κ This is (3.19). Hence repeating the argument in the proof of Theorem 3.2 we get that ωw (xn ) ⊂ F. Then by virtue of (5.4) and Lemma 2.4, we conclude that xn → q, where q = PF x0 .  kxn − An xn k ≤

Regarding the cyclic algorithm (4.1), we have the following modification which has strong convergence. Theorem 5.2. Let C be a closed convex subset of a Hilbert space H . Given a positive integer N ≥ 1, let, for each 0 ≤ i ≤ N − 1, Ti : C → C be a κi -strict pseudo-contraction for some 0 ≤ κi < 1. Let κ = max{κi : 1 ≤ i ≤ N }. N −1 N −1 Assume the common fixed point set F := ∩i=0 F(Ti ) of {Ti }i=0 is nonempty. Given x0 ∈ C, let {xn }∞ n=1 be the sequence generated by the following algorithm:   x0 ∈ C chosen arbitrarily,    yn = αn xn + (1 − αn )T[n] xn , (5.11) Cn = {z ∈ C : kyn − zk2 ≤ kxn − zk2 − (1 − αn )(αn − κ)kxn − T[n] xn k2 },   Q = {z ∈ C : hx − z, x − x i ≥ 0},  n n n 0   xn+1 = PCn ∩Q n x0 . Assume that the control sequence {αn }∞ n=0 is chosen so that 0 ≤ αn < 1 for all n. Then {x n } converges strongly to PF x0 . Proof. The proof of this theorem is similar to that of Theorem 5.1. The main points include (i) (ii) (iii) (iv) (v) (vi)

xn is well defined for all n ≥ 1. kxn − x0 k ≤ kq − x0 k for all n, where q = PF x0 . kxn+1 − xn k → 0. kxn − T[n] xn k → 0. ωn (xn ) ⊂ F. xn → q.

To prove (i)–(iv), one simply replaces An with T[n] in the proof of Theorem 5.1. One can prove (v) by repeating the argument in the proof of Theorem 4.1. Finally the strong convergence to q of {xn } is the consequence of (ii), (v) and Lemma 2.4.  Acknowledgements Genaro Lopez Acedo is supported in part by DGES, Grant BFM2003-03893-C02-01, and Junta de Andalucia, Grant FQM-127. Hong-Kun Xu is supported in part by the National Research Foundation of South Africa. References [1] H. Bauschke, The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150–159. [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. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems 20 (2004) 103–120. [4] A. Genel, J. Lindenstrauss, An example concerning fixed points, Israel J. Math. 22 (1975) 81–86. [5] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, in: Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, 1990. [6] K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, 1984. [7] O. G¨uler, On the convergence of the proximal point algorithm for convex optimization, SIAM J. Control Optim. 29 (1991) 403–419. [8] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967) 957–961. [9] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974) 147–150.

G.L. Acedo, H.-K. Xu / Nonlinear Analysis 67 (2007) 2258–2271

2271

[10] S. Kamimura, W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (2003) 938–945. [11] T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60. [12] T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140–1152. [13] P.L. Lions, Approximation de points fixes de contractions, C. R. Acad. Sci. S`er. A–B Paris 284 (1977) 1357–1359. [14] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506–510. [15] G. Marino, H.K. Xu, Convergence of generalized proximal point algorithms, Comm. Pure Appl. Anal. 3 (2004) 791–808. [16] G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. (2006), in press (doi:10.1016/j.jmaa.2006.06.055). Available online 27 July 2006. [17] C. Matinez-Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point processes, Nonlinear Anal. 64 (2006) 2400–2411. [18] K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379. [19] J.G. O’Hara, P. Pillay, H.K. Xu, Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417–1426. [20] J.G. O’Hara, P. Pillay, H.K. Xu, Iterative approaches to convex feasibility problems in Banach spaces, Nonlinear Anal. 64 (2006) 2022–2042. [21] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979) 274–276. [22] S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287–292. [23] O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, J. Math. Anal. Appl. 194 (1991) 911–933. [24] N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997) 3641–3645. [25] M.V. Solodov, B.F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program. Ser. A 87 (2000) 189–202. [26] K.K. Tan, H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (2) (1993) 301–308. [27] K.K. Tan, H.K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122 (1994) 733–739. [28] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486–491. [29] H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002) 240–256. [30] H.K. Xu, Remarks on an iterative method for nonexpansive mappings, Comm. Appl. Nonlinear Anal. 10 (1) (2003) 67–75. [31] H.K. Xu, Strong convergence of an iterative method for nonexpansive mappings and accretive operators, J. Math. Anal. Appl. 314 (2006) 631–643. [32] H.K. Xu, Strong convergence of approximating fixed point sequences for nonexpansive mappings, Bull. Austral. Math. Soc. 74 (2006) 143–151.