Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups

Nonlinear Analysis 64 (2006) 1140 – 1152 www.elsevier.com/locate/na

Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups Tae-Hwa Kima,1 , Hong-Kun Xub,∗,2 a Division of Mathematical Sciences, Pukyong National University, Pusan 608-737, Korea b School of Mathematical Sciences, University of KwaZulu-Natal, Westville Campus, Private Bag X54001,

Durban 4000, South Africa Received 10 May 2005; accepted 25 May 2005

Abstract The Mann iterations for nonexpansive mappings have in general only weak convergence in a Hilbert space. We modify an iterative method of Mann’s type introduced by Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379] for nonexpansive mappings and prove strong convergence of our modified Mann’s iteration processes for asymptotically nonexpansive mappings and semigroups. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: primary 47H09; secondary 65J15 Keywords: Strong convergence; Modified Mann iteration; Asymptotically nonexpansive mapping; Asymptotically nonexpansive semigroup

∗ Corresponding author. Tel.: +27 31 260 7418; fax: +27 31 260 7806.

E-mail addresses: [email protected] (T.-H. Kim), [email protected] (H.-K. Xu). 1 Supported by Pukyong National University Research Abroad Fund in 2004. 2 Supported in part by the National Research Foundation of South Africa.

0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.05.059

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1. Introduction Let X be a real Banach space, C a nonempty closed convex subset of X, and T : C → C a mapping. Recall that T is nonexpansive if T x − T y
x − y for all x, y ∈ C, and T is asymptotically nonexpansive [4] if there exists a sequence {kn } of positive real numbers with limn→∞ kn = 1 and such that T n x − T n y
kn x − y for all integers n 1 and x, y ∈ C. A point x ∈ C is a fixed point of T provided T x = x. Denote by Fix(T ) the set of fixed points of T; that is, Fix(T ) = {x ∈ C : T x = x}. Recall also that a one-parameter family S={T (t) : t 0} of self-mappings of a nonempty closed convex subset C of a Hilbert space H is said to be a (continuous) Lipschitzian semigroup on C (see, e.g., [15]) if the following conditions are satisfied: (i) (ii) (iii) (iv)

T (0)x = x, x ∈ C, T (t + s)x = T (t)T (s)x, t, s 0, x ∈ C, for each x ∈ C, the map t  → T (t)x is continuous on [0, ∞), there exists a bounded measurable function L : (0, ∞) → [0, ∞) such that, for each t > 0, T (t)x − T (t)y
Lt x − y,

x, y ∈ C.

A Lipschitzian semigroup S is called nonexpansive (or a contraction semigroup) if Lt =1 for all t > 0, and asymptotically nonexpansive if lim supt→∞ Lt
1, respectively. We use Fix(S) to denote the common fixed point set of the semigroup S; that is, Fix(S) = {x ∈ C : T (s)x = x, ∀s > 0}. Note that for an asymptotically nonexpansive semigroup S, we can always assume that the Lipschitzian constants {Lt }t>0 are such that Lt 1 for all t > 0, L is nonincreasing in t, and limt→∞ Lt = 1; otherwise we replace Lt , for each t > 0, with L˜ t := max{sups  t Ls , 1}. Construction of fixed points of nonexpansive mappings (and of common fixed points of nonexpansive semigroups) is an important subject in the theory of nonexpansive mappings and finds application in a number of applied areas, in particular, in image recovery and signal processing (see, e.g., [2,8,11,16,17]). However, the sequence {T n x}∞ n=0 of iterates of the mapping T at a point x ∈ C may not converge even in the weak topology. Thus averaged iterations prevail. Indeed, Mann’s iterations do have weak convergence. More precisely, a Mann’s iteration procedure is a sequence {xn } which is generated in the following recursive way: xn+1 =
n xn + (1 −
n )T x n ,

n0,

(1.1)

where the initial guess x0 ∈ C is chosen arbitrarily. For example, Reich [9] proved that if X is a uniformly
convex Banach space with a Fréchet differentiable norm and if {
n } is chosen such that ∞ n=1
n (1 −
n ) = ∞, then the sequence {xn } defined by (1.1) converges weakly to a fixed point of T. However we note that Mann’s iterations have only weak convergence even in a Hilbert space [3]. Attempts to modify the Mann iteration method (1.1) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [7] proposed the following

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modification of the Mann iteration method (1.1) for a single nonexpansive mapping T in a Hilbert space H: ⎧ x0 ∈ C chosen arbitrarily, ⎪ ⎪ ⎪ yn =
n xn + (1 −
n )T x n , ⎨ Cn = {z ∈ C : yn − z
xn − z}, ⎪ ⎪ Q ⎪ ⎩ n = {z ∈ C : xn − z, x0 − xn 0}, xn+1 = PCn ∩Qn (x0 ),

(1.2)

where PK denotes the metric projection from H onto a closed convex subset K of H. They also proposed the following iteration process for a nonexpansive semigroup S = {T (s) : 0
s < ∞} in a Hilbert space H: ⎧ x0 ∈ C chosen arbitrarily, ⎪ ⎪ ⎪ 1  tn ⎪ ⎪ T (s)xn ds, ⎨ yn =
n xn + (1 −
n ) tn 0 Cn = {z ∈ C : yn − z
xn − z}, ⎪ ⎪ ⎪ ⎪ Q ⎪ ⎩ n = {z ∈ C : xn − z, x0 − xn 0}, xn+1 = PCn ∩Qn (x0 ).

(1.3)

They proved that if the sequence {
n } is bounded above from one and if {tn } is a positive real divergent sequence, then the sequence {xn } generated by (1.2) (resp. (1.3)) converges strongly to PF ix(T ) (x0 ) (resp. PF ix(S) (x0 )). The adaptation of Mann’s iteration (1.1) to asymptotically nonexpansive mappings T is given below xn+1 =
n xn + (1 −
n )T n xn ,

n0.

(1.4)

Weak convergence of the sequence {xn } generated by (1.4) is proved by Schu [10] (see also Tan and Xu [14]). It is the purpose of this paper to adapt the iteration (1.2) and (1.3) to asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups. More precisely, we introduce the following iteration processes for asymptotically nonexpansive mappings T and asymptotically nonexpansive semigroups S = {T (t) : t 0}, respectively, with C a closed convex bounded subset of a Hilbert space H: ⎧ x0 ∈ C chosen arbitrarily, ⎪ ⎪ ⎪ ⎨ yn =
n xn + (1 −
n )T n xn , Cn = {z ∈ C : yn − z2
xn − z2 + n }, ⎪ ⎪ ⎪ ⎩ Qn = {z ∈ C : xn − z, x0 − xn 0}, xn+1 = PCn ∩Qn (x0 ), where n = (1 −
n )(kn2 − 1)(diam C)2 → 0

as n → ∞,

(1.5)

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and ⎧ x0 ∈ C chosen arbitrarily, ⎪ ⎪ ⎪ 1  tn ⎪ ⎪ T (s)xn ds, ⎨ yn =
n xn + (1 −
n ) tn 0 Cn = {z ∈ C : yn − z2
xn − z2 + ˜ n }, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Qn = {z ∈ C : xn − z, x0 − xn 0}, xn+1 = PCn ∩Qn (x0 ),

(1.6)

where ˜ n = (1 −
n )



1 tn



tn 0

2 Ls ds

 − 1 (diam C)2 → 0

as n → ∞.

We shall prove that both iteration processes (1.5) and (1.6) converge strongly to a fixed point of T and a common fixed point of S, respectively, provided the sequence {
n } is bounded from above. Another modified Mann’s iteration process for nonexpansive mappings T has recently been introduced by the authors [5] which also has strong convergence provided the sequences {
n } and {n } satisfy certain conditions which are as follows:

x0 ∈ C chosen arbitrarily, yn =
n xn + (1 −
n )T x n , xn+1 = n u + (1 − n )yn .

(1.7)

Apparently, the iteration method (1.7) is simpler than (1.2). However, we do not know if we can adapt the method (1.7) to asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups similar to what we shall adapt in the next sections, the iteration methods (1.2) and (1.3) to asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups. We will use the notation:
for weak convergence and → for strong convergence. We will also use the fact that in a Hilbert space, if xn
x and if xn  → x, then xn → x.

2. Convergence to a fixed point of asymptotically nonexpansive mappings In this section we propose a modification of the Mann iteration method (1.2) to have strong convergence for asymptotically nonexpansive mappings. But before presenting the main result of this section, we include the following lemma which is known as the demiclosedness principle for asymptotically nonexpansive mappings and which is a special case of [6, Theorem 3.1]. Lemma 2.1 (Lin et al. [6]). Let T be an asymptotically nonexpansive mapping defined on a bounded closed convex subset C of a Hilbert space H. Assume that {xn } is a sequence in

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C with the properties (i) xn
z and (ii) T x n − xn → 0. Then z ∈ Fix(T ). Theorem 2.2. Let C be a bounded closed convex subset of a Hilbert space H and let T : C → C be an asymptotically nonexpansive mapping. Assume that {
n }∞ n=0 is a sequence in (0, 1) such that
n
a for all n and for some 0 < a < 1. Define a sequence {xn }∞ n=0 in C by the following algorithm: ⎧ x0 ∈ C chosen arbitrarily, ⎪ ⎪ ⎪ ⎨ yn =
n xn + (1 −
n )T n xn , Cn = {z ∈ C : yn − z2
xn − z2 + n }, ⎪ ⎪ ⎪ ⎩ Qn = {z ∈ C : xn − z, x0 − xn 0}, xn+1 = PCn ∩Qn (x0 ),

(2.1)

where n = (1 −
n )(kn2 − 1)(diam C)2 → 0

as n → ∞.

Then {xn } converges in norm to PF ix(T ) (x0 ). Proof. First note that T has a fixed point in C [4]; that is, Fix(T ) is nonempty. Next observe that Cn is convex. Indeed, the defining inequality in Cn is equivalent to the inequality 2(xn − yn ), z
xn 2 − yn 2 + n which is affine (and hence convex) in z. Next observe that Fix(T ) ⊂ Cn for all n. Indeed, we have, for all p ∈ Fix(T ), yn − p2 = 
n (xn − p) + (1 −
n )(T n xn − p)2

n xn − p2 + (1 −
n )T n xn − p2

n xn − p2 + (1 −
n )kn2 xn − p2 = xn − p2 + [
n + (1 −
n )kn2 − 1]xn − p2
xn − p2 + n . So p ∈ Cn for all n. Next we show that Fix(T ) ⊂ Cn ∩ Qn

for all n0.

(2.2)

It suffices to show that Fix(T ) ⊂ Qn for all n0. We prove this by induction. For n = 0, we have Fix(T ) ⊂ C = Q0 . Assume that Fix(T ) ⊂ Qn . Since xn+1 is the projection of x0

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onto Cn ∩ Qn , we have xn+1 − z, x0 − xn+1 0

∀z ∈ Cn ∩ Qn .

As Fix(T ) ⊂ Cn ∩ Qn , the last inequality holds, in particular, for all z ∈ Fix(T ). This together with the definition of Qn+1 implies that Fix(T ) ⊂ Qn+1 . Hence (2.2) holds for all n 0. Next we show that xn+1 − xn  → 0.

(2.3)

Indeed, by the definition of Qn , we have xn = PQn (x0 ) which together with the fact that xn+1 ∈ Cn ∩ Qn ⊂ Qn implies that x0 − xn 
x0 − xn+1 . This shows that the sequence {xn − x0 } is increasing. Since C is bounded, we obtain that the limn→∞ xn − x0  exists. Noticing again that xn = PQn (x0 ) and xn+1 ∈ Qn which imply that xn+1 − xn , xn − x0 0, and also noticing the identity u − v2 = u2 − v2 − 2u − v, v ∀u, v ∈ H , we obtain xn+1 − xn 2 = (xn+1 − x0 ) − (xn − x0 )2 = xn+1 − x0 2 − xn − x0 2 − 2xn+1 − xn , xn − x0
xn+1 − x0 2 − xn − x0 2 → 0 as n → ∞. We now claim that T x n − xn  → 0. We first observe that xn − T n xn  → 0. Indeed, by definition of yn we have 1 yn − xn  1 −
n 1
(yn − xn+1  + xn+1 − xn ). 1 −
n

T n xn − xn 


(2.4)

2 2 Since xn+1 ∈ C√ n , yn − xn+1 
xn − xn+1  + n which implies that yn − xn+1 
xn − xn+1  + n . Also since
n
a for all n, it follows from (2.4) that

T n xn − xn 


1 (2xn+1 − xn  + n ) → 0. 1−a

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Putting k∞ = sup{kn : n 1} < ∞, we deduce that T x n − xn 
T x n − T n+1 xn  + T n+1 xn − T n+1 xn+1  + T n+1 xn+1 − xn+1  + xn+1 − xn 
k∞ xn − T n xn  + T n+1 xn+1 − xn+1  + (1 + k∞ )xn − xn+1  → 0. Assume that a {xni } is a subsequence of {xn } such that xni
x. ˜ By Lemma 2.1, x˜ ∈ Fix(T ). We now prove that x˜ = PFix(T ) (x0 ) and the convergence is strong. Put x = PF ix(T ) (x0 ) and consider the sequence {x0 − xni }. Then we have x0 − xni
x0 − x˜ and by the weak lower semicontinuity of the norm and by the fact that x0 − xn+1 
x0 − x  for all n0 which is implied by the fact that xn+1 = PCn ∩Qn (x0 ), we obtain ˜ x0 − x 
x0 − x
lim inf x0 − xni 
lim sup x0 − xni  i→∞


x0 − x .

i→∞

˜ (hence x˜ = x by the uniqueness of the nearest point This implies that x0 − x  = x0 − x projection of x0 onto Fix(T )) and that x0 − xni  → x0 − x . It follows that x0 − xni → x0 − x ; hence, xni → x . Since {xni } is an arbitrary (weakly convergent) subsequence of {xn }, we conclude that xn → x .  Remark 2.3. In the weak convergence result of the iteration process (1.4) for asymptotically nonexpansive mappings T (cf. [10,13]), there is a restriction on the sequence {kn } of the Lipschtzian constants of the mappings {T n } which is the assumption ∞ 

(kn − 1) < ∞.

n=1

While in our Theorem 2.2 we do not need this assumption.

3. Convergence to a common fixed point of asymptotically nonexpansive semigroups Assume in this section that S = {T (t) : t 0} is an asymptotically nonexpansive semigroup defined on a nonempty closed convex bounded subset C of a Hilbert space H. Recall that we use Lt to denote the Lipschitzian constant of the mapping T (t) and assume that Lt

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t is bounded and measurable so that the integral 0 Ls ds exists for all t > 0. Recall also that Lt 1 for all t > 0, Lt is nonincreasing in t, and limt→∞ Lt = 1. In the rest of this section, we put L∞ = sup{Lt : 0 < t < ∞} < ∞. Recall furthermore that we use Fix(S) to denote the common fixed point set of S, i.e., the set {x ∈ C : T (t)x = x, t > 0}. Note that the boundedness of C implies that Fix(S) is nonempty (cf. [13]). To prove our strong convergence result (Theorem 3.3 below) we first establish some technical lemmas. Lemma 3.1. Let C be a nonempty bounded closed convex subset of H and S = {T (t) : 0
t < ∞} be an asymptotically nonexpansive semigroup on C. If {xn } is a sequence in C satisfying the properties (a) xn
z; and (b) lim supt→∞ lim supn→∞ T (t)xn − xn  = 0, then z ∈ Fix(S). Proof. This lemma is the continuous version of Lemma 2.3 of Tan and Xu [12]. The proof given in [12] is easily extended to the continuous case.  Lemma 3.2. Let C be a nonempty bounded closed convex subset of H and S = {T (t) : 0
t < ∞} be an asymptotically nonexpansive semigroup on C. Then it holds that  t  t

 1  1  = 0. lim sup lim sup sup  T (u)x du − T (s) T (u)x du   t 0 s→∞ t→∞ x∈C t 0 Proof. For x ∈ C and t > 0, put 1 t (x) := t



t

T (u)x du.

0

Now fix s > 0 and let t > s. Then, we can choose a divergent subsequence {in } of {n} such that t t in
s
(in + 1) n n

for each integer n 1.

(For such an example, consider in = [n(s/t)], where [ · ] means the Gauss function.) Consequently we have limn→∞ (t/n)in = s. For x ∈ C, we put  xi,n = T

t i x n

and

n 1  xi,n . n = n i=1

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In the identity (cf. [1]) n − v2 =

n n 1  1  xi,n − n 2 , xi,n − v2 − n n

v ∈ H,

i=1

i=1

taking v = T ((t/n)in ) n , we infer that 

2  in    n − T t in n  = 1 xi,n − v2   n n i=1  

2  n  t 1   t  in n  T + i x−T   n n n −


1 n

1 n

i=in +1 n 

xi,n − n 2

i=1 i n 

xi,n − v2 +

i=1

n L2(t/n)in n−i n

n 1  xi,n − n 2 − n

xi,n − n 2

i=1

i=1

in n−i n 1  1 = xi,n − v2 + (L2(t/n)in − 1) xi,n − n 2 n n i=1

i=1

n 1  − xi,n − n 2 n i=n−in +1   n − in in 2 + (L(t/n)in − 1) (diam(C))2
n n

 in 2 + L(t/n)in − 1 (diam(C))2 .
n

Hence 

     in t n − T in n  + L2(t/n)in − 1 · (diam C).
  n n

(3.1)

Noticing n =



n t 1 t 1  t T i x→ T (u)x du = t (x), t n n t 0 i=1

(3.2)

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we obtain that



    t  in n  t (x) − T (s)t (x)
t (x) − n  + n − T  n   



   t t  + T n in n − T n in t (x)   

  t  + T in t (x) − T (s)t (x)  n 

    t  i  − T
(L∞ + 1)n − t (x) +  n n   n n   

  t  + T n in t (x) − T (s)t (x) .

(3.3)

Since limn→∞ ti n /n = s, we have ti n /n > s − 1 for all n large enough. Also since the function t → Lt is nonincreasing, we get lim sup Lti n /n
Ls−1 .

(3.4)

n→∞

Now taking the lim sup as n → ∞ in (3.3) and noticing (3.1), (3.2) and (3.4), we obtain  s (3.5) + L2s−1 − 1 · (diam C). t (x) − T (s)t (x)
t We therefore conclude from (3.5) that lim sup lim sup sup t (x) − T (s)t (x) = 0. s→∞

t→∞

x∈C



Now we present the strong convergence of as asymptotically nonexpansive semigroup on C in a Hilbert space. Theorem 3.3. Let C be a nonempty bounded closed convex subset of H and S = {T (t) : 0
t < ∞} be an asymptotically nonexpansive semigroup on C. Assume also that {
n }∞ n=0 is a sequence in (0, 1) such that
n
a for all n and for some 0 < a < 1 and {tn } is a positive real divergent sequence. Define a sequence {xn }∞ n=0 in C by the algorithm ⎧ x0 ∈ C chosen arbitrarily, ⎪ ⎪ ⎪ 1  tn ⎪ ⎪ T (u)xn du, ⎨ yn =
n xn + (1 −
n ) tn 0 (3.6) Cn = {z ∈ C : yn − z2
xn − z2 + ˜ n }, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Qn = {z ∈ C : xn − z, x0 − xn 0}, xn+1 = PCn ∩Qn (x0 ), where ˜ n = (1 −
n )



1 tn



tn 0

2 Ls ds

 − 1 (diam C)2 → 0

Then {xn } converges in norm to PF ix(S) (x0 ).

as n → ∞.

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Proof. First observe that Fix(S) ⊂ Cn for all n. Indeed, we have for all p ∈ Fix(S),   1 yn − p = 
x + (1 −
) n n n  t 2

2  T (s)xn ds − p   n 0  tn 2 1  

n xn − p2 + (1 −
n ) T (s)x ds − p n t  n 0  tn

2 1 2 T (s)xn − p ds

n xn − p + (1 −
n ) tn 0  tn

2 1 2

n xn − p + (1 −
n ) Ls ds xn − p2 tn 0 = xn − p2 + ˜ n . tn

So p ∈ Cn for all n. As in the proof of Theorem 2.2, {xn } is well defined and Fix(S) ⊂ Cn ∩ Qn

for all n 0.

Also, similar to the proof of Theorem 2.2, we can show that xn+1 − xn  → 0.

(3.7)

We can deduce that for all 0
s < ∞, 

  tn   1  T (s)xn − xn 
T (s)xn − T (s) T (u)xn du   tn 0    tn

tn   1 1  + T (s) T (u)x du − T (u)x du n n   tn 0 tn 0  tn  1  + T (u)xn du − xn  t  n 0   tn  1 
(L∞ + 1)  T (u)xn du − xn   t    n 0tn

tn   1 1  + T (s) T (u)xn du − T (u)xn du  tn 0 tn 0 := (L∞ + 1)An + Bn (s). We claim that (i) limn→∞ An = 0; and (ii) lim sups→∞ lim supn→∞ Bn (s) = 0.

(3.8)

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As a matter of fact, that (ii) holds is guaranteed by Lemma 3.2, while (i) is verified by the following argument. By the definition of yn we have   tn  1  T (u)xn du − xn  An =   tn 0 1 = yn − xn  1 −
n 1
(yn − xn+1  + xn+1 − xn ). (3.9) 1−a Note that the fact xn+1 ∈ Cn implies that yn − xn+1 2
xn − xn+1 2 + ˜ n which in turn implies that  yn − xn+1 
xn − xn+1  + ˜ n . It follows from (3.9) that  

1 2xn+1 − xn  + ˜ n → 0. An
1−a We thus conclude from (3.8) that lim sup lim sup T (s)xn − xn  = 0. s→∞

n→∞

An application of Lemma 3.1 implies that every weak limit point of {xn } is a member of Fix(S). Repeating the last part of the proof of Theorem 2.2, we can prove that PF ix(S) (x0 ) is the only weak limit point of {xn }; hence {xn } weakly converges to PF ix(S) (x0 ), and that the convergence is moreover in the strong topology.  Acknowledgements The authors thank the referee for his comments and suggestions which improved the presentation of this manuscript. References [1] H. Brézis, F.E. Browder, Nonlinear ergodic theorems, Bull. Am. Math. Soc. 82 (1976) 959–961. [2] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems 20 (2004) 103–120. [3] A. Genel, J. Lindenstrass, An example concerning fixed points, Israel J. Math. 22 (1975) 81–86. [4] K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Am. Math. Soc. 35 (1972) 171–174. [5] T.W. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60. [6] P.K. Lin, K.K. Tan, H.K. Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal. 24 (1995) 929–946.

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[7] K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379. [8] C.I. Podilchuk, R.J. Mammone, Image recovery by convex projections using a least-squares constraint, J. Opt. Soc. Am. A 7 (1990) 517–521. [9] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979) 274–276. [10] J. Schu, Weak, strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991) 153–159. [11] M.I. Sezan, H. Stark, Applications of convex projection theory to image recovery in tomography and related areas, in: H. Stark (Ed.), Image Recovery Theory and Applications, Academic Press, Orlando, 1987, pp. 415–462. [12] K.K. Tan, H.K. Xu, The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces, Proc. Am. Math. Soc. 114 (1992) 399–404. [13] K.K. Tan, H.K. Xu, Fixed point theorems for Lipschitzian semigroups in Banach spaces, Nonlinear Anal. 20 (1993) 395–404. [14] K.K. Tan, H.K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Am. Math. Soc. 122 (1994) 733–739. [15] H.K. Xu, Strong asymptotic behavior of almost-orbits of nonlinear semigroups, Nonlinear Anal. 46 (2001) 135–151. [16] D. Youla, Mathematical theory of image restoration by the method of convex projections, in: H. Stark (Ed.), Image Recovery Theory and Applications, Academic Press, Orlando, 1987, pp. 29–77. [17] D. Youla, On deterministic convergence of iterations of relaxed projection operators, J. Visual Comm. Image Representation 1 (1990) 12–20.