Convergence of a hybrid algorithm for a reversible semigroup of nonlinear operators in Banach spaces

Nonlinear Analysis 73 (2010) 3413–3419

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Convergence of a hybrid algorithm for a reversible semigroup of nonlinear operators in Banach spaces Kyung Soo Kim Department of Mathematics Education, Kyungnam University, Masan, Kyungnam, 631-701, Republic of Korea

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Article history: Received 27 February 2010 Accepted 12 July 2010 MSC: 26A18 47H10 47H20

abstract The purpose of this paper is to study hybrid iterative schemes of Halpern types for a semigroup = = {T (s) : s ∈ S } of relatively nonexpansive mappings on a closed and convex subset C of a Banach space with respect to a sequence {µn } of asymptotically left invariant means defined on an appropriate invariant subspace of l∞ (S ). We prove that given a certain sequence {αn } in [0, 1], x ∈ C , we can generate an iterative sequence {xn } which converges strongly to ΠF (=) x where ΠF (=) x is the generalized projection from C onto the fixed point set F (=). Our main result is even new for the case of a Hilbert space. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Amenable Asymptotically left invariant mean Common fixed point Generalized projection operator Hybrid iterative scheme Metric projection Relatively nonexpansive Reversible semigroup

1. Introduction Let E be a real Banach space with the topological dual E ∗ and let C be a closed and convex subset of E. A mapping T of C into itself is called nonexpansive if kTx − Tyk ≤ kx − yk for each x, y ∈ C . In 1967, Halpern [1] first introduced the following iterative scheme for nonexpansive mappings:



x0 = u ∈ C , chosen arbitrary, xn+1 = αn u + (1 − αn )Txn , ∀n ≥ 1,

(1.1)

where {αn } is a sequence in [0, 1]. He pointed out that the conditions limn→∞ αn = 0 and n=1 αn = ∞ are necessary in the sense that, if the iteration (1.1) converges to a fixed point of T , then these conditions must be satisfied. To find a common fixed point of a family of nonexpansive mappings, Aoyama et al. [2] introduced the following iterative sequence. Let x1 = x ∈ C and

P∞

xn+1 = αn x + (1 − αn )Tn xn ,

(1.2)

for all n ∈ N, where C is a nonempty, closed and convex subset of a Banach space, {αn } is a sequence of [0, 1], and {Tn } is a sequence of nonexpansive mappings. Then they proved that, under some suitable conditions, the sequence {xn } defined by (1.2) converges to a common fixed point of {Tn } (cf. [3]).

E-mail address: [email protected]. 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.07.027

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Let S be a semigroup. The purpose of this paper is to study hybrid iterative schemes of Halpern type (1.2) for a semigroup

= = {T (s) : s ∈ S } of relatively nonexpansive mappings on a closed and convex subset C of a Banach space with respect to a sequence {µn } of asymptotically left invariant means defined on an appropriate invariant subspace of l∞ (S ). We prove that given a certain sequence {αn } in [0, 1], x ∈ C , we can generate an iterative sequence {xn } which converges strongly to ΠF (=) x where ΠF (=) x is the generalized projection from C onto the fixed point set F (=). 2. Preliminaries Recently, many authors have tried to extend the above result from Hilbert spaces to Banach spaces. x +y A real Banach space E is said to be strictly convex if k 2 k < 1 for all x, y ∈ E with kxk = kyk = 1 and x 6= y. It is said to be uniformly convex if limn→∞ kxn − yn k = 0 for any two sequences {xn } and {yn } in E such that kxn k = kyn k = 1 and x +y limn→∞ k n 2 n k = 1. It is known that a uniformly convex Banach space is reflexive and strictly convex. Let U = {x ∈ E : kxk = 1} be the unit sphere of E. Then the Banach space E is said to be smooth if lim

n→∞

kx + tyk − kxk t

exists for each x, y ∈ U. It is said to be uniformly smooth if the limit is attained uniformly for x, y ∈ E. ∗ Let E be a real Banach space, E ∗ the dual space of E. We denote by J the normalized duality mapping from E to 2E defined by Jx = {f ∗ ∈ E ∗ : hx, f ∗ i = kxk2 = kf ∗ k2 }, where h·, ·i denotes the generalized duality pairing. It is well known that if E ∗ is strictly convex then J is single valued and if E is uniformly smooth then J is uniformly norm-to-norm continuous on bounded subsets of E. Moreover, if E is a reflexive and strictly convex Banach space with a strict dual, then J −1 is single valued, one-to-one, surjective and it is the duality mapping from E ∗ into E and thus JJ −1 = IE ∗ and J −1 J = IE (see [4]). We note that in a Hilbert space H, J is the identity operator. As we know, if C is a nonempty, closed and convex subset of a Hilbert space H and PC : H → C is the metric projection of H onto C , then PC is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not apposite to more general Banach spaces. In this connection, Alber [5] introduced a generalized projection operator ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces. Consider the functional defined by

φ(x, y) = kxk − 2hx, Jyi + kyk2 for x, y ∈ E .

(2.1)

Observe that, in a Hilbert space H, (2.1) reduces to

φ(x, y) = kx − yk2 ,

x, y ∈ H .

The generalized projection ΠC : E → C is a map that assigns to an arbitrary point x ∈ E the minimum point of the functional φ(x, y), that is, Πc x = x¯ , where x¯ is the solution to the minimization problem

φ(¯x, x) = inf φ(y, x). y∈C

(2.2)

The existence and uniqueness of the operator ΠC follow from the properties of the functional φ(x, y) and strict monotonicity of the mapping J (see, for example, [5] and [6]). In a Hilbert space, ΠC = PC . It is obvious from the definition of the function φ that: (1) (2) (3) (4)

(kxk − kyk)2 ≤ φ(x, y) ≤ (kxk + kyk)2 for all x, y ∈ E . φ(x, y) = φ(x, z ) + φ(z , y) + 2hx − z , Jz − Jyi for all x, y, z ∈ E . φ(x, y) = hx, Jx − Jyi + hy − x, Jyi ≤ kxk k Jx − Jyk + ky − xk kyk for all x, y ∈ E. If E is a reflexive, strictly convex and smooth Banach space, then, for all x, y ∈ E, φ(x, y) = 0 if and only if x = y.

For more details see [4]. Let C be a closed and convex subset of E, and let T be a mapping from C into itself. We denote by F (T ) the set of fixed points of T . A point p in C is said to be an asymptotic fixed point of T [7] if C contains a sequence {xn } which converges weakly to p such that the strong limn→∞ (Txn − xn ) = 0. The set of asymptotic fixed points of T will be denoted by Fˆ (T ). A mapping T from C into itself is called relatively nonexpansive [8–10] if Fˆ (T ) = F (T ) and φ(p, Tx) ≤ φ(p, x) for all x ∈ C and p ∈ F (T ). The asymptotic behaviors of relatively nonexpansive mappings were studied in [8,9,11]. Remark 2.1. In [11], it is shown that if H is a Hilbert space, C is a nonempty, closed and convex subset of H and T is a nonexpansive mapping from C into itself, then T is a relatively nonexpansive mapping from C onto itself. Remark 2.2 (cf. [11]). Let E be a strictly convex and smooth Banach space, let C be a closed and convex subset of E, and let T be a relatively nonexpansive mapping from C into itself. Then F (T ) is closed and convex.

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Let S be a semigroup. We denote by l∞ (S ) the Banach space of all bounded real-valued functionals on S with supremum norm. For each s ∈ S, we define the left and right translation operators l(s) and r (s) on l∞ (S ) by

(l(s)f )(t ) = f (st ) and (r (s)f )(t ) = f (ts) for each t ∈ S and f ∈ l∞ (S ), respectively. Let X be a subspace of l∞ (S ) containing 1. An element µ in the dual space X ∗ of X is said to be a mean on X if kµk = µ(1) = 1. For s ∈ S, we can define a point evaluation δs by δs (f ) = f (s) for each f ∈ X . It is well known that µ is the mean on X if and only if inf f (s) ≤ µ(f ) ≤ sup f (s) s∈S

s∈S

for each f ∈ X . Let X be a translation invariant subspace of l∞ (S ) (i.e. l(s)X ⊂ X and r (s)X ⊂ X for each s ∈ S) containing 1. Then a mean µ on X is said to be left invariant (resp. right invariant) if

µ(l(s)f ) = µ(f ) (resp. µ(r (s)f ) = µ(f )) for each s ∈ S and f ∈ X . A mean µ on X is said to be invariant if µ is both left and right invariant [12–15]. S is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. S is amenable if S is left and right amenable. In this case, l∞ (S ) also has an invariant mean. Moreover, l∞ (S ) is amenable when S is a commutative semigroup or a solvable group. However, the free group or semigroup of two generators is not left or right amenable (see [16–18]). A net {µα } of means on X is said to be asymptotically left (resp. right) invariant if





lim(µα (l(s)f ) − µα (f )) = 0 resp. lim(µα (r (s)f ) − µα (f )) = 0 α

α

for each f ∈ X and s ∈ S, and it is said to be left (resp. right) strongly asymptotically invariant (or strong regular) if



lim kl∗ (s)µα − µα k = 0 resp. lim kr ∗ (s)µα − µα k = 0 α



α

for each s ∈ S, where l∗ (s) and r ∗ (s) are the adjoint operators of l(s) and r (s), respectively. Such nets were first studied by Day in [16] and called weak* invariant and norm invariant, respectively. From now on S denotes a semigroup with an identity e. S is called left reversible if any two closed right ideals of S have non-void intersection, i.e., aS ∩ bS 6= ∅ for a, b ∈ S. In this case, (S , ) is a directed system when the binary relation ‘‘’’ on S is defined by a  b if and only if {a} ∪ aS ⊇ {b} ∪ bS for a, b ∈ S. The class of left reversible semigroups includes all groups and commutative semigroups. If a semigroup S is left amenable, then S is left reversible. But the converse is not true [19–23]. Let S be a semigroup and let C be a closed and convex subset of E. A family = = {T (s) : s ∈ S } is called a representation of S as relatively nonexpansive mappings on C if T (s) is relatively nonexpansive and T (st ) = T (s)T (t ) for each s, t ∈ S. We denote by F (=) the set of common fixed points of =, i.e. F (=) =

\

F (T (s)) =

s∈S

\ {x ∈ C : T (s)x = x}. s∈S

We assume that X is a subspace of l∞ (S ) containing 1 and the function s 7→ hT (s)x, x∗ i is contained in X for each x ∈ C and x∗ ∈ E ∗ . Then there exists a unique point x0 of E such that µhT (·)x, x∗ i = hx0 , x∗ i for a mean µ on X , x ∈ C and x∗ ∈ E ∗ . We denote such a point x0 by Tµ x. Note that Tµ x is contained in the closure of the convex hull of {T (s)x : s ∈ S } for each x ∈ C (cf. [24]). 3. Lemmas We need the following lemmas for the proof of our main result. Lemma 3.1 (cf. [25]). Let E be a uniformly convex and smooth Banach space and let {xn }, {yn } be two sequences of E. If limn→∞ φ(xn , yn ) = 0 and either {xn } or {yn } is bounded, then limn→∞ kxn − yn k = 0. Lemma 3.2 (cf. [5]). Let C be a nonempty, closed and convex subset of a smooth Banach space E and x ∈ E. Then x0 = ΠC x if and only if

hx0 − y, Jx − Jx0 i ≥ 0 for all y ∈ C . Lemma 3.3 (cf. [5]). Let E be a reflexive, strictly convex and smooth Banach space, and let C be a nonempty, closed and convex subset of E and x ∈ E. Then

φ(y, ΠC x) + φ(ΠC x, x) ≤ φ(y, x) for all y ∈ C . To obtain the main result, we assume that the following lemma holds. Lemma 3.4 (cf. [26]). Let µ be a left invariant mean on X . Then F (=) = F (T (µ))∩ Ca , where Ca denotes the set of almost periodic elements in C , i.e. all x ∈ C such that {T (s)x : s ∈ S } is relatively compact in the norm topology of E.

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Lemma 3.5 (cf. [26]). Let {µn } be an asymptotically left invariant sequence of means on X . If z ∈ Ca and lim infn→∞ kTµn z −z k = 0, then z is a common fixed point of =. Proof. From lim infn→∞ kTµn z − z k = 0, there exists a subsequence {Tµnk z } of {Tµn z } that converges strongly to z. Since the set of means on X is compact in the weak* topology, there exists a subnet {µnkα : α ∈ Λ} of {µnk } such that {µnkα } converges to µ in the weak* topology. Then, it is easy to show that µ is a left invariant mean on X . On the other hand, for each x∗ ∈ E ∗ , we have

hTµnk z , x∗ i = µnkα hT (·)z , x∗ i → µhT (·)z , x∗ i = hTµ z , x∗ i. α

Now, since {Tµnk z } converges strongly to z, we have hz , xi = hTµ z , x∗ i and hence z = Tµ z. It follows from Lemma 3.4 that z is a common fixed point of =.  4. The main result In this section, we establish a strong convergence theorem for finding common fixed points of relatively nonexpansive mappings in uniformly convex and uniformly smooth Banach spaces. Theorem 4.1. Let S be a left reversible semigroup and = = {T (s) : s ∈ S } be a representation of S as relatively nonexpansive mappings from a nonempty, closed and convex subset C of a uniformly convex and uniformly smooth Banach space E into C with F (=) 6= ∅. Let X be a subspace of l∞ (S ) and let {µn } be an asymptotically left invariant sequence of means on X . Let {αn } be a sequence in [0, 1] such that 0 < αn < 1 and limn→∞ αn = 0. Let {xn } be a sequence generated by the following algorithm:

 x0 ∈ C , chosen arbitrary,   C1 = C ,    x1 = ΠC1 x0 , yn = J −1 (αn Jx1 + (1 − αn )JTµn xn ),      Cn+1 = {z ∈ Cn : φ(z , yn ) ≤ αn φ(z , x1 ) + (1 − αn )φ(z , xn )}, xn+1 = ΠCn+1 x1 , ∀n ≥ 1. Then {xn } converges strongly to ΠF (=) x1 , where ΠF (=) is the generalized projection from C onto F (=). Proof. We divide the proof into three steps. (I) We show first that the sequence {xn } is well defined. First, we prove by induction that Cn is closed and convex for all n ≥ 1. It is obvious that C1 = C is closed and convex. Suppose that Ck is closed and convex for some k ∈ N. For all z ∈ Ck , one knows that

φ(z , yk ) ≤ αk φ(z , x1 ) + (1 − αk )φ(z , xk ) is equivalent to 2αk hz , Jx1 i + 2(1 − αk )hz , Jxk i − 2hz , Jyk i ≤ αk kx1 k2 + (1 − αk )kxk k2 − kyk k2 . It can be seen easily that Ck+1 is bounded. Now we prove that Ck+1 is convex. In fact, let z1 , z2 ∈ Ck+1 , λ ∈ (0, 1). Put z 0 = λz1 + (1 − λ)z2 . Since z1 , z2 ∈ Ck+1 , we have 2αk hλz1 , Jx1 i + 2(1 − αk )hλz1 , Jxk i − 2hλz1 , Jyk i ≤ λ[αk kx1 k2 + (1 − αk )kxk k2 − kyk k2 ] and 2αk h(1 − λ)z2 , Jx1 i + 2(1 − αk )h(1 − λ)z2 , Jxk i − 2h(1 − λ)z2 , Jyk i ≤ (1 − λ)[αk kx1 k2 + (1 − αk )kxk k2 − kyk k2 ]. Add these two inequalities; we obtain 2αk hz 0 , Jx1 i + 2(1 − αk )hz 0 , Jxk i − 2hz 0 , Jyk i ≤ αk kx1 k2 + (1 − αk )kxk k2 − kyk k2 . This is equivalent to

φ(z 0 , yk ) ≤ αk φ(z 0 , x1 ) + (1 − αk )φ(z 0 , xk ). Since Ck is convex, we have z 0 ∈ Ck . Then we obtain z 0 ∈ Ck+1 . It follows that Ck+1 is closed and convex. Therefore, for all n ≥ 1, Cn is closed and convex. Next, we prove by induction that F (=) ⊂ Cn for all n ≥ 1. In fact, F (=) ⊂ C = C1 is obvious. Suppose that F (=) ⊂ Cn for some n ∈ N. We have

kTµn xn k = = ≤ =

sup{|hTµn xn , x∗ i| : x∗ ∈ E ∗ , kx∗ k = 1} sup{|µn hT (·)xn , x∗ i| : x∗ ∈ E ∗ , kx∗ k = 1} sup{µn (kT (·)xn k kx∗ k) : x∗ ∈ E ∗ , kx∗ k = 1}

µn kT (·)xn k.

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Then, for all p ∈ F (=) ⊂ Cn , we have

φ(p, Tµn xn ) = kpk2 − 2hp, JTµn xn i + kTµn xn k2 = kpk2 − 2µn hp, JT (·)xn i + µn kT (·)xn k2 = µn φ(p, T (·)xn ) ≤ µn φ(p, xn ) = φ(p, xn ).

(4.1)

2

From the convexity of k · k and (4.1), we get

φ(p, yn ) = φ(p, J −1 (αn Jx1 + (1 − αn )JTµn xn )) = kpk2 − 2hp, αn Jx1 + (1 − αn )JTµn xn i + kαn Jx1 + (1 − αn )JTµn xn k2 ≤ kpk2 − 2αn hp, Jx1 i − 2(1 − αn )hp, JTµn xn i + αn kx1 k2 + (1 − αn )kTµn xn k2 = αn φ(p, x1 ) + (1 − αn )φ(p, Tµn xn ) ≤ αn φ(p, x1 ) + (1 − αn )φ(p, xn ) for all p ∈ F (=) ⊂ Cn . So, p ∈ Cn+1 . That is F (=) ⊂ Cn+1 . Consequently, F (=) ⊂ Cn for all n ≥ 1. This implies that {xn } is well defined. (II) We show that xn → p ∈ F (=). From xn = ΠCn x1 and Lemma 3.2, we have

hxn − u, Jx1 − Jxn i ≥ 0,

∀u ∈ Cn .

It follows from F (=) ⊂ Cn for all n ≥ 1 that

hxn − z , Jx1 − Jxn i ≥ 0,

∀z ∈ F (=).

From Lemma 3.3, we have

φ(xn , x1 ) = φ(ΠCn x1 , x1 ) ≤ φ(z , x1 ) − φ(z , xn ) ≤ φ(z , x1 ), for each z ∈ F (=) ⊂ Cn for all n ≥ 1. Therefore the sequence {φ(xn , x1 )} is bounded. Furthermore, since xn = ΠCn x1 and xn+1 = ΠCn+1 x1 ∈ Cn+1 ⊂ Cn , we have

φ(xn , x1 ) ≤ φ(xn+1 , x1 ),

for all n ≥ 1.

This implies that {φ(xn , x1 )} is nondecreasing and hence limn→∞ φ(xn , x1 ) exists. Similarly, by Lemma 3.3, we have

φ(xn+m , xn ) = φ(xn+m , ΠCn x1 ) ≤ φ(xn+m , x1 ) − φ(ΠCn x1 , x1 ) = φ(xn+m , x1 ) − φ(xn , x1 ),

∀n ≥ 1,

(4.2)

for any positive integer m. The existence of limn→∞ φ(xn , x1 ) implies that φ(xn+m , xn ) → 0 as n → ∞. From Lemma 3.1, we have lim kxn+m − xn k = 0.

n→∞

Hence {xn } is a Cauchy sequence. Since E is a Banach space and Cn is closed and convex, there exists a point p ∈ C such that xn → p as n → ∞. Now, we show that p ∈ F (=). By taking m = 1 in (4.2), we obtain lim φ(xn+1 , xn ) = 0.

n→∞

(4.3)

It follows from Lemma 3.1 that lim kxn+1 − xn k = 0.

n→∞

Since xn+1 ∈ Cn+1 , we obtain

φ(xn+1 , yn ) ≤ αn φ(xn+1 , x1 ) + (1 − αn )φ(xn+1 , xn ). It follows from (4.3) and limn→∞ αn = 0 that lim φ(xn+1 , yn ) = 0.

n→∞

(4.4)

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From Lemma 3.1, we have lim kxn+1 − yn k = 0.

(4.5)

n→∞

Combining (4.4) and (4.5), we have

kxn − yn k ≤ kxn − xn+1 k + kxn+1 − yn k → 0 as n → ∞. Since J is uniformly norm-to-norm continuous on any bounded sets, we have lim k Jxn − Jyn k = 0.

n→∞

On the other hand, we have k Jyn − JTµn xn k = αn k Jx1 − JTµn xn k. From the condition for {αn }, we have lim k Jyn − JTµn xn k = 0.

n→∞

Since J −1 is also uniformly norm-to-norm continuous on bounded sets, we obtain lim kyn − Tµn xn k = 0.

(4.6)

n→∞

Moreover, we have

kxn − Tµn xn k ≤ kxn − xn+1 k + kxn+1 − yn k + kyn − Tµn xn k. From (4.4)–(4.6), lim kxn − Tµn xn k = 0.

n→∞

From Lemma 3.5, we have xn ∈ F (=). Since F (=) is closed and limn→∞ xn = p, we have p ∈ F (=). (III) Finally, we show that p = ΠF (=) x1 . From xn = Πcn x1 , we obtain

hxn − z , Jx1 − Jxn i ≥ 0,

∀z ∈ F (=) ⊂ Cn .

(4.7)

Taking the limit as n → ∞ in (4.7), we have

hp − z , Jx1 − Jpi ≥ 0,

∀z ∈ F (=),

and hence p = ΠF (=) x1 by Lemma 3.2. This completes the proof.



In a Hilbert space, Theorem 4.1 is reduced to the following. Corollary 4.1. Let S be a left reversible semigroup and = = {T (s) : s ∈ S } be a representation of S as relatively nonexpansive mappings from a nonempty, closed and convex subset C of a Hilbert space H into C with F (=) 6= ∅. Let X be a subspace of l∞ (S ) and let {µn } be an asymptotically left invariant sequence of means on X . Let {αn } be a sequence in [0, 1] such that 0 < αn < 1 and limn→∞ αn = 0. Let {xn } be a sequence generated by the following algorithm:

 x0 ∈ C , chosen arbitrary,     C = C,   1 x1 = PC1 x0 , yn = αn x1 + (1 − αn )Tµn xn ,   2 2 2    Cn+1 = {z ∈ Cn : kz − yn k ≤ αn kz − x1 k + (1 − αn )kz − xn k }, xn+1 = PCn+1 x1 , ∀n ≥ 1. Then {xn } converges strongly to PF (=) x1 , where PF (=) is a metric projection. A point p in C is said to be strong asymptotic fixed point of T if C contains a sequence {xn } which converges strongly to p such that the strong limn→∞ (Txn − xn ) = 0. The set of strong asymptotic fixed points of T will be denoted by F˜ (T ). A mapping T from C into itself is called relatively weak nonexpansive if F˜ (T ) = F (T ) and φ(p, Tx) ≤ φ(p, x) for all x ∈ C and p ∈ F (T ). T is called relatively weak quasi-nonexpansive if F (T ) 6= ∅ and φ(p, Tx) ≤ φ(p, x) for all x ∈ C and p ∈ F (T ). Remark 4.1. The class of relatively weak quasi-nonexpansive mappings is more general than the class of relatively weak nonexpansive mappings [8,9,11] which requires the strong restriction F˜ (T ) = F (T ). Example 4.1 (cf. [27]). Let E be a uniformly and strictly convex Banach space and A ⊂ E × E ∗ be a maximal monotone mapping such that its zero set A−1 0 is nonempty. Then the resolvent Jr = (J + rA)−1 J is a closed and relatively weak quasiasymptotic mapping from E onto D(A) and F (Jr ) = A−1 0.

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Example 4.2 (cf. [27]). Let ΠC be the generalized projection from a smooth, strictly convex and reflexive Banach space E onto a nonempty, closed and convex subset C of E. Then ΠC is a closed and relatively weak quasi-nonexpansive mapping from E onto C with F (ΠC ) = C . Remark 4.2. It is obvious that a relatively nonexpansive mapping is a relatively weak nonexpansive mapping. In fact, for any mapping T : C → C , we have F (T ) ⊂ F˜ (T ) ⊂ Fˆ (T ). Therefore, if T is a relatively nonexpansive mapping, then F (T ) = F˜ (T ) = Fˆ (T ). Remark 4.3. Do Theorem 4.1 and Corollary 4.1 remain true with the weaker condition of a semigroup of relatively weak quasi-nonexpansive mappings? Acknowledgements The author would like to thank Prof. Anthony To-Ming Lau for his helpful suggestions. Also, special thanks are due to the referee for his/her deep insight which improved the paper. References [1] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967) 957–961. [2] K. 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