JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.11,NO.3,451-460,2009,COPYRIGHT 2009 EUDOXUS PRESS, LLC

Journal of Computational Analysis and Applications, Vol. 10, No. 0, 000-000, 2008. COPYRIGHT 2008 EUDOXUS PRESS, LLC

AVERAGED ITERATES FOR NON-EXPANSIVE NONSELF-MAPPINGS IN BANACH SPACES YISHENG SONG AND YEOL JE CHO

Abstract. Let E be a uniformly convex Banach space with a uniformly Gˆ ateaux differentiable norm and K be a closed convex subset of E which is also a sunny non-expansive retract of E. Assume that T : K → E is a nonexpansive mappings with a fixed point. The two averaged iterative sequences {xn } are given by xn+1 = xn+1 =

n 1  (αn u + (1 − αn )(P T )j xn ), n + 1 j=0

n 1  P (αn u + (1 − αn )(T P )j xn ), n + 1 j=0

∀n ≥ 0, ∀n ≥ 0,

where P is sunny non-expansive retraction of E onto K and αn ∈ (0, 1) sat∞  lim αn = 0 and αn = ∞. We prove that {xn }

isfying the conditions:

n→∞

n=0

converges strongly to some fixed point of T and, furthermore, as applications, we obtain the viscosity averaged approximation results for T . Key Words and Phrases: Non-expansive nonself-mappings, averaged iterates; uniformly convex Banach space. 2000 AMS Subject Classification: 47H05, 47H10, 47H17.

1. Introduction Let T be a mapping with domain D(T ) and range R(T ) in Banach space E. T is called non-expansive if T x − T y ≤ x − y,

∀x, y ∈ D(T ).

In 1967, Halpern [5] firstly introduced the following iteration scheme in Hilbert space: for a non-expansive self-mapping T and u = 0, x0 ∈ K, xn+1 = αn u + (1 − αn )T xn ,

∀n ≥ 0.

∞ He pointed out that the control conditions limn→∞ αn = 0 and n=1 αn = ∞ are necessary for the convergence of the iteration scheme to a fixed point of T . In last decades, many authors have studied the iterative algorithms for nonexpansive nonself-mappings and obtained a series of good results. For example, for any given u ∈ K, define the following implicit iterative sequences {xt } as follows: xt = tu + (1 − t)P T xt ,

∀t ∈ (0, 1),

The corresponding author: [email protected] (Yeol Je Cho). The second author was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2007-313-C00040). 1

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YISHENG SONG AND YEOL JE CHO

and xt = P (tu + (1 − t)T xt ), ∀t ∈ (0, 1), where P is the projector (or non-expansive retraction) from E to K. In 1995, Xu and Yin [23] showed that, as t → 0, {xt } converges strongly to some fixed point of T in Hilbert spaces. In 1997, Xu [20] proved and extended the results of Xu and Yin [23] from Hilbert spaces to uniformly smooth Banach spaces. In 1998, Jung and Kim [6] obtained the same results in uniformly convex Banach spaces with the uniformly Gˆ ateaux differentiable norm. In the same year, in the frame of reflexive Banach spaces with the uniformly Gˆ ateaux differentiable norm, Takahashi and Kim [18] also gained the strong convergent results of the sequence {xt }. But they all dealt with the implicit iteration of T , not the explicit iteration. Recently, in a reflexive Banach space E with a weakly sequentially continuous duality mapping, Song and Chen [11] studied the following explicit iterative schemes for non-expansive nonself-mapping T defined by xn+1 = P (αn f (xn ) + (1 − αn )T xn ),

∀n ≥ 0,

and showed that {xn } converges strongly to a fixed point p of T , which is also the unique solution of a variational inequality. In 2002, Xu [21] also obtained the strong convergence of {xn } given by (1.1) for a non-expansive mapping in uniformly convex and uniformly smooth Banach spaces: n 1  j xn+1 = αn u + (1 − αn ) T xn , ∀n ≥ 0. (1.1) n + 1 j=0 In 2004, Matsushita and Kuroiwa [7] investigated the following explicit averaging iterates of non-expansive nonself-mappings in Hilbert spaces and gained the strongly convergent outcomes of {xn } defined by (1.2) and (1.3), respectively, n

xn+1 = and

1  (αn u + (1 − αn )(P T )j xn ), n + 1 j=0

∀n ≥ 0,

(1.2)

n

xn+1 =

1  P (αn u + (1 − αn )(T P )j xn ), n + 1 j=0

∀n ≥ 0.

(1.3)

In the above two results, ∞the control conditions for the iterative schemes only need limn→∞ αn = 0 and n=1 αn = ∞. In this paper, our purpose is to extend the main results of Xu [21], Matsushita and Kuroiwa [7] to uniformly convex Banach spaces with the uniformly Gˆ ateaux differentiable norm for nonexpansive non-self mappings, which also develop and complement the main corresponding results of [22, 20, 23, 11, 13, 14, 6, 18] and many others. 2. Preliminaries Let E be a Banach space and J denote the normalized duality mapping from E ∗ into 2E given by J(x) = {f ∈ E ∗ , x, f  = xf , x = f },

∀ x ∈ E,

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AVERAGED ITERATES FOR NON-EXPANSIVE NONSELF-MAPPINGS

3

where E ∗ is the dual space of E and ·, · denotes the generalized duality pairing. In the sequel, we denote the single-valued duality mapping by j and denote F (T ) = {x ∈ D(T ); x = T x} by the fixed point set of T , where D(T ) is the domain ∗ of T . We write xn  x (respectively, xn  x) to indicate that the sequence {xn } converges weakly (respectively, weak star) to x. As usual, xn → x denotes the strong convergence. If K ⊂ E, K stands for the closure of K. Recall that the norm of E is said to be Gˆ ateaux differentiable (or E is said to be smooth) if the limit x + ty − x lim (1.4) t→0 t exists for each x, y on the unit sphere S(E) = {x ∈ E : x = 1} of E. Moreover, if for each y in S(E) the limit defined by (1.4) is uniformly attained for x in S(E), we say that the norm of E is uniformly Gˆ ateaux differentiable. A Banach space E is said to uniformly smooth if the limit (1.4) is attained uniformly for (x, y) ∈ S(E)×S(E). It is well known that the normalized duality mapping J in a smooth Banach space is single-valued and norm topology to weak star topology continuous on any bounded sets of E [17, Theorems 4.3.1, 4.3.2, 4.3.3]. Also, see [8, 3]. A Banach space E is said to strictly convex if x = y = 1, x = y implies

x + y < 1. 2

A Banach space E is said to uniformly convex if, for all ε ∈ [0, 2], ∃δε > 0 such that x + y x = y = 1 with x − y ≥ ε implies < 1 − δε . 2 If C and D are nonempty subsets of a Banach space E such that C is nonempty closed convex and D ⊂ C, then a mapping P : C → D is called a retraction from C to D if P is continuous with F (P ) = D. A mapping P : C → D is called sunny if P (P x + t(x − P x)) = P x,

∀x ∈ C

whenever P x + t(x − P x) ∈ C and t > 0. A subset D of Cis said to be a sunny non-expansive retract of C if there exists a sunny non-expansive retraction of C onto D. For more details, see [17, 20, 18, 4]. The following Lemma is well known [17]: Lemma 2.1. Let K be a nonempty convex subset of a smooth Banach space E, ∅ = D ⊂ K, J : E → E ∗ the normalized duality mapping of E and P : K → D a retraction. Then the following are equivalent: (i) x − P x, J(y − P x) ≤ 0 for all x ∈ K and y ∈ D. (ii) P is both sunny and non-expansive. Hence there exists at most a sunny non-expansive retraction P from K onto D. In 1980, Reich [9] showed that, if E is uniformly smooth and F (T ) is the fixed point set of a non-expansive mapping T from K into itself, then there is the unique sunny non-expansive retraction from K onto F (T ).

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Theorem R. Let K be a nonempty closed convex subset of a uniformly smooth Banach space E and T : K → K a non-expansive mapping with a fixed point. Then F (T ) is a sunny non-expansive retract of K. In 1984, Takahashi and Ueda [19] obtained the same conclusion as Reich’s in uniformly convex Banach space with a uniformly Gˆ ateaux differentiable norm. Recently, in a reflexive Banach spaces E with a weakly sequentially continuous duality mapping, Song and Chen [11] (f (x) ≡ u ∈ K) constructed the similar results to Reich’s and Takahashi and Ueda’s results. Recently, Song et al. [10, 13, 14] also obtained the same outcomes in reflexive and strictly convex Banach space with a uniformly Gˆ ateaux differentiable norm. Lemma 2.2 [16, Lemmas 3.1, 3.3] Let E be a real smooth and strictly convex Banach space and K be a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E. Assume that T : K → E is a nonexpansive mapping and P is a sunny nonexpansive retraction of E onto K, then F (T ) = F (P T ). Using the similar technique of Song and Chen [10, Theorem 3.1] (also see [11, 12, 13]), we easily testify the following results (taking f (x) ≡ u in [10, Theorem 3.1]). Lemma 2.3. Let E be a reflexive and strictly convex Banach space with the uniformly Gˆ ateaux differentiable norm. Suppose that K is a nonempty closed convex subset of E and T : K → K is a non-expansive mapping satisfying F (T ) = ∅. For any t ∈ (0, 1), let {xt } be a sequence defined by the following equation: xt = tu + (1 − t)T xt . Then, as t → 0, {xt } converges strongly to some fixed point PF (T ) u of T , where PF (T ) is a sunny non-expansive retraction from K to F (T ). In the proof of our main theorems, we also need the following lemmas which can be found in [22, 21]: Lemma 2.4. [21, Lemma 2.5] Let {an } be a sequence of nonnegative real numbers satisfying the property: an+1 ≤ (1 − γn )an + γn βn ,

∀n ≥ 0,

where {γn } ⊂ (0, 1) and {βn } ⊂ R such that (i)

∞  n=0

γn = ∞,

(ii) lim sup βn ≤ 0. n→∞

Then {an } converges to zero as n → ∞. 3. The Main Results The following theorem was proved by Bruck in [1, 2]: Theorem B. [2, Corollary 1.1] Let K be a nonempty bounded closed convex subset of a uniformly convex Banach space E and T : K → K a non-expansive

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AVERAGED ITERATES FOR NON-EXPANSIVE NONSELF-MAPPINGS

mapping. Then, for any x ∈ K and the Ces` aro means An x =

5

 j 1 n−1 T x, we have n j=0

lim sup An x − T (An x) = 0.

n→∞ x∈K

Theorem 3.1. Let E be a uniformly convex Banach space with the uniformly Gˆ ateaux differentiable norm. Suppose that K is a nonempty closed convex subset of E, which is also a sunny non-expansive retract of E, and T : K → E is a non-expansive mapping with F (T ) = ∅. Let {xn } be a sequence defined by n

xn+1 =

1  P (αn u + (1 − αn )(T P )j xn ), n + 1 j=0

∀n ≥ 0,

(3.1)

where P is a sunny non-expansive retract from E to K. If αn ∈ (0, 1) satisfies the following conditions: (i) lim αn = 0, n→∞ ∞  (ii) αn = ∞. n=0

Then, as n → ∞, {xn } converges strongly to some fixed point PF (T ) u of T , where PF (T ) is a sunny non-expansive retract from K to F (T ). Proof. For fixed y ∈ F (T ), we have P y = y = T y = P T y by Lemma 2.2 and the definition of the sunny non-expansive retraction. Moreover, it follows that xn+1 − y n

≤

1  P (αn u + (1 − αn )(T P )j xn ) − P T y n + 1 j=0

≤

1  (αn u − y + (1 − αn )(T P )j xn − (T P )j y) n + 1 j=0

≤

1  (αn u − y + (1 − αn )xn − y) n + 1 j=0

n

n

≤ max{u − y, xn − y} ··· ≤ max{u − y, x0 − y}. This implies the boundedness of {xn } and so are {T xn } and {(T P )j xn } for any fixed n  1 (P T )j and M is a constant such that (T P )j xn −u ≤ j ≥ 0. If we set An = n+1 j=0

M , then we have

n

xn+1 − An xn  ≤

1  P (αn u + (1 − αn )(T P )j xn ) − (P T )j xn  n + 1 j=0 n

≤

1  (αn u − (T P )j xn ) n + 1 j=0

≤ αn M → 0 (n → ∞).

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Therefore, it follows that lim xn+1 − An xn  = 0.

(3.2)

n→∞

Take v ∈ F (T ) and define a subset D of K by D = {x ∈ K : x − v ≤ r}, where r = max{v − x0 , v − u, v − p} and p = PF (T ) u. Then D is a nonempty closed bounded convex subset of K, P T (D) ⊂ D and {xn } ⊂ D. It follows from Theorem B for a nonexpansive self-mapping P T on D that lim An xn − P T (An xn ) ≤ lim sup An x − P T (An x) = 0.

n→∞

n→∞ x∈D

(3.3)

On the other hand, xn+1 − P T xn+1  ≤ xn+1 − An xn  + An xn − P T (An xn ) + P T (An xn ) − P T xn+1  ≤ 2xn+1 − An xn  + P T (An xn ) − An xn . Combining (3.2) and (3.3), we have lim xn+1 − P T xn+1  = 0.

(3.4)

n→∞

By Lemmas 2.2 and 2.3, there exists a sunny non-expansive retraction from K to F (T ), say PF (T ) . In order to xn → p = PF (T ) u, that is, xn − p → 0, the application of Lemma 2.4 is desired. Since xn+1 − P (αn u + (1 − αn )p) n

≤

1  P (αn u + (1 − αn )(T P )j xn ) − P (αn u + (1 − αn )p) n + 1 j=0

≤

1  (1 − αn )(T P )j xn − p n + 1 j=0

n

≤ (1 − αn )xn − p, then we have xn+1 − p2 = xn+1 − (αn u + (1 − αn )p), J(xn+1 − p) + αn u − p, J(xn+1 − p) ≤ xn+1 − P (αn u + (1 − αn )p)xn+1 − p + αn u − p, J(xn+1 − p)) ≤ (1 − αn )xn − pxn+1 − p + αn u − p, J(xn+1 − p) xn − p2 + xn+1 − p2 + αn u − p, J(xn+1 − p). 2 Therefore, we have ≤ (1 − αn )

xn+1 − p2 ≤ (1 − αn )xn − p2 + 2αn u − p, J(xn+1 − p). If we employ Lemma 2.4 into (3.5) and use the condition

∞  n=0

remains to prove that lim sup f (p) − p, J(xn+1 − p) ≤ 0. n→∞

(3.5)

αn = ∞, then it (3.6)

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AVERAGED ITERATES FOR NON-EXPANSIVE NONSELF-MAPPINGS

7

For this purpose, if we consider zt = tu+(1−t)P T zt , then it follows from Lemma 2.3 and Lemma 2.2 that p = PF (T ) u = PF (P T ) u = lim zt . Hence we have t→0

zt − xn+1 2 = (1 − t) P T zt − xn+1 , J(zt − xn+1 ) + t u − xn+1 , J(zt − xn+1 ) = (1 − t)( P T zt − P T xn+1 , J(zt − xn+1 ) + P T xn+1 − xn+1 , J(zt − xn+1 )) + t u − p, J(zt − xn+1 ) + t p − zt , J(zt − xn+1 ) + t zt − xn+1 , J(zt − xn+1 ) ≤ xn+1 − zt 2 + P T xn+1 − xn+1 M + t u − p, J(zt − xn+1 ) + zt − pM and so

xn+1 − P T xn+1  (3.7) M + M zt − p, t where M is a constant such that M ≥ xn+1 − zt  by the boundedness of {xn } and {zt }. Therefore, taking the upper limit as n → ∞ firstly and then, as t → 0 in (3.7), using (3.4) and zt → p, we get

u − p, J(xn+1 − zt ) ≤

lim sup lim sup u − p, J(xn+1 − zt ) ≤ 0. t→0

n→∞

(3.8)

On the other hand, since {xn+1 − zt } converges strongly to (xn+1 − p) as t → 0 and the duality mapping J is single-valued and the norm topology to the weak star topology is uniformly continuous in any bounded subset of E with the uniformly Gˆ ateaux differentiable norm, it follows that, for all n ≥ 0, | u − p, J(xn+1 − p) − J(xn+1 − zt )| → 0 uniformly (t → 0). Therefore, for any ε > 0, there exists δ > 0 such that, for all t ∈ (0, δ) and n ≥ 0,

u − p, J(xn+1 − p) < u − p, J(xn+1 − zt ) + ε. Hence, noting (3.8), we have lim sup u − p, J(xn+1 − p) ≤ lim sup lim sup( u − p, J(xn+1 − zt ) + ε) n→∞

≤ 0.

t→0

n→∞

Namely, (3.6) is proved. Therefore, Lemma 2.4 is satisfied and so the theorem is proved. This completes the proof.  Corollary 3.2. Suppose that E, K, T, αn , P are as Theorem 3.1. Let {xn } be a sequence defined by the following equation: n

xn+1 =

1  (αn u + (1 − αn )(P T )j xn ), n + 1 j=0

∀n ≥ 0.

(3.9)

Then, as n → ∞, {xn } converges strongly to some fixed point PF (T ) u of T , where PF (T ) is a sunny non-expansive retraction of K onto F (T ). Proof. Because P is a sunny nonexpansive retraction of E onto K, P T x ∈ K for all x ∈ K and P T x − P T y ≤ x − y for all x, y ∈ K. Let S = P T . Then S is a non-expansive self-mapping on K and Sx = P T x = P T P x = SP x since P x = x for all x ∈ K. On the other hand, (P T )j = (P T )(P P T )(P P T )j−2 = · · · = (SP )j−1 (P T ) = (SP )j

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YISHENG SONG AND YEOL JE CHO

since P 2 = P . Therefore, from (3.9), we have n

xn+1 =

1  (αn u + (1 − αn )S j xn ) n + 1 j=0 n

=

1  P (αn u + (1 − αn )(SP )j xn ). n + 1 j=0

Consequently, from Theorem 3.1, the conclusion is proved. This completes the proof.  Remark 1. (1) Since every Hilbert space is a uniformly convex Banach space and the sunny non-expansive retraction in Hilbert space coincides with the metric projection, then Theorems 3.1 and 3.2 contain Theorems 2 and Theorem 1 of Matsushita and Kuroiwa [7] as a special case, respectively. (2) The conclusions of Theorem 3.2 and 3,1 still hold if E is assumed to be a uniformly smooth Banach space instead of to have the uniformly Gˆ ateaux differentiable norm since a uniformly smooth Banach space have the uniformly Gˆ ateaux differentiable norm. In particular, if T is a non-expansive self-mapping on K and take P = I (: the identity operator), then our result contains Theorem 3.2 in Xu [21]. 4. Applications As applications of Theorem 3.1, we present the following viscosity approximation results in virtue of Lemma 2.1 and the proof technique of Suzuki [15, Theorems 5, 6]: Theorem 4.1. Suppose that E, K, T, αn , P are as Theorem 3.1. Let {xn } be a sequence defined by the following equation: n

xn+1 =

1  P (αn f (xn ) + (1 − αn )(T P )j xn ), n + 1 j=0

∀n ≥ 0,

(4.1)

where f is a contractive self-mapping on K with the contractive coefficient β ∈ (0, 1). Then, as n → ∞, {xn } converges strongly to p, which is an unique solution in F (T ) to the following variational inequality:

f (p) − p, J(y − p) ≤ 0,

∀y ∈ F (T ).

(4.2)

Proof. Theorem 3.1 of Song and Chen [10] guarantees that the variational inequality (4.2) has the unique solution p in F (T ). (also, see [22, 11, 12, 13]). Then it follows from Lemma 2.1 that PF (T ) f (p) = p. Define a sequence {yn } in K by y1 ∈ K and n

yn+1 =

1  P (αn f (p) + (1 − αn )(T P )j yn ), n + 1 j=0

∀n ≥ 0.

Then, by Theorem 3.1, {yn } converges strongly to p = PF (T ) f (p) ∈ F (T ).

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AVERAGED ITERATES FOR NON-EXPANSIVE NONSELF-MAPPINGS

9

Next, we testify xn → p as n → ∞. Indeed, it follows that xn+1 − yn+1  n

≤

1  (αn f (xn ) − f (p) + (1 − αn )(T P )j xn − (T P )j yn ) n + 1 j=0

≤ αn βxn − p + (1 − αn )xn − yn  ≤ [1 − (1 − β)αn ]xn − yn  + βαn yn − p. Hence, from Lemma 2.4, we obtain xn − yn  → 0. Therefore, lim xn − p = 0. n→∞ This completes the proof.  Corollary 4.2. Suppose that E, K, T, αn , P are as Theorem 3.1. Let {xn } be a sequence defined by the following equation: n 1  xn+1 = (αn f (xn ) + (1 − αn )(P T )j xn ), ∀n ≥ 0, n + 1 j=0 where f is a contractive self-mapping on K with the contractive coefficient β ∈ (0, 1). Then, as n → ∞, {xn } converges strongly to p, which is the unique solution in F (T ) to the variational inequality (4.2). References 1. R. E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math. 32 (1979) 107-116. 2. R. E. Bruck,On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, Israel J. Math. 38(1981) 304-314. 3. S. S. Chang, Y. J. Cho and H. Y. Zhou, Iterative methods fo nonlinear opeator equations in Banach spaces, Nova Science Publishers, Inc., Huntington, Now York, 2002 (U.S.A). 4. J. P. Gossez and E. L. Dozo, Some geometric properties related to the fixed point theory for non-expansive mappings, Pacfic J. Math. 40(1972), 565-573. 5. B. Halpern, Fixed points of non-expansive maps. Bull. Amer. Math. Soc. 73(1967) 957-961. 6. J. S. Jung and S. S. Kim, Strong convergence theorems for non-expansive nonself-mappings in Banach spaces, Nonlinear Anal. 33(1998) 321-329. 7. S. Matsushita and D. Kuroiwa, Strong convergence of averaging iterations of non-expansive nonself-mappings, J. Math. Anal. Appl. 294(2004) 206-214. 8. R. E. Megginson, An introduction to Banach space theory , 1998 Springer-Verlag New Tork, Inc. 9. S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces J. Math. Anal Appl. 75(1980) 287-292. 10. Y. S. Song and R. D. Chen, Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings, Appl. Math. Comput. 180(2006) 275-287. 11. Y. S. Song and R. D. Chen, Viscosity approximation methods for non-expansive nonselfmappings, J. Math. Anal. Appl. 321(2006) 316-326. 12. Y. S. Song and R. D. Chen, Convergence theorems of iterative algorithms for continuous pseudocontractive mappings, Nonlinear Anal., 67(2007) 486-497. 13. Y. S. Song and R. D. Chen, Viscosity approximative methods to Ces` aro means for nonexpansive mappings, Applied Mathematics and Computation, 186(2007) 1120-1128. 14. Y. S. Song and Q. C. Li, Viscosity approximation for nonexpansive nonself-mappings in reflexive Banach spaces, J. of Systems Science and Complexity, in press(2007). 15. T. Suzuki, Moudafi’s viscosity approximations with Meir-Keeler contractions, J. Math. Anal. Appl. 325(2007) 342-352. 16. S. Matsushita and W. Takahashi, Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions, Nonlinear Analysis (2006), doi:10.1016/j.na.2006.11.007

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17. W. Takahashi, Nonlinear Functional Analysis– Fixed Point Theory and its Applications, Yokohama Publishers inc, Yokohama, 2000 (Japanese). 18. W. Takahashi and G. E. Kim, Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces, Nonlinear Anal. 32(1998) 447-454. 19. W. Takahashi and Y. Ueda, On Reich’s strong convergence for resolvents of accretive operators, J. Math. Anal. Appl. 104(1984) 546-553. 20. H. K. Xu, Approximating curves of non-expansive nonself-mappings in Banach spaces, Comptes Rendus de l’Acadmie des Sciences - Series I - Mathematics, 325(1997) 151-156. 21. H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66(2002) 240256. 22. H. K. Xu, Viscosity approximation methods for non-expansive mappings, J. Math. Anal. Appl. 298(2004) 279-291. 23. H. K. Xu and X. M. Yin, Strong convergence theorems for non-expansive nonself-mappings, Nonlinear Anal. 24(1995) 223-228. Yisheng Song College of Mathematics and Information Science, Henan Normal University, P.R. China, 453007. E-mail address: [email protected] Yeol Je Cho Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Korea E-mail address: [email protected]