Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces

J. Math. Anal. Appl. 341 (2008) 1–11 www.elsevier.com/locate/jmaa

Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces Abdul Rahim Khan a,∗ , Abdul-Aziz Domlo b , Hafiz Fukhar-ud-din c a Department of Mathematical Sciences, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia b Department of Mathematics, Taibah University, P.O. Box 30002, Madinah Munawarah, Saudi Arabia c Department of Mathematics, Islamia University, Bahawlpur 63100, Pakistan

Received 24 January 2006 Available online 14 July 2007 Submitted by R. Curto

Abstract In this paper, we introduce a general iteration scheme for a finite family of asymptotically quasi-nonexpansive mappings. The new iterative scheme includes the modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor and Khan and Takahashi scheme as special cases. Our results are generalizations as well as refinement of several known results in the current literature. © 2007 Elsevier Inc. All rights reserved. Keywords: Modified Mann and Ishikawa iterations; Asymptotically quasi-nonexpansive mapping; Common fixed point; Weak and strong convergence; Banach space

1. Introduction and preliminaries Let C be a nonempty subset of a real Banach space X and T a selfmapping of C. Denote by F (T ), the set of fixed points of T . Throughout this paper, we assume that F (T ) = ∅. The mapping T is said to be (i) nonexpansive if T x − T y  x − y, for all x, y ∈ C; (ii) quasi-nonexpansive if T x − p  x − p, for all x ∈ C and p ∈ F (T ); (iii) asymptotically nonexpansive if there exists a sequence {un } in [0, +∞) with limn→∞ un = 0 and T n x −T n y  (1 + un )x − y, for all x, y ∈ C and n = 1, 2, 3, . . . ; (iv) asymptotically quasi-nonexpansive if there exists a sequence {un } in [0, +∞) with limn→∞ un = 0 and T n x − p  (1 + un )x − p, for all x ∈ C, p ∈ F (T ) and n = 1, 2, . . . ; (v) uniformly L-Lipschitzian if there exists a constant L > 0 such that T n x − T n y  Lx − y, for all x, y ∈ C and n = 1, 2, 3, . . . ; * Corresponding author.

E-mail addresses: [email protected] (A.R. Khan), [email protected] (A.-A. Domlo), [email protected] (H. Fukhar-ud-din). 0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.06.051

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(vi) (L − γ ) uniform Lipschitz if there are constants L > 0 and γ > 0 such that T n x − T n y  Lx − yγ for all x, y ∈ C and n = 1, 2, 3, . . . (cf. Qihou [21], p. 468); (vii) semi-compact if for a sequence {xn } in C with limn→∞ xn − T xn  = 0, there exists a subsequence {xni } of {xn } such that xni → p ∈ C. From the above definitions, it follows that: (i) a nonexpansive mapping must be quasi-nonexpansive and asymptotically nonexpansive; (ii) an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive; (iii) a uniformly L-Lipschitzian mapping is (L − 1) uniform Lipschitz. However, the converse of these statements is not true, in general. The map T : C → X is said to be demiclosed at 0 if for each sequence {xn } in C converging weakly to x and {T xn } converging strongly to 0, we have T x = 0. A Banach space X is said to satisfy Opial’s property if for each x ∈ X and each sequence {xn } weakly convergent to x, the following condition holds for all x = y: lim inf xn − x < lim inf xn − y. n→∞

n→∞

It is well known that all Hilbert spaces and p (1 < p < ∞) spaces are Opial spaces while Lp spaces (p = 2) are not. Schu [23], in 1991, considered the following modified Mann iteration process (cf. Mann [13]): xn+1 = (1 − αn )xn + αn T n xn ,

n = 1, 2, 3, . . .

(1.1)

where {αn } is a sequence in (0, 1) which is bounded away from 0 and 1, i.e., a  αn  b for all n and some 0 < a  b < 1. In 1994, Tan and Xu [28] studied the modified Ishikawa iteration process (cf. Ishikawa [9]):   (1.2) xn+1 = (1 − αn )xn + αn T n (1 − βn )xn + βn T n xn , n = 1, 2, 3, . . . where {αn } and {βn } are two sequences in (0, 1) such that {αn } is bounded away from 0 and 1 and {βn } is bounded away from 1. Xu and Noor [29], in 2002, introduced a three-step iterative scheme as follows: zn = (1 − γn )xn + γn T n xn , yn = (1 − βn )xn + βn T n zn , xn+1 = (1 − αn )xn + αn T n yn ,

n = 1, 2, 3, . . . ,

where {αn }, {βn }, {γn } are sequences of real numbers in [0, 1]. For essential background and motivation of the three-step iterative scheme, we refer to the fundamental work of Noor [14]. It is well known that the three-step iterative scheme includes the Mann (one-step) and Ishikawa (two-step) iterations as special cases. The theory of three-step iterative scheme is very rich, and this scheme, in the context of one or more mappings, has been extensively studied (for example, see Cho et al. [3], Fukhar-ud-din and Khan [4], Noor et al. [17], Petrot [18] and Suantai [25]). It has been shown in [1], that three-step method performs better than two-step and one-step methods for solving variational inequalities. The three-step schemes are natural generalization of the splitting methods to solve partial differential equations (inclusions); for details, see, Noor [14–16]. This signifies that Noor three-step methods are robust and more efficient than the Mann (one-step) and Ishikawa (two-step) type iterative methods to solve problems of pure and applied sciences. Finding common fixed points of a finite family {Ti : i = 1, 2, . . . , k} of mappings acting on a Hilbert space is a problem that often arises in applied mathematics. In fact, many algorithms have been introduced for different classes of mappings with a nonempty set of common fixed points. Unfortunately, the existence results of common fixed points of a family of mappings are not known in many situations. Therefore, it is natural to consider approximation results for these classes of mappings. Approximating common fixed points of a finite family of nonexpansive mappings by iteration has been studied by many authors (see, for example, Kuhfittig [12], Rhoades [22] and Takahashi and

A.R. Khan et al. / J. Math. Anal. Appl. 341 (2008) 1–11

3

Shimoji [27]). Ghosh and Debnath [6] proved some convergence results for common fixed points of families of quasinonexpansive mappings. Goeble and Kirk [8], in 1972, introduced the notion of an asymptotically nonexpansive mapping and established that if C is a nonempty closed convex bounded subset of a uniformly convex Banach space X and T is an asymptotically nonexpansive selfmapping of C, then T has a fixed point. Bose [2] initiated in 1978, the study of iterative construction for fixed points of asymptotically nonexpansive mappings. Xu and Ori [30], in 2001, introduced an implicit iteration process for a finite family of nonexpansive mappings. Sun [26], in 2003, modified this implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings. Khan and Takahashi [10] have approximated common fixed points of two asymptotically nonexpansive mappings by the modified Ishikawa iteration. Recently, Shahzad and Udomene [24] established convergence theorems for the modified Ishikawa iteration process of two asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings. For related results with error terms, we refer to [4,11,21] and [24]. For a finite family of mappings, it is desirable to devise a general iteration scheme which extends the modified Mann iteration (1.1), the modified Ishikawa iteration (1.2), Khan and Takahashi scheme [10] and the three-step iteration by Xu and Noor [29], simultaneously. Thereby, to achieve this goal, we introduce a new iteration process for a finite family {Ti : i = 1, 2, . . . , k} of asymptotically quasi-nonexpansive mappings as follows: Let C be a convex subset of a Banach space X and x1 ∈ C. Suppose that αin ∈ [0, 1], n = 1, 2, 3, . . . and i = 1, 2, . . . , k. Let {Ti : i = 1, 2, . . . , k} be a family of selfmappings of C. The iteration scheme is defined as follows: xn+1 = (1 − αkn )xn + αkn Tkn y(k−1)n , n y(k−1)n = (1 − α(k−1)n )xn + α(k−1)n Tk−1 y(k−2)n , n y(k−2)n = (1 − α(k−2)n )xn + α(k−2)n Tk−2 y(k−3)n , .. .

y2n = (1 − α2n )xn + α2n T2n y1n , y1n = (1 − α1n )xn + α1n T1n y0n ,

(1.3)

where y0n = xn for all n. Clearly, the iteration process (1.3) generalizes the modified Mann iteration (1.1), the modified Ishikawa iteration (1.2) and the three-step iteration scheme  from one mapping to the finite family of mappings {Ti : i = 1, 2, . . . , k}. In the sequel, we assume that F = ki=1 F (Ti ). The main purpose of this paper is to: (i) establish a necessary and sufficient condition for convergence of the iteration scheme (1.3) to a common fixed point of a finite family of asymptotically quasi-nonexpansive mappings in a Banach space; (ii) prove weak and strong convergence results of the iteration scheme (1.3) to a common fixed point of a finite family of (L − γ ) uniform Lipschitz and asymptotically quasi-nonexpansive mappings in a uniformly convex Banach space. Our work is a significant generalization of the corresponding results of Khan and Takahashi [10], Qihou [20], Schu [23], Shahzad and Udomene [24], Tan and Xu [28] and Xu and Noor [29]. Moreover, these results provide analogue of the results of Sun [26], for the iteration scheme (1.3) instead of the implicit iteration. We need the following useful known lemmas for the development of our convergence results. Lemma 1.1. (Cf. [26, Lemma 2.2].) Let the sequences {an } and {un } of real numbers satisfy: an+1  (1 + un )an , where an  0, un  0, for all n = 1, 2, 3, . . . ∞ and n=1 un < +∞. Then (i) limn→∞ an exists; (ii) if lim infn→∞ an = 0, then limn→∞ an = 0.

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Lemma 1.2. (See [23, Lemma 1.3].) Let X be a uniformly convex Banach space. Assume that 0 < b  tn  c < 1, n = 1, 2, 3, . . . . Let the sequences {xn } and {yn } in X be such that lim supn→∞ xn   a, lim supn→∞ yn   a and limn→∞ tn xn + (1 − tn )yn  = a, where a  0. Then limn→∞ xn − yn  = 0. 2. Convergence theorems in Banach spaces The aim of this section is to prove some results for the iterative process (1.3) to converge to a common fixed point of a finite family of asymptotically quasi-nonexpansive mappings in a Banach space. We begin with the following: Lemma 2.1. Let C be a nonempty closed convex subset of a Banach space, and {Ti : i = 1, 2, . . . , k} a family of asymptotically quasi-nonexpansive selfmappings of C, i.e., Tin x − pi   (1 + uin )x − pi  for all x ∈ C and pi ∈ F i ), i = 1, 2, . . . , k where {uin } are sequences in [0, +∞) with limn→∞ uin = 0 for each i. Assume that F = ∅ and (T ∞ n=1 uin < +∞ for each i. Define the sequence {xn } as in (1.3). Then  k (a) there exists a sequence {νn } in [0, +∞) such that ∞ n=1 νn < +∞ and xn+1 − p  (1 + νn ) xn − p, for all p ∈ F and all n; (b) there exists a constant M > 0 such that xn+m − p  Mxn − p, for all p ∈ F and n, m = 1, 2, 3, . . . . Proof. (a) Let p ∈ F and νn = max1ik uin , for all n. Since Now we have   y1n − p  (1 − α1n )xn − p + α1n T1n xn − p 

∞

n=1 uin

< +∞ for each i, therefore

∞

n=1 νn

< +∞.

 (1 − α1n )xn − p + α1n (1 + u1n )xn − p = (1 + α1n u1n )xn − p  (1 + νn )xn − p. Assume that yj n − p  (1 + νn )j xn − p holds for some 1  j  k − 2. Then   y(j +1)n − p  (1 − α(j +1)n )xn − p + α(j +1)n Tjn+1 yj n − p   (1 − α(j +1)n )xn − p + α(j +1)n (1 + u(j +1)n )yj n − p  (1 − α(j +1)n )xn − p + α(j +1)n (1 + u(j +1)n )(1 + νn )j xn − p  (1 − α(j +1)n )xn − p + α(j +1)n (1 + νn )j +1 xn − p  

 j +1 (j + 1)j · · · (j + 2 − r) r νn xn − p = 1 − α(j +1)n + α(j +1)n 1 + r! r=1   j +1 (j + 1)j · · · (j + 2 − r) r = 1 + α(j +1)n νn xn − p r! r=1

j +1

 (1 + νn )

xn − p.

Thus, by induction, we have yin − p  (1 + νn )i xn − p,

for all i = 1, 2, . . . , k − 1.

Now, by (2.1), we obtain   xn+1 − p  (1 − αkn )xn − p + αkn Tkn y(k−1)n − p   (1 − αkn )xn − p + αkn (1 + ukn )y(k−1)n − p  (1 − αkn )xn − p + αkn (1 + ukn )(1 + νn )k−1 xn − p  (1 − αkn )xn − p + αkn (1 + νn )k xn − p

(2.1)

A.R. Khan et al. / J. Math. Anal. Appl. 341 (2008) 1–11





= 1 − αkn + αkn 1 +

k k(k − 1) · · · (k − r + 1)

r!

r=1

 = 1 + αkn

k k(k − 1) · · · (k − r + 1)

r!

r=1

5

 xn − p

νnr

 νnr

xn − p

 (1 + νn )k xn − p. This completes the proof of (a). (b) If t  0, then 1 + t  et and so, (1 + t)k  ekt , k = 1, 2, . . . . Thus, from part (a), we get xn+m − p  (1 + νn+m−1 )k xn+m−1 − p



 exp{kνn+m−1 }xn+m−1 − p  · · ·  exp k  exp k Setting M = exp{k

∞

∞

n+m−1

νi xn − p

i=1

νi xn − p.

i=1

i=1 νi },

completes the proof.

2

The above lemma generalizes Theorem 3.1 for two asymptotically quasi-nonexpansive mappings by Shahzad and Udomene [24] to the case of any finite family of such mappings. The next result is the main result of this section. It deals with a necessary and sufficient condition for the convergence of {xn } generated by the iteration process (1.3) to a point of F ; for this we follow the arguments of Qihou [20, Theorem 1]. Theorem 2.2. Let C be a nonempty closed convex subset of a Banach space X, and {Ti : i = 1, 2, . . . , k} a famn ily of asymptotically quasi-nonexpansive selfmappings of C, i.e., T i∞x − pi   (1 + uin )x − pi , for all x ∈ C and pi ∈ F (Ti ), i = 1, 2, . . . , k. Suppose that F = ∅, x1 ∈ C and n=1 uin < +∞ for all i. Then the iterative sequence {xn }, defined by (1.3), converges strongly to a common fixed point of the family of mappings if and only if lim infn→∞ d(xn , F ) = 0, where d(x, F ) = infp∈F x − p. Proof. We will only prove the sufficiency; the necessity is obvious. From Lemma 2.1(a), we have xn+1 − p  (1 + νn )k xn − p, for all p ∈ F and all n. Therefore,



d(xn+1 , F )  (1 + νn ) d(xn , F ) = 1 + k

k k(k − 1) · · · (k − r + 1) r=1

r!

νnr

d(xn , F ).

∞ k k(k−1)···(k−r+1) r  νn < +∞. By Lemma 1.1 and lim infn→∞ d(xn , F ) = 0, we get As ∞ n=1 νn < +∞, so n=1 r=1 r! that limn→∞ d(xn , F ) = 0. Next, we prove that {xn } is a Cauchy sequence. From Lemma 2.1(b), we have xn+m − p  Mxn − p,

for all p ∈ F and all n, m = 1, 2, 3, . . . .

Since limn→∞ d(xn , F ) = 0, therefore for each  > 0, there exists a natural number n1 such that  , for all n  n1 . d(xn , F )  3M Hence, there exists z1 ∈ F such that  . xn1 − z1   2M From (2.2) and (2.3), for all n  n1 , we have

(2.2)

(2.3)

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xn+m − xn   xn+m − z1  + xn − z1   Mxn1 − z1  + Mxn1 − z1        +M = . M 2M 2M Thus, {xn } is a Cauchy sequence and so converges to q ∈ X. Finally, we show that q ∈ F . For any  > 0, there exists a natural number n2 such that xn − q 

 , 2(2 + ν1 )

for all n  n2 .

(2.4)

Again, limn→∞ d(xn , F ) = 0 implies that there exists a natural number n3  n2 such that d(xn , F ) 

 , 3(4 + 3ν1 )

for all n  n3 .

Thus, there exists z2 ∈ F such that xn3 − z2  

 . 2(4 + 3ν1 )

(2.5)

From (2.4) and (2.5), for any Ti , i = 1, 2, . . . , k, we get Ti q − q = Ti q − z2 + z2 − Ti xn3 + Ti xn3 − z2 + z2 − xn3 + xn3 − q  Ti q − z2  + 2Ti xn3 − z2  + xn3 − z2  + xn3 − q  (1 + ν1 )q − z2  + 2(1 + ν1 )xn3 − z2  + xn3 − z2  + xn3 − q  (1 + ν1 )xn3 − q + (1 + ν1 )xn3 − z1  + 2(1 + ν1 )xn3 − z2  + xn3 − z2  + xn3 − q = (2 + ν1 )xn3 − q + (4 + 3ν1 )xn3 − z2    + (4 + 3ν1 ) = .  (2 + ν1 ) 2(2 + ν1 ) 2(4 + 3ν1 ) Since  is arbitrary, therefore Ti q − q = 0, for all i, i.e., Ti q = q, i = 1, 2, . . . , k. Thus q ∈ F . 2 Remark 2.3. Theorem 2.2 contains as special cases, Theorem 3.2 of Shahzad and Udomene [24] and Theorem 1 by Qihou [20] together with its Corollaries 1 and 2, which are themselves extensions of the results of Ghosh and Debnath [7] and Petryshyn and Williamson [19]. An asymptotically nonexpansive mapping is asymptotically quasi-nonexpansive, so we have: Corollary 2.4. Let C be a nonempty closed convex subset of a Banach space X, and {Ti : i = 1, 2, . . . , k} a family of asymsptotically nonexpansive selfmappings ofC, i.e., Tin x − Tin y  (1 + uin )x − y, for all x, y ∈ C and i = 1, 2, . . . , k. Suppose that F = ∅, x1 ∈ C and ∞ n=1 uin < +∞, for all i. Then the iterative sequence {xn }, defined by (1.3), converges strongly to a point p ∈ F if and only if lim infn→∞ d(xn , F ) = 0. Corollary 2.5. Let C, {Ti : i = 1, 2, . . . , k}, F and uin be as in Theorem 2.2. Then the iterative sequence {xn }, defined by (1.3), converges strongly to a point p ∈ F if and only if there exists a subsequence {xnj } of {xn } which converges to p. Theorem 2.6. Let C be a nonempty closed convex subset of a Banach space X, and  {Ti : i = 1, 2, . . . , k} a family of asymptotically nonexpansive selfmappings of C. Suppose that F = ∅, x1 ∈ C and ∞ n=1 uin < +∞ for all i. Let {xn } be the sequence defined by (1.3). If limn→∞ xn −Ti xn  = 0, i = 1, 2, . . . , k and one of the mappings is semi-compact, then {xn } converges strongly to p ∈ F .

A.R. Khan et al. / J. Math. Anal. Appl. 341 (2008) 1–11

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Proof. Let T be semi-compact for some 1    k. Then there exists a subsequence {xnj } of {xn } such that xnj → p ∈ C. Hence p − Ti p = lim xnj − Ti xnj  = 0, nj →∞

for all i = 1, 2, . . . , k.

Thus, p ∈ F and by Corollary 2.5, {xn } converges strongly to a common fixed point of the family of mappings.

2

Theorem 2.7. Let C, {Ti : i = 1, 2, . . . , k}, F and uin be as in Theorem 2.2. Suppose that there exists a map Tj which satisfies the following conditions: (i) limn→∞ xn − Tj xn  = 0; (ii) there exists a constant M such that xn − Tj xn   Md(xn , F ), for all n. Then the sequence {xn }, defined by (1.3), converges strongly to a point p ∈ F . Proof. From (i) and (ii), it follows that limn→∞ d(xn , F ) = 0. By Theorem 2.2, {xn } converges strongly to a common fixed point of the family of mappings. 2 3. Results in uniformly convex Banach spaces In this section, we establish some weak and strong convergence results for the iterative scheme (1.3) by removing the condition lim infn→+∞ d(xn , F ) = 0 from the results obtained in Section 2; for this we have to consider the class of (L − γ ) uniform Lipschitz and asymptotically quasi-nonexpansive mappings on a uniformly convex Banach space. Lemma 3.1. Let C be a nonempty closed convex subset of a uniformly convex Banach space X, and {Ti : i = 1, 2, 3, . . . , k} a family of (L − γ ) uniform Lipschitz and asymptotically quasi-nonexpansive selfmappings of C, n i.e., T i∞x − pi   (1 + uin )x − pi  for all x ∈ C and pi ∈ F (Ti ), where {uin } are sequences in [0, ∞) with n=1 uin < ∞, for each i ∈ {1, 2, 3, . . . , k}. Assume that F = ∅ and the sequence {xn } is as in (1.3) with αin ∈ [δ, 1 − δ] for some δ ∈ (0, 12 ). Then (i) limn→∞ xn − p exists for all p ∈ F ; (ii) limn→∞ xn − Tjn y(j −1)n  = 0, for each j = 1, 2, . . . , k; (iii) limn→∞ xn − Tj xn  = 0, for each j = 1, 2, . . . , k. Proof. Let p ∈ F and νn = max1ik uin , for all n. (i) By Lemma 1.1(i) and Lemma 2.1(a), it follows that limn→∞ xn − p exists for all p ∈ F . Assume that lim xn − p = c.

(3.1)

n→∞

(ii) The inequality (2.1) and (3.1) give that lim sup yj n − p  c, n→∞

for 1  j  k − 1.

We also note that:

   xn+1 − p = (1 − αkn )(xn − p) + αkn Tkn y(k−1)n − p   (1 − αkn )xn − p + αkn (1 + vn )y(k−1)n − p .. .  (1 − αkn α(k−1)n · · · α(j +1)n )(1 + vn )k−j xn − p + αkn α(k−1)n · · · α(j +1)n (1 + vn )k−j yj n − p.

Therefore, xn − p 

xn − p xn+1 − p − k−j + yj n − p k−j δ δ (1 + vn )k−j

(3.2)

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A.R. Khan et al. / J. Math. Anal. Appl. 341 (2008) 1–11

and hence c  lim inf yin − p,

for 1  j  k − 1.

n→∞

(3.3)

From (3.2) and (3.3), we have lim yj n − p = c,

n→∞

for each j = 1, 2, 3, . . . , k − 1.

That is,

   lim (1 − αj n )(xn − p) + αj n Tjn y(j −1)n − p  = c,

n→∞

for each j = 1, 2, 3, . . . , k − 1. Also, from (3.2), we obtain   lim supT n y(j −1)n − p   c, j

n→∞

By Lemma 1.2, we get   lim T n y(j −1)n − xn  = 0, n→∞

j

for each j = 1, 2, 3, . . . , k − 1.

for each j = 1, 2, 3, . . . , k − 1.

(3.4)

For the case j = k, by (2.1), we have  n  T y(k−1)n − p   (1 + ukn )y(k−1)n − p  (1 + ukn )(1 + νn )k−1 xn − p. k

But limn→∞ xn − p = c, by part (i). So,   lim supT n y(k−1)n − p   c. k

n→∞

Moreover,

   lim (1 − αkn )(xn − p) + αkn Tkn y(k−1)n − p  = lim xn+1 − p = c.

n→∞

n→∞

Again by Lemma 1.2, we get   lim xn − T n y(k−1)n  = 0. Thus, (3.4) and (3.5) imply that   lim T n y(j −1)n − xn  = 0, n→∞

(3.5)

k

n→∞

j

for each j = 1, 2, 3, . . . , k.

(3.6)

(iii) For j = 1, from part (ii), we have   lim T n xn − xn  = 0. n→∞

1

If j = 2, 3, 4, . . . , k, then we have  n      T xn − xn  =  T n xn − T n y(j −1)n + T n y(j −1)n − xn  j j j j   = Lxn − y(j −1)n γ + Tjn y(j −1)n − xn   γ    = L α(j −1)n xn − Tjn−1 y(j −2)n  + Tjn y(j −1)n − xn  → 0. Hence,  n  T xn − xn  → 0 as n → ∞, for 1  j  k. j Let us observe that:

(3.7)

      xn − Tj xn   xn − xn+1  + xn+1 − Tjn+1 xn+1  + Tjn+1 xn+1 − Tjn+1 xn  + Tjn+1 xn − Tj xn       γ γ    αkn xn − Tkn y(k−1)n  + xn+1 − Tjn+1 xn+1  + L αkn xn − Tkn y(k−1)n  + LTjn xn − xn  .

Using (3.6) and (3.7), we get lim xn − Tj xn  = 0,

n→∞

for 1  j  k.

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Theorem 3.2. Let C be a nonempty closed convex subset of a uniformly convex Banach space X satisfying the Opial property and let {Ti : i = 1, 2, 3, . . . , k} be a family of (L − γ ) uniform Lipschitz and asymptotically quasinonexpansive selfmappings of C, i.e., Tin x − pi   (1 + uin )x − pi  for all x ∈ C and pi ∈ F (Ti ), i = 1, 2, 3, . . . , k where {uin } are sequences in [0, ∞) with ∞ n=1 uin < ∞ for each i = 1, 2, 3, . . . , k. Let the sequence {xn } be as in 1 (1.3) with αin ∈ [δ, 1 − δ] for some δ ∈ (0, 2 ). If F = ∅ and each I − Ti , i = 1, 2, 3, . . . , k, is demiclosed at 0, then {xn } converges weakly to a common fixed point of the family {Ti : i = 1, 2, 3, . . . , k}. Proof. Let p ∈ F . Then limn→∞ xn − p exists as proved in Lemma 3.1(i) and hence {xn } is bounded. Since a uniformly convex Banach space is reflexive, there exists a subsequence {xnj } of {xn } converging weakly to some z1 ∈ C. By Lemma 3.1, limn→∞ xn − Ti xn  = 0 and I − Ti is demiclosed at 0 for i = 1, 2, 3, . . . , k, so we obtain Ti z1 = z1 . That is, z1 ∈ F . In order to show that {xn } converges weakly to z1 , take another subsequence {xnk } of {xn } converging weakly to some z2 ∈ C. Again, as above, we can prove that z2 ∈ F . Next, we show that z1 = z2 . Assume z1 = z2 . Then by the Opial property lim xn − z1  = lim xnj − z1 

n→∞

nj →∞

< lim xnj − z2  nj →∞

= lim xn − z2  n→∞

= lim xnk − z2  nk →∞

< lim xnk − z1  nk →∞

= lim xn − z1 . n→∞

This contradiction proves that {xn } converges weakly to a common fixed point of the family {Ti : i = 1, 2, 3, . . . , k}. 2 Theorem 3.3. Under the hypotheses of Lemma 3.1, assume that, for some 1  i  k, Tim is semi-compact for some positive integer m. Then {xn } converges strongly to some common fixed point of the family {Tj : j = 1, 2, 3, . . . , k}. Proof. By Lemma 3.1(iii), we have lim xn − Tj xn  = 0,

n→∞

for 1  j  k.

(3.8)

Fix i ∈ {1, 2, 3, . . . , k} and suppose Tim to be semi-compact for some m  1. From (3.8), we obtain   m       T xn − xn   T m xn − T m−1 xn  + T m−1 xn − T m−2 xn  + · · · + T 2 xn − Ti xn  + Ti xn − xn  i i i i i i  Ti xn − xn  + (m − 1)LTi xn − xn γ → 0. Since {xn } is bounded and Tim is semi-compact, {xn } has a convergent subsequence {xnj } such that xnj → q ∈ C. Hence, from (3.8), we have q − Ti q = lim xnj − Ti xnj  = 0, n→∞

for all i = 1, 2, 3, . . . , k.

Thus q ∈ F and by Corollary 2.5, {xn } converges strongly to a common fixed point q of the family {Ti : i = 1, 2, 3, . . . , k}. 2 4. Concluding remarks 1. The family of (L − γ ) uniform Lipschitz and asymptotically quasi-nonexpansive mappings in Lemma 3.1 can be replaced by a family of asymptotically nonexpansive mappings. We state this result as follows; the proof is similar to that of Lemma 3.1.

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Lemma 4.1. Let C be a nonempty closed convex subset of a uniformly convex Banach space X,and {Ti : i = 1, 2, . . . , k} a family of asymptotically nonexpansive selfmappings of C. Assume that F = ∅ and ∞ i=1 uin < ∞ 1 for each i = 1, 2, . . . , k. Let {xn } be as in (1.3) with αin ∈ [δ, 1 − δ] for some δ ∈ (0, 2 ). Then (i), (ii) and (iii) of Lemma 3.1 hold. 2. Lemma 3.1(i) extends Lemma 2.1 of Tan and Xu [28]. Lemma 3.1(ii) extends Theorem 3.3 of Shahzad and Udomene [24] for two uniformly continuous asymptotically quasi-nonexpansive mappings to any finite family of (L − γ ) uniform Lipschitz and asymptotically quasi-nonexpansive mappings. 3. Lemma 4.1(ii) and Lemma 4.1(iii) contain as special cases, Lemma 2.2 of Xu and Noor [29] and Lemma 1.5 of Schu [23], respectively. 4. On the lines of the proof of Theorem 3.2 and using Lemma 1.6 of Schu [23] and Lemma 4.1, the following result can be easily proved. Theorem 4.2. Under the hypotheses of Lemma 4.1, assume that the space X satisfies the Opial property. Then the sequence {xn } converges weakly to a common fixed point of the family of mappings. 5. The special cases of Theorem 4.2 are Theorems 3.1–3.2 of Tan and Xu [28] and Theorem 2.1 due to Schu [23]. 6. Following the arguments of the proof of Theorem 3.3, it is now easy to prove: Theorem 4.3. Under the assumptions of Lemma 4.1, suppose that, for some 1  i  k, and a positive integer m, Tim is semi-compact. Then {xn } converges strongly to some common fixed point of the family of mappings. The above theorem contains as a special case, Theorem 2.2 of Schu [23]. 7. Theorem 2.2, Corollary 2.5, Theorems 3.3 and 4.3 about the iteration scheme (1.3) are analogues of Theorem 3.1, Corollary 3.2, Theorems 3.3 and 3.4 in the context of implicit iteration process by Sun [26], respectively (see also [5]). Acknowledgments The authors thank the referee for useful suggestions to improve presentation of the paper. The author A.R. Khan gratefully acknowledges support provided by King Fahd University of Petroleum and Minerals during this research.

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