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A new hybrid iteration method for a finite family of asymptotically nonexpansive mappings in Banach spaces

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Abstract

In this paper, we first introduce a new hybrid iteration method for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in Banach spaces, and then we discuss the strong and weak convergence for the iterative processes. The results presented in this paper extend and improve the corresponding results of Wang and Osilike.

MSC:47H05, 47H09, 49M05.

1 Introduction and preliminaries

Throughout this paper we assume that E is a real Banach space and T:EE is a mapping. We denote by F(T) and D(T) the set of fixed points and the domain of T, respectively.

Recently, the convergence problems of an implicit (or non-implicit) iterative process to a common fixed point for a finite family of asymptotically nonexpansive mappings (or nonexpansive mappings) in Hilbert spaces or uniformly convex Banach spaces have been considered by several authors (see, e.g., [124]).

Recall that E is said to satisfy Opial’s condition [11] if for each sequence { x n } in E, the condition that the sequence x n x weakly implies that

lim sup n x n x< lim sup n x n y

for all yE with yx.

Definition 1.1 Let D be a closed subset of E and T:DD be a mapping.

  1. (1)

    T is said to be demi-closed at the origin if for each sequence { x n } in D, the conditions x n x 0 weakly and T x n 0 strongly imply T x 0 =0.

  2. (2)

    T is said to be semi-compact if for any bounded sequence x n in D such that x n T x n 0 (n), there exists a subsequence { x n i }{ x n } such that x n i x D.

  3. (3)

    T is said to be asymptotically nonexpansive [3] if there exists a sequence { k n }[1,) with lim n k n =1 such that

    T n x T n y k n xy,x,yD,n1;
  4. (4)

    T is said to be L-Lipschitzian if there exists a constant L>0 such that TxTyLxy for all x,yD.

Proposition 1.1 Let K be a nonempty subset of E, and let { T i } i = 1 m :KK be m asymptotically nonexpansive mappings. Then there exists a sequence { k n }[1,) with k n 1 such that

T i n x T i n y k n xy,n1,x,yK,i=1,2,,m.
(1.1)

Proof Since for each i=1,2,,m, T i :KK is an asymptotically nonexpansive mapping, there exists a sequence { k n ( i ) }[1,) with k n ( i ) 1 (n) such that

T i n x T i n y k n ( i ) xy,x,yK,n1,i=1,2,,m.

Letting

k n =max { k n ( 1 ) , k n ( 2 ) , , k n ( m ) } ,

we have that { k n }[1,) with k n 1 (n) and

T i n x T i n y k n ( i ) xy k n xy,n1

for all x,yK and for each i=1,2,,m. □

In 2007, for studying the strong and weak convergence of fixed points of nonexpansive mappings in a Hilbert space H, Wang [19] introduced the following hybrid iteration scheme:

x n + 1 = α n x n +(1 α n ) T λ n + 1 x n ,n0,
(1.2)

where T λ n + 1 x n =T x n λ n + 1 μF(T x n ) for all x n H, x 0 H is an initial point, F:HH is an η-strongly monotone and k-Lipschitzian mapping, μ is a positive fixed constant.

In the same year, Osilike et al. [13] extended the results of Wang from Hilbert spaces to arbitrary Banach spaces and proved those theorems by Wang without the strong monotonicity condition.

In this paper, we introduce the following new hybrid iteration method in Banach spaces:

x n + 1 = α n x n +(1 α n ) [ i = 1 m τ i T i n x n λ n + 1 μ f ( i = 1 m τ i T i n x n ) ] ,n0
(1.3)

for a finite family of asymptotically nonexpansive mappings { T i } i = 1 m :KK, where f:KK is an L-Lipschitzian mapping, μ is a positive fixed constant, { α n } is a sequence in (0,1), { λ n }[0,1) and { τ i } i = 1 m (0,1) such that i = 1 m τ i =1.

Especially, if { T i } i = 1 m :KK are m nonexpansive mappings, f:KK is an L-Lipschitzian mapping, μ is a positive fixed constant, { α n } is a sequence in (0,1), { λ n }[0,1) and { τ i } i = 1 m (0,1) such that i = 1 m τ i =1, then the sequence { x n } defined by

x n + 1 = α n x n +(1 α n ) [ i = 1 m τ i T i x n λ n + 1 μ f ( i = 1 m τ i T i x n ) ] ,n0,
(1.4)

is called the hybrid iteration scheme for a finite family of nonexpansive mappings { T i } i = 1 N .

The purpose of this paper is to study the weak and strong convergence of an iterative sequence { x n } defined by (1.3) and (1.4) to a common fixed point for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the main results in [13] and [19].

In order to prove the main results of this paper, we need the following lemmas.

Lemma 1.1 [17]

Let { a n }, { b n }, { c n } be three nonnegative real sequences satisfying the following condition:

a n + 1 (1+ b n ) a n + c n ,n0.

If n = 1 b n < and n = 1 c n <, then the limit lim n a n exists.

Lemma 1.2 [15]

Let E be a uniformly convex Banach space, and let b, c be two constants with 0<b<c<1. Suppose that { t n } is a sequence in [b,c] and { x n }, { y n } are two sequences in E. Then the conditions

{ lim n ( 1 t n ) x n + t n y n = d , lim sup n x n d , lim sup n y n d ,

imply that lim n x n y n =0, where d0 is some constant.

Lemma 1.3 [4]

Let E be a uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let T:KK be an asymptotically nonexpansive mapping with F(T). Then IT is semi-closed at zero, where I is the identity mapping of E, that is, for each sequence { x n } in K, if { x n } converges weakly to qK and {(IT) x n } converges strongly to 0, then (IT)q=0.

2 Main results

We are now in a position to prove our main results in this paper.

Theorem 2.1 Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let { T 1 , T 2 ,, T m }:KK be m asymptotically nonexpansive mappings with F= i = 1 m F( T i ) (the set of common fixed points of { T 1 , T 2 ,, T m }); f:KK is an L-Lipschitzian mapping. Let the hybrid iteration { x n } be defined by (1.3), where { α n } and { λ n } are real sequences in [0,1), let { k n } be the sequence defined by (1.1) satisfying the following conditions:

  1. (i)

    α α n β for some α,β(0,1);

  2. (ii)

    n = 1 ( k n 1)<;

  3. (iii)

    n = 1 λ n <.

Then

  1. (1)

    lim n x n p exists for each pF,

  2. (2)

    lim n x n T l x n =0, l=1,2,3,,m,

  3. (3)

    { x n } converges strongly to a common fixed point of { T 1 , T 2 , T 3 ,, T m } if and only if lim n d( x n ,F)=0.

Proof (1) Since F= i = 1 m F( T i ), for each pF, it follows from Proposition 1.1 that

x n + 1 p = α n x n + ( 1 α n ) [ i = 1 m τ i T i n x n λ n + 1 μ f ( i = 1 m τ i T i n x n ) ] p α n x n p + ( 1 α n ) i = 1 m τ i ( T i n x n T i n p ) + ( 1 α n ) λ n + 1 μ f ( i = 1 m τ i T i n x n ) α n x n p + ( 1 α n ) i = 1 m τ i T i n x n T i n p + ( 1 α n ) λ n + 1 μ f ( i = 1 m τ i T i n x n ) f ( p ) + ( 1 α n ) λ n + 1 μ f ( p ) α n x n p + ( 1 α n ) k n x n p + ( 1 α n ) λ n + 1 μ k n L x n p + ( 1 α n ) λ n + 1 μ f ( p ) .
(2.1)

Since k n 1 (n), we know that { k n } is bounded, and there exists M 1 1 such that k n M 1 . Let u n = k n 1, n1, by condition (ii) we have n = 1 u n <. Therefore we have

x n + 1 p α n x n p + ( 1 α n ) ( 1 + u n ) x n p + ( 1 α n ) λ n + 1 μ M 1 L x n p + ( 1 α n ) λ n + 1 μ f ( p ) α n x n p + ( 1 α n + u n ) x n p + λ n + 1 μ M 1 L x n p + λ n + 1 μ f ( p ) ( 1 + u n + λ n + 1 μ M 1 L ) x n p + λ n + 1 μ f ( p ) .
(2.2)

Taking a n = x n p, b n = u n + λ n + 1 μ M 1 L, c n = λ n + 1 μf(p) and by using condition (iii) and n = 1 u n <, it is easy to see that

n = 1 b n <; n = 1 c n <.

It follows from Lemma 1.1 that lim n x n p exists.

(2) Since { x n p} is bounded, there exists M 2 >0 such that

x n p M 2 ,n1.
(2.3)

We can assume that

lim n x n p=d,
(2.4)

where d0 is some number. Since { x n p} is a convergent sequence, so { x n } is a bounded sequence in K. Let

σ n = i = 1 m τ i T i n x n λ n + 1 μf ( i = 1 m τ i T i n x n ) ,

then

x n + 1 p= α n ( x n p ) + ( 1 α n ) ( σ n p ) .
(2.5)

By (2.4) we have that

lim sup n x n p=d.
(2.6)

From (2.1) and (2.3) we have

f ( i = 1 m τ i T i n x n ) L k n x n p + f ( p ) L M 1 M 2 + f ( p ) .
(2.7)

By condition (iii), k n M 1 , n = 1 u n < and (2.3), (2.4), (2.6), we have that

lim sup n σ n p lim sup n { i = 1 m τ i T i n x n p + λ n + 1 μ f ( i = 1 m τ i T i n x n ) } lim sup n { k n x n p + λ n + 1 μ ( L M 1 M 2 + f ( p ) ) } = lim sup n { ( 1 + u n ) x n p + λ n + 1 μ ( L M 1 M 2 + f ( p ) ) } lim sup n { x n p + u n M 2 + λ n + 1 μ ( L M 1 M 2 + f ( p ) ) } d .
(2.8)

Thus from (2.4), (2.5), (2.6), (2.8) and Lemma 1.2 we know that

lim n σ n x n =0.
(2.9)

By (2.9), we have that

x n + 1 x n = ( α n 1 ) x n + ( 1 α n ) σ n ( 1 α n ) σ n x n 0 ( n ) .
(2.10)

From (2.10) we obtain that

lim n x n + j x n =0,j=1,2,3,,m.
(2.11)

It follows from (2.7) and (2.9) that

lim n x n i = 1 m τ i T i n x n lim n ( x n σ n + σ n i = 1 m τ i T i n x n ) lim n ( x n σ n + λ n + 1 μ f ( i = 1 m τ i T i n x n ) ) lim n ( x n σ n + λ n + 1 μ ( L M 1 M 2 + f ( p ) ) ) = 0 .
(2.12)

Let ξ l , n = x n T l n x n , l{1,2,3,,m}, then from (2.12) we have

ξ l , n = x n T l n x n = 1 τ l τ l x n T l n x n 1 τ l i = 1 m τ i x n T i n x n = 1 τ l x n i = 1 m τ i T i n x n 0 ( n ) .
(2.13)

It follows from (2.10) and (2.13) that

x n T l x n x n T l n x n + T l n x n T l x n ξ l , n + k 1 T l n 1 x n x n ξ l , n + k 1 ( T l n 1 x n T l n 1 x n 1 + T l n 1 x n 1 x n 1 + x n 1 x n ) ξ l , n + k 1 ( k n 1 x n x n 1 + ξ l , n 1 + x n 1 x n ) = ξ l , n + k 1 ( 1 + k n 1 ) x n x n 1 + ξ l , n 1 0 ( n ) .
(2.14)

(3) From (2.2) and (2.3), we have that

x n + 1 p(1+ b n ) x n p+ c n ,n1,
(2.15)

where b n = u n + λ n + 1 μ M 1 L and c n = λ n + 1 μf(p) with n = 1 b n < and n = 1 c n <. Hence, we have

d( x n ,F)(1+ b n )d( x n 1 ,F)+ c n ,n1.
(2.16)

It follows from (2.16) and Lemma 1.1 that the limit lim n d( x n ,F) exists.

If { x n } converges strongly to a common fixed point p of { T 1 , T 2 , T 3 ,, T m }, then it follows from (2.3) and Lemma 1.2 that the limit lim n x n p=0. Since 0d( x n ,F) x n p, we know that lim n d( x n ,F)=0, and so lim sup n d( x n ,F)=0.

Conversely, suppose lim sup n d( x n ,F)=0, then lim n d( x n ,F)=0.

Next we prove that the sequence { x n } is a Cauchy sequence in K. In fact, since n = 1 b n <, 1+texp{t} for all t>0, from (2.15) we have

x n pexp{ b n } x n 1 p+ c n .
(2.17)

Hence, for any positive integers n, m, from (2.17) it follows that

x n + m p exp { b n + m } x n + m 1 p + c n + m exp { b n + m } [ exp { b n + m 1 } x n + m 2 p + c n + m 1 ] + c n + m = exp { b n + m + b n + m 1 } x n + m 2 p + exp { b n + m } c n + m 1 + c n + m exp { i = n n + m b i } x n p + exp { i = n + 1 n + m b i } i = n + 1 n + m c i W x n p + W i = n + 1 c i ,

where W=exp{ n = 1 b n }<.

Since lim n d( x n ,F)=0 and n = 1 c n <, for any given ϵ>0, there exists a positive integer n 0 such that

d( x n ,F)< ϵ 4 ( W + 1 ) , i = n + 1 c i < ϵ 2 W ,n n 0 .

Therefore there exists p 1 F such that

d( x n , p 1 )< ϵ 2 ( W + 1 ) ,n n 0 .

Consequently, for any n n 0 and for all m1, we have

x n + m x n x n + m p 1 + x n p 1 < ( 1 + W ) x n p 1 + W i = n + 1 c i ϵ 2 ( W + 1 ) ( 1 + W ) + W ϵ 2 W = ϵ .

This implies that { x n } is a Cauchy sequence in K. By the completeness of K, we can assume that x n x K. Then from (2) and Lemma 1.3 we have x F, and so x is a common fixed point of T 1 , T 2 , T 3 ,, T m . This completes the proof of Theorem 2.1. □

Theorem 2.2 Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let { T 1 , T 2 , T 3 ,, T m }:KK be m asymptotically nonexpansive mappings with F= i = 1 m F( T i ), and at least there exists T l , 1lm, which is semi-compact. f:KK is an L-Lipschitzian mapping. Let { α n } and { λ n } be real sequences in [0,1), { k n } be the sequence defined by (1.1) satisfying the following conditions:

  1. (i)

    α α n β for some α,β(0,1);

  2. (ii)

    n = 1 ( k n 1)<;

  3. (iii)

    n = 1 λ n <.

Then the hybrid iterative process { x n } defined by (1.3) converges strongly to a common fixed point of { T 1 , T 2 , T 3 ,, T m } in K.

Proof From the proof of Theorem 2.1, { x n } is bounded, and lim n x n T l x n =0, l=1,2,3,,m. Especially, we have

lim n x n T 1 x n =0.
(2.18)

By the assumption of Theorem 2.2, we may assume that T 1 is semi-compact, without loss of generality. Then it follows from (2.18) that there exists a subsequence { x n k } of { x n } such that { x n k } converges strongly to pK, and we have

p T l p= lim n k x n k T l x n k = lim n x n T l x n =0,l=1,2,3,,m.

This implies that pF. In addition, since lim n x n p exists, therefore lim n x n p=0, that is, { x n } converges strongly to a fixed point of { T 1 , T 2 , T 3 ,, T m } in K. This completes the proof of Theorem 2.2. □

Theorem 2.3 Under the conditions of Theorem  2.1, if E satisfies Opial’s condition, then the hybrid iterative process { x n } defined by (1.3) converges weakly to a common fixed point of { T 1 , T 2 , T 3 ,, T m } in K.

Proof From the proof of Theorem 2.1, we know that { x n } is a bounded sequence in K. Since E is uniformly convex, it must be reflexive, so every bounded subset of E is weakly compact. Therefore, there exists a subsequence { x n j }{ x n } such that { x n j } converges weakly to x K. From (2.14) we have

lim j x n j T l x n j =0,l=1,2,3,,m.
(2.19)

By Lemma 1.3, we know that x F( T l ). By the arbitrariness of l{1,2,3,,m}, we have that x l = 1 m F( T i ).

Suppose that there exists some subsequence { x n k }{ x n } such that x n k y K weakly and y x . From Lemma 1.3, y F. By (2.2) we know that lim n x n x and lim n x n y exist. By the virtue of Opial’s condition of E, we have

lim n x n x = lim j x n j x < lim j x n j y = lim n x n y = lim k x n k y < lim k x k x = lim n x n x ,

which is a contraction. Hence x = y . This implies that { x n } converges weakly to a common fixed point of { T 1 , T 2 , T 3 ,, T m } in K. This completes the proof of Theorem 2.3. □

Remark 2.1 Theorems 2.1, 2.2 and 2.3 extend the results of [13] and [19] from a nonexpansive mapping to a finite family of asymptotically nonexpansive mappings.

Theorem 2.4 Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let { T 1 , T 2 , T 3 ,, T m }:KK be m nonexpansive mappings with F= i = 1 m F( T i ); f:KK is an L-Lipschitzian mapping. Let a hybrid iterative sequence { x n } be defined by (1.4), where { α n } and { λ n } are real sequences in [0,1) satisfying the following conditions:

  1. (i)

    α α n β for some α,β(0,1);

  2. (ii)

    n = 1 λ n <.

Then

  1. (1)

    lim n x n p exists for each pF,

  2. (2)

    lim n x n T l x n =0, l=1,2,3,,m,

  3. (3)

    { x n } converges strongly to a common fixed point of { T 1 , T 2 , T 3 ,, T m } if and only if lim n d( x n ,F)=0.

Theorem 2.5 Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let { T 1 , T 2 , T 3 ,, T m }:KK be m nonexpansive mappings with F= i = 1 m F( T i ), and at least there exists T l , 1lm, which is semi-compact. f:KK is an L-Lipschitzian mapping. Let { α n } and { λ n } be real sequences in [0,1) satisfying the following conditions:

  1. (i)

    α α n β for some α,β(0,1);

  2. (ii)

    n = 1 λ n <.

Then the hybrid iterative process { x n } defined by (1.4) converges strongly to a common fixed point of { T 1 , T 2 , T 3 ,, T m } in K.

Theorem 2.6 Under the conditions of Theorem  2.4, if E satisfies Opial’s condition, then the hybrid iterative process { x n } defined by (1.4) converges weakly to a common fixed point of { T 1 , T 2 , T 3 ,, T m } in K.

The proofs of Theorems 2.4, 2.5 and 2.6 can be obtained from those of Theorems 2.1, 2.2 and 2.3 with the condition that { T 1 , T 2 , T 3 ,, T m }:KK are m nonexpansive mappings.

Remark 2.2 Theorems 2.4, 2.5 and 2.6 extend the results of [13] and [19] from a nonexpansive mapping to a finite family of nonexpansive mappings.

References

  1. 1.

    Bauschke HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 1996, 202: 150–159. 10.1006/jmaa.1996.0308

  2. 2.

    Chang SS, Cho YJ: The implicit iterative processes for asymptotically nonexpansive mappings. 1. Nonlinear Analysis and Applications 2003, 369–382.

  3. 3.

    Chang SS, Cho YJ, Zhou HY: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. J. Korean Math. Soc. 2001, 38: 1245–1260.

  4. 4.

    Cho YJ, Zhou HY, Guo G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 2004, 47: 707–717. 10.1016/S0898-1221(04)90058-2

  5. 5.

    Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3

  6. 6.

    Gornicki J: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Comment. Math. Univ. Carol. 1989, 301: 249–252.

  7. 7.

    Gu F: Implicit iterative process for common fixed point of a finite family of asymptotically nonexpansive mappings. Studia Sci. Math. Hung. 2008, 45(2):235–250.

  8. 8.

    Halpern B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0

  9. 9.

    Lions PL: Approximation de points fixes de contractions. C. R. Acad. Sci. Paris, Sér. A 1977, 284: 1357–1359.

  10. 10.

    Marino G, Xu H-K: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028

  11. 11.

    Opial Z: Weak convergence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0

  12. 12.

    Osilike MO: Implicit iteration process for common fixed point of a finite family of strictly pseudocontractive maps. J. Math. Anal. Appl. 2004, 294: 73–81. 10.1016/j.jmaa.2004.01.038

  13. 13.

    Osilike MO, Isiogugu FO, Nwokoro PU: Hybrid iteration method for fixed points of nonexpansive mappings in arbitrary Banach spaces. Fixed Point Theory Appl. 2007., 2007: Article ID 64306

  14. 14.

    Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 1980, 75: 287–292. 10.1016/0022-247X(80)90323-6

  15. 15.

    Schu J: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884

  16. 16.

    Sun ZH: Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings. J. Math. Anal. Appl. 2003, 286: 351–358. 10.1016/S0022-247X(03)00537-7

  17. 17.

    Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iterations process. J. Math. Anal. Appl. 1993, 178: 301–308. 10.1006/jmaa.1993.1309

  18. 18.

    Tan KK, Xu HK: The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 1992, 114: 399–404. 10.1090/S0002-9939-1992-1068133-2

  19. 19.

    Wang L: An iteration method for nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2007., 2007: Article ID 28619

  20. 20.

    Wittmann R: Approximation of fixed points of nonexpansive mappings. Arch. Math. 1992, 58: 486–491. 10.1007/BF01190119

  21. 21.

    Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 179–291.

  22. 22.

    Xu HK, Ori MG: An implicit iterative process for nonexpansive mappings. Numer. Funct. Anal. Optim. 2001, 22: 767–773. 10.1081/NFA-100105317

  23. 23.

    Yao Y, Yao J-C: On modified iterative method for nonexpansive mappings and monotone mappings. Appl. Math. Comput. 2007, 186: 1551–1558. 10.1016/j.amc.2006.08.062

  24. 24.

    Zhou YY, Chang SS: Convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in Banach spaces. Numer. Funct. Anal. Optim. 2002, 23: 911–921. 10.1081/NFA-120016276

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Acknowledgements

The study was supported by the National Natural Science Foundation of China (11271105, 11071169) and the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030).

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Correspondence to Feng Gu.

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Gu, F. A new hybrid iteration method for a finite family of asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2013, 322 (2013) doi:10.1186/1687-1812-2013-322

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Keywords

  • asymptotically nonexpansive mapping
  • nonexpansive mapping
  • hybrid iteration method
  • common fixed point
  • Opial condition
  • demi-closed principle
  • semi-compactness