MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2014/379523 379523 Research Article Strong Convergence Theorems for Two Total Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces He Xi-bing 1 Hui Xiao-jian 2 http://orcid.org/0000-0003-4862-9150 Xing Hui 1 Sadarangani Kishin 1 School of Mathematics and Statistics Xi'an Jiaotong University Xi'an, Shaanxi 710049 China xjtu.edu.cn 2 Department of Foundation Xijing University Xi'an, Shaanxi 710123 China xijing.edu.cn 2014 1672014 2014 28 03 2014 27 06 2014 16 7 2014 2014 Copyright © 2014 Xi-bing He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The purpose of this paper is to establish some strong convergence theorems for a common fixed point of two total quasi- ϕ -asymptotically nonexpansive mappings in Banach space by means of the hybrid method in mathematical programming. The results presented in this paper extend and improve on the corresponding ones announced by Martinez-Yanes and Xu (2006), Plubtieng and Ungchittrakool (2007), Qin et al. (2009), and many others.

1. Introduction

Let E be a smooth Banach space and let E * be the topological dual of E . Let the function ϕ : E × E R + be defined by (1) ϕ ( x , y ) = x 2 - 2 x , J y + y 2 , x , y E , where J is the normalized duality mapping from E to E * . Let K be a nonempty subset of E and T : K K a nonlinear mapping. Recall that a mapping T is called nonexpansive if T x - T y x - y , x , y K . A point x K is said to be a fixed point of T  if T x = x . Let F ( T ) : = { x K : T x = x } be the set of fixed points of T . A point p K is said to be an asymptotic fixed point of T  if K contains a sequence { x n } which converges weakly to p such that the strong lim n ( x n - T x n ) = 0 . The set of asymptotic fixed points of T will be denoted by F ^ ( T ) .

Recall that a mapping T : K K is called ( i ) relatively nonexpansive  if F ^ ( T ) = F ( T ) ϕ and ϕ ( p , T x ) ϕ ( p , x ) , x K , p F ( T ) ; ( ii ) closed if for any sequence { x n } K with x n x and T x n y , T x = y ; ( iii ) quasi- ϕ -nonexpansive  if F ( T ) ϕ and ϕ ( p , T x ) ϕ ( p , x ) , x K , p F ( T ) ; ( iv ) quasi- ϕ -asymptotically nonexpansive  if F ( T ) ϕ and there exists a sequence { μ n } [ 0 , ) with lim n μ n = 0 such that ϕ ( p , T n x ) ( 1 + μ n ) ϕ ( p , x ) , x K , p F ( T ) , n N ; ( v ) total quasi- ϕ -asymptotically nonexpansive  if F ( T ) ϕ and there exist nonnegative real sequences { μ n } { ν n } with lim n μ n = lim n ν n = 0 and a strictly increasing continuous function ζ : R + R + such that ϕ ( p , T n x ) ϕ ( p , x ) + μ n ζ ( ϕ ( p , x ) ) + ν n , x K , p F ( T ) , n N ; ( vi ) uniformly L-Lipschitz continuous  if there exists a constant L > 0 such that T n x - T n y L x - y , x , y K , n 1 .

It is well known that total quasi- ϕ -asymptotically nonexpansive mappings contain relatively nonexpansive mappings, quasi- ϕ -nonexpansive mappings, and quasi- ϕ -asymptotically nonexpansive mappings as its special cases; see [4, 6, 7] for more details.

Iterative approximation of fixed points for nonexpansive mappings has been considered by many papers for either the Mann iteration  or the Ishkawa iteration ; see, for example,  and the references therein.

In 1967, Halpern  considered the following explicit iteration which was referred to as Halpern iteration: (2) x 0 K arbitrarily chosen , x n + 1 = α n x 0 + ( 1 - α n ) T x n , n 0 , where T is nonexpansive. He proves the strong convergence of { x n } to a point of T provided that α n = n - θ , where θ ( 0,1 ) .

Recently, many authors improved and generalized the results of Halpern  by means of different methods; see, for example, [2, 4, 10, 1217] and the references therein. In general, there are two ways as follows.

One of the methods is to combine Halpern iteration with Mann iteration; see, for example, C.E. Chidume and C.O. Chidume  and Hu .

Another method is to use the hybrid projection algorithm method; see, for example, Martinez-Yanes and Xu , Plubtieng and Ungchittrakool , and Qin et al. .

Motivated and inspired by [2, 4, 10, 1217], the purpose of this paper is to modify Halpern iteration (2) by means of both methods above for two total quasi- ϕ -asymptotically nonexpansive mappings and then to prove the strong convergence in the framework of Banach spaces. The results presented in this paper extend and improve the corresponding results of Martinez-Yanes and Xu , Plubtieng and Ungchittrakool , Qin et al. , and others.

Throughout the paper, we denote for weak convergence and for strong convergence.

2. Preliminaries

Let E be a real Banach space endowed with the norm · and let E * be the topological dual of E . For all x E and x * E * , the value of f at x is denoted by x , f and is called the duality pairing. Then, the normalized duality mapping J : E 2 E * is defined by (3) J ( x ) = { f E * : x , f = x 2 = f 2 } , x E . It is well known that the operator J is well defined and J is the identity mapping if and only if E is a Hilbert space. But in general, J is nonlinear and multiple-valued.

The following basic properties for a Banach space E can be found in .

If E is uniformly smooth, then J is unformly continuous on each bounded subset of E .

If E is reflexive and strictly convex, then J - 1 is norm-weak-continuous.

If E is a smooth, strictly convex, and reflexive Banach space, then J is single-valued, one-to-one, and onto.

A Banach space E is uniformly smooth if and only if E * is uniformly convex.

Each uniformly convex Banach space E has the Kadec-Klee property; that is, for any sequence { x n } E , if x n x E and x n x , then x n x .

Every uniformly smooth Banach space is reflexive.

Moreover, we have the Lyapunov functional ϕ defined by (1). It is obvious from the definition of the function ϕ that we have the following property.

Property 1 (see [<xref ref-type="bibr" rid="B13">19</xref>]).

If the function ϕ is defined by (1), then we have

( x - y ) 2 ϕ ( x , y ) ( x - y ) 2 , x , y E ;

ϕ ( x , J - 1 ( λ J y + ( 1 - λ ) J z ) ) λ ϕ ( x , y ) + ( 1 - λ ) ϕ ( x , z ) , x , y , z E ;

ϕ ( x , y ) = ϕ ( x , z ) + ϕ ( z , y ) + 2 x - z , J z - J y , x , y , z E ;

ϕ ( x , y ) = x , J x - J y + y - x , J y x J x - J y + y - x J y , x , y E .

It follows Albert  that the generalized projection Π K : E K is defined by Π K ( x ) = arg inf y K ϕ ( y , x ) , x E . So we have the following lemmas.

Lemma 1 (see [<xref ref-type="bibr" rid="B2">20</xref>]).

Let E be a smooth, strictly convex, and reflexive Banach space and let K be a nonempty closed convex subset of E . Then the following conclusions hold:

ϕ ( x , Π K y ) + ϕ ( Π K y , y ) ϕ ( x , y ) , x K , y E ;

if x K and z K , then z = Π K x z - y , J x - J z 0 , y K ;

for x , y E , ϕ ( x , y ) = 0 if and only if x = y .

Remark 2.

If H is a real Hilbert space, then ϕ ( x , y ) = x - y 2 and Π K = P K (the metric projection of H onto a closed convex subset K ).

Lemma 3 (see [<xref ref-type="bibr" rid="B2">20</xref>]).

Let E be a uniformly convex and smooth Banach space and let { x n } , { y n } be sequences of E . If ϕ ( x n , y n ) 0 (as n ) and either { x n } or { y n } is bounded, then x n - y n 0 (as n ).

Lemma 4 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

Let E be a smooth, strictly convex, and reflexive Banach space with Kadec-Klee property, and let K be a nonempty closed convex subset of E . Let T : K K be a closed and totally quasi- ϕ -asymptotically nonexpansive mapping with nonnegative real sequences { μ n } , { ν n } and a strictly increasing continuous function ζ : R + R + such that lim n μ n = lim n ν n = 0 and ζ ( 0 ) = 0 . If ν 1 = 0 , then the fixed point set F ( T ) of T is a closed and convex subset of K .

Lemma 5 (see [<xref ref-type="bibr" rid="B9">21</xref>]).

Let E be a uniformly convex Banach space and let B r ( 0 ) = { x E : x r } be a closed ball of E . Then there exists a continuous strictly increasing convex function g : [ 0 , ) [ 0 , ) with g ( 0 ) = 0 such that (4) λ x + μ y + γ z 2 λ x 2 + μ y 2 + γ z 2 - λ μ g ( x - y ) , for all x , y , z B r ( 0 ) and λ , μ , γ [ 0,1 ] with λ + μ + γ = 1 .

3. Main Results

To prove the main results, we need the following hypotheses for the sequences { α n } and { β n ( j ) }    ( j = 1,2 , 3 ) :

0 < α n < 1 for all n N { 0 } and lim n α n = 0 ,

0 β n ( 1 ) , β n ( 2 ) , β n ( 3 ) 1 , β n ( 1 ) + β n ( 2 ) + β n ( 2 ) = 1 for all n N { 0 } , lim n β n ( 1 ) = 0 and lim in f n β n ( 2 ) β n ( 3 ) > 0 .

The main results of this paper are stated as follows.

Theorem 6.

Let E be a real strictly convex and uniformly smooth Banach space with Kadec-Klee property, and let K be a nonempty closed convex subset of E . Let S , T : K K be two closed, uniformly L-Lipschitz continuous and totally quasi- ϕ -asymptotically nonexpansive mappings with nonnegative real sequences { μ n ( j ) } , { ν n ( j ) } , j = 1,2 and a strictly increasing continuous function ζ : R + R + such that lim n μ n ( j ) = lim n ν n ( j ) = 0 , j = 1,2 and ζ ( 0 ) = 0 . Let { x n } be a sequence defined by (5) x 0 C 1 = K    a r b i t r a r i l y c h o s e n , y n = J - 1 ( α n J x 0 + ( 1 - α n ) J z n ) , z n = J - 1 ( β n ( 1 ) J x n + β n ( 2 ) J T n x n + β n ( 3 ) J S n x n ) , C n + 1 = { z C n : ϕ ( z , y n ) α n ϕ ( z , x 0 ) + ( 1 - α n ) ϕ ( z , x n ) + ( 1 - α n ) ξ n } , x n + 1 = Π C n + 1 x 0 , n 0 , where ξ n = ( β n ( 2 ) μ n ( 2 ) + β n ( 3 ) μ n ( 1 ) ) sup p F ζ ( ϕ ( p , x n ) ) + ( β n ( 2 ) ν n ( 2 ) + β n ( 3 ) ν n ( 1 ) ) . Assume that the sequences { α n } and { β n ( j ) }    ( j = 1,2 , 3 ) satisfy hypotheses (i) and (ii). If F : = F ( S ) F ( T ) ϕ is bounded and ν 1 = 0 , then the sequence { x n } converges strongly to Π F x 0 .

Proof.

We divide the proof of Theorem 6 into six steps.

We first show that F is a closed and convex subset in K .

By Lemma 4, it is trivial to show that F ( S ) and F ( T ) are two closed and convex subsets of K . Therefore F is closed and convex in K .

Next we prove that C n is a closed and convex subset in K for all n 1 .

As a matter of fact, by hypothesis, C 1 = K is closed and convex. Suppose that C k is closed and convex for some k 1 . From the definition of ϕ , we may know that (6) C k + 1 = { z C k : ϕ ( z , y k ) α k ϕ ( z , x 0 ) + ( 1 - α k ) ϕ ( z , x k ) + ( 1 - α k ) ξ k } , = { y k 2 + ( 1 - α k ) ξ k z : 2 z , J x k - J y k + 2 α k z , J x 0 α k x 0 2 + ( 1 - α k ) x k 2 - y k 2 + ( 1 - α k ) ξ k } C k and thus C k + 1 is closed and convex. Therefore, by induction principle, C n are closed and convex for all n 1 . This also shows that Π F x 0 is well defined.

Now we show that F C n , n 1 .

Indeed, it is obvious that F C 1 = K . Suppose that F C n for some n 1 . Hence for any u F C n , by Lyapunov functional (1) and Lemma 5, we have (7) ϕ ( u , z n ) = ϕ ( u , J - 1 ( β n ( 1 ) J x n + β n ( 2 ) J T n x n + β n ( 3 ) J S n x n ) ) = u 2 - 2 u , β n ( 1 ) J x n + β n ( 2 ) J T n x n + β n ( 3 ) J S n x n + β n ( 1 ) J x n + β n ( 2 ) J T n x n + β n ( 3 ) J S n x n 2 u 2 - 2 β n ( 1 ) u , J x n - 2 β n ( 2 ) u , J T n x n - 2 β n ( 3 ) u , J S n x n + β n ( 1 ) x n 2 + β n ( 2 ) T n x n 2 + β n ( 3 ) S n x n 2 - β n ( 2 ) β n ( 3 ) g ( T n x n - S n x n ) β n ( 1 ) ϕ ( u , x n ) + β n ( 2 ) ϕ ( u , T n x n ) + β n ( 3 ) ϕ ( u , S n x n ) - β n ( 2 ) β n ( 3 ) g ( T n x n - S n x n ) β n ( 1 ) ϕ ( u , x n ) + β n ( 2 ) [ ϕ ( u , x n ) + μ n ( 2 ) ζ ( ϕ ( u , x n ) ) + ν n ( 2 ) ] + β n ( 3 ) [ ϕ ( u , x n ) + μ n ( 1 ) ζ ( ϕ ( u , x n ) ) + ν n ( 1 ) ] - β n ( 2 ) β n ( 3 ) g ( J T n x n - J S n x n ) ϕ ( u , x n ) + ξ n - β n ( 2 ) β n ( 3 ) g ( J T n x n - J S n x n ) ϕ ( u , x n ) + ξ n . Moreover, it follows from Property 1(ii) that we have (8) ϕ ( u , y n ) = ϕ ( u , J - 1 ( α n J x 0 + ( 1 - α n ) J z n ) ) α n ϕ ( u , x 0 ) + ( 1 - α n ) ϕ ( u , z n ) . Combining (7) with (8), we have (9) ϕ ( u , y n ) α n ϕ ( u , x 0 ) + ( 1 - α n ) ( ϕ ( u , x n ) + ξ n ) . This shows that u C n + 1 . Therefore, by induction principle, we have F C n for all n 1 .

Next we prove that { x n } converges strongly to some point p * K .

In fact, since x n = Π C n x 0 , it follows from Lemma 1(ii) that we have (10) x n - y , J x 0 - J x n 0 , y C n . Again since F C n for all n 1 , we have (11) x n - u , J x 0 - J x n 0 , u F . It follows from Lemma 1(i) that for each u F and for each n 1 we have (12) ϕ ( x n , x 0 ) = ϕ ( Π C n x 0 , x 0 ) ϕ ( u , x 0 ) - ϕ ( u , x n ) ϕ ( u , x 0 ) . Therefore { ϕ ( x n , x 0 ) } is bounded, and hence { x n } is bounded.

On the other hand, since x n = Π C n x 0 and x n + 1 = Π C n + 1 x 0 C n + 1 C n , we have ϕ ( x n , x 0 ) ϕ ( x n + 1 , x 0 ) for all n 1 . This implies that { ϕ ( x n , x 0 ) } is nondecreasing, and so lim n ϕ ( x n , x 0 ) exists. By the construction of { ϕ ( x n , x 0 ) } , we have C m C n and x m = Π C m x 0 C n for any positive integer m > n . Therefore, using Lemma 1(i), we have (13) ϕ ( x n + m , x n ) = ϕ ( x n + m , Π C n x 0 ) ϕ ( x n + m , x 0 ) - ϕ ( x n , x 0 ) for all n 1 . Since lim n ϕ ( x n , x 0 ) exists, we obtain that (14) ϕ ( x n + m , x n ) 0 ( as n ) , m 1 . Thus, by Lemma 3, we have (15) lim n x n + m - x n = 0 , m 1 . This implies that the sequence { x n } is a Cauchy sequence in K . Since K is a nonempty closed subset of Banach space E , this implies that it is complete. Hence there exists an p * K such that (16) lim n x n = p * . By the way, it is easy from (16) to see that (17) lim n ξ n = lim n [ ( β n ( 2 ) μ n ( 2 ) + β n ( 3 ) μ n ( 1 ) ) sup p F ζ ( ϕ ( p , x n ) ) + ( β n ( 2 ) ν n ( 2 ) + β n ( 3 ) ν n ( 1 ) ) ( β n ( 2 ) μ n ( 2 ) + β n ( 3 ) μ n ( 1 ) ) sup p F ζ ( ϕ ( p , x n ) ) ] = 0 .

Now we prove that p * F .

Since x n + 1 C n + 1 and by the structure of C n + 1 , we have (18) ϕ ( x n + 1 , y n ) α n ϕ ( x n + 1 , x 0 ) + ( 1 - α n ) ( ϕ ( x n + 1 , x n ) + ξ n ) . Hence, by means of limits (14), (17) and lim n α n = 0 and using Lemma 3, we get that lim n x n + 1 - y n = 0 . But (19) x n - y n x n - x n + 1 + x n + 1 - y n . Thus lim n x n - y n = 0 . This implies that (20) lim n y n = p * .

Meanwhile, it follows from (7) and (8) that we have (21) ( 1 - α n ) β n ( 2 ) β n ( 3 ) g ( J T n x n - J S n x n ) α n ϕ ( u , x 0 ) + ( 1 - α n ) ( ϕ ( u , x n ) + ξ n ) - ϕ ( u , y n ) α n ( ϕ ( u , x 0 ) - ϕ ( u , y n ) ) + ( 1 - α n ) ξ n + ( 1 - α n ) ( ϕ ( u , x n ) - ϕ ( u , y n ) ) , where g : [ 0 , ) [ 0 , ) is a continuous strictly increasing convex function with g ( 0 ) = 0 . Thus, by virtue of (16), (17), (20), (21), lim n α n = 0 , and lim in f n β n ( 2 ) β n ( 3 ) > 0 , we have lim n g ( J T n x n - J S n x n ) = 0 , and hence lim n J T n x n - J S n x n = 0 . Since J - 1 is also uniformly norm-to-norm continuous on bounded sets, we get (22) lim n T n x n - S n x n = lim n J - 1 ( J T n x n - J S n x n ) = 0 . Next, using the convexity of · 2 , Property 1(iv), and hypothesis (ii), we obtain that (23) ϕ ( T n x n , z n ) = ϕ ( T n x n , J - 1 ( β n ( 1 ) J x n + β n ( 2 ) J T n x n + β n ( 3 ) J S n x n ) ) = T n x n 2 - 2 T n x n , β n ( 1 ) J x n + β n ( 2 ) J T n x n + β n ( 3 ) J S n x n + β n ( 1 ) J x n + β n ( 2 ) J T n x n + β n ( 3 ) J S n x n 2 T n x n 2 - 2 β n ( 1 ) T n x n , J x n - 2 β n ( 2 ) T n x n , J T n x n - 2 β n ( 3 ) T n x n , J S n x n + β n ( 1 ) x n 2 + β n ( 2 ) T n x n 2 + β n ( 3 ) S n x n 2 = β n ( 1 ) ϕ ( T n x n , x n ) + β n ( 3 ) ϕ ( T n x n , S n x n ) 0 ( as    n ) . Hence, from Property 1(ii), hypothesis (i), and (23), we have (24) ϕ ( T n x n , y n ) = ϕ ( T n x n , J - 1 ( α n J x 0 + ( 1 - α n ) J z n ) ) α n ϕ ( T n x n , x 0 ) + ( 1 - α n ) ϕ ( T n x n , z n ) 0    hhhhhhhhhhhhhihhhhhhhh ( as    n ) . Thus, by Lemma 3, we have lim n T n x n - y n = 0 and hence (25) T n x n - x n T n x n - y n + y n - x n 0 m m m m m m m m m i i i i i i i u u u k k k k k k i ( as    n ) . Moreover, we observe that (26) S n x n - x n S n x n - T n x n + T n x n - x n 0 k o k o k o k o k o k k k k k k k k k k i i i i u u u y u y u y u ( as    n ) . Since lim n x n = p * and E is uniformly smooth, it follows from (25) and (26) that we have (27) lim n T n x n = p * , (28) lim n S n x n = p * . Next, by the assumption that T is uniformly L -Lipschitz continuous, we have (29) T n + 1 x n - T n x n T n + 1 x n - T n + 1 x n + 1 + T n + 1 x n + 1 - x n + 1 + x n + 1 - x n + x n - T n x n ( L + 1 ) x n + 1 - x n + T n + 1 x n + 1 - x n + 1 + x n - T n x n . Combining (15) and (25) with (29), we have (30) lim n T n + 1 x n - T n x n = 0 . Hence, it follows from (27) that we have (31) lim n T n + 1 x n = p * , that is, (32) lim n T T n x n = p * . Furthermore, in view of (27) and the closeness of T , it yields that T p * = p * . Similarly, we have S p * = p * . Therefore, this implies that p * F .

Finally we prove that p * = Π F x 0 and so x n Π F x 0 .

Let w = Π F x 0 . Since w F C n and x n Π F x 0 , we have ϕ ( x n , x 0 ) ϕ ( w , x 0 ) , n 1 . This implies that (33) ϕ ( p * , x 0 ) = lim n ϕ ( x n , x 0 ) ϕ ( w , x 0 ) . Since w = Π F x 0 , this implies that p * = w . Therefore, x n Π F x 0 . This completes the proof.

If S = T , β n ( 1 ) = 0 or β n ( 1 ) = β n ( 3 ) = 0 for all n 1 in Theorem 6, then we have the following corollary.

Corollary 7.

Let E , K be the same as Theorem 6. Let T : K K be closed, uniformly L-Lipschitz continuous, and totally quasi- ϕ -asymptotically nonexpansive mappings with nonnegative real sequences { μ n } , { ν n } and a strictly increasing continuous function ζ : R + R + such that lim n μ n = lim n ν n = 0 and ζ ( 0 ) = 0 . Let { x n } be a sequence defined by (34) x 0 C 1 = K    a r b i t r a r i l y c h o s e n , y n = J - 1 ( α n J x 0 + ( 1 - α n ) J T n x n ) , C n + 1 = { z C n : ϕ ( z , y n ) α n ϕ ( z , x 0 ) + ( 1 - α n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π C n + 1 x 0 , n 0 , where ξ n = ( 1 - α n ) ( μ n sup p F ζ ( ϕ ( p , x n ) ) + ν n ) and Π C n + 1 is the generalized projection from E onto C n + 1 . Assume that the sequence { α n } satisfies hypothesis (i). If F : = F ( T ) ϕ is bounded and ν 1 = 0 , then the sequence { x n } converges strongly to Π F x 0 .

It is well known that the quasi- ϕ -asymptotically nonexpansive mappings are a special case of the total quasi- ϕ -asymptotically nonexpansive mappings. So we obtain the following theorem.

Theorem 8.

Let E , K be the same as Theorem 6. Let S , T : K K be two closed, uniformly L-Lipschitz continuous, and quasi- ϕ -asymptotically nonexpansive mappings with nonnegative real sequences { μ n ( 1 ) } , { μ n ( 2 ) } such that lim n μ n ( 1 ) = lim n μ n ( 2 ) = 0 . Let { x n } be a sequence defined by (35) x 0 C 1 = K    a r b i t r a r i l y c h o s e n , y n = J - 1 ( α n J x 0 + ( 1 - α n ) J z n ) , z n = J - 1 ( β n ( 1 ) J x n + β n ( 2 ) J T n x n + β n ( 3 ) J S n x n ) , C n + 1 = { z C n : ϕ ( z , y n ) α n ϕ ( z , x 0 ) + ( 1 - α n ) ϕ ( z , x n ) + ( 1 - α n ) ξ n } , x n + 1 = Π C n + 1 x 0 , n 0 , where ξ n = ( β n ( 2 ) μ n ( 2 ) + β n ( 3 ) μ n ( 1 ) ) sup p F ϕ ( p , x n ) . Assume that the sequences { α n } and { β n ( j ) }    ( j = 1,2 , 3 ) satisfy hypotheses (i) and (ii). If F : = F ( S ) F ( T ) ϕ is bounded and ν 1 = 0 , then the sequence { x n } converges strongly to Π F x 0 .

Proof.

The proof is similar to that of Theorem 6 and hence we omit it. This completes the proof.

If S = T , β n ( 1 ) = 0 or β n ( 1 ) = β n ( 3 ) = 0 for all n 1 in Theorem 8, then we have the following corollary.

Corollary 9.

Let E , K be the same as Theorem 6. Let T : K K be closed, uniformly L-Lipschitz continuous, and quasi- ϕ -asymptotically nonexpansive mappings with nonnegative real sequences { μ n } such that lim n μ n = 0 . Let { x n } be a sequence defined by (36) x 0 C 1 = K    a r b i t r a r i l y c h o s e n , y n = J - 1 ( α n J x 0 + ( 1 - α n ) J T n x n ) , C n + 1 = { z C n : ϕ ( z , y n ) α n ϕ ( z , x 0 ) + ( 1 - α n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π C n + 1 x 0 , n 0 , where ξ n = ( 1 - α n ) μ n sup p F ϕ ( p , x n ) and Π C n + 1 is the generalized projection from E onto C n + 1 . Assume that the sequence { α n } satisfies hypothesis ( i ) . If F : = F ( T ) ϕ is bounded, then the sequence { x n } converges strongly to Π F x 0 .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.

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