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Strong convergence theorems for a countable family of totally quasi-ϕ-asymptotically nonexpansive nonself mappings in Banach spaces with applications

Abstract

In this paper, we introduce a class of totally quasi-ϕ-asymptotically nonexpansive nonself mappings and study the strong convergence under a limit condition only in the framework of Banach spaces. Meanwhile, our results are applied to study the approximation problem of a solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:7864-7870, 2012).

MSC:47J05, 47H09, 49J25.

1 Introduction

Assume that X is a real Banach space with the dual X , D is a nonempty closed convex subset of X. We also denote by J the normalized duality mapping from X to 2 X which is defined by

J(x)= { f X : x , f = x 2 = f 2 } ,xX,

where , denotes the generalized duality pairing.

Let D be a nonempty closed subset of a real Banach space X. A mapping T:DD is said to be nonexpansive if TxTyxy for all x,yD. An element pD is called a fixed point of T:DD if p=T(p). The set of fixed points of T is represented by F(T).

A Banach space X is said to be strictly convex if x + y 2 1 for all x,yX with x=y=1 and xy. A Banach space is said to be uniformly convex if lim n x n y n =0 for any two sequences { x n },{ y n }X with x n = y n =1 and lim n x n + y n 2 =0.

The norm of a Banach space X is said to be Gâteaux differentiable if for each x,yS(x), the limit

lim t 0 x + t y x t
(1.1)

exists, where S(x)={x:x=1,xX}. In this case, X is said to be smooth. The norm of a Banach space X is said to be Fréchet differentiable if for each xS(x), the limit (1.1) is attained uniformly for yS(x) and the norm is uniformly Fréchet differentiable if the limit (1.1) is attained uniformly for x,yS(x). In this case, X is said to be uniformly smooth.

A subset D of X is said to be a retract of X if there exists a continuous mapping P:XD such that Px=x for all xX. It is well known that every nonempty closed convex subset of a uniformly convex Banach space X is a retract of X. A mapping P:XD is said to be a retraction if P 2 =P. It follows that if a mapping P is a retraction, then Py=y for all y in the range of P. A mapping P:XD is said to be a nonexpansive retraction if it is nonexpansive and it is a retraction from X to D.

Next, we assume that X is a smooth, strictly convex and reflexive Banach space and D is a nonempty closed convex subset of X. In the sequel, we always use ϕ:X×X R + to denote the Lyapunov functional defined by

ϕ(x,y)= x 2 2x,Jy+ y 2 ,x,yX.
(1.2)

It is obvious from the definition of the function ϕ that

(1.3)
(1.4)

and

ϕ ( x , J 1 ( λ J y + ( 1 λ ) J z ) ) λϕ(x,y)+(1λ)ϕ(x,z)
(1.5)

for all λ[0,1] and x,y,zX.

Following Alber [1], the generalized projection Π D :XD is defined by

Π D (x)=arg inf y D ϕ(y,x),xX.
(1.6)

Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.

In the sequel, we denote the strong convergence and weak convergence of the sequence { x n } by x n x and x n x, respectively.

Lemma 1.1 (see [1])

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

  1. (a)

    ϕ(x,y)=0 if and only if x=y;

  2. (b)

    ϕ(x, Π D y)+ϕ( Π D y,y)ϕ(x,y), x,yD;

  3. (c)

    if xX and zD, then z= Π D x if and only if zy,JxJz0, yD.

Remark 1.1 (see [2])

Let Π D be the generalized projection from a smooth, reflexive and strictly convex Banach space X onto a nonempty closed convex subset D of X. Then Π D is a closed and quasi-ϕ-nonexpansive from X onto D.

Remark 1.2 (see [2])

If H is a real Hilbert space, then ϕ(x,y)= x y 2 , and Π D is the metric projection of H onto D.

Definition 1.1 Let P:XD be a nonexpansive retraction.

  1. (1)

    A nonself mapping T:DX is said to be quasi-ϕ-nonexpansive if F(T)Φ, and

    ϕ ( p , T ( P T ) n 1 x ) ϕ(p,x),xD,pF(T),n1;
    (1.7)
  2. (2)

    A nonself mapping T:DX is said to be quasi-ϕ-asymptotically nonexpansive if F(T)Φ, and there exists a real sequence k n [1,+), k n 1 (as n), such that

    ϕ ( p , T ( P T ) n 1 x ) k n ϕ(p,x),xD,pF(T),n1;
    (1.8)
  3. (3)

    A nonself mapping T:DX is said to be totally quasi-ϕ-asymptotically nonexpansive if F(T)Φ, and there exist nonnegative real sequences { v n }, { μ n } with v n , μ n 0 (as n) and a strictly increasing continuous function ζ: R + R + with ζ(0)=0 such that

    ϕ ( p , T ( P T ) n 1 x ) ϕ(p,x)+ v n ζ [ ϕ ( p , x ) ] + μ n ,xD,n1,pF(T).
    (1.9)

Remark 1.3 From the definitions, it is obvious that a quasi-ϕ-nonexpansive nonself mapping is a quasi-ϕ-asymptotically nonexpansive nonself mapping, and a quasi-ϕ-asymptotically nonexpansive nonself mapping is a totally quasi-ϕ-asymptotically nonexpansive nonself mapping, but the converse is not true.

Next, we present an example of a quasi-ϕ-nonexpansive nonself mapping.

Example 1.1 (see [2])

Let H be a real Hilbert space, D be a nonempty closed and convex subset of H and f:D×DR be a bifunction satisfying the conditions: (A1) f(x,x)=0, xD; (A2) f(x,y)+f(y,x)0, x,yD; (A3) for each x,y,zD, lim t 0 f(tz+(1t)x,y)f(x,y); (A4) for each given xD, the function yf(x,y) is convex and lower semicontinuous. The so-called equilibrium problem for f is to find an x D such that f( x ,y)0, yD. The set of its solutions is denoted by EP(f).

Let r>0, xH and define a mapping T r :DDH as follows:

T r (x)= { z D , f ( z , y ) + 1 r y z , z x 0 , y D } ,xDH,
(1.10)

then (1) T r is single-valued, and so z= T r (x); (2) T r is a relatively nonexpansive nonself mapping, therefore it is a closed quasi-ϕ-nonexpansive nonself mapping; (3) F( T r )=EP(f) and F( T r ) is a nonempty and closed convex subset of D; (4) T r :DD is nonexpansive. Since F( T r ) is nonempty, and so it is a quasi-ϕ-nonexpansive nonself mapping from D to H, where ϕ(x,y)= x y 2 , x,yH.

Now, we give an example of a totally quasi-ϕ-asymptotically nonexpansive nonself mapping.

Example 1.2 (see [2])

Let D be a unit ball in a real Hilbert space l 2 , and let T:D l 2 be a nonself mapping defined by

T:( x 1 , x 2 ,) ( 0 , x 1 2 , a 2 x 2 , a 3 x 3 , ) l 2 ,( x 1 , x 2 ,)D,

where { a i } is a sequence in (0,1) such that i = 2 a i = 1 2 .

It is proved in Goebal and Kirk [3] that

  1. (i)

    TxTy2xy, x,yD;

  2. (ii)

    T n x T n y2 j = 2 n a j , x,yD, n2.

Let k 1 =2, k n =2 j = 2 n a j , n2, then lim n k n =1. Letting ν n = k n 1 (n2), ζ(t)=t (t0) and { μ n } be a nonnegative real sequence with μ n 0, then from (i) and (ii) we have

T n x T n y 2 x y 2 + ν n ζ ( x y 2 ) + μ n ,x,yD.

Since D is a unit ball in a real Hilbert space l 2 , it follows from Remark 1.2 that ϕ(x,y)= x y 2 , x,yD. The above inequality can be written as

ϕ ( T n x , T n y ) ϕ(x,y)+ ν n ζ ( ϕ ( x , y ) ) + μ n ,x,yD.

Again, since 0D and 0F(T), this implies that F(T)Φ. From above inequality, we get that

ϕ ( p , T ( P T ) n 1 x ) ϕ(p,x)+ ν n ζ ( ϕ ( p , x ) ) + μ n ,pF(T),xD,

where P is the nonexpansive retraction. This shows that the mapping T defined as above is a totally quasi-ϕ-asymptotically nonexpansive nonself mapping.

Lemma 1.2 (see [4])

Let X be a uniformly convex and smooth Banach space, and let { x n } and { y n } be two sequences of X such that { x n } and { y n } are bounded; if ϕ( x n , y n )0, then x n y n 0.

Lemma 1.3 Let X be a smooth, strictly convex and reflexive Banach space and D be a nonempty closed convex subset of X. Let T:DX be a totally quasi-ϕ-asymptotically nonexpansive nonself mapping with μ 1 =0, then F(T) is a closed and convex subset of D.

Proof Let { x n } be a sequence in F(T) such that x n p. Since T is a totally quasi-ϕ-asymptotically nonexpansive nonself mapping, we have

ϕ( x n ,Tp)ϕ( x n ,p)+ v 1 ζ ( ϕ ( x n , T p ) )

for all nN. Therefore,

ϕ(p,Tp)= lim n ϕ( x n ,Tp) lim n ϕ( x n ,p)+ v 1 ζ ( ϕ ( x n , p ) ) =ϕ(p,p)=0.

By Lemma 1.2, we obtain Tp=p. So, we have pF(T). This implies F(T) is closed.

Let p,qF(T) and t(0,1), and put w=tp+(1t)q. We prove that wF(T). Indeed, in view of the definition of ϕ, let { u n } be a sequence generated by u 1 =Tw, u 2 =T(PT)w, u 3 =T ( P T ) 2 w,, u n =T ( P T ) n 1 w=TP u n 1 , we have

ϕ ( w , u n ) = w 2 2 w , J u n + u n 2 = w 2 2 t p + ( 1 t ) q , J u n + u n 2 = w 2 + t ϕ ( p , u n ) + ( 1 t ) ϕ ( q , u n ) t p 2 ( 1 t ) q 2 .
(1.11)

Since

(1.12)

Substituting (1.10) into (1.11) and simplifying it, we have

ϕ(w, u n )t v n ζ [ ϕ ( p , w ) ] +(1t) v n ζ [ ϕ ( q , w ) ] + μ n 0(as n).

Hence, we have u n w. This implies that u n + 1 w. Since TP is closed and u n + 1 =T ( P T ) n w=TP u n , we have TPw=w. Since wC, and so Tw=w, i.e., wF(T). This implies F(T) is convex. This completes the proof of Lemma 1.3. □

Definition 1.2 (1) (see [5]) A countable family of nonself mappings { T i }:DX is said to be uniformly quasi-ϕ-asymptotically nonexpansive if i = 1 F( T i )Φ, and there exist nonnegative real sequences k n [1,+), k n 1, such that for each i1,

ϕ ( p , T i ( P T i ) n 1 x ) k n ϕ(p,x),xD,n1,pF(T).
(1.13)
  1. (2)

    A countable family of nonself mappings { T i }:DX is said to be uniformly totally quasi-ϕ-asymptotically nonexpansive if i = 1 F( T i )Φ, and there exist nonnegative real sequences { v n }, { μ n } with v n , μ n 0 (as n) and a strictly increasing continuous function ζ: R + R + with ζ(0)=0 such that for each i1,

    ϕ ( p , T i ( P T i ) n 1 x ) ϕ(p,x)+ v n ζ [ ϕ ( p , x ) ] + μ n ,xD,n1,pF(T).
    (1.14)

(3) (see [5]) A nonself mapping T:DX is said to be uniformly L-Lipschitz continuous if there exists a constant L>0 such that

T ( P T ) n 1 x T ( P T ) n 1 y Lxy,x,yD,n1.
(1.15)

Considering the strong and weak convergence of asymptotically nonexpansive self or nonself mappings, relatively nonexpansive, quasi-ϕ-nonexpansive and quasi-ϕ-asymptotically nonexpansive self or nonself mappings have been considered extensively by several authors in the setting of Hilbert or Banach spaces (see [419]).

The purpose of this paper is to modify the Halpern and Mann-type iteration algorithm for a family of totally quasi-ϕ-asymptotically nonexpansive nonself mappings to have the strong convergence under a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang et al. [5, 6, 20, 21], Su et al. [16], Kiziltunc et al. [10], Yildirim et al. [11], Yang et al. [22], Wang [18, 19], Pathak et al. [14], Thianwan [17], Qin et al. [15], Hao et al. [9], Guo et al. [7], Nilsrakoo et al. [13] and others.

2 Main results

Theorem 2.1 Let X be a real uniformly smooth and uniformly convex Banach space, D be a nonempty closed convex subset of X. Let { T i }:DX be a family of uniformly totally quasi-ϕ-asymptotically nonexpansive nonself mappings with sequences { v n }, { μ n }, with v n , μ n 0 (as n), and a strictly increasing continuous function ζ: R + R + with ζ(0)=0 such that for each i1, { T i }:DX is uniformly L i -Lipschitz continuous. Let { α n } be a sequence in [0,1] and { β n } be a sequence in (0,1) satisfying the following conditions:

  1. (i)

    lim n α n =0;

  2. (ii)

    0< lim n inf β n lim n sup β n <1.

Let x n be a sequence generated by

{ x 1 X  is arbitrary ; D 1 = D , y n , i = J 1 [ α n J x 1 + ( 1 α n ) ( β n J x n + ( 1 β n ) J T i ( P T i ) n 1 x n ) ] ( i 1 ) , D n + 1 = { z D n : sup i 1 ϕ ( z , y n , i ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π D n + 1 x 1 ( n = 1 , 2 , ) ,
(2.1)

where ξ n = v n sup p F ζ(ϕ(p, x n ))+ μ n , F= i = 1 F( T i ), Π D n + 1 is the generalized projection of X onto D n + 1 . If is nonempty, then { x n } converges strongly to Π F x 1 .

Proof (I) First, we prove that and D n are closed and convex subsets in D.

In fact, by Lemma 1.3 for each i1, F( T i ) is closed and convex in D. Therefore, is a closed and convex subset in D. By the assumption that D 1 =D is closed and convex, suppose that D n is closed and convex for some n1. In view of the definition of ϕ, we have

D n + 1 = { z D n : sup i 1 ϕ ( z , y n , i ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } = i 1 { z D : sup i 1 ϕ ( z , y n , i ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } D n = i 1 { z D : 2 α n z , J x 1 + 2 ( 1 α n ) z , J x n 2 z , J y n , i α n x 1 2 + ( 1 α n ) x n 2 y n , i 2 } D n .

This shows that D n + 1 is closed and convex. The conclusions are proved.

  1. (II)

    Next, we prove that F D n for all n1.

In fact, it is obvious that F D 1 . Suppose that F D n .

Let w n , i = J 1 ( β n J x n +(1 β n )J T i ( P T i ) n 1 x n ). Hence for any uF D n , by (1.5), we have

ϕ ( u , y n , i ) = ϕ ( u , J 1 ( α n J x 1 + ( 1 α n ) J w n , i ) ) α n ϕ ( u , x 1 ) + ( 1 α n ) ϕ ( u , w n , i )
(2.2)

and

ϕ ( u , w n , i ) = ϕ ( u , J 1 ( β n J x n + ( 1 β n ) J T i ( P T i ) n 1 x n ) ) β n ϕ ( u , x n ) + ( 1 β n ) ϕ ( u , T i ( P T i ) n 1 x n ) β n ϕ ( u , x n ) + ( 1 β n ) { ϕ ( u , x n ) + v n ζ [ ϕ ( u , x n ) ] + μ n } = ϕ ( u , x n ) + ( 1 β n ) v n ζ [ ϕ ( u , x n ) ] + ( 1 β n ) μ n .
(2.3)

Therefore, we have

sup i 1 ϕ ( u , y n , i ) α n ϕ ( u , x 1 ) + ( 1 α n ) [ ϕ ( u , x n ) + ( 1 β n ) v n ζ [ ϕ ( u , x n ) ] + ( 1 β n ) μ n ] α n ϕ ( u , x 1 ) + ( 1 α n ) ϕ ( u , x n ) + v n sup p F ζ [ ϕ ( p , x n ) ] = α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n ,
(2.4)

where ξ n = v n sup p F ζ(ϕ(p, x n ))+ μ n . This shows that uF D n + 1 and so F D n . The conclusion is proved.

  1. (III)

    Now, we prove that { x n } converges strongly to some point p .

Since x n = Π D n x 1 , from Lemma 1.1(c), we have

x n y,J x 1 J x n 0,y D n .

Again since F D n , we have

x n u,J x 1 J x n 0,uF.

It follows from Lemma 1.1(b) that for each uF and for each n1,

ϕ( x n , x 1 )=ϕ( Π D n x 1 , x 1 )ϕ(u, x 1 )ϕ(u, x n )ϕ(u, x 1 ).
(2.5)

Therefore, {ϕ( x n , x 1 )} is bounded, and so is { x n }. Since x n = Π D n x 1 and x n + 1 = Π D n + 1 x 1 D n + 1 D n , we have ϕ( x n , x 1 )ϕ( x n + 1 , x 1 ). This implies that {ϕ( x n , x 1 )} is nondecreasing. Hence, lim n ϕ( x n , x 1 ) exists.

By the construction of { D n }, for any mn, we have D m D n and x m = Π D m x 1 D n . This shows that

ϕ( x m , x n )=ϕ( x m , Π D n x 1 )ϕ( x m , x 1 )ϕ( x n , x 1 )0(as n).

It follows from Lemma 1.2 that lim n x m x n =0. Hence, { x n } is a Cauchy sequence in D. Since D is complete, without loss of generality, we can assume that lim n x n = p (some point in D).

By the assumption, it is easy to see that

lim n ξ n = lim n [ v n sup p F ζ ( ϕ ( p , x n ) ) + μ n ] =0.
(2.6)
  1. (IV)

    Now, we prove that p F.

Since x n + 1 D n + 1 , from (2.1) and (2.6), we have

sup i 1 ϕ( x n + 1 , y n , i ) α n ϕ( x n + 1 , x 1 )+(1 α n )ϕ( x n + 1 , x n )+ ξ n 0.
(2.7)

Since x n p , it follows from (2.7) and Lemma 1.2 that

y n , i p .
(2.8)

Since { x n } is bounded and { T i } is a family of uniformly total quasi-ϕ-asymptotically nonexpansive nonself mappings, we have

ϕ ( p , T i ( P T i ) n 1 x n ) ϕ(p, x n )+ v n ζ [ ϕ ( p , x n ) ] + μ n ,xD,n,i1,pF( T i ).

This implies that { T i ( P T i ) n 1 x n } is uniformly bounded.

Since

w n , i = J 1 ( β n J x n + ( 1 β n ) J T i ( P T i ) n 1 x n ) β n x n + ( 1 β n ) T i ( P T i ) n 1 x n x n + T i ( P T i ) n 1 x n ,

this implies that { w n , i } is also uniformly bounded.

In view of α n 0, from (2.1), we have that

lim n J y n , i J w n , i = lim n α n J x 1 J w n , i =0
(2.9)

for each i1.

Since J 1 is uniformly continuous on each bounded subset of X , it follows from (2.8) and (2.9) that

w n , i p
(2.10)

for each i1. Since J is uniformly continuous on each bounded subset of X, we have

0 = lim n J w n , i J P = lim n ( β n J x n + ( 1 β n ) J T i ( P T i ) n 1 x n ) J p = lim n β n ( J x n J p ) + ( 1 β n ) ( J T i ( P T i ) n 1 x n J p ) = lim n ( 1 β n ) J T i ( P T i ) n 1 x n J p .
(2.11)

By condition (ii), we have that

lim n J T i ( P T i ) n 1 x n J P =0.

Since J is uniformly continuous, this shows that

lim n T i ( P T i ) n 1 x n = P
(2.12)

for each i1. Again, by the assumption that { T i }:DX is uniformly L i -Lipschitz continuous for each i1, thus we have

(2.13)

for each i1.

We get lim n T i ( P T i ) n x n T i ( P T i ) n 1 x n =0. Since lim n T i ( P T i ) n 1 x n = P and lim n x n = p , we have lim n T i P T i ( P T i ) n 1 x n = p .

In view of the continuity of T i P, it yields that T i P p = p . Since p C, it implies that T i p = p . By the arbitrariness of i1, we have p F.

  1. (V)

    Finally, we prove that p = Π F x 1 and so x n Π F x 1 = p .

Let w= Π F x 1 . Since wF D n and x n = Π D n x 1 , we have ϕ( x n , x 1 )ϕ(w, x 1 ). This implies that

ϕ ( p , x 1 ) = lim n ϕ( x n , x 1 )ϕ(w, x 1 ),
(2.14)

which yields that p =w= Π F x 1 . Therefore, x n Π F x 1 . The proof of Theorem 3.1 is completed. □

By Remark 1.3, the following corollary is obtained.

Corollary 2.1 Let X, D, { α n }, { β n } be the same as in Theorem  2.1. Let { T i }:DX be a family of uniformly quasi-ϕ-asymptotically nonexpansive nonself mappings with the sequence k n [1,+), k n 1, such that for each i1, { T i }:DX is uniformly L i -Lipschitz continuous.

Let x n be a sequence generated by

{ x 1 X  is arbitrary ; D 1 = D , y n , i = J 1 [ α n J x 1 + ( 1 α n ) ( β n J x n + ( 1 β n ) J T i ( P T i ) n 1 x n ) ] ( i 1 ) , D n + 1 = { z D n : sup i 1 ϕ ( z , y n , i ) α n ϕ ( z , x 1 ) + ( 1 α n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π D n + 1 x 1 ( n = 1 , 2 , ) ,
(2.15)

where ξ n =( k n 1) sup p F ϕ(p, x n ), F= i = 1 F( T i ), Π D n + 1 is the generalized projection of X onto D n + 1 . If is nonempty, then { x n } converges strongly to Π F x 1 .

3 Application

In this section we utilize Corollary 2.1 to study a modified Halpern iterative algorithm for a system of equilibrium problems. We have the following result.

Theorem 3.1 Let H be a real Hilbert space, D be a nonempty closed and convex subset of H. { α n }, ( β n ) be the same as in Theorem  2.1. Let { f i }:D×DR be a countable family of bifunctions satisfying conditions (A1)-(A4) as given in Example  1.1. Let { T r , i :DDH} be the family of mappings defined by (1.9), i.e.,

T r , i (x)= { z D , f i ( z , y ) + 1 r y z , z x 0 , y D } ,xDH.

Let { x n } be the sequence generated by

{ x 1 D  is arbitrary ; D 1 = D , f i ( u n , i , y ) + 1 r y u n , i , u n , i x n 0 , y D , r > 0 , i 1 , y n , i = α n x 1 + ( 1 α n ) [ β n x n + ( 1 β n ) u n , i ] , D n + 1 = { z D n : sup i 1 z y n , i 2 α n z x 1 2 + ( 1 α n ) z x n 2 } , x n + 1 = Π D n + 1 x 1 ( n = 1 , 2 , ) .
(3.1)

If F= i = 1 F( T r , i )Φ, then { x n } converges strongly to Π F x 1 , which is a common solution of the system of equilibrium problems for f.

Proof In Example 1.1, we have pointed out that u n , i = T r , i ( x n ), F( T r , i )=EP( f i ) is nonempty and convex for all i1, T r , i is a countable family of quasi-ϕ-nonexpansive nonself mappings. Since F( T r , i ) is nonempty, so T r , i is a countable family of quasi-ϕ-nonexpansive mappings and for all i1, T r , i is a uniformly 1-Lipschitzian mapping. Hence, (3.1) can be rewritten as follows:

{ x 1 H  is arbitrary ; D 1 = D , y n , i = α n x 1 + ( 1 α n ) [ β n x n + ( 1 β n ) T r , i x n ] , D n + 1 = { z D n : sup i 1 z y n , i 2 α n z x 1 2 + ( 1 α n ) z x n 2 } , x n + 1 = Π D n + 1 x 1 ( n = 1 , 2 , ) .
(3.2)

Therefore, the conclusion of Theorem 3.1 can be obtained from Corollary 2.1. □

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Acknowledgements

The author is very grateful to both reviewers for careful reading of this paper and for their comments.

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Yi, L. Strong convergence theorems for a countable family of totally quasi-ϕ-asymptotically nonexpansive nonself mappings in Banach spaces with applications. J Inequal Appl 2012, 268 (2012). https://doi.org/10.1186/1029-242X-2012-268

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Keywords

  • generalized projection
  • quasi-ϕ-asymptotically nonexpansive nonself mapping
  • totally quasi-ϕ-asymptotically nonexpansive nonself mapping
  • iterative sequence
  • nonexpansive retraction