CONVERGENCE OF CR-ITERATION TO COMMON FIXED POINTS OF THREE G-NONEXPANSIVE MAPPINGS IN BANACH SPACES WITH GRAPHS

1859-3100 Tập 16, Số 3 (2019): 76-90 Vol. 16, No. 3 (2019): 76-90 Email: tapchikhoahoc@hcmue.edu.vn; Website: http://tckh.hcmue.edu.vn

CONVERGENCE OF CR-ITERATION

TO COMMON FIXED POINTS OF THREE G-NONEXPANSIVE MAPPINGS IN BANACH SPACES WITH GRAPHS

Nguyen Trung Hieu, Pham Thi Ngoc Mai Dong Thap University

Corresponding author: Nguyen Trung Hieu – Email: ngtrunghieu@dthu.edu.vn

Received: 30/01/2019; Revised: 20/3/2019; Accepted: 21/3/2019

ABSTRACT

The research introduces CR-iteration process and establishes some results about the weak and strong convergence of CR-iteration process to common fixed points of three G-nonexpansive mappings in uniformly convex Banach spaces with graphs. In addition, a numerical example is provided to illustrate for the convergence of CR-iteration process to common fixed points three G- nonexpansive mappings.

Keywords: G-nonexpansive mapping, CR-iteration process, Banach space with graph.

1. Introduction

In fixed point theory, nonexpansive mappings play an important role in studying the existence and approximation of fixed points of nonlinear mappings in Banach spaces. In recent times, the nonexpansive mappings were extended and generalized in many various ways. In 2012, Aleomraninejad, Rezapour, and Shahzad introduced the notion of G- nonexpansive mappings in metric spaces with directed graphs and stated the convergence for this class mapping in complete metric spaces with directed graphs. In 2015, Tiammee, Kaewkhao, and Suantai proved Browder 's convergence theorem for G-nonexpansive mappings and studied the convergence of Halpern iteration to projecting of initial point onto the set of fixed points of G-nonexpansive mappings in Hilbert spaces with directed graphs. In 2016, Tripak proved the convergence of Ishikawa iteration to some common fixed points of two G-nonexpansive mappings in Banach spaces with directed graphs. In 2018, Kangtunyakarn generalized the results in (Tripak, 2016) to proving Halpern iteration for a finite family of G-nonexpansive mappings in Banach spaces with directed graphs. After that, Suparatulatorn, Cholamjiak, and Suantai generalized the results in (Kangtunyakarn, 2018) and proposed the convergence of S-iteration to some common fixed points of two G-nonexpansive mappings in Banach spaces with directed graphs. In 2018, Sridarat, Suparatulatorn, Suantai, and Cho established the convergence of SP- iteration to common fixed points of three G-nonexpansive mappings in uniformly Banach spaces with directed graphs. An interesting work naturally rises is to continue studying the

convergence to common fixed points of G-nonexpansive mappings by some generalized iterations in Banach spaces with directed graphs.

In recent years, several iteration methods were proposed for approximating fixed points of nonexpansive mappings. In 2012, Chugh, Kumar, and Kumar introduced CR iterative scheme and studied the convergence of this iteration to fixed points of quasi- contraction in Banach spaces. In addition, the authors also showed that CR iterative scheme is faster than Picard, Mann, Ishikawa, Agarwal, Noor and SP iterative schemes by some numerical examples. However, the constructing of CR-type iteration process and studying the convergence of them to common fixed points of G-nonexpansive mappings in Banach spaces with directed graphs has not been considered yet. Thus, in this study, we introduce CR-iteration process and establish some convergence results of CR-iteration to common fixed points of three G-nonexpansive mappings in uniformly Banach spaces with directed graphs. First, we recall some notions and lemmas which will be useful in what follows.

Let C be a nonempty subset of a real normed space X. Let G = ( ( ), ( ))V G E G be a directed graph, where V G( ) is a set of vertices of graph G such that V G( )=C, E G( ) is a set of its edges such that ( , )x x Î E G( ) for all x Î C, and G has no parallel edges.

Definition 1.1.

[(Tripak, 2016), Definition 2.4] Let X be a normed space, C be a nonempty subset of X, and G = ( ( ), ( ))V G E G be a directed graph such that ( )V G =C. Then G is said to be transitive if for all x y z, , ÎV G( ) such that ( , ),( , )x y y z Î E G( ), _then( , )x z Î E G( ).

Definition 1.2.

[(Tiammee, Kaewkhao, and Suantai, 2015), p.4] Let X be a normed space, C be a nonempty subset of X, and G = ( ( ), ( ))V G E G be a directed graph such that ( )V G = C. Then C is said to have the property G if for any sequence { }x_n in C with (x x_n, _n+1)Î E G( ) for all n Î ¥^* and {x_n}converges weakly to x Î C, there exists a subsequence {x_{n k( )}} of {x_n} such that (x_{n k( )}, )x Î E G( ) for all k Î ¥^*.

Definition 1.3.

[(Shahzad & Al-Dubiban, 2006), p.534] Let X be a normed space, C be a nonempty subset of X, and T C: ®C be a mapping. Then T is called semicompact if for any bounded sequence { }x_n in C with lim || _{n n} || 0,

n x T x

® ¥ - = there exists a subsequence {x_{n k( )}} of {x_n}such that {x_{n k( )}} converges to xÎ C.

Definition 1.4.

[(Tiammee et al., 2015), p.2] Let X be a normed space, C be a nonempty subset of X, ( ( ), ( ))

G = V G E G be a directed graph such that V G( )=C, _and T C: ® C be a mapping.

Then T is called G-nonexpansive if the following conditions hold:

(1) T is edge-preserving, that is, for all ( , )x y Î E G( ), we have (T x T y, )Î E G( ).

(2) ||T x- T y || ||£ x- y||for all ( , )x y Î E G( ).

We denote that F T( )= {xÎ C T x: = x} is the set of fixed points of a mapping

: .

T C ® C The following result shows that the sufficient condition for the closed convex property of the set F T( ) with T is a G-nonexpansive mapping.

Proposition 1.5.

[(Tiammee et al., 2015), Theorem 3.2] Let X be a normed space, C be a nonempty subset of X, G = ( ( ), ( ))V G E G be a directed graph with V G( )= C, E G being convex, C ( ) have the property G, and T C: ® C be a G-nonexpansive mapping such that

( ) ( ) ( ).

F T ´ F T Ì E G Then F T( )is closed and convex.

Definition 1.6.

[(Dozo, 1973), Definition 1.1] Let X be a normed space. Then X is said to satisfy Opial’s condition if for any x Î X and {x_n} converges weakly to ,x we have

lim inf || _n || lim inf || _n ||

n x x n x y

® ¥ - < ® ¥ - for all yÎ X y, ¹ x. Proposition 1.7.

[(Sridarat, Suparatulatorn, Suantai, and Cho, 2018), Proposition 3.5] Let X be a Banach space satisfying the Opial’s condition, C be a nonempty subset of X,

( ( ), ( ))

G = V G E G be a directed graph such that V G( )=C, C have the property G, :

T C ® C be a G-nonexpansive, {x_n} be a sequence in C such that {x_n} converges weakly to pÎ C x x, ( ,_{n n+1)})Î E G( ) and lim || _{n n} || 0.

n x T x

® ¥ - = Then T p= p. Definition 1.8.

[(Sridarat et al., 2018), p.13] Let X be a normed space, C be a nonempty subset of X and T T T₁, ₂, ₃:C ® C be three mappings. ThenT T T₁, ₂, ₃ are said to satisfy the condition (C) if there exists a nondecreasing function f : [0, )¥ ® [0, )¥ with f(0)= 0, ( )f r > 0 for all

0

r > and for all x > 0 such that

{

1 2 3

}

max ||x- T x|| , ||x- T x ||, ||x- T x| | ³ f d x F( ( , )),

where F = F T( ₁)ÇF T( ₂)ÇF T( ₃) and d x F( , )= inf{||x- y||:y Î F}.

Lemma 1.9.

[(Schu, 1991), Lemma 1.3] Let X be a uniformly convex Banach space, {a_n} be a sequence in [ ,1d - d] with dÎ (0, 1)and { }, { }x_n y_n be two sequences in X such that

lim sup || _n || , lim sup || _n ||

n n

x r y r

® ¥ ® ¥

£ £ and lim || _{n n} (1 _n) _n ||

n a x a y r

® ¥ + - = withr ³ 0.

Then lim || _{n n} || 0.

n x y

® ¥ - =

2. Main results

By combining CR iterative scheme in (Chugh, Kumar, and Kumar, 2012) with SP- iteration process in (Sridarat et al., 2018), we introduce CR-iteration process {x_n} for three G-nonexpansive mappings as follows:

1 ,

x Î C

1

1 2

1 3

(1 )

(1 )

(1 )

n n n n n

n n n n n

n n n n n

z x T x

y T x T z

x y T y

g g

b b

a a

+

ìï = - +

ïïï = - +

íïï = - +

ïïî

for all nÎ ¥^*, (2.1)

where {a_n}, {b_n}, {g_n} are three sequences in [0,1], C is a nonempty closed convex subset of a Banach space X and T T T₁, ₂, ₃:C ® C are three G-nonexpansive mappings.

Next, we establish some properties of CR-iteration.

Proposition 2.1.

Let X be a normed space, C be a nonempty convex subset of X, G = ( ( ), ( ))V G E G be a directed graph which is transitive with V G( )=C E G, ( ) being convex, T T T₁, ₂, ₃:C ® C

be three G-nonexpansive mappings ,{ }x_n be a sequence defined by recursion (2.1) satisfying ( , ),x p₁ ( ,p x₁)Î E G( ) with pÎ F. Then (x p_n, ), ( , ),y p_n ( , ),z_n p ( ,p x_n),( ,p y_n), ( ,p z_n),(x y_n, _n),(x z_n, _n),(x_n,x_n+1)Î E G( ) for all n Î ¥^*.

Proof.

We will prove ( ,p x_n), ( ,p y_n),( ,p z_n)Î E G( )by using mathematical introduction. First, we prove that ( ,p y₁), ( , )p z₁ Î E G( ). Indeed, since pÎ F, we have pÎ F T( )_i and hence T pi = p for all i = 1, 3. Since ( ,p x₁)Î E G( ) and T₁ is edge-preserving, we get ( ,p T x1 1)Î E G( ). Moreover,

1 1 1 1 1 1 1 1 1 1 1

( ,p z )= ( , (1p - g)x + gT x )= (1- g)( ,p x )+ g( ,p T x ). (2.2) Since ( ,p x₁) and ( ,p T x_{1 1})Î E G( ),from (2.2), we get ( ,p z₁)Î E G( ). Combining this with the edge-preserving property of T₂, we obtain ( ,p T z_{2 1})Î E G( ). Furthermore,

1 1 1 1 1 2 1 1 1 1 1 2 1

( ,p y )= ( , (1p - b)T x + bT z )= (1- b )( ,p T x)+ b( ,p T z ). (2.3)

Then, combining (2.3) with( ,p T x_{1 1}) and( ,p T z_{2 1})Î E G( ), we get that ( ,p y₁)Î E G( ).

Next, suppose that ( ,p x_k)Î E G( ) for all k³ 1. We will prove that ( ,p x_k+1), ( ,p y_k+1), ( ,p z_k+1)Î E G( ). Indeed, since T₁ is edge-preserving, we obtain ( ,p T x_{1 k})Î E G( ). Moreover,

1 1

( ,p z_k)= ( , (1p - g_k)x_k + g_kT x_k)= (1- g_k)( ,p x_k)+ g_k( ,p T x_k). (2.4) Thus, combining (2.4) with ( ,p x_k) and ( ,p T x_{1 k})Î E G( ), we have ( ,p z_k)Î E G( ).

Then, since T₂ is edge-preserving, we obtain ( ,p T z_{2 k})Î E G( ). Furthermore,

1 2 1 2

( ,p y_k)= ( , (1p - b_k)T x_k + b_kT z_k)= (1- b_k)( ,p T x_k)+ b_k( ,p T z_k). (2.5) By combining (2.5) with ( ,p T x_{1 k}) and ( ,p T z_{2 k})Î E G( ),we obtain ( ,p y_k)Î E G( ).

Since T₃ is edge-preserving, we get ( ,p T y_{3 k})Î E G( ). We also have

1 3 3

( ,p x_k+ )=( ,(1p - a_k)y_k +a_kT y_k)= (1- a_k)( ,p y_k)+a_k( ,p T y_k). (2.6) Thus, from ( ,p y_k), ( ,p T y_{3 k})Î E G( ) and (2.6), we obtain ( ,p x_k+1)Î E G( ). Combining this with the edge-preserving property of T₁, we conclude that ( ,p T x_{1 k+1})Î E G( ).

Furthermore,

1 1 1 1 1 1 1 1 1 1 1

( ,p z_k+ )=( ,(1p - g_k+ )x_k+ + g_k+T x_k+)=(1- g_k+ )( ,p x_k+ )+g_k+ ( ,p T x_k+ ). (2.7) It follows from ( ,p x_k+1),( ,p T x_{1 k+1})Î E G( ) and (2.7), we obtain( ,p z_k+1)Î E G( ). Since T2 is edge-preserving, we get ( ,p T z_{2 k+1})Î E G( ). Moreover,

1 1 1 1 1 2 1 1 1 1 1 2 1

( ,p y_k+ )= ( ,(1p - b_k+ )T x_k+ +b_k+T z_k+ )= (1- b_k+ )( ,p T x_k+ )+ b_k+ ( ,p T z_k+ ). (2.8) Then, by combining (2.8) with ( ,p T x_{1 k+1}) and ( ,p T z_{2 k+1})Î E G( ), we obtain ( ,p y_k+1)Î E G( ). Therefore, by induction, we conclude that ( ,p x_n), ( ,p y_n), ( ,p z_n)Î E G( ) for all n Î ¥^*. Next, by using similar arguments as in the above proofs, we also see that ( , ), ( , ),x p_n y p_n (z_n, )p Î E G( )for all n Î ¥^*.

Finally, we shall prove that ( ,x y_{n n}),( ,x z_{n n}),( ,x x_{n n+1})Î E G( ). In fact, by using the transitive property of G and ( , ),x p_n ( ,p y_n),( , ),x p_n ( ,p z_n),( , ),x p_n ( ,p x_n+1)Î E G( ), we conclude that ( ,x y_{n n}), ( ,x z_{n n}), ( ,x x_{n n+1})Î E G( ). 

Proposition 2.2.

Let X be a normed space, C be a nonempty closed convex subset of X, ( ( ), ( ))

G = V G E G be a directed graph which is transitive with V G( )=C E G, ( ) being convex, T T T₁, ₂, ₃:C ® C be three G-nonexpansive mappings such thatF ¹ Æ { }, x_n be a sequence defined by recursion (2.1) satisfying ( , ),x p₁

( ,p x1)Î E G( ) with pÎ F. Then { }x_n is bounded and lim || _n ||

n x p

® ¥ - exists.

Proof.

It follows from pÎ F p x, ( , ₁), ( , )x p₁ Î E G( ) and Proposition 2.1, we conclude that ( , ), ( , ), ( , )x p_n y p_n z p_n Î E G( ). Since ( , )x p_n Î E G T p( ), ₁ = p and T₁ is a G-nonexpansive mapping, we have

1 1

||z_n - p|| (1£ - g_n) ||x_n- p||+g_n||T x_n- T p||

£ (1- g_n) ||x_n - p||+g_n ||x_n- p||

||= x_n- p|| .

(2.9) Since ( , ), ( , )x p_n z p_n Î E G T p( ), ₁ =T p₂ = p and T T₁, ₂ are two G-nonexpansive mappings, by using (2.9), we obtain

1 1 2 2

||y_n- p|| (1£ - b_n) ||T x_n- T p||+b_n ||T z_n - T p||

^{£ (1- bn) ||xn- p| |+bn ||zn- p||}

£ (1- b_n) ||x_n- p||+b_n ||x_n - p||

^{=||xn- p|| .} (2.10) Since ( , )y p_n Î E G T p( ), ₃ = p and T₃ is a G-nonexpansive mapping, by using (2.10), we get

1 3 3

||x_n+ - p|| (1£ - a_n) ||y_n- p||+a_n ||T y_n- T p||

£ (1- a_n) ||y_n- p||+a_n ||y_n- p||

£||y_n - p||

||£ x_n- p|| .

(2.11)

It follows from (2.11), we conclude that {x_n} is bounded and lim|| _n ||

n x p

®¥ - exists. 

Proposition 2.3.

Let X be a uniformly convex Banach space, C be a nonempty closed convex subset of X, {a_n}, {b_n}, {g_n}Ì [ , 1e - e] with eÎ (0, 1),G = ( ( ), ( ))V G E G be a directed graph which is transitive with V G( )=C E G, ( ) being convex, T T T₁, ₂, ₃:C ® C be three G-nonexpansive mappings such that F ¹ Æ, { }x_n be a sequence defined by recursion (2.1) satisfying

1 1

( ,p x ), ( , )x p Î E G( )withp Î F. Then

1 2 3

lim || _{n n} || lim || _{n n} || lim || _{n n} || 0.

n x T x n x T x n x T x

® ¥ - = ® ¥ - = ® ¥ - =

Proof.

It follows from pÎ F p x,( , ),( , )₁ x p₁ Î E G( ) and Proposition 2.1, we conclude that ( ,p x_n), ( , ), ( , ), ( , ), ( ,x p_n y p_n z p_n z x_{n n}), ( ,y x_{n n})Î E G( ) for all n Î ¥^*.

Furthermore, from Proposition 2.2, we get that lim || _n ||

n x p

® ¥ - exists. Put lim || _n || .

n x p c

® ¥ - = Then, from (2.9), we obtain lim sup || _n || lim sup || _n || .

n n

z p x p c

® ¥ ® ¥

- £ - = (2.12)

Since ( , )x p_n Î E G( ) and T₁ is a G-nonexpansive mapping, we obtain

1 1 1

||T x_n - p|| ||£ T x_n - T p|| ||£ x_n- p|| . Therefore,

lim sup || 1 _n || lim sup || _n || .

n n

T x p x p c

® ¥ ® ¥

- £ - =

(2.13) Since ( , ), ( , ), ( , )x p_n y p_n z p_n Î E G( ),and T T₁, ₂ are two G-nonexpansive mappings, by (2.10) and (2.11), we conclude that

||x_n+1- p|| ||£ y_n- p|| (1£ - b_n) ||x_n - p||+b_n||z_n- p|| .

This implies that b_n||x_n - p|| ||£ x_n- p||- ||x_n+1- p||+b_n ||z_n- p|| . Therefore,

(

1

)

|| _n || 1 | | _n || || _n || || _n || .

n

x p x p x p z p

b ⁺

- £ - - - + -

(

1

)

1 ||x_n p|| ||x_n p|| ||z_n p|| .

e ⁺

£ - - - + -

By combining this with lim inf(|| _n || || _{n 1} ||) 0,

n

x p x ₊ p

® ¥

- - - = we conclude that

lim inf || _n || .

n

c z p

® ¥

£ - It follows from (2.12), we obtain lim || _n || .

n z p c

® ¥ - = Therefore, lim sup || (1 _n)( _n ) _n( 1 _n ) || lim sup || _n || .

n n

x p T x p z p c

g g

® ¥ ® ¥

- - + - = - =

By combining this with lim sup || _{1 n} || , lim sup || _n || ,

n n

T x p c x p c

® ¥ ® ¥

- £ - £ from Lemma

1.9, we obtain that lim || 1 _{n n} || 0.

n T x x

® ¥ - = (2.14)

Furthermore, since ||z_n- x_n||= g_n ||T x_{1 n} - x_n||, by (2.14), we have lim || _{n n} || 0.

n z x

® ¥ - = (2.15)

Next, from (2.10), we see that

lim sup || _n || lim sup || _n || .

n n

y p x p c

® ¥ ® ¥

- £ - = (2.16)

Since ( , )x p_n Î E G( ) and T₂ is a G-nonexpansive, we get

2 2 2

||T z_n - p|| ||= T z_n- T p|| ||£ z_n- p|| ||£ x_n- p|| . This implies thatlimsup || _2n || limsup || _n ||

n n

T z p x p c

® ¥ ®¥

- £ - = . Moreover, by (2.11), we have

lim inf || _n 1 || lim inf || _n ||

n x ₊ p n y p

® ¥ - £ ® ¥ - .

By combining the above with liminf || _{n 1} || ,

n

x ₊ p c

® ¥

- = we have

lim inf || _n || .

c n y p

£ ® ¥ - By combining this with (2.16), we conclude that lim || _n || .

n y p c

® ¥ - = Thus,

1 2

limsup || (1 _n)( _n ) _n( _n ) || limsup || _n || .

n n

T x p T z p y p c

b b

® ¥ ® ¥

- - + - = - =

By combining this with limsup || _{1 n} || , limsup || _{2 n} || ,

n n

T x p c T z p c

® ¥ ® ¥

- £ - £ byLemma 1.9,

we conclude that

1 2

lim || _{n n} || 0.

n T x T z

® ¥ - = (2.17)

Since ( ,z x_{n n})Î E G( ) and T₂ is a G-nonexpansive mapping, we see that

2 1 1 2 2 2

||x_n- T x_n || ||£ x_n- T x_n||+ ||T x_n - T z_n||+ ||T z_n- T x_n ||

£||x_n- T x_{1 n} ||+ ||T x_{1 n} - T z_{2 n} | |+ ||z_n - x_n || .

By the above inequality, (2.14), (2.15) and (2.17), we conclude that lim || _n 2 _n || 0.

n x T x

® ¥ - = Furthermore, from ||y_n - T x_{1 n} ||= b_n ||T z_{2 n} - T x_{1 n} || and (2.17), we obtain

lim || _n 1 _n || 0.

n y T x

® ¥ - =

(2.18)

Then, from ||y_n- x_n || ||£ y_n- T x_{1 n} ||+ ||T x_{1 n}- x_n||, (2.14) and (2.18), we have

lim || _{n n} || 0.

n y x

® ¥ - =

(2.19) Furthermore, by ( , )y p_n Î E G( ) and T₃ is a G-nonexpansive, we get

3 3 3

||T y_n - p|| ||£ T y_n- T p|| ||£ y_n - p|| ||£ x_n- p|| .

This implies that lim sup || _{3 n} || lim sup || _n || .

n n

T y p x p c

® ¥ ® ¥

- £ - = Then, by Lemma

1.9 and using the following inequality: lim sup || _{3 n} || , lim sup || _n ||

n n

T y p c y p c

® ¥ ® ¥

- £ - £ and

3 1

lim sup || (1 _n)( _n ) _n( _n ) || lim sup || _n || ,

n n

y p T y p x p c

a a ₊

® ¥ ® ¥

- - + - = - =

we conclude that lim || 3 _{n n} || 0.

n T y y

® ¥ - = (2.20) We also have ||x_n+1- x_n || || (1= - a_n)y_n+ a_nT y_{3 n}- x_n || ||£ y_n- x_n ||+a_n ||T y_{3 n} - y_n || . This implies that ||x_n+1- x_n || ||£ y_n - x_n ||+(1- e) ||T y_{3 n} - y_n || .Therefore, from the above inequality, (2.19) and (2.20), we obtain

lim || _n 1 _n || 0.

n x ₊ x

® ¥ - = (2.21)

Then, by the following inequality ||x_n+1-T y_3n|| ||£ x_n+1- x_n||+||x_n- y_n||+||y_n-T y_3n||, (2.19), (2.20) and (2.21), we have

1 3

lim || _{n n} || 0.

n x ₊ T y

® ¥ - =

(2.22) Since ( ,y x_{n n})Î E G( ) and T₃ is a G-nonexpansive, we get

3 1 1 3 3 3

||x_n- T x_n|| ||£ x_n - x_n+ ||+||x_n+ - T y_n ||+||T y_n - T x_n||

£||x_n - x_n+1 ||+ ||x_n+1- T y_{3 n} ||+ ||y_n - x_n || . (2.23)

Therefore, by (2.19), (2.21), (2.22), (2.23), we conclude that lim || _{n 3 n}|| 0

n x T x

®¥ - =

 The following result is a sufficient condition for the weak convergence of iteration process (2.1) to common fixed points of three G-nonexpansive mappings in uniformly convex Banach spaces.

Theorem 2.4.

Let X be a uniformly convex Banach space satisfying the Opial’s condition, C be a nonempty closed convex subset of X, {a_n}, {b_n}, {g_n}Ì [ , 1e - e] with e Î (0, 1),

( ( ), ( ))

G = V G E G be a directed graph which is transitive with V G( )=C E G, ( ) being convex, C have the property G, T T T₁, ₂, ₃:C ® C be three G-nonexpansive mappings such

that F ¹ Æ, { }x_n be a sequence defined by recursion (2.1) satisfying ( ,p x₁), ( , )x p₁ Î E G( ) withp Î F. Then { }x_n converges weakly to qÎ F.

Proof.

Since X is a uniformly convex Banach space, we see that X is a reflexive Banach space. Moreover, by Proposition 2.2, we get that {x_n} is bounded. Therefore, there exists a subsequence {x_{n i( )}} of {x_n}such that {x_{n i( )}}converges weakly to qÎ C. Then, by Proposition 2.3, we conclude that

( ) 1 ( ) ( ) 2 ( ) ( ) 3 ( )

lim || _{n i n i} || lim || _{n i n i} || lim || _{n i n i} || 0.

i x T x i x T x i x T x

® ¥ - = ® ¥ - = ® ¥ - =

Thus, from the above and by Proposition 1.7, we conclude that T q₁ =T q₂ =T q₃ = q and hence qÎ F = F T( )₁ ÇF T( )₂ ÇF T( ).₃

Suppose that {x_n} doesnot converges weakly to q. Then, there exists a subsequence {x_{n k}( )} of {x_n} such that {x_{n k( )}} converges weakly to q₁Î C with q ¹ q₁. By using similar arguments as in the above proofs, from Proposition 1.7, we conclude that q₁ Î F. Furthermore, from Proposition 2.2, we get that lim || _n ||

n x q

® ¥ - and lim || _{n 1}||

n x q

® ¥ - exist.

Then, by the Opial’s condition, we see that

( ) ( ) 1 1

lim || _n || lim inf || _{n j} || lim || _{n j} || lim || _n ||

n x q j x q j x q n x q

® ¥ - = ® ¥ - < ® ¥ - = ® ¥ -

lim inf || _{n k( ) 1}|| lim || _{n k( )} || lim || _n ||,

k x q k x q n x q

® ¥ ® ¥ ® ¥

= - < - = -

which is a contradiction. This implies that { }x_n converges weakly to qÎ F.  Next, we prove some strong convergence results of iteration (2.1) to common fixed points of three G-nonexpansive mappings in uniformly convex Banach spaces.

Theorem 2.5.

Let X be a uniformly convex Banach space, C be a nonempty closed convex subset of X, {a_n}, {b_n}, {g_n}Ì [ , 1e - e] with eÎ (0, 1),G = ( ( ), ( ))V G E G be a directed graph which is transitive with V G( )=C E G, ( ) being convex, C have the property G, T T T₁, ₂, ₃:C ® C be three G-nonexpansive mappings such that F ¹ Æ, F T( _i)´ F T( _i)Ì E G( ) for all i = 1, 2, 3 and satisfying the condition (C), { }x_n be a sequence defined by recursion (2.1) such that

1 1

( ,p x ), ( , )x p Î E G( )withp Î F. Then{ }x_n converges strongly to q Î F.

Proof.

It follows from pÎ F p x, ( , ₁), ( , )x p₁ Î E G( ) and Proposition 2.1, we conclude that (x_n, ), (p y_n, ), ( , )p z_n p Î E G( ) for all n Î ¥^*.

Then, by Proposition 2.2, we see that lim || _n ||

n x p

® ¥ - exists and { }x_n is bounded. On the other hand, by (2.11), we obtain||x_n+1- p|| ||£ x_n - p|| for all nÎ ¥^*.This implies that

( _n 1, ) ( _n, )

d x ₊ F £ d x F and hence lim ( , )_n

n d x F

® ¥ exists. Next, by Proposition 2.3, we have

1 2 3

lim || _{n n} || lim || _{n n} || lim || _{n n} || 0.

n x T x n x T x n x T x

® ¥ - = ® ¥ - = ® ¥ - = (2.24)

Since T T T₁, ₂, ₃ satisfy the condition (C), there exists a non-decreasing function : [0, ) [0, )

f ¥ ® ¥ with f(0)= 0, ( )f r > 0for all r > 0 and

1 2 3

max{||x_n - T x_n ||, ||x_n- T x_n ||, ||x_n - T x_n ||}³ f d x F( ( , ))._n (2.25) By combining (2.24) with (2.25), we obtain lim ( ( , ))_n 0.

n f d x F

® ¥ = Suppose that

lim ( , )_n 0.

n d x F

® ¥ > Then for every e> 0, there exists n₀ Î ¥^* such that for all n ³ n₀, we have d x F( , )_n > e. This implies that f d x F( ( , ))_n ³ f e( ) for all n ³ n₀. Therefore,

lim ( ( , ))_n ( ) (0) 0,

n f d x F f e f

® ¥ ³ ³ = which contradicts to lim ( ( , ))_n 0.

n f d x F

® ¥ = So

lim ( , )_n 0.

n d x F

® ¥ = Then, there exists a subsequence {x_{n k( )}} of {x_n}and a sequence { }

k

p in Fsuch that ||x_{n k( )}- p_k || 2 .£ ^-k Therefore, from (2.11), we conclude that

( 1) ( )

||x_{n k+} - p_k || ||£ x_{n k} - p_k || 2 .£ ^-k This implies that

( 1) ( 1)

1 1 ( 1) ( 1)

||p_k+ - p_k || ||£ p_k+ - x_{n k+} ||+ ||x_{n k+} - p_k || 2£ ^{- k+} + 2^-k £ 2^{- k-} .

It follows that {p_k} is a Cauchy sequence inF. Furthermore, by Proposition 1.5, we see that F = F T( )₁ ÇF T( )₂ ÇF T( ₃) is closed in Banach spaces. Thus, there exits qÎ F such

that lim .

k k

p q

® ¥

= By combining this with ||x_{n k( )}- q|| ||£ x_{n k( )}- p_k||+||p_k- q|| 2£ ^-k+||p_k- q||, we obtain lim || _{n k( )} || 0.

k x q

® ¥ - = Moreover, since lim || _n ||

n x q

® ¥ - exists, we conclude that lim || _n || 0

n x q

® ¥ - = and hence { }x_n strongly converges to qÎ F.  In Theorem 2.5, by replacing the assumption “satisfying the condition (C) of three G- nonexpansive mappings” by the assumption “one of three G-nonexpansive mappings is semicompact”, we obtain the following result.

Theorem 2.6.

Let X be a uniformly convex Banach space, C be a nonempty closed convex subset of X, {a_n}, {b_n}, {g_n}Ì [ , 1e - e] with eÎ (0, 1),G = ( ( ), ( ))V G E G be a directed graph which is transitive with V G( )=C E G, ( ) being convex, C have the property G, T T T₁, ₂, ₃:C ® C be three G-nonexpansive mappings such that F ¹ Æ, F T( _i)´ F T( _i)Ì E G( ) for each i = 1, 2, 3 and one of T T₁, ₂ and T₃ is semicompact, { }x_n be a sequence defined by recursion (2.1) such that ( ,p x₁), ( , )x p₁ Î E G( ) withp Î F. Then { }x_n converges strongly to qÎ F.

Proof.

By Proposition 2.3, we obtain lim || _{n 1 n} || 0

n x T x

® ¥ - = for each i = 1, 2, 3. By Proposition 2.2, we conclude that { }x_n is bounded. By the semicompactness of one of T T₁, ₂ and T₃, there exists a subsequence {x_{n k( )}} of { }x_n such that {x_{n k( )}} converges strongly to qÎ C. Then, by the property G of C and the transitive property of G, there exists a subsequence

( ( ))

{x_{n k i} } of {x_{n k( )}} such that (x_{n k i( ( ))}, )q Î E G( ).Therefore, for each j = 1, 2, 3, we have

( ( )) ( ( )) ( ( )) ( ( ))

||q- T q_j || ||£ q- x_{n k i} ||+ ||x_{n k i} - T x_{j n k i} ||+ ||T x_{j n k i} - T q_j ||

^{£||q- xn k i( ( ))||+ ||xn k i( ( ))- T xj n k i( ( ))||+ ||xn k i( ( ))- q|| .}

This implies that T q_j = qfor each j = 1, 2, 3 and hence qÎ F. As in the proof of Theorem 2.5, we conclude that lim ( , )_n

n d x F

® ¥ exists. Furthermore, from

( ) ( )

( _{n k}, ) || _{n k} ||,

d x F £ x - q we see that lim ( _{n k( )}, ) 0

k d x F

® ¥ = and hence lim ( , )_n 0.

n d x F

® ¥ = By using

similar arguments as in the proof of Theorem 2.5, we conclude that { }x_n converges strongly to qÎ F. 

Finally, we provide an example to illustrate for the convergence to common fixed points of three G-nonexpansive mappings by CR-iteration process which generated by (2.1). In addition, the example also shows that the convergence to common fixed points of given mappings by CR-iteration process is faster than SP-iteration process in (Sridarat et al., 2018).

Example 2.7.

Let X = ¡ be a Banach space with norm given by ||x|| |= x| for all xÎ ¡ , [0, 2],

C = G = ( ( ), ( ))V G E G be a directed graph defined by V G( )=C,( , )x y Î E G( ) if and only if 0.45£ x y, £ 1.7 and ,x y Î C. Define three mappings T T T₁, ₂, ₃ :C ® C by