Approximation of Common Fixed Points for Nonself-Asymptotically Nonexpansive Mappings

Approximation of Common Fixed Points for Nonself-Asymptotically Nonexpansive Mappings

Esra YOLACAN Hukmi KIZILTUNC

AtaturkUniversity Faculty of Science

Department of Mathematics 25240 Erzurum, Turkey

Abstract.

Suppose that K is a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T_i:K E (i=1,2,3,4,5,6) be six of weakly inward and asymptotically nonexpansive mappings with respect to P with common sequence

  k

_n

  1,  

satisfying

   



^n=1

k

n

^{1 <}

and

 

_i 

i

T F F



⁶

1

=

= , respectively. For any given

x

₁

 K ,

suppose that

 

xn is a sequence generated iteratively by

   

   

   

















 

,

=

,

=

,

=

6 3 5

3 3

4 2 3

2 2

2 1 1

1 1

1

n n n

n n n

n n n

n n n

n n n

n n n

n n n

n n n n n n

x PT c x PT b x a z

z PT c z PT b x a y

y PT c y PT b x a x

where

  a

_ni

,   b

_ni and

  c

_ni

 i = 1,2,3 

are sequences in

  ,1   

for some

    0,1

which satisfy condition

1

ni

=

ni

ni

b c

a    i = 1,2,3  .

Under some suitable conditions, strong convergence theorems of

 

xn to a common fixed point of

  T

_i ⁶_i=1 are obtained.

Keywords and phrases: Nonself-Asymptotically Nonexpansive Mappings, common fixed point, strongconvergence.

2000 Mathematics Subject Classification: 47H09, 47H10, 46B20.

1. Introduction

Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. A mapping

T : K  K

is called

nonexpansi ve

if

Tx  Ty  x  y

for all x, yK . A mapping

T : K  K

is called

ally

asymptotic

nonexpansi ve

if there exists a sequence

  k

_n

  1,  

with

k

_n

 1

such that y

x k y T x

Tⁿ  ⁿ  _n  (1.1)

for all x, yK and

n  1

.

A mapping

T : K  K

is called uniformly LLipschitzian if there exists constant

L  0

such that y

x L y T x

Tⁿ  ⁿ   (1.2)

for all x, yK and

n  1

.

A subset K of E is said to be a retract of E if there exists a continuous map

P : E  K

such that

Px = x

, for all

x  K

. Every closed convex subset of a uniformly convex Banach space is a retract. A mapping

K E

P : 

is said to be a retraction if

P =

²

P

. It follows that if a map P is a retraction, then Py = y for all )

(P R

y , the range of P.

It is well-known that every closed convex subset of a uniformly convex Banach space is a retract. The concept of asymptotically nonexpansive nonself-mappings was firstly introduced by Chidume et al. [2] as the generalization of asymptotically nonexpansive self-mappings. The asymptotically nonexpansive nonself-mapping is defined as follows:

Let K be a nonempty subset of real normed linear space

E .

Let

P : E  K

be the nonexpansive retraction of E onto

K .

A nonself mapping

T : K  E

is called asymptotically

nonexpansi ve

if there exists sequence

  k

_n

  1,  

,

k

_n

 1

(n) such that

1.

, , all for )

( )

(PT ^1xT PT ^1y k xy x yK n

T ^{n n} _n (1.3)

Let K be a nonempty subset of real normed linear space

E .

Let

P : E  K

be the nonexpansive retraction of E onto

K .

A nonself mapping

T : K  E

is called uniformly LLipschitzian if there exists constant

 0

L

such that

1.

, , all for )

( )

(PT ^1xT PT ^1y L xy x yK n

T ^{n n} (1.4)

As a matter of fact, if T is self-mapping, then P becomes the identity mapping, so that (1.3) and (1.4) reduces to (1.1) and (1.2), respectively. In addition, if

T : K  E

is asymptotically nonexpansive and

P : E  K

is a nonexpansive retraction, then

PT : K  K

is asymptotically nonexpansive. Indeed, for all x,yK and





n

, it follows that

y PT PT x PT PT y PT x

PT)ⁿ ( )ⁿ = ( )^{n 1} ( )^{n 1}

(  ^  ^

y PT T x PT

T( )^n1  ( )^n1



. y x k

_n





The converse, however, may not be true. Therefore, Zhou et al. [4] introduced the following generalized definition recently.

Definition 1.[4] Let K be a nonempty subset of real normed linear space

E .

Let

P : E  K

be the nonexpansive retraction of E into

K .

(i)A nonself mapping

T : K  E

is called asymptotically nonexpansive with respect to P if there exists sequences

 

k_n 



1,



with

k

_n

 1

as

n  

such that

. , , , )

( )

(PT ⁿx PT ⁿy k_n xy x yK n

(ii)A nonself mapping

T : K  E

is said to be uniformly L-Lipschitzian with respect to P if there exists a constant

L  0

such that

. , , , )

( )

(PT ⁿx PT ⁿy L xy x yK n

Zhou et al. [4] obtained some strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings with respect to P in uniformly convex Banach spaces. As a consequence, the main results of Chidume et al. [2] were deduced.

Inspired and motivated by this facts, we study three step iteration scheme for approximating common fixed points of six nonself-asymptotically nonexpansive mappings with respect to P and to prove some strong convergence theorems for such mappings in uniformly convex Banach spaces. The scheme (1.5) is defined as follows.

Let K be a nonempty closed convex subset of a real normed linear space E with retraction P. Let T_i:K E 6)

1,2,3,4,5,

=

(i be six nonself-asymptotically nonexpansive mappings with respect to

P .

For any given

K

x

₁



and

n  1

, suppose that

 

xn is a sequence generated iteratively by

   

   

   

















 

,

=

,

=

,

=

6 3 5

3 3

4 2 3

2 2

2 1 1

1 1

1

n n n

n n n

n n n

n n n

n n n

n n n

n n n

n n n n n n

x PT c x PT b x a z

z PT c z PT b x a y

y PT c y PT b x a x

(1.5)

where

  a

_ni

,   b

ni and

  c

ni

 i = 1,2,3 

are sequences in

  ,1   

for some

    0,1

which satisfy condition

1

ni

=

ni

ni

b c

a    i = 1,2,3  .

If b_n3 =b_n2 =c_n3 =c_n2 0 for all n1, then (1.5) reduces to the iteration defined by Zhou et al. [4]

, , ) ( )

(

=

, _{1 1 1 1 1 2}

1K x _ a x b PT x c PT x n

x _{n n n n} ⁿ _{n n} ⁿ _n (1.6)

where

   a

_n1

, b

_n1 and

  c

n1 are three sequences in

 ^{ ,1  } 

for some

    0,1 ,

satisfying a_n1b_n1c_n1 =1.

2.Preliminaries

For the sake of convenience, we restate the following concepts and results.

Let E be a Banach space with its dimension greater than or equal to

2.

The modulus of E is the function

  0 , 1 (0,2]

: )

(  



_E defined by

.

= 1,

= 1,

= : ) 2 (

1 1 inf

= )

(   

 

  x  y x y x  y

E

 



A Banach space E is uniformly convex if and only if



_E

(  ) > 0

for all



(0,2].

Let E be a Banach space and

^{S ( E ) =}  ^{x  E : x = 1}  ^.

The space E is said to be smooth if

t

x ty x

t



 lim



0

exists for all x,yS(E).

Let

C

and K be subsets of a Banach space E . A mapping P from

C

into K is called sunny if Px

Px x t Px

P(  (  ))= for

x  C

with Pxt(xPx)C and

t  0.

For any xK, the inward set

I

_K

(x )

is defined as follows:

 : = ( ), , 0  .

= )

( x y  E y x   z  x z  K   I

_K

A mapping

T : K  E

is said to satisfy the inward condition if

Tx  I

_K

(x )

for all

x  K .

T is said to be weakly inward if

Tx  clI

_K

(x )

for each xK,where

clI

_K

(x )

is the closure of

I

_K

(x )

.

A mapping

T : K  K

is said to be completely continuous if for every bounded sequence

  x

n

^,

there exists a subsequence say

 

xn_j of

 

x_n such that

 

Txn_j converges to some element of the range

T .

A mapping

T : K  K

is said to be demicompact if any sequence

 

xn in K satisfying

x

_n

 Tx

_n

 0

as





n

has a convergent subsequence.

We need the following lemmas for our main results.

Lemma 1.[5] If

  r

_n ,

 

t_n are two sequences of nonnegative real numbers such that



1



, 1

1   

 t r n

r_{n n n}

and



^{ <}

1

= n n

t , then

lim

n

r

_n exists.

Lemma 2.[3] Let E be real smooth Banach space, let K be nonempty closed convex subset of E with P as a sunny nonexpansive retraction, and let

T : K  E

be a mapping satisfying weakly inward condition. Then

).

(

= ) (PT F T F

Lemma 3.[1] Let E be a real uniformly convex Banach space and

^B

_R

⁼  ^{x  E : x  R} 

. Then there exists a continuous, strictly increasing, and convex function

g :  0, ^  ^  0, ^ 

, g(0)=0 such that

 

^,

2 2

2

2 x y z g x y

z y

x



























for all x, y,

z  B

_R, and all

 ,  ,     0,1

with











=1. 3.Main Results

In this section, we present some several strong convergence theorems of the three step iteration scheme (1.5) to a common fixed point for six nonself-asymptotically nonexpansive mappings with respect to P in a real uniformly convex Banach spaces. We shall make use of the following lemmas.

Lemma 4.Let E be a real normed space and K a nonempty closed convex subset of E which is also a nonexpansive retract of E . Let T_i:K E (i=1,2,3,4,5,6) be six nonself-asymptotically nonexpansive mappings with respect to P with common sequence

  k

_n

  1,  

satisfying



^n=1

 k

n

 ¹  ^< 

. Suppose that

 

xn is defined by (1.5)

.

If

 

_i 

i

T F F



⁶

1

=

= , then

lim

_n

x

_n

 p

exists for all pF.

Proof. Let pF. From (1.5), we have

  PT x c   PT x p b

x a p

z

_n

 =

_{n3 n}



_{n3 5} ⁿ _n



_{n3 6} ⁿ _n



(3.1)

p x k c p x k b p x

a

_{n n}

 

_{n n n}

 

_{n n n}





_{3 3 3}

. p x k

_{n n}





By (1.5) and (3.1), we obtain

  ^{PT z c}   ^{PT z p}

b x a p

y

_n

 =

_{n2 n}



_{n2 3} ⁿ _n



_{n2 4} ⁿ _n



(3.2)

p z k c p z k b p x

a

_{n n}

 

_{n n n}

 

_{n n n}





_{2 2 2}

p x k c p x k b p x

a

_{n n}

 

_{n n n}

 

_{n n n}





_{2 2} ² ₂ ²

2

.

p x k

_{n n}





Therefore, it follows from (1.5) and (3.2) that

 ^{x p}  ^b    ^{PT y p}  ^c  ^{  PT y p} 

a p

x

_n1

 =

_{n1 n}

 

_{n1 1} ⁿ _n

 

_{n1 2} ⁿ _n



(3.3)

p y k c p y k b p x

a

_{n n}

 

_{n n n}

 

_{n n n}





_{1 1 1}

p x k c p x k b p x

a

_{n n}

 

_{n n n}

 

_{n n n}





_{1 1} ³ ₁ ³

p x k

_{n n}





³

 1  .

=  

_n

x

_n

 p

Note that



^n=1

 k

n

 ¹  ^< 

is equivalent to

_

^n₌₁

 k

n³

^{ 1}  ^{<  .}

Thus, by (3.3) and Lemma 1,

lim

_n

x

_n

 p

exists for allpF. This completes the proof.

Lemma 5.Let E be a real uniformly convex Banach space and K a nonempty closed convex subset of E which is also a nonexpansive retract of E . Let T_i:K E (i=1,2,3,4,5,6) be six nonself-asymptotically nonexpansive mappings with respect to P with common sequence

  k

_n

  1,  

satisfying



^n=1

 k

n

 ¹  ^< 

.

Suppose that

 

xn is defined by (1.5), where

  a

_ni

,   b

ni and

  c

ni

 i = 1,2,3 

are sequences in

  ,1   

for some

^{ }   ^{0,1 .}

^If

 

_i 

i

T F F



⁶

1

=

= , then

lim

n

x

n

   PT

i

x

n

^{= 0}

for each i=1,2,3,4,5,6.

Proof. From (1.5), by the property of P, and Lemma 3 , we have

 

_{5 3}

 

₆ ²

3 3

2

= a x b PT x c PT x p

p

z

_n



_{n n}



_n ⁿ _n



_n ⁿ _n



(3.4)

 

₅ ² ₃

 

₆ ²

3 2

3

x p b PT x p c PT x p

a

_{n n}

 

_n ⁿ _n

 

_n ⁿ _n





  

n



n n

n

n

b g x PT x

a

_{3 3 1}



₅



2 2 3 2 2

3 2

3 x p b k x p c k x p

a_{n n}   _{n n n}  _{n n n} 



  

n



n

n

PT x

x

g

_{1 5}

2



 

  

n



n n

n

n

x p g x PT x

k

²



²



² ₁



₅

 

which implies that

g

1

 x

_n

   PT

5 ⁿ

x

_n

  ⁰

as

n  

. Since

g

1

^:     ^{0,   0, }

with

g

₁

  0 = 0

is a continuous strictly increasing convex function, it follows that

  = 0.

lim

5 n

n n n

x PT x 



 (3.5)

Similarly, we obtain

  = 0.

lim

6 n

n n n

x PT x 



 (3.6)

It follows from (1.5), (3.4) and Lemma 3 that

 

_{3 2}

 

₄ ²

2 2

2

= a x b PT z c PT z p

p

y

_n



_{n n}



_n ⁿ _n



_n ⁿ _n



(3.7)

 

₃ ² ₂

 

₄ ²

2 2

2

x p b PT z p c PT z p

a

_{n n}

 

_n ⁿ _n

 

_n ⁿ _n





  

n



n n

n

n

b g x PT z

a

_{2 2 2}



₃



2 2 2 2 2

2 2

2 x p b k z p c k z p

a_{n n}  _{n n n}  _{n n n}



  

n



n n

n

n

b g x PT z

a

_{2 2 2}



₃



4 2 2 4 2

2 2

2 x p b k x p c k x p

a_{n n}  _{n n n}   _{n n n}



       ^{ }

n



n n

n n n n n

n

n

c g x PT x a b g x PT z

b

₂



₂ ² ₁



₅



_{2 2 2}



₃

 

  

3

 ^,

2 2 2

4

n n n

n

n

x p g x PT z

k   

 

which implies that

g

2

 x

_n

   PT

3 ⁿ

z

_n

  ⁰

as

n  

. Since

g

2

^:  ^{0, }  ^  ^{0, } 

with

g

2

  ^{0 = 0}

is a continuous strictly increasing convex function, it follows that

  = 0.

lim

_{n 3} ⁿ _n

n

z PT x 



 (3.8)

Similarly, we have

  = 0.

lim

_{n 4} ⁿ _n

n

z PT x 



 (3.9)

Similarly, it follows from (1.5), (3.7) and Lemma 3 that

 

_{1 1}

 

₂ ²

1 1 2

1

p = a x b PT y c PT y p

x

_n



_{n n}



_n ⁿ _n



_n ⁿ _n



 

₁ ² ₁

 

₂ ²

1 2

1

x p b PT y p c PT y p

a

_{n n}

 

_n ⁿ _n

 

_n ⁿ _n





  

n



n n

n

n

b g x PT y

a

_{1 1 3}



₁



2 2 1 2 2

1 2

1 x p b k y p c k y p

a_{n n}  _{n n n}  _{n n n}



  

n



n

n

PT y

x

g

_{3 1}

2



 

6 2 1 6 2

1 2

1 x p b k x p c k x p

a_{n n}  _{n n n}   _{n n n}



       ^{ }

n



n n

n n n

n

n

c g x PT z g x PT y

b

₁



₁ ² ₂



₃



² ₃



₁

  

  

1

 ^,

3 2 2 6

n n n

n

n

x p g x PT y

k   

 

which implies that

g

3

 x

_n

   PT

1 ⁿ

y

_n

  ⁰

as

n  

. Since g₃:



0,







0,



with g₃

 

0 =0 is a continuous strictly increasing convex function, it follows that

  = 0.

lim

_{n 1} ⁿ _n

n

y PT x 



 (3.10)

Similarly, we have

  = 0.

lim

2 n

n n n

y PT x 



 (3.11)

It follows from (1.5), (3.5) and (3.6) that

   

n n n n

n n n

n n n

n

x a x b PT x c PT x x

z  =

₃



_{3 5}



_{3 6}



(3.12)

   

n

n n

n n n n

n

x PT x c x PT x

b

₃



₅



₃



₆







0,asn

It follows from (3.8) and (3.12) that

  PT

₃ ⁿ

z

_n

 z

_n

 x

_n

   PT

₃ ⁿ

z

_n

 z

_n

 x

_n (3.13)





0,asn Similarly we have

  = 0.

lim

4 n n

n n

z z

PT 



 (3.14)

Noting that

       ^{ }

n n



n n

n n n n

n n n n

n

z a x z b PT z z c PT z z

y  =

₂

 

_{2 3}

 

_{2 4}



, we have

   

n n

n n

n n n n

n n n n

n

z a x z b PT z z c PT z z

y  

₂

 

_{2 3}

 

_{2 4}



This with (3.12), (3.13) and (3.14) implies that

0 lim

n n

=

n

z y 



 (3.15)

From (3.12) and (3.15)

0 lim

n n

=

n

y x 



 (3.16)

It follows from (3.10) and (3.16) that

   

n n n

n n

n n

n

y y x PT y x y

PT

₁

  

₁

 

(3.17)





0,asn Similarly we have

  = 0.

lim

₂ ⁿ _{n n}

n

y y

PT 



 (3.18)

From (3.10) and (3.11), we have

   

n n n n

n n n n n n

n

x a x b PT y c PT y x

x

_1

 =

₁



_{1 1}



_{1 2}



(3.19)

   

n

n n

n n n n

n

x PT y c x PT y

b

₁



₁



₁



₂







0,asn Noting that

       

n

n n

n n

n n n

n n n

n

PT x x z PT z z PT z PT x

x 

₃

  

₃

 

₃



₃

   

n n

n n

n

n

x z PT z z

k   



 1

₃

This with (3.12) and (3.13) implies that

  = 0

lim

3 n

n n n

x PT x 



 (3.20)

Similarly we have

  = 0

lim

_{n 4} ⁿ _n

n

x PT x 



 (3.21)

Noting that

       

n

n n

n n

n n n

n n n

n

PT x x y PT y y PT y PT x

x 

₁

  

₁

 

₁



₁

   

n n

n n

n

n

x y PT y y

k   



 1

₁

This with (3.16) and (3.17) implies that

  = 0

lim

1 n

n n n

x PT x 



 (3.22)

Similarly we have

  = 0.

lim

2 n

n n n

x PT x 



 (3.23)

Since an asymptotically nonexpansive mapping with respect to P must be uniformly L-Lipschitzian with respect to P,where

L = sup

_n1

  k

_n

 ¹

, then we have

     

1

 

1

1 1

1 1

1

1  















_{i n}



_n



_i ⁿ _n



_i ⁿ _n



_{i n}

n

PT x x PT x PT x PT x

x

 

1 1

 

1 1

1   





  

 x

_n

PT

_i ⁿ

x

_n

L x

_n

PT

_i ⁿ

x

_n

   

n

n i n n

n n

n i

n

PT x L x x L x PT x

x     



_1 ^1 _{1 1}

    

1

 L PT

_i ⁿ

x

_n

PT

_i ⁿ

x

_n

   

n

n i n n

n i

n

PT x L x PT x

x   



_1 ^1 _1

 L 1  x

_{n 1}

x

_n

.

L  



_

Consequently, by (3.5), (3.6), (3.20), (3.21), (3.22), (3.23) and (3.19), it can be obtained that

  = 0( = 1,2,3,4,5, 6) lim x

_n

PT

_i

x

_n

i

n





 (3.24)

This completes the proof.

Theorem 1.Let E be a real uniformly convex Banach space and K a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T_i:K E (i=1,2,3,4,5,6) be six nonself-asymptotically nonexpansive mappings with respect to P with common sequence

  k

_n

  1,  

satisfying



^n=1

 k

n

^{ 1}  ^{< }

. Suppose that

 

x_n is defined by (1.5), where

  a

_ni

,   b

_ni and

  c

_ni

 i = 1,2,3 

are sequences in

  ,1   

for some

    0,1

and

 

_i 

i

T F F



⁶

1

=

= . Then

 

xn converges strongly to a common fixed point of

  ^T

_i ⁶_i=1 ^{if and}

only if

lim inf

_n

d   x

_n

^, F ^{= 0}

, where

^d   ^x

n

^{, F = inf}  ^x

n

 ^{p : p}  ^F  ^.

Proof. The necessity is obvious. Thus, we need only prove the sufficiency. Suppose that

lim inf

_n

d   x

_n

^, F ^{= 0}

. From (3.3), we obtain



x 1^,F

 

¹



k^{3 1}

 

d

 

x ^,F ^.

d _n   _n  _n (3.25)

As

_

^n₌₁

 k

n³

^{ 1}  ^{< }

, therefore

lim

n

^d   ^x

n

, ^F

exists by Lemma 1. But by hypothesis

  ^{, = 0,}

lim inf

_n

d x

_n

F

hence we must have

lim

_n

d   x

_n

^, F ^{= 0.}

Next we shall prove that

 

xn is a Cauchy sequence. It follows from (3.3) that for any n,mn₀

  x p

p

x

_nm

  1  

_{nm1 nm1}





_{n m}

 x

_{n m}

 p

 exp 

_{ 1  1}

. exp

1

=

p x

_n

k m n

n k



 

 



 

 ^ 

^{ }

^

Let

M ^{= exp}  

^k₌₁



k

 ^,

^then

^{M > 0}

^and

. ,

, n m n

₀

p

x M p

x

_nm

 

_n

  

(3.26)

For an arbitrary



>0, since

lim

_n

d   x

_n

^, F ^{= 0}

, there exists a positive integer

N

₁ such that

 

F M x d _n

< 4

,



for all

n  N

₁

.

So, we have

 

^.

< 4

1,F M

x

d _N



This means that there exists a x^ F such that 4 .

1 x M

x_{N   }



It follows from (3.26) that for all

n  N

₁ and m1,







 x  x x  x x x_{n m n n m n}