Open Access

Strong convergence for asymptotically nonexpansive mappings in the intermediate sense

Fixed Point Theory and Applications20142014:162

https://doi.org/10.1186/1687-1812-2014-162

Received: 13 February 2014

Accepted: 4 July 2014

Published: 23 July 2014

Abstract

In this paper, let C be a nonempty closed convex subset of a strictly convex Banach space. Then we prove strong convergence of the modified Ishikawa iteration process when T is an ANI self-mapping such that T ( C ) is contained in a compact subset of C, which generalizes the result due to Takahashi and Kim (Math. Jpn. 48:1-9, 1998).

MSC:47H05, 47H10.

Keywords

strong convergence fixed point Mann and Ishikawa iteration process ANI

1 Introduction

Let C be a nonempty closed convex subset of a Banach space E, and let T be a mapping of C into itself. Then T is said to be asymptotically nonexpansive [1] if there exists a sequence { k n } , k n 1 , with lim n k n = 1 , such that
T n x T n y k n x y
for all x , y C and n 1 . In particular, if k n = 1 for all n 1 , T is said to be nonexpansive. T is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such that
T n x T n y L x y
for all x , y C and n 1 . T is said to be asymptotically nonexpansive in the intermediate sense (in brief, ANI) [2] provided T is uniformly continuous and
lim sup n sup x , y C ( T n x T n y x y ) 0 .
We denote by F ( T ) the set of all fixed points of T, i.e., F ( T ) = { x C : T x = x } . We define the modulus of convexity for a convex subset of a Banach space; see also [3]. Let C be a nonempty bounded convex subset of a Banach space E with d ( C ) > 0 , where d ( C ) is the diameter of C. Then we define δ ( C , ϵ ) with 0 ϵ 1 as follows:
δ ( C , ϵ ) = 1 r inf { max ( x z , y z ) z x + y 2 : x , y , z C , x y r ϵ } ,
where r = d ( C ) . When { x n } is a sequence in E, then x n x will denote strong convergence of the sequence { x n } to x. For a mappings T of C into itself, Rhoades [4] considered the following modified Ishikawa iteration process (cf. Ishikawa [5]) in C defined by x 1 C :
x n + 1 = α n T n y n + ( 1 α n ) x n , y n = β n T n x n + ( 1 β n ) x n ,
(1.1)

where { α n } and { β n } are two real sequences in [ 0 , 1 ] . If β n = 0 for all n 1 , then the iteration process (1.1) reduces to the modified Mann iteration process [6] (cf. Mann [7]).

Takahashi and Kim [8] proved the following result: Let E be a strictly convex Banach space and C be a nonempty closed convex subset of E and T : C C be a nonexpansive mapping such that T ( C ) is contained in a compact subset of C. Suppose x 1 C , and the sequence { x n } is defined by x n + 1 = α n T [ β n T x n + ( 1 β n ) x n ] + ( 1 α n ) x n , where { α n } and { β n } are chosen so that α n [ a , b ] and β n [ 0 , b ] or α n [ a , 1 ] and β n [ a , b ] for some a, b with 0 < a b < 1 . Then { x n } converges strongly to a fixed point of T. In 2000, Tsukiyama and Takahashi [9] generalized the result due to Takahashi and Kim [8] to a nonexpansive mapping under much less restrictions on the iterative parameters { α n } and { β n } .

In this paper, let C be a nonempty closed convex subset of a strictly convex Banach space. We prove that if T : C C is an ANI mapping such that T ( C ) is contained in a compact subset of C, then the iteration { x n } defined by (1.1) converges strongly to a fixed point of T, which generalizes the result due to Takahashi and Kim [8].

2 Strong convergence theorem

We first begin with the following lemma.

Lemma 2.1 [9]

Let C be a nonempty compact convex subset of a Banach space E with r = d ( C ) > 0 . Let x , y , z C and suppose x y ϵ r for some ϵ with 0 ϵ 1 . Then, for all λ with 0 λ 1 ,
λ ( x z ) + ( 1 λ ) ( y z ) max ( x z , y z ) 2 λ ( 1 λ ) r δ ( C , ϵ ) .

Lemma 2.2 [9]

Let C be a nonempty compact convex subset of a strictly convex Banach space E with r = d ( C ) > 0 . If lim n δ ( C , ϵ n ) = 0 , then lim n ϵ n = 0 .

Lemma 2.3 [10]

Let { a n } and { b n } be two sequences of nonnegative real numbers such that n = 1 b n < and
a n + 1 a n + b n

for all n 1 . Then lim n a n exists.

Lemma 2.4 Let C be a nonempty compact convex subset of a Banach space E, and let T : C C be an ANI mapping. Put
c n = sup x , y C ( T n x T n y x y ) 0 ,

so that n = 1 c n < . Suppose that the sequence { x n } is defined by (1.1). Then lim n x n z exists for any z F ( T ) .

Proof The existence of a fixed point of T follows from Schauder’s fixed theorem [11]. For a fixed z F ( T ) , since
T n y n z y n z + c n = β n T n x n + ( 1 β n ) x n z + c n β n T n x n z + ( 1 β n ) x n z + c n β n x n z + c n + ( 1 β n ) x n z + c n x n z + 2 c n ,
we obtain
x n + 1 z = α n T n y n + ( 1 α n ) x n z α n T n y n z + ( 1 α n ) x n z α n ( x n z + 2 c n ) + ( 1 α n ) x n z x n z + 2 c n .

By Lemma 2.3, we readily see that lim n x n z exists. □

Theorem 2.5 Let C be a nonempty compact convex subset of a strictly convex Banach space E with r = d ( C ) > 0 . Let T : C C be an ANI mapping. Put
c n = sup x , y C ( T n x T n y x y ) 0 ,

so that n = 1 c n < . Suppose x 1 C , and the sequence { x n } defined by (1.1) satisfies α n [ a , b ] and lim sup n β n = b < 1 or lim inf n α n > 0 and β n [ a , b ] for some a, b with 0 < a b < 1 . Then lim n x n T x n = 0 .

Proof The existence of a fixed point of T follows from Schauder’s fixed theorem [11]. For any fixed z F ( T ) , we first show that if α n [ a , b ] and lim sup n β n = b < 1 for some a , b ( 0 , 1 ) , then we obtain lim n T x n x n = 0 . In fact, let ϵ n = T n y n x n r . Then we have 0 ϵ n 1 since T n y n x n r . As in the proof of Lemma 2.4, we obtain
T n y n z x n z + 2 c n .
(2.1)
Since
T n y n x n = r ϵ n ,
and by (2.1) and Lemma 2.1, we have
x n + 1 z = α n ( T n y n z ) + ( 1 α n ) ( x n z ) x n z + 2 c n 2 α n ( 1 α n ) r δ ( C , ϵ n ) .
Thus
2 α n ( 1 α n ) r δ ( C , ϵ n ) x n z x n + 1 z + 2 c n .
Since
2 r n = 1 a ( 1 b ) δ ( C , T n y n x n r ) < ,
we obtain
lim n δ ( C , T n y n x n r ) = 0 .
By using Lemma 2.2, we obtain
lim n T n y n x n = 0 .
(2.2)
Since
T n x n x n T n x n T n y n + T n y n x n x n y n + c n + T n y n x n = β n T n x n x n + c n + T n y n x n ,
we obtain
( 1 β n ) T n x n x n c n + T n y n x n .
(2.3)
Since lim sup n β n = b < 1 , we have
lim inf n ( 1 β n ) = 1 b > 0 .
(2.4)
From (2.2), (2.3) and (2.4), we obtain
lim n T n x n x n = 0 .
(2.5)
Since
x n + 1 x n = ( 1 α n ) x n + α n T n y n x n = α n T n y n x n b T n y n x n ,
and by (2.2), we obtain
lim n x n + 1 x n = 0 .
(2.6)
Since
x n T x n x n x n + 1 + x n + 1 T n + 1 x n + 1 + T n + 1 x n + 1 T n + 1 x n + T n + 1 x n T x n 2 x n x n + 1 + c n + 1 + x n + 1 T n + 1 x n + 1 + T n + 1 x n T x n
and by the uniform continuity of T, (2.5) and (2.6), we have
lim n x n T x n = 0 .
(2.7)
Next, we show that if lim inf n α n > 0 and β n [ a , b ] , then we also obtain (2.7). In fact, let ϵ n = T n x n x n r . Then we have 0 ϵ n 1 . From lim inf n α n > 0 , there are some positive integer n 0 and a positive number a such that α n > a > 0 for all n n 0 . Since
x n + 1 z = α n ( T n y n z ) + ( 1 α n ) ( x n z ) α n T n y n z + ( 1 α n ) x n z α n y n z + α n c n + ( 1 α n ) x n z ,
and hence
x n + 1 z x n z α n y n z x n z + c n .
So, we obtain
x n z y n z x n z x n + 1 z α n + c n x n z x n + 1 z a + c n .
(2.8)
Since
T n x n z x n z + c n ,
from Lemma 2.1, we obtain
y n z = β n T n x n + ( 1 β n ) x n z = β n ( T n x n z ) + ( 1 β n ) ( x n z ) x n z + c n 2 β n ( 1 β n ) r δ ( C , ϵ n ) .
(2.9)
By using (2.8) and (2.9), we obtain
2 β n ( 1 β n ) r δ ( C , ϵ n ) x n z y n z + c n x n z x n + 1 z a + 2 c n .
Hence
2 r n = 1 a ( 1 b ) δ ( C , T n x n x n r ) < .
We also obtain
lim n x n T n x n = 0
(2.10)
similarly to the argument above. Since
y n x n = β n T n x n + ( 1 β n ) x n x n β n T n x n x n b T n x n x n ,
and by using (2.10), we obtain
lim n x n y n = 0 .
(2.11)
Since
T n y n x n T n y n T n x n + T n x n x n y n x n + c n + T n x n x n ,
by using (2.10) and (2.11), we obtain
lim n T n y n x n = 0 .
(2.12)
Since
T n y n y n T n y n x n + x n y n ,
by using (2.11) and (2.12), we obtain
lim n T n y n y n = 0 .
(2.13)
Since
x n x n 1 = ( 1 α n 1 ) x n 1 + α n 1 T n 1 y n 1 x n 1 = α n 1 T n 1 y n 1 x n 1 T n 1 y n 1 y n 1 + y n 1 x n 1 ,
by (2.11) and (2.13), we get
lim n x n x n 1 = 0 .
(2.14)
From
T n 1 x 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 2 x n x n 1 + c n 1 + T n 1 x n 1 x n 1
and by (2.10) and (2.14), we obtain
lim n T n 1 x n x n = 0 .
(2.15)
Since
x n T x n x n y n + y n T n y n + T n y n T n x n + T n x n T x n y n T n y n + 2 x n y n + c n + T n x n T x n
and by the uniform continuity of T, (2.11), (2.13) and (2.15), we have
lim n x n T x n = 0 .

 □

Our Theorem 2.6 carries over Theorem 3 of Takahashi and Kim [8] to an ANI mapping.

Theorem 2.6 Let C be a nonempty closed convex subset of a strictly convex Banach space E, and let T : C C be an ANI mapping, and let T ( C ) be contained in a compact subset of C. Put
c n = sup x , y C ( T n x T n y x y ) 0 ,

so that n = 1 c n < . Suppose x 1 C , and the sequence { x n } defined by (1.1) satisfies α n [ a , b ] and lim sup n β n = b < 1 or lim inf n α n > 0 and β n [ a , b ] for some a, b with 0 < a b < 1 . Then { x n } converges strongly to a fixed point of T.

Proof By Mazur’s theorem [12], A : = c o ¯ ( { x 1 } T ( C ) ) is a compact subset of C containing { x n } which is invariant under T. So, without loss of generality, we may assume that C is compact and { x n } is well defined. The existence of a fixed point of T follows from Schauder’s fixed theorem [11]. If d ( C ) = 0 , then the conclusion is obvious. So, we assume d ( C ) > 0 . From Theorem 2.5, we obtain
lim n x n T x n = 0 .
(2.16)

Since C is compact, there exist a subsequence { x n k } of the sequence { x n } and a point p C such that x n k p . Thus we obtain p F ( T ) by the continuity of T and (2.16). Hence we obtain lim n x n p = 0 by Lemma 2.4. □

Corollary 2.7 Let C be a nonempty closed convex subset of a strictly convex Banach space E, and let T : C C be an asymptotically nonexpansive mapping with { k n } satisfying k n 1 , n = 1 ( k n 1 ) < , and let T ( C ) be contained in a compact subset of C. Suppose x 1 C , and the sequence { x n } defined by (1.1) satisfies α n [ a , b ] and lim sup n β n = b < 1 or lim inf n α n > 0 and β n [ a , b ] for some a, b with 0 < a b < 1 . Then { x n } converges strongly to a fixed point of T.

Proof Note that
n = 1 c n = n = 1 ( k n 1 ) diam ( C ) < ,

where diam ( C ) = sup x , y C x y < . The conclusion now follows easily from Theorem 2.6. □

We give an example which satisfies all assumptions of T in Theorem 2.6, i.e., T : C C is an ANI mapping which is not Lipschitzian and hence not asymptotically nonexpansive.

Example 2.8 Let E : = R and C : = [ 0 , 2 ] . Define T : C C by
T x = { 1 , x [ 0 , 1 ] ; 2 x , x [ 1 , 2 ] .
Note that T n x = 1 for all x C and n 2 and F ( T ) = { 1 } . Clearly, T is uniformly continuous, ANI on C, but T is not Lipschitzian. Indeed, suppose not, i.e., there exists L > 0 such that
| T x T y | L | x y |
for all x , y C . If we take y : = 2 and x : = 2 1 ( L + 1 ) 2 > 1 , then
2 x L ( 2 x ) 1 L 2 2 x = 1 ( L + 1 ) 2 L + 1 L .

This is a contradiction.

We also give an example of an ANI mapping which is not a Lipschitz function.

Example 2.9 Let E = R and C = [ 3 π , 3 π ] and let | h | < 1 . Let T : C C be defined by
T x = h x sin n x
for each x C and for all n N , where denotes the set of all positive integers. Clearly F ( T ) = { 0 } . Since
T ( x ) = h x sin n x , T 2 x = h 2 x sin n x sin n h x sin n ( sin n x ) ,
we obtain { T n x } 0 uniformly on C as n . Thus
lim sup n { T n x T n y x y 0 } = 0
for all x , y C . Hence T is an ANI mapping, but it is not a Lipschitz function. In fact, suppose that there exists h > 0 such that | T x T y | h | x y | for all x , y C . If we take x = 5 π 2 n and y = 3 π 2 n , then
| T x T y | = | h 5 π 2 n sin n 5 π 2 n h 3 π 2 n sin n 3 π 2 n | = 4 h π n ,
whereas
h | x y | = h | 5 π 2 n 3 π 2 n | = h π n .

Declarations

Acknowledgements

The author would like to express their sincere appreciation to the anonymous referee for useful suggestions which improved the contents of this manuscript.

Authors’ Affiliations

(1)
Department of Applied Mathematics, Pukyong National University

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© Kim; licensee Springer. 2014

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