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Strong convergence for asymptotically nonexpansive mappings in the intermediate sense

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.

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 xy

for all x,yC and n1. In particular, if k n =1 for all n1, 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 Lxy

for all x,yC and n1. 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)={xC:Tx=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 0ϵ1 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 n1, 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:CC 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<ab<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:CC 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,zC and suppose xyϵr for some ϵ with 0ϵ1. Then, for all λ with 0λ1,

λ ( x z ) + ( 1 λ ) ( y z ) max ( x z , y z ) 2λ(1λ)rδ(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 n1. 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:CC 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 zF(T).

Proof The existence of a fixed point of T follows from Schauder’s fixed theorem [11]. For a fixed zF(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:CC 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<ab<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 zF(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 )rδ(C, ϵ n ) x n z x n + 1 z+2 c n .

Since

2r n = 1 a(1b)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 )=1b>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

2r n = 1 a(1b)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:CC 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<ab<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 pC such that x n k p. Thus we obtain pF(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:CC 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<ab<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 xy<. 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:CC 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:CC by

Tx= { 1 , x [ 0 , 1 ] ; 2 x , x [ 1 , 2 ] .

Note that T n x=1 for all xC and n2 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

|TxTy|L|xy|

for all x,yC. If we take y:=2 and x:=2 1 ( L + 1 ) 2 >1, then

2 x L(2x) 1 L 2 2x= 1 ( L + 1 ) 2 L+1L.

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=[3π,3π] and let |h|<1. Let T:CC be defined by

Tx=hxsinnx

for each xC and for all nN, where ℕ 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,yC. Hence T is an ANI mapping, but it is not a Lipschitz function. In fact, suppose that there exists h>0 such that |TxTy|h|xy| for all x,yC. If we take x= 5 π 2 n and y= 3 π 2 n , then

|TxTy|= | 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|xy|=h | 5 π 2 n 3 π 2 n | = h π n .

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Acknowledgements

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

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Kim, G.E. Strong convergence for asymptotically nonexpansive mappings in the intermediate sense. Fixed Point Theory Appl 2014, 162 (2014). https://doi.org/10.1186/1687-1812-2014-162

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Keywords

  • strong convergence
  • fixed point
  • Mann and Ishikawa iteration process
  • ANI