Open Access

Strong convergence to a fixed point of a total asymptotically nonexpansive mapping

Fixed Point Theory and Applications20132013:302

https://doi.org/10.1186/1687-1812-2013-302

Received: 16 May 2013

Accepted: 22 October 2013

Published: 19 November 2013

Abstract

In this paper, we prove strong convergence for the modified Ishikawa iteration process of a total asymptotically nonexpansive mapping satisfying condition (A) in a real uniformly convex Banach space. Our result generalizes the results due to Rhoades (J. Math. Anal. Appl. 183:118-120, 1994).

MSC:47H05, 47H10.

Keywords

strong convergence fixed point modified Ishikawa iteration process total asymptotically nonexpansive mapping

1 Introduction

Let X be a real Banach space, let C be a nonempty closed convex subset of X, and let T be a mapping of C into itself. Then T is said to be asymptotically nonexpansive [2] 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
(1.1)
for all x , y C and n 1 . 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 . If T is asymptotically nonexpansive, then it is uniformly L-Lipschitzian. We denote by the set of all positive integers. T is said to be total asymptotically nonexpansive (in brief, TAN) [3] if there exist two nonnegative real sequences { c n } and { d n } with c n , d n 0 as n , ϕ Γ ( R + ) such that
T n x T n y x y + c n ϕ ( x y ) + d n ,
(1.2)
for all x , y C and n 1 , where R + : = [ 0 , ) and ϕ Γ ( R + ) if and only if ϕ is strictly increasing, continuous on R + and ϕ ( 0 ) = 0 . It is clear that if we take ϕ ( t ) = t for all t 0 and d n = 0 for all n 1 in (1.2), it is reduced to (1.1). Approximating fixed points of the modified Ishikawa iterative scheme under total asymptotically nonexpansive mappings has been investigated by several authors; see, for example, Chidume and Ofoedu [4, 5], Kim [6], Kim and Kim [7] and others. For a mapping T of C into itself in a Hilbert space, Schu [8] considered the following modified Ishikawa iteration process (cf. Ishikawa [9]) in C defined by
x 1 C , x n + 1 = ( 1 α n ) x n + α n T n y n , y n = ( 1 β n ) x n + β n T n x n ,
(1.3)
where { α n } and { β n } are two real sequences in [ 0 , 1 ] . If β n = 0 for all n 1 , then iteration process (1.3) becomes the following modified Mann iteration process (cf. Mann [10]):
x 1 C , x n + 1 = ( 1 α n ) x n + α n T n x n ,
(1.4)

where { α n } is a real sequence in [ 0 , 1 ] .

Rhoades [1] proved the following results which extended Theorems 1.5 and 2.3 of Schu [8] to uniformly convex Banach spaces.

Theorem 1.1 Let X be a uniformly convex Banach space, let C be a nonempty bounded closed convex subset of X, and let T : C C be a completely continuous asymptotically nonexpansive mapping with { k n } satisfying k n 1 , n = 1 ( k n r 1 ) < , r = max { 2 , p } . Then, for any x 1 C , the sequence { x n } defined by (1.4), where { α n } satisfies a α n 1 a for all n 1 and some a > 0 , converges strongly to some fixed point of T.

Theorem 1.2 Let X be a uniformly convex Banach space, let C be a nonempty bounded closed convex subset of E, and let T : C C be a completely continuous asymptotically nonexpansive mapping with { k n } satisfying k n 1 , n = 1 ( k n r 1 ) < , r = max { 2 , p } . Then, for any x 1 C , the sequence { x n } defined by (1.3), where { α n } , { β n } satisfy a ( 1 α n ) , ( 1 β n ) 1 a for all n 1 and some a > 0 , converges strongly to some fixed point of T.

On the other hand, Kim [11] proved the following result which generalized Theorem 1 of Senter and Dotson [12].

Theorem 1.3 Let X be a real uniformly convex Banach space, let C be a nonempty closed convex subset of X, and let T be a nonexpansive mapping of C into itself satisfying condition (A) with F ( T ) . Suppose that for any x 1 in C, the sequence { x n } is defined by x n + 1 = ( 1 α n ) x n + α n [ β n x n + ( 1 β n ) T x n ] , for all n 1 , where { α n } and { β n } are sequences in [ 0 , 1 ] such that n = 1 α n ( 1 α n ) = and n = 1 β n < . Then { x n } converges strongly to some fixed point of T.

In this paper, we prove that if T is a total asymptotically nonexpansive self-mapping satisfying condition (A), the iteration { x n } defined by (1.3) converges strongly to some fixed point of T, which generalizes the results due to Rhoades [1].

2 Preliminaries

Throughout this paper, we denote by X a real Banach space. Let C be a nonempty closed convex subset of X, and let T be a mapping from C into itself. Then we denote by F ( T ) the set of all fixed points of T, i.e., F ( T ) = { x C : T x = x } . We also denote by a b : = max { a , b } . A Banach space X is said to be uniformly convex if the modulus of convexity δ X = δ X ( ϵ ) , 0 < ϵ 2 , of X defined by
δ X ( ϵ ) = inf { 1 x + y 2 : x , y X , x 1 , y 1 , x y ϵ }

satisfies the inequality δ X ( ϵ ) > 0 for every ϵ ( 0 , 2 ] . When { x n } is a sequence in X, then x n x will denote strong convergence of the sequence { x n } to x.

Definition 2.1 [12]

A mapping T : C C with F ( T ) is said to satisfy condition (A) if there exists a nondecreasing function f : [ 0 , ) [ 0 , ) with f ( 0 ) = 0 and f ( r ) > 0 for all r ( 0 , ) such that
x T x f ( d ( x , F ( T ) ) )

for all x C , where d ( x , F ( T ) ) = inf z F ( T ) x z .

3 Strong convergence theorem

We first begin with the following lemma.

Lemma 3.1 [13]

Let { a n } , { b n } and { c n } be sequences of nonnegative real numbers such that n = 1 b n < , n = 1 c n < and
a n + 1 ( 1 + b n ) a n + c n

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

Lemma 3.2 [14]

Let X be a uniformly convex Banach space. Let x , y X . If x 1 , y 1 and x y ϵ > 0 , then λ x + ( 1 λ ) y 1 2 λ ( 1 λ ) δ ( ϵ ) for 0 λ 1 .

Lemma 3.3 Let C be a nonempty closed convex subset of a uniformly convex Banach space X, and let T : C C be a TAN mapping with F ( T ) . Suppose that { c n } , { d n } and ϕ satisfy the following two conditions:
  1. (I)

    α , β > 0 such that ϕ ( t ) α t for all t β .

     
  2. (II)

    n = 1 c n < , n = 1 d n < .

     

Suppose that the sequence { x n } is defined by (1.3). Then lim n x n z exists for any z F ( T ) .

Proof For any z F ( T ) , we set
M : = 1 ϕ ( β ) < .
From (I) and strict increasing of ϕ, we obtain
ϕ ( t ) ϕ ( β ) + α t , t 0 .
(3.1)
By using (3.1), we have
T n x n z x n z + c n ϕ ( x n z ) + d n x n z + c n { ϕ ( β ) + α x n z } + d n ( 1 + α c n ) x n z + κ n M ,
where κ n = c n + d n and n = 1 κ n < . Since
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 β n { ( 1 + α c n ) x n z + κ n M } + ( 1 β n ) x n z ( 1 + α c n ) x n z + κ n M ,
and thus
y n z + c n ϕ ( y n z ) ( 1 + α c n ) x n z + κ n M + c n { ϕ ( β ) + α y n z } ( 1 + α c n ) x n z + κ n M + c n ϕ ( β ) + α c n ( 1 + α c n ) x n z + α c n κ n M ( 1 + σ n ) x n z + δ n M ,
where σ n = 2 α c n + α 2 c n 2 , δ n = κ n + c n + α c n κ n , n = 1 σ n < and n = 1 δ n < . So, we have
T n y n z y n z + c n ϕ ( y n z ) + d n ( 1 + σ n ) x n z + δ n M + d n ( 1 + σ n ) x n z + η n M ,
where η n = δ n + d n and n = 1 η n < . Hence
x n + 1 z = ( 1 α n ) x n + 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 { ( 1 + σ n ) x n z + η n M } ( 1 + σ n ) x n z + η n M .

By Lemma 3.1, we see that lim n x n z exists. □

Theorem 3.4 Let X be a uniformly convex Banach space, and let C be a nonempty closed convex subset of X. Let T : C C be a uniformly continuous and TAN mapping with F ( T ) . Suppose that { c n } , { d n } and ϕ satisfy the following two conditions:
  1. (I)

    α , β > 0 such that ϕ ( t ) α t for all t β .

     
  2. (II)

    n = 1 c n < , n = 1 d n < .

     

Suppose that for any x 1 in C, the sequence { x n } defined by (1.3) satisfies n = 1 α n ( 1 α n ) = and lim β n = 0 . Then { x n } converges strongly to some fixed point of T.

Proof For any z F ( T ) , by Lemma 3.3, { x n } is bounded. We set
M : = 1 ϕ ( β ) sup n 1 x n z < .
By Lemma 3.3, we see that lim n x n z ( r ) exists. Without loss of generality, we assume r > 0 . As in the proof of Lemma 3.3, we obtain
T n y n z ( 1 + σ n ) x n z + η n M x n z + ν n M ,
where ν n = σ n + η n and n = 1 ν n < . By using Lemma 3.2 and Takahashi [15], we obtain
x n + 1 z = ( 1 α n ) x n + α n T n y n z = ( 1 α n ) ( x n z ) + α n ( T n y n z ) ( x n z + ν n M ) [ 1 2 α n ( 1 α n ) δ X ( T n y n x n x n z + ν n M ) ] .
Hence we obtain
2 α n ( 1 α n ) ( x n z + ν n M ) δ X ( T n y n x n x n z + ν n M ) x n z x n + 1 z + ν n M .
Thus
2 α n ( 1 α n ) ( x n z + ν n M ) δ X ( T n y n x n x n z + ν n M ) < .
Since δ X is strictly increasing, continuous and n = 1 α n ( 1 α n ) = , we obtain
lim inf n T n y n x n = 0 .
(3.2)
By using (3.1) in the proof of Lemma 3.3, we have
T n 1 x n 1 z x n 1 z + c n 1 ϕ ( x n 1 z ) + d n 1 x n 1 z + c n 1 { ϕ ( β ) + α x n 1 z } + d n 1 ( 1 + α c n 1 ) x n 1 z + ρ n 1 M ,
where ρ n 1 = c n 1 + d n 1 and n = 2 ρ n 1 < . Thus
y n 1 z = β n 1 T n 1 x n 1 + ( 1 β n 1 ) x n 1 z β n 1 T n 1 x n 1 z + ( 1 β n 1 ) x n 1 z β n 1 { ( 1 + α c n 1 ) x n 1 z + ρ n 1 M } + ( 1 β n 1 ) x n 1 z ( 1 + α c n 1 ) x n 1 z + ρ n 1 M ,
and hence
y n 1 z + c n 1 ϕ ( y n 1 z ) ( 1 + α c n 1 ) x n 1 z + ρ n 1 M + c n 1 { ϕ ( β ) + α y n 1 z } ( 1 + α c n 1 ) x n 1 z + ρ n 1 M + c n 1 ϕ ( β ) + α c n 1 ( 1 + α c n 1 ) x n 1 z + α c n 1 ρ n 1 M ( 1 + μ n 1 ) x n 1 z + φ n 1 M ,
where μ n 1 = 2 α c n 1 + α 2 c n 1 2 , φ n 1 = ρ n 1 + c n 1 + α c n 1 ρ n 1 , n = 2 μ n 1 < and n = 2 φ n 1 < . So, we have
T n 1 y n 1 z y n 1 z + c n 1 ϕ ( y n 1 z ) + d n 1 ( 1 + μ n 1 ) x n 1 z + φ n 1 M + d n 1 x n 1 z + ω n 1 M ,
where ω n 1 = μ n 1 + φ n 1 + d n 1 and n = 2 ω n 1 < . By using Lemma 3.2 and Takahashi [15], we obtain
x n z = ( 1 α n 1 ) x n 1 + α n 1 T n 1 y n 1 z = ( 1 α n 1 ) ( x n 1 z ) + α n 1 ( T n 1 y n 1 z ) ( x n 1 z + ω n 1 M ) [ 1 2 α n ( 1 α n ) δ X ( T n 1 y n 1 x n 1 x n 1 z + ω n 1 M ) ] .
By the same method as above, we obtain
lim inf n T n 1 y n 1 x n 1 = 0 .
(3.3)
Since { x n } is bounded and T is a TAN mapping, we obtain
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 β n M ,
where M = sup n 1 T n x n x n < . By using lim β n = 0 , we have
lim n x n y n = 0 .
(3.4)
Since
T n y n y n T n y n x n + x n y n ,
by (3.2) and (3.4), we obtain
lim inf n T n y n y n = 0 .
(3.5)
By using (3.3) and (3.4), we obtain
lim inf n T n 1 y n 1 y n 1 = 0 .
(3.6)
Since
T n 1 x n 1 x n 1 T n 1 x n 1 T n 1 y n 1 + T n 1 y n 1 x n 1 x n 1 y n 1 + c n 1 ϕ ( x n 1 y n 1 ) + d n 1 + T n 1 y n 1 x n 1 ,
by using (3.3) and (3.4), we have
lim inf n T n 1 x n 1 x n 1 = 0 .
(3.7)
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 (3.4) and (3.6), we get
lim inf n x n x n 1 = 0 .
(3.8)
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 ϕ ( x n x n 1 ) + d n 1 + T n 1 x n 1 x n 1 ,
by (3.7) and (3.8), we obtain
lim inf n T n 1 x n x n = 0 .
(3.9)
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 ϕ ( x n y n ) + d n + T n x n T x n
and by the uniform continuity of T, (3.4), (3.5) and (3.9), we have
lim inf n x n T x n = 0 .
(3.10)
By using condition (A), we obtain
f ( d ( x n , F ( T ) ) ) x n T x n
(3.11)
for all n 1 . As in the proof of Lemma 3.3, we obtain
x n + 1 z ( 1 + σ n ) x n z + η n M .
(3.12)
Thus
inf z F ( T ) x n + 1 z ( 1 + σ n ) inf z F ( T ) x n z + η n M .
By using Lemma 3.1, we see that lim n d ( x n , F ( T ) ) ( c ) exists. We first claim that lim n d ( x n , F ( T ) ) = 0 . In fact, assume that c = lim n d ( x n , F ( T ) ) > 0 . Then we can choose n 0 N such that 0 < c 2 < d ( x n , F ( T ) ) for all n n 0 . By using condition (A), (3.10) and (3.11), we obtain
0 < f ( c 2 ) f ( d ( x n i , F ( T ) ) ) x n i T x n i 0
as i . This is a contradiction. So, we obtain c = 0 . Next, we claim that { x n } is a Cauchy sequence. Since n = 1 σ n < , we obtain n = 1 ( 1 + σ n ) : = U < . Let ϵ > 0 be given. Since lim n d ( x n , F ( T ) ) = 0 and n = 1 η n < , there exists n 0 N such that for all n n 0 , we obtain
d ( x n , F ( T ) ) < ϵ 4 U + 4 and i = n 0 η i < ϵ 4 M .
(3.13)
Let n , m n 0 and p F ( T ) . Then, by (3.12), we obtain
x n x m x n p + x m p i = n 0 n 1 ( 1 + σ i ) x n 0 p + M i = n 0 n 1 η i + i = n 0 m 1 ( 1 + σ i ) x n 0 p + M i = n 0 m 1 η i 2 [ i = n 0 ( 1 + σ i ) x n 0 p + M i = n 0 η i ] .
Taking the infimum over all p F ( T ) on both sides and by (3.13), we obtain
x n x m 2 [ i = n 0 ( 1 + σ i ) d ( x n 0 , F ( T ) ) + M i = n 0 η i ] < 2 [ ( U + 1 ) ϵ 4 U + 4 + M ϵ 4 M ] = ϵ

for all n , m n 0 . This implies that { x n } is a Cauchy sequence. Let lim n x n = q . Then d ( q , F ( T ) ) = 0 . Since F ( T ) is closed, we obtain q F ( T ) . Hence { x n } converges strongly to some fixed point of T. □

Remark 3.5 If T : C C is completely continuous, then it satisfies demicompact and, if T is continuous and demicompact, it satisfies condition (A); see Senter and Dotson [12].

Remark 3.6 If { α n } is bounded away from both 0 and 1, i.e., a α n b for all n 1 and some a , b ( 0 , 1 ) , then n = 1 α n ( 1 α n ) = and lim n β n = 0 hold. However, the converse is not true. For example, consider α n = 1 n .

We give an example of a mapping T : C C which satisfies all the assumptions of T in Theorem 3.4, i.e., T : C C is a uniformly continuous mapping with F ( T ) which is TAN on C, not Lipschitzian and hence not asymptotically nonexpansive.

Example 3.7 Let X : = R and C : = [ 0 , 2 ] . Define T : C C by
T x = { 1 , x [ 0 , 1 ] ; 1 3 4 x 2 , x [ 1 , 2 ] .
Note that T n x = 1 for all x C and n 2 and F ( T ) = { 1 } . Clearly, T is both uniformly continuous and TAN on C. We show that T satisfies condition (A). In fact, if x [ 0 , 1 ] , then | x 1 | = | x T x | . Similarly, if x [ 1 , 2 ] , then
| x 1 | = x 1 x 1 3 4 x 2 = | x T x | .
So, we get d ( x , F ( T ) ) = | x 1 | | x T x | for all x 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 x = 2 1 3 ( L + 1 ) 2 > 1 and y = 2 , then
1 3 4 x 2 L ( 2 x ) 1 3 L 2 2 x 2 + x = 1 12 L 2 + 24 L + 1 .

This is a contradiction.

Declarations

Acknowledgements

The author would like to express their sincere appreciation to the anonymous referees 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. 2013

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