Open Access

Convergence of the modified Mann's iterative method for asymptotically κ-strictly pseudocontractive mappings

Fixed Point Theory and Applications20112011:100

https://doi.org/10.1186/1687-1812-2011-100

Received: 4 May 2011

Accepted: 9 December 2011

Published: 9 December 2011

Abstract

Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm, and K a nonempty, closed, and convex subset of E. Let T : KK be an asymptotically κ-strictly pseudocontractive mapping with a nonempty fixed point set. We prove that (I - T) is demiclosed at 0 and obtain a weak convergence theorem of the modified Mann's algorithm for T under suitable control conditions. Moreover, we also elicit a necessary and sufficient condition that guarantees strong convergence of the modified Mann's iterative sequence to a fixed point of T in a real Banach spaces with the Fréchet differentiable norm.

2000 AMS Subject Classification: 47H09; 47H10.

Keywords

asymptotically κ-strictly pseudocontractive mappingsdemiclosedness principlethe modified Mann's algorithmfixed points

1 Introduction

Let E and E* be a real Banach space and the dual space of E, respectively. Let K be a nonempty subset of E. Let J denote the normalized duality mapping from E into 2 E * given by J(x) = {f E* : 〈x, f〉 = ||x||2 = ||f||2}, for all x E, where 〈·,·〉 denotes the duality pairing between E and E*. In the sequel, we will denote the set of fixed points of a mapping T : K → K by F (T) = {x K : Tx = x}.

A mapping T : KK is called asymptotically κ-strictly pseudocontractive with sequence { κ n } n = 1 [ 1 , ) such that lim n→∞ κ n = 1 (see, e.g., [13]) if for all x, y K, there exist a constant κ [0, 1) and j(x - y) J(x - y) such that
T n x - T n y , j ( x - y ) κ n x - y 2 - κ x - y - ( T n x - T n y ) 2 , n 1 .
(1)
If I denotes the identity operator, then (1) can be written as
( I - T n ) x - ( I - T n ) y , j ( x - y ) κ ( I - T n ) x - ( I - T n ) y 2 - ( κ n - 1 ) x - y 2 , n 1 .
(2)
The class of asymptotically κ-strictly pseudocontractive mappings was first introduced in Hilbert spaces by Qihou [3]. In Hilbert spaces, j is the identity and it is shown by Osilike et al. [2] that (1) (and hence (2)) is equivalent to the inequality
T n x - T n y 2 λ n x - y 2 + λ x - y - ( T n x - T n y ) 2 ,

where lim n→∞ λ n = lim n→∞ [1 + 2(κ n - 1)] = 1, λ = (1 - 2κ) [0, 1).

A mapping T with domain D(T) and range R(T) in E is called strictly pseudocontractive of Browder-Petryshyn type [4], if for all x, y D(T), there exists κ [0, 1) and j(x - y) J(x - y) such that
T x - T y , j ( x - y ) x - y 2 - κ x - y - ( T x - T y ) 2 .
(3)
If I denotes the identity operator, then (3) can be written as
( I - T ) x - ( I - T ) y , j ( x - y ) κ ( I - T ) x - ( I - T ) y 2 .
(4)
In Hilbert spaces, (3) (and hence (4)) is equivalent to the inequality
T x - T y 2 x - y 2 + k x - y - ( T x - T y ) 2 , k = ( 1 - 2 κ ) < 1 ,

It is shown in [5] that the class of asymptotically κ-strictly pseudocontractive mappings and the class of κ-strictly pseudocontractive mappings are independent.

A mapping 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 , n 1

for all x, y K and is said to be demiclosed at a point p if whenever {x n } D(T) such that {x n } converges weakly to x D(T ) and {Tx n } converges strongly to p, then Tx = p.

Kim and Xu [6] studied weak and strong convergence theorems for the class of asymptotically κ-strictly pseudocontractive mappings in Hilbert space. They obtained a weak convergence theorem of modified Mann iterative processes for this class of mappings. Moreover, a strong convergence theorem was also established in a real Hilbert space by hybrid projection method. They proved the following.

Theorem KX [6] Let K be a closed and convex subset of a Hilbert space H. Let T : KK be an asymptotically κ-strictly pseudocontractive mapping for some 0 κ < 1 with sequence {κ n } [1, ∞) such that n = 1 ( κ n - 1 ) < and F(T ) ≠ . Let { x n } n = 1 be a sequence generated by the modified Mann's iteration method:
x n + 1 = α n x n + ( 1 - α n ) T n x n , n 1 ,

Assume that the control sequence { α n } n = 1 is chosen in such a way that κ + λ ≤ α n 1 - λ for all n, where λ (0, 1) is a small enough constant. Then, {x n } converges weakly to a fixed point of T.

The modified Mann's iteration scheme was introduced by Schu [7, 8] and has been used by several authors (see, for example, [13, 911]). One question is raised naturally: is the result in Theorem KX true in the framework of the much general Banach space?

Osilike et al. [5] proved the convergence theorems of modified Mann iteration method in the framework of q-uniformly smooth Banach spaces which are also uniformly convex. They also obtained that a modified Mann iterative process {x n } converges weakly to a fixed point of T under suitable control conditions. However, the control sequence {α n } [0,1] depended on the Lipschizian constant L and excluded the natural choice α n = 1 n , n 1 . These are motivations for us to improve the results. We prove the demiclosedness principle and weak convergence theorem of the modified Mann's algorithm for T in the framework of uniformly convex Banach spaces which have the Fréchet differentiable norm. Moreover, we also elicit a necessary and sufficient condition that guarantees strong convergence of the modified Mann's iterative sequence to a fixed point of T in a real Banach spaces with the Fréchet differentiable norm.

We will use the notation:
  1. 1.

    for weak convergence.

     
  2. 2.

    ω W ( x n ) = { x : x n j x } denotes the weak ω-limit set of {x n }.

     

2 Preliminaries

Let E be a real Banach space. The space E is called uniformly convex if for each ε > 0, there exists a δ > 0 such that for x, y E with ||x|| ≤ 1, ||y|| ≤ 1, ||x - y|| ≥ ε, we have 1 2 ( x + y ) 1 - δ . The modulus of convexity of E is defined by
δ E ( ε ) = inf { 1 - 1 2 ( x + y ) : x 1 , y 1 , x - y ε , } x , y E
for all ε [0,2]. E is uniformly convex if δ E (0) = 0 and δ E (ε) > 0 for all ε (0, 2]. The modulus of smoothness of E is the function ρ E : [0, ∞) [0, ∞) defined by
ρ E ( τ ) = sup { 1 2 ( x + y + x - y ) - 1 : x 1 , y τ } , x , y E .

E is uniformly smooth if and only if lim τ 0 ρ E ( τ ) τ = 0 .

E is said to have a Fréchet differentiable norm if for all x U = {x E : ||x|| = 1}
lim t 0 x + t y - x t
exists and is attained uniformly in y U. In this case, there exists an increasing function b : [0, ∞) [0, ∞) with lim t 0 [ b ( t ) t ] = 0 such that for all x, h E
1 2 x 2 + h , j ( x ) 1 2 x + h 2 1 2 x 2 + h , j ( x ) + b ( h ) .
(5)

It is well known (see, for example, [[12], p. 107]) that uniformly smooth Banach space has a Fréchet differentiable norm.

Lemma 2.1 [2, p. 80] Let { a n } n = 1 , { b n } n = 1 , { δ n } n = 1 be nonnegative sequences of real numbers satisfying the following inequality
a n + 1 ( 1 + δ n ) a n + b n , n 1 .

If n = 1 δ n < and n = 1 b n < , then lim n →∞ a n exists. If in addition { a n } n = 1 has a subsequence which converges strongly to zero, then lim n →∞ a n = 0.

Lemma 2.2 [2, p. 78] Let E be a real Banach space, K a nonempty subset of E, and T : KK an asymptotically κ-strictly pseudocontractive mapping. Then, T is uniformly L-Lipschitzian.

Lemma 2.3 [[13], p. 29] Let K be a nonempty, closed, convex, and bounded subset of a uniformly convex Banach space E, and let T : KE be a nonexpansive mappings. Let {x n } be a sequence in K such that {x n } converges weakly to some point x K. Then, there exists an increasing continuous function h : [0, ∞) → [0, ∞) with h(0) = 0 depending on the diameter of K such that
h ( x - T x ) liminf n x n - T x n .

Lemma 2.4 [[14], p. 9] Let E be a real Banach space with the Fréchet differentiable norm.

For x E, let β*(t) be defined for 0 < t < ∞ by
β * ( t ) = sup y U x + t y 2 - x 2 t - 2 y , j ( x ) .
Then, lim t →0 + β*(t) = 0 and
x + h 2 x 2 + 2 h , j ( x ) + h β * ( h ) , h E \ { 0 } .
(6)
Remark 2.5 In a real Hilbert space, we can see that β*(t) = t for t > 0. In our more general setting, throughout this article we will still assume that
β * ( t ) 2 t ,

where β* is a function appearing in (6).

Then, we prove the demiclosedness principle of T in the uniformly convex Banach space which has the Fréchet differentiable norm.

Lemma 2.6 Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm. Let K be a nonempty, closed, and convex subset of E and T : KK an asymptotically κ-strictly pseudocontractive mapping with F(T) ≠ . Then, (I - T) is demiclosed at 0.

Proof. Let {x n } be a sequence in K which converges weakly to p K and {x n - Tx n } converges strongly to 0. We prove that (I - T)(p) = 0. Let x* F(T). Then, there exists a constant r > 0 such that ||x n - x*|| ≤ r, n ≥ 1. Let B ̄ r = { x E : x - x * r } , and let C = K B ̄ r . Then, C is nonempty, closed, convex, and bounded, and {x n } C. Choose any α (0, κ) and let T α,n : KK be defined for all x K by
T α , n x = ( 1 - α ) x + α T n x , n 1 ,
Then for all x, y K,
T α , n x - T α , n y 2 = ( x - y ) - α [ ( I - T n ) x - ( I - T n ) y ] 2 x - y 2 - 2 α ( I - T n ) x - ( I - T n ) y , j ( x - y ) + α x - y - ( T n x - T n y ) β * [ α x - y - ( T n x - T n y ) ] x - y 2 - 2 α [ κ x - y - ( T n x - T n y ) 2 - ( κ n - 1 ) x - y 2 ] + 2 α 2 x - y - ( T n x - T n y ) 2 = [ 1 + 2 α ( κ n - 1 ) ] x - y 2 - 2 α ( κ - α ) x - y - ( T n x - T n y ) 2 τ n 2 x - y 2 ,
(7)
where τ n = [ 1 + 2 α ( κ n - 1 ) ] 1 2 . (In fact, in (7) the domain of β*(·) requires ||x - y - (T n x-T n y)|| ≠ 0. But when ||x - y - (T n x-T n y)|| = 0, we have ||T α,n x-T α,n y||2 = ||x - y||2, which still satisfies the inequality T α , n x - T α , n y 2 τ n 2 x - y 2 . So we do not specially emphasize the situation that the argument of β*(·) equals 0 in this inequality and the following proof of Theorem 3.1.) Define G α,m : KE by
G α , m x = 1 τ m T α , m x , m 1 .
Then, G α,m is nonexpansive and it follows from Lemma 2.3 that there exists an increasing continuous function h : [0, ∞) → [0, ∞) with h(0) = 0 depending on the diameter of K such that
h ( p - G α , m p ) liminf n x n - G α , m x n .
(8)
Observe that
x n - G α , m x n = x n - 1 τ m T α , m x n x n - T α , m x n + ( 1 - 1 τ m ) ( τ m x n - x * + x * ) x n - T α , m x n + ( 1 - 1 τ m ) ( τ m r + x * ) ,
(9)
and as n → ∞
x n - T α , m x n = α x n - T m x n j = 1 m T j - 1 x n - T j x n [ 1 + L ( m - 1 ) ] x n - T x n 0 .
(10)
Thus, it follows from (9) and (10) that
limsup n x n - G α , m x n ( 1 - 1 τ m ) ( τ m r + x * ) ,
so that (8) implies that
h ( p - G α , m p ) ( 1 - 1 τ m ) ( τ m r + x * ) .
Observe that
p - G α , m p p - T α , m p - ( 1 - 1 τ m ) T α , m p p - T α , m p - ( 1 - 1 τ m ) ( τ m r + x * ) ,
so that
p - T α , m p p - G α , m p + ( 1 - 1 τ m ) ( τ m r + x * ) h - 1 [ ( 1 - 1 τ m ) ( τ m r + x * ) ] + ( 1 - 1 τ m ) ( τ m r + x * ) 0 , a s m .

Since T is continuous, we have (I - T)(p) = 0, completing the proof of Lemma 2.6.    □

Lemma 2.7 Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm, and let K be a nonempty, closed, and convex subset of E. Let T : KK be an asymptotically κ-strictly pseudocontractive mapping with F(T) ≠ . Let { x n } n = 1 be the sequence satisfying the following conditions:
  1. (a)

    lim n x n - p exists for every p F(T );

     
  2. (b)

    lim n x n - T x n = 0 ;

     
  3. (c)

    lim n t x n + ( 1 - t ) p 1 - p 2 exists for all t [0, 1] and for all p1, p2 F (T ).

     

Then, the sequence {x n } converges weakly to a fixed point of T.

Proof. Since lim n →∞ ||x n - p|| exists, then {x n } is bounded. By (b) and Lemma 2.6, we have ω W ( x n ) F ( T ) . Assume that p 1 , p 2 ω W ( x n ) and that { x n i } and { x m j } are subsequences of {x n } such that x n i p 1 and x m j p 2 , respectively. Since E has the Fréchet differentiable norm, by setting x = p1 - p2, h = t(x n - p1) in (5) we obtain
1 2 p 1 - p 2 2 + t x n - p 1 , j ( p 1 - p 2 ) 1 2 t x n + ( 1 - t ) p 1 - p 2 2 1 2 p 1 - p 2 2 + t x n - p 1 , j ( p 1 - p 2 ) + b ( t x n - p 1 ) ,
where b is an increasing function. Since ||x n - p1|| ≤ M, n ≥ 1, for some M > 0, then
1 2 p 1 - p 2 2 + t x n - p 1 , j ( p 1 - p 2 ) 1 2 t x n + ( 1 - t ) p 1 - p 2 2 1 2 p 1 - p 2 2 + t x n - p 1 , j ( p 1 - p 2 ) + b ( t M ) .
Therefore,
1 2 p 1 - p 2 2 + t limsup n x n - p 1 , j ( p 1 - p 2 ) 1 2 lim n t x n + ( 1 - t ) p 1 - p 2 2 1 2 p 1 - p 2 2 + t liminf n x n - p 1 , j ( p 1 - p 2 ) + b ( t M ) .

Hence, lim sup n x n - p 1 , j ( p 1 - p 2 ) lim inf n x n - p 1 , j ( p 1 - p 2 ) + b ( t M ) t . Since lim t 0 + b ( t M ) t = 0 , then lim n →∞x n - p1, j(p1 - p2)〉 exists. Since lim n →∞x n - p1, j(p1 - p2)〉 = 〈p - p1, j(p1 - p2)〉, for all p ω W ( x n ) . Set p = p2. We have 〈p2 - p1, j(p1 - p2)〉 = 0, that is, p2 = p1. Hence, ω W ( x n ) is singleton, so that {x n } converges weakly to a fixed point of T.    □

3 Main results

Theorem 3.1 Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm, and let K be a nonempty, closed, and convex subset of E. Let T : KK be an asymptotically κ-strictly pseudocontractive mapping for some 0 ≤ κ < 1 with sequence { κ n } n = 1 [ 1 , ) , such that n = 1 ( κ n - 1 ) < , and let F(T) ≠ . Assume that the control sequence { α n } n = 1 is chosen so that

  1. (i*)

    0 < α n < κ, n ≥ 1;

     
  2. (ii*)

    n = 1 α n ( κ - α n ) = . (11)

     
Given x1 K, then the sequence { x n } n = 1 is generated by the modified Mann's algorithm:
x n + 1 = ( 1 - α n ) x n + α n T n x n ,
(12)

converges weakly to a fixed point of T.

Proof. Pick a p F(T). We firstly show that lim n →∞ ||x n - p|| exists. To see this, using (2) and (6), we obtain
x n + 1 - p 2 = ( x n - p ) - α n ( x n - T n x n ) 2 x n - p 2 - 2 α n x n - T n x n , j ( x n - p ) + α n x n - T n x n β * ( α n x n - T n x n ) x n - p 2 - 2 α n [ κ x n - T n x n 2 - ( κ n - 1 ) x n - p 2 ] + 2 α n 2 x n - T n x n 2 = [ 1 + 2 α n ( κ n - 1 ) ] x n - p 2 - 2 α n ( κ - α n ) x n - T n x n 2 .
(13)
Obviously,
x n + 1 - p 2 [ 1 + 2 α n ( κ n - 1 ) ] x n - p 2 .
(14)
Let δ n = 1 + 2α n (κ n - 1). Since n = 1 ( κ n - 1 ) < , we have
n = 1 ( δ n - 1 ) 2 n = 1 ( κ n - 1 ) < ,

then (14) implies lim n →∞ ||x n - p|| exists by Lemma 2.1 (and hence the sequence {||x n - p||} is bounded, that is, there exists a constant M > 0 such that ||x n - p|| < M ).

Then, we prove lim n →∞ ||x n - Tx n || = 0. In fact, it follows from (13) that
n = 1 j 2 α n ( κ - α n ) x n - T n x n 2 n = 1 j ( x n - p 2 - x n + 1 - p 2 ) + n = 1 j [ 2 α n ( κ n - 1 ) ] x n - p 2 n = 1 j ( x n - p 2 - x n + 1 - p 2 ) + n = 1 j ( δ n - 1 ) M 2 .
Then,
n = 1 2 α n ( κ - α n ) x n - T n x n 2 < x 1 - p 2 + M 2 n = 1 ( δ n - 1 ) < .
(15)

Since n = 1 α n ( κ - α n ) = , then (15) implies that lim inf n →∞ ||x n - T n x n || = 0. Thus lim n →∞ ||x n - T n x n || = 0.

By Lemma 2.2 we know that T is uniformly L-Lipschitzian, then there exists a constant L > 0, such that
x n - T x n x n - T n x n + T n x n - T x n x n - T n x n + L T n - 1 x n - x n x n - T n x n + L T n - 1 x n - T n - 1 x n - 1 + L T n - 1 x n - 1 - x n x n - T n x n + L 2 x n - x n - 1 + L T n - 1 x n - 1 - x n - 1 + L x n - x n - 1 x n - T n x n + L ( 2 + L ) T n - 1 x n - 1 - x n - 1

Hence, lim n →∞ ||x n - Tx n || = 0.

Now we prove that for all p1, p2 F(T), lim n →∞ ||tx n + (1 - t)p1 - p2|| exists for all t [0, 1]. Let σ n (t) = ||tx n + (1 - t)p1 - p2||. It is obvious that lim n →∞ σ n (0) = ||p1 - p2|| and lim n →∞ σ n (1) = lim n →∞ ||x n - p2|| exist. So, we only need to consider the case of t (0, 1).

Define T n : KK by
T n x = ( 1 - α n ) x + α n T n x , x K .
Then for all x, y K,
T n x - T n y 2 = ( x - y ) - α n [ ( I - T n ) x - ( I - T n ) y ] 2 x - y 2 - 2 α n ( I - T n ) x - ( I - T n ) y , j ( x - y ) + α n x - y - ( T n x - T n y ) β * [ α n x - y - ( T n x - T n y ) ] x - y 2 - 2 α n [ κ x - y - ( T n x - T n y ) 2 - ( κ n - 1 ) x - y 2 ] + 2 α n 2 x - y - ( T n x - T n y ) 2 = [ 1 + 2 α n ( κ n - 1 ) ] x - y 2 - 2 α n ( κ - α n ) x - y - ( T n x - T n y ) 2 .

By the choice of α n , we have ||T n x - T n y||2 ≤ [1 + 2α n n - 1)]||x - y||2. For the convenience of the following discussing, set λ n = [ 1 + 2 α n ( κ n - 1 ) ] 1 2 , then ||T n x - T n y|| ≤ λ n ||x - y||.

Set S n,m = T n + m -1T n + m -2 ··· T n , m ≥ 1. We have
S n , m x - S n , m y ( j = n n + m - 1 λ j ) x - y f o r a l l x , y K ,
and
S n , m x n = x n + m , S n , m p = p f o r a l l p F ( T ) .
Set b n,m = ||S n,m (tx n + (1 - t)p1) - tS n,m x n - (1 - t)S n,m p1||. If ||x n - p1|| = 0 for some n0, then x n = p1 for any nn0 so that lim n →∞ ||x n - p1|| = 0, in fact {x n } converges strongly to p1 F(T). Thus, we may assume ||x n - p1|| > 0 for any n ≥ 1. Let δ denote the modulus of convexity of E. It is well known (see, for example, [[15], p. 108]) that
t x + ( 1 - t ) y 1 - 2 min { t , ( 1 - t ) } δ ( x - y ) 1 - 2 t ( 1 - t ) δ ( x - y )
(16)
for all t [0, 1] and for all x, y E such that ||x|| ≤ 1, ||y|| ≤ 1. Set
w n , m = S n , m p 1 - S n , m ( t x n + ( 1 - t ) p 1 ) t j = n n + m - 1 λ j x n - p 1 z n , m = S n , m ( t x n + ( 1 - t ) p 1 ) - S n , m x n ( 1 - t ) j = n n + m - 1 λ j x n - p 1
Then, ||w n , m || ≤ 1 and ||z n , m || ≤ 1 so that it follows from (16) that
2 t ( 1 - t ) δ ( w n , m - z n , m ) 1 - t w n , m + ( 1 - t ) z n , m .
(17)
Observe that
w n , m - z n , m = b n , m t ( 1 - t ) ( j = n n + m - 1 λ j ) x n - p 1
and
t w n , m + ( 1 - t ) z n , m = S n , m x n - S n , m p 1 ( j = n n + m - 1 λ j ) x n - p 1 ,
it follows from (17) that
2 t ( 1 - t ) j = n n + m - 1 λ j x n - p 1 δ b n , m t ( 1 - t ) ( j = n n + m - 1 λ j ) x n - p 1 j = n n + m - 1 λ j x n - p 1 - S n , m x n - S n , m p 1 = j = n n + m - 1 λ j x n - p 1 - x n + m - p 1 .
(18)
Since E is uniformly convex, then δ ( s ) s is nondecreasing, and since ( j = n n + m - 1 λ j ) x n - p 1 ( j = n n + m - 1 λ j ) λ n - 1 x n - 1 - p 1 ( j = n n + m - 1 λ j ) ( j = 1 n - 1 λ j ) x 1 - p 1 = ( j = 1 n + m - 1 λ j ) x 1 - p 1 , hence it follows from (18) that
j = 1 n + m - 1 λ j x 1 - p 1 2 δ 4 j = 1 n + m - 1 λ j x 1 - p 1 b n , m j = n n + m - 1 λ j x n - p 1 - x n + m - p 1 since t ( 1 - t ) 1 4 for all t [ 0 , 1 ] .
Since lim n j = 1 n + m - 1 λ j = 1 and since δ(0) = 0 and lim n →∞ ||x n - p1|| exists, then the continuity of δ yields lim n →∞ b n,m = 0 uniformly for all m ≥ 1. Observe that
σ n + m ( t ) t x n + m + ( 1 - t ) p 1 - p 2 + ( S n , m ( t x n + ( 1 - t ) p 1 ) - t S n , m x n - ( 1 - t ) S n , m p 1 ) + S n , m ( t x n + ( 1 - t ) p 1 ) - t S n , m x n - ( 1 - t ) S n , m p 1 = S n , m ( t x n + ( 1 - t ) p 1 ) - S n , m p 2 + b n , m j = n n + m - 1 λ j t x n + ( 1 - t ) p 1 - p 2 + b n , m = j = n n + m - 1 λ j σ n ( t ) + b n , m . (6) 

Hence, lim sup n →∞ σ n (t) ≤ lim inf n →∞ σ n (t), this ensures that lim n →∞ σ n (t) exists for all t (0, 1).

Now, apply Lemma 2.7 to conclude that {x n } converges weakly to a fixed point of T.    □

Theorem 3.2 Let E be a real Banach space with the Fréchet differentiable norm, and let K be a nonempty, closed, and convex subset of E. Let T : KK be an asymptotically κ-strictly pseudocontractive mapping for some 0 ≤ κ < 1 with sequence {κ n } [1, ∞) such that n = 1 ( κ n - 1 ) < , let F(T) ≠ . Let {α n } be a real sequence satisfying the condition (11). Given x1 K, let { x n } n = 1 be the sequence generated by the modified Mann's algorithm (12). Then, the sequence {x n } converges strongly to a fixed point of T if and only if
liminf n d ( x n , F ( T ) ) = 0 ,

where d(x n , F(T)) = inf p F ( T )||x n - p||.

Proof. In the real Banach space E with the Fréchet differentiable norm, we still have
x n + 1 - p 2 δ n x n - p 2 .
(19)

as we have already proved in Theorem 3.1. Thus, [d(x n +1 - p)]2δ n [d(x n - p)]2 and it follows from Lemma 2.1 that lim n -∞ d(x n , F(T )) exists.

Now if {x n } converges strongly to a fixed point p of T, then lim n →∞ ||x n - p|| = 0. Since
0 d ( x n , F ( T ) ) x n - p ,

we have lim inf n →∞ d(x n , F(T )) = 0.

Conversely, suppose lim inf n →∞ d(x n , F(T)) = 0, then the existence of lim n →∞ d(x n , F (T)) implies that lim n →∞ d(x n , F(T)) = 0. Thus, for arbitrary ε > 0 there exists a positive integer n0 such that d ( x n , F ( T ) ) < ε 2 for any nn0.

From (19), we have
x n + 1 - p 2 x n - p 2 + M 2 ( δ n - 1 ) , n 1 ,
and for some M > 0, ||x n - p|| < M. Now, an induction yields
x n - p 2 x n - 1 - p 2 + M 2 ( δ n - 1 - 1 ) x n - 2 - p 2 + M 2 ( δ n - 2 - 1 ) + M 2 ( δ n - 1 - 1 ) x l - p 2 + M 2 j = l n - 1 ( δ j - 1 ) , n - 1 l 1 ,
Since n = 1 ( δ n - 1 ) < , then there exists a positive integer n1 such that j = n ( δ j - 1 ) < ( ε 2 M ) 2 , nn1. Choose N = max{n0, n1}, then for all n, mN + 1 and for all p F (T ) we have
x n - x m x n - p + x m - p [ x N - p 2 + M 2 j = N n - 1 ( δ j - 1 ) ] 1 2 + [ x N - p 2 + M 2 j = N m - 1 ( δ j - 1 ) ] 1 2 [ x N - p 2 + M 2 j = N ( δ j - 1 ) ] 1 2 + [ x N - p 2 + M 2 j = N ( δ j - 1 ) ] 1 2 .
Taking infimum over all p F(T), we obtain
| | x n x m | | { [ d ( x N , F ( T ) ) ] 2 + M 2 j = N ( δ j 1 ) } 1 2 + { [ d ( x N , F ( T ) ) ] 2 + M 2 j = N ( δ j 1 ) } 1 2 < 2 [ ( ε 2 ) 2 + M 2 ( ε 2 M ) 2 ] 1 2 < 2 ε .
Thus, { x n } n = 0 is Cauchy. We can also prove lim n →∞ ||x n - Tx n || = 0 as we have done in Theorem 3.1. Suppose lim n →∞ x n = u. Then,
0 u - T u u - x n + x n - T x n + L x n - u 0 , a s n .

Thus, u F(T).    □

Declarations

Acknowledgements

This study was supported by the Youth Teacher Foundation of North China Electric Power University.

Authors’ Affiliations

(1)
School of Mathematics and Physics, North China Electric Power University
(2)
School of Economics, Renmin University of China
(3)
Easyway Company Limited

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© Zhang and Xie; licensee Springer. 2011

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