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General composite implicit iteration process for a finite family of asymptotically pseudo-contractive mappings

Abstract

In this paper, a modified general composite implicit iteration process is used to study the convergence of a finite family of asymptotically nonexpansive mappings. Weak and strong convergence theorems have been proved, in the framework of a Banach space.

MSC:47H09, 47H10.

1 Introduction

Let K be a nonempty subset of a real Banach space E and let J:E→ 2 E ∗ is the normalized duality mapping defined by

J(x)= { f ∈ E ∗ : 〈 x , f 〉 = ∥ x ∥ ∥ f ∥ ; ∥ x ∥ = ∥ f ∥ } ,∀x∈E,

where E ∗ denotes the dual space of E and 〈⋅,⋅〉 denotes the generalized duality pairing.

It is well known that if E ∗ is strictly convex, then J is single valued.

In the sequel, we shall denote the single valued normalized duality mapping by j.

Let K be a nonempty subset of E. A mapping T:K→K is said to be L-Lipschitzian if there exists a constant L>0 such that for all x,y∈K, we have ∥Tx−Ty∥≤L∥x−y∥. It is said to be nonexpansive if ∥Tx−Ty∥≤∥x−y∥, for all x,y∈K. T is called asymptotically nonexpansive [1] if there exists a sequence { h n }⊆[1,∞) with lim n → ∞ h n =1 such that ∥ T n x− T n y∥≤ h n ∥x−y∥, for all integers n≥1 and all x,y∈K.

A mapping T is said to be pseudo-contractive [2, 3], if there exists j(x−y)∈J(x−y) such that 〈Tx−Ty,j(x−y)〉≤ ∥ x − y ∥ 2 , for all x,y∈K. T is called strongly pseudo-contractive, if there exists a constant β∈(0,1), j(x−y)∈J(x−y) such that 〈Tx−Ty,j(x−y)〉≤β ∥ x − y ∥ 2 , for all x,y∈K. It is said to be asymptotically pseudo-contractive [4] if there exists a sequence { h n }⊆[1,∞) with lim n → ∞ h n =1 and j(x−y)∈J(x−y) such that

〈 T n x − T n y , j ( x − y ) 〉 ≤ h n ∥ x − y ∥ 2 ,∀x,y∈K,∀n≥1.
(1.1)

It follows from Kato [5] that

∥x−y∥≤ ∥ x − y + r [ ( h n I − T n ) x − ( h n I − T n ) y ] ∥ ,∀x,y∈K,∀n≥1,r>0.
(1.2)

We use F(T) to denote the set of fixed points of T; that is, F(T)={x∈K:x=Tx}.

It follows from the definition that if T is asymptotically nonexpansive, then for all j(x−y)∈J(x−y),

〈 T n x − T n y , j ( x − y ) 〉 =∥x−y∥ ∥ T n x − T n y ∥ ≤ h n ∥ x − y ∥ 2 .

Hence every asymptotically nonexpansive mapping is asymptotically pseudo-contractive.

It can be observed from the definition that an asymptotically nonexpansive mapping is uniformly L-Lipschitzian, where L= sup n ≥ 1 { h n }.

Now consider an example of non-Lipschitzian mapping due to Rhoades [6]. Define a mapping T:[0,1]→[0,1] by the formula Tx= { 1 − x 2 3 } 3 2 , for x∈[0,1]. Schu [4] used this example to show that the class of asymptotically nonexpansive mappings is a subclass of the class of pseudo-contractive mappings. Since T is not Lipschitzian, it cannot be asymptotically nonexpansive. Also T 2 is the identity mapping and T is monotonically decreasing, and it follows that

|x−y|| T n x− T n y|= | x − y | 2 for all n=2m,m∈N

and

( x − y ) ( T n x − T n y ) = ( x − y ) ( T x − T y ) ≤ 0 ≤ | x − y | 2 for all  n = 2 m − 1 , m ∈ N .

Hence T is asymptotically pseudo-contractive mapping with constant sequence {1}.

The iterative approximation problems for a nonexpansive mapping, an asymptotically nonexpansive mapping, and an asymptotically pseudo-contractive mapping were studied extensively by Browder [7], Kirk [8], Goebel and Kirk [1], Schu [4], Xu [9, 10], Liu [11] in the setting of Hilbert space or uniformly convex Banach space.

In 2001, Xu and Ori [12] introduced the following implicit iteration process for a finite family of nonexpansive self-mappings in Hilbert space:

{ x 0 ∈ K arbitrary , x n = α n x n − 1 + ( 1 − α n ) T n x n , n ≥ 1 ,
(1.3)

where { α n } be a sequence in (0,1) and T n = T n mod N . They proved in [12] that the sequence { x n } converges weakly to a common fixed point of T n , n=1,2,…,N.

Later on Osilike and Akuchu [13], and Chen et al. [14] extended the iteration process (1.3) to a finite family of asymptotically pseudo-contractive mapping and a finite family of continuous pseudo-contractive self-mapping, respectively. Zhou and Chang [15] studied the convergence of a modified implicit iteration process to the common fixed point of a finite family of asymptotically nonexpansive mappings. Then Su and Li [16], and Su and Qin [17] introduced the composite implicit iteration process and the general iteration algorithm, respectively, which properly include the implicit iteration process. Recently, Beg and Thakur [18] introduced a modified general composite implicit iteration process for a finite family of random asymptotically nonexpansive mapping and proved strong convergence theorems.

The purpose of this paper is to consider a finite family { T i } i = 1 N of asymptotically pseudo-contractive mappings and to establish convergence results in Banach spaces based on the modified general composite implicit iteration:

For x 0 ∈K, construct a sequence { x n } by

x n = α n x n − 1 + ( 1 − α n ) T i ( n ) k ( n ) y n , y n = r n x n + s n x n − 1 + t n T i ( n ) k ( n ) x n + w n T i ( n ) k ( n ) x n − 1
(1.4)

for each n≥1, which can be written as n=(k(n)−1)N+i(n), where i(n)=1,2,…,N and k(n)≥1 is a positive integer, with k(n)→∞ as n→∞. The sequences { α n }, { r n }, { s n }, { t n } and { w n } are in (0,1) such that r n + s n + t n + w n =1 for all n≥1.

2 Preliminaries

In what follows we shall use the following results.

Lemma 2.1 [19]

Let E be a Banach space, K be a nonempty closed convex subset of E, and T:K→K be a continuous and strong pseudo-contraction. Then T has a unique fixed point.

Lemma 2.2 [20]

Let { a n }, { b n }, and { c n } be three nonnegative sequences satisfying the following condition:

a n + 1 ≤(1+ b n ) a n + c n for all n≥ n 0 ,

where n 0 is some nonnegative integer, ∑ n = 0 ∞ b n <∞ and ∑ n = 0 ∞ c n <∞.

Then

  1. (i)

    lim n → ∞ a n exists;

  2. (ii)

    if, in addition, there exists a subsequence { a n i }⊂{ a n } such that a n i →0, then a n →0 as n→∞.

Lemma 2.3 [21]

Let E be a uniformly convex Banach space and let a, b be two constants with 0<a<b<1. Suppose that { t n }⊂[a,b] is a real sequence and { x n }, { y n } are two sequences in E. Then the conditions

lim sup n → ∞ ∥ x n ∥≤d, lim sup n → ∞ ∥ y n ∥≤dand lim n → ∞ ∥ t n x n + ( 1 − t n ) y n ∥ =d

imply that lim n → ∞ ∥ x n − y n ∥=0, where d≥0 is some constant.

Lemma 2.4 [22]

Let E be a reflexive smooth Banach space with a weakly sequential continuous duality mapping J. Let K be a nonempty bounded and closed convex subset of E and T:K→K be a uniformly L-Lipschitzian and asymptotical pseudo-contraction. Then I−T is demiclosed at zero, where I is the identical mapping.

We shall denote weak convergence by ⇀ and strong convergence by →.

A Banach space E is said to satisfy Opial’s condition if for any sequence { x n }∈E, x n ⇀x as n→∞ implies

lim sup n → ∞ ∥ x n −x∥< lim sup n → ∞ ∥ x n −y∥,∀y∈E with x≠y.

We know that a Banach space with a sequentially continuous duality mapping satisfies Opial’s condition (for details, see [23]).

3 The main results

Throughout this section, E is a uniformly convex Banach space, K a nonempty closed convex subset of E. ℕ denotes the set of natural numbers and I={1,2,…,N}, the set of the first N natural numbers. T i (i∈I) are N uniformly Lipschitzian asymptotically pseudo-contractive self-mappings on K. Let F= ⋂ i ∈ I F( T i )≠∅.

Since T i (i∈I) are uniformly Lipschitzian, there exist constants L i >0 such that ∥ T i n x− T i n y∥≤ L i ∥x−y∥, for all x,y∈K, n∈N and i∈I. Also, since T i (i∈I) are asymptotically pseudo-contractive; therefore there exist sequences { h n ( i ) } such that 〈 T i n x− T i n y,j(x−y)〉≤ h n ( i ) ∥ x − y ∥ 2 for all x,y∈K and i∈I.

Take L= max i ∈ I ( L i ) and h n = max i ∈ I ( h n ( i ) ).

Before presenting the main results, we first show that the proposed iteration (1.4) is well defined.

Let T be uniformly Lipschitzian asymptotically pseudo-contractive mapping. For every fixed u∈K and α∈( L + L 2 L + L 2 + 1 ,1), define a mapping S n :K→K by the formula

S n x = α u + ( 1 − α ) T n a , a = r x + s u + t T n x + w T n u  for all  x ∈ K ,
(3.1)

where α,r,s,t,w∈(0,1), with (1−α)(L+ L 2 )<1.

Then, for all x,y∈K, j(x−y)∈J(x−y), we have

S n y = α u + ( 1 − α ) T n b , b = r y + s u + t T n y + w T n u  for all  x ∈ K .
(3.2)

Now

〈 T n a − T n b , j ( x − y ) 〉 = ∥ T n a − T n b ∥ ∥ x − y ∥ ≤ L ∥ a − b ∥ ∥ x − y ∥ = L ∥ r ( x − y ) + t ( T n x − T n y ) ∥ ∥ x − y ∥ ≤ L ( r ∥ x − y ∥ + t L ∥ x − y ∥ ) ∥ x − y ∥ = ( L r + t L 2 ) ∥ x − y ∥ 2 ≤ ( L + L 2 ) ∥ x − y ∥ 2 ,

so

〈 S n x − S n y , j ( x − y ) 〉 = ( 1 − α ) 〈 T n a − T n b , j ( x − y ) 〉 ≤ ( 1 − α ) ( L + L 2 ) ∥ x − y ∥ 2 .

Since (1−α)(L+ L 2 )∈(0,1), S n is strongly pseudo-contractive, which is also continuous, by Lemma 2.1, S n has a unique fixed point x ∗ ∈K, i.e.

S n x ∗ = α u + ( 1 − α ) T n a , a = r x ∗ + s u + t T n x ∗ + w T n u  for all  x ∈ K .
(3.3)

Thus the implicit iteration (1.4) is defined in K for a finite family { T i } of uniformly Lipschitzian asymptotically pseudo-contractive self-mappings on K, provided α n ∈(α,1), where α= L + L 2 L + L 2 + 1 , for all n∈N, L= max i ∈ I ( L i ).

Lemma 3.1 Let E, K, and T i (i∈I) be as defined above and let { x n } be the sequence defined by (1.4), where { α n } is a sequence of real numbers such that 0<α< α n ≤β<1 for α= L + L 2 L + L 2 + 1 and β is some constant and satisfying the conditions ∑ n = 1 ∞ (1− α n )<∞ and lim n → ∞ h n − 1 1 − α n =0. Let b>0 be a real number such that t n + w n ≤b/L<1. Then

  1. (i)

    lim n → ∞ ∥ x n −p∥ exists, for all p∈F,

  2. (ii)

    lim n → ∞ d( x n ,F) exists, where d( x n ,F)= inf p ∈ F ∥ x n −p∥,

  3. (iii)

    lim n → ∞ ∥ x n − T l x n ∥=0, ∀l∈I.

Proof Let p∈F. Using (1.4), we have

∥ x n − p ∥ 2 = 〈 x n − p , j ( x n − p ) 〉 ≤ α n 〈 x n − 1 − p , j ( x n − p ) 〉 + ( 1 − α n ) 〈 T i ( n ) k ( n ) y n − T i ( n ) k ( n ) x n , j ( x n − p ) 〉 + ( 1 − α n ) h k ( n ) ∥ x n − p ∥ 2 = α n ∥ x n − 1 − p ∥ ∥ x n − p ∥ + ( 1 − α n ) L ∥ y n − x n ∥ ∥ x n − p ∥ + ( 1 − α n ) h k ( n ) ∥ x n − p ∥ 2 .
(3.4)

Using (1.4), we obtain

∥ y n − x n ∥ = ∥ s n ( x n − 1 − x n ) + t n ( T i ( n ) k ( n ) x n − x n ) + w n ( T i ( n ) k ( n ) x n − 1 − x n ) ∥ ≤ s n ∥ x n − 1 − p ∥ + s n ∥ x n − p ∥ + t n L ∥ x n − p ∥ + t n ∥ x n − p ∥ + w n L ∥ x n − 1 − p ∥ + w n ∥ x n − p ∥ .
(3.5)

Substituting (3.5) in (3.4), we get

∥ x n − p ∥ 2 ≤ ( α n + ( 1 − α n ) L ( s n + w n L ) ) ∥ x n − 1 − p ∥ ∥ x n − p ∥ + ( 1 − α n ) [ ( s n + t n + w n + t n L ) L + h k ( n ) ] ∥ x n − p ∥ 2 ≤ ( α n + ( 1 − α n ) ( 1 + L ) L ) ∥ x n − 1 − p ∥ ∥ x n − p ∥ + ( 1 − α n ) [ ( 1 + L ) L + h k ( n ) ] ∥ x n − p ∥ 2 ≤ ( α n + ( 1 − α n ) ( 1 + L ) L ) ∥ x n − 1 − p ∥ ∥ x n − p ∥ + [ ( 1 − α n ) ( 1 + L ) L + ( 1 − α n + μ k ( n ) ) ] ∥ x n − p ∥ 2 ,
(3.6)

where μ k ( n ) = h k ( n ) −1 for all n≥1, by condition ∑ n = 1 ∞ ( h k ( n ) −1)<∞, we have ∑ n = 1 ∞ μ k ( n ) <∞.

Therefore, we have

∥ x n − p ∥ ≤ ( α n + ( 1 − α n ) ( 1 + L ) L ) α n − μ k ( n ) − ( 1 − α n ) ( 1 + L ) L ∥ x n − 1 − p ∥ ≤ [ 1 + μ k ( n ) + 2 ( 1 − α n ) ( 1 + L ) L α n − μ k ( n ) − ( 1 − α n ) ( 1 + L ) L ] ∥ x n − 1 − p ∥ ≤ [ 1 + μ k ( n ) + 2 ( 1 − α n ) ( 1 + L ) L 1 − ( 1 − α n + μ k ( n ) + ( 1 − α n ) ( 1 + L ) L ) ] ∥ x n − 1 − p ∥ .
(3.7)

Since lim n → ∞ h k ( n ) − 1 1 − α n = lim n → ∞ μ k ( n ) 1 − α n =0, there exists a M such that μ k ( n ) 1 − α n <M.

Now, we consider the second term on the right side of (3.7). We have

( 1 − α n + μ k ( n ) + ( 1 − α n ) ( 1 + L ) L ) ≤(1− α n ) [ 1 + M + ( 1 + L ) L ] .

By condition ∑ n = 1 ∞ (1− α n )<∞, we have lim n → ∞ (1− α n )=0, then there exists a natural number N 1 such that if n> N 1 , then

1− ( 1 − α n + μ k ( n ) + ( 1 − α n ) ( 1 + L ) L ) ≥ 1 2 .

Therefore, it follows from (3.7) that

∥ x n − p ∥ ≤ [ 1 + 2 { μ k ( n ) + 2 ( 1 − α n ) ( 1 + L ) L } ] ∥ x n − 1 − p ∥ = ( 1 + σ n ) ∥ x n − 1 − p ∥ ,
(3.8)

where σ n =2{ μ k ( n ) +2(1− α n )(1+L)L}.

Taking the infimum over p∈F, we have

d( x n ,F)≤(1+ σ n )d( x n − 1 ,F).
(3.9)

Since ∑ n = 1 ∞ μ k ( n ) <∞ and ∑ n = 1 ∞ (1− α n )<∞, we have

∑ n = 1 ∞ σ n <∞.

Thus, by Lemma 2.2, lim n → ∞ ∥ x n −p∥ and lim n → ∞ d( x n ,F) exist.

Without loss of generality, we assume

lim n → ∞ ∥ x n −p∥= d 1 .
(3.10)

Set v k ( n ) = h k ( n ) − 1 h k ( n ) , and from (1.2), we have

∥ x n − p ∥ ≤ ∥ x n − p + 1 − α n 2 α n h k ( n ) [ ( h k ( n ) I − T i ( n ) k ( n ) ) x n − ( h k ( n ) I − T i ( n ) k ( n ) ) p ] ∥ ≤ ∥ x n − p + 1 − α n 2 α n [ α n ( x n − 1 − T i ( n ) k ( n ) x n ) + ( 1 − α n ) ( T i ( n ) k ( n ) y n − T i ( n ) k ( n ) x n ) ] ∥ + ( 1 − α n 2 α n ) ( h k ( n ) − 1 h k ( n ) ) ∥ T i ( n ) k ( n ) x n − p ∥ = ∥ x n − p + 1 − α n 2 ( x n − 1 − T i ( n ) k ( n ) x n ) + ( 1 − α n ) 2 2 α n ( T i ( n ) k ( n ) y n − T i ( n ) k ( n ) x n ) ∥ + ( 1 − α n 2 α n ) v k ( n ) ∥ T i ( n ) k ( n ) x n − p ∥ ≤ ∥ x n − p + 1 2 ( x n − 1 − x n ) ∥ + ( 1 − α n 2 α n ) v k ( n ) ∥ T i ( n ) k ( n ) x n − p ∥ + ( 1 − α n ) 2 2 α n L ∥ y n − x n ∥ ≤ ∥ 1 2 ( x n − p ) + 1 2 ( x n − 1 − p ) ∥ + ( 1 − α n 2 α n ) v k ( n ) ∥ T i ( n ) k ( n ) x n − p ∥ + ( 1 − α n ) 2 2 α n L ∥ y n − x n ∥ .

Thus

lim inf n → ∞ ∥ x n − p ∥ ≤ lim inf n → ∞ ∥ 1 2 ( x n − p ) + 1 2 ( x n − 1 − p ) ∥ + lim inf n → ∞ ( 1 − α n 2 α n ) v k ( n ) ∥ T i ( n ) k ( n ) x n − p ∥ + lim inf n → ∞ ( 1 − α n ) 2 2 α n L ∥ y n − x n ∥ .

Since v k ( n ) = h k ( n ) − 1 h k ( n ) ∈(0,1), we have lim n → ∞ v k ( n ) =0 and from ∑ n = 1 ∞ (1− α n )<∞, we have lim n → ∞ (1− α n )=0 and using (3.10), we have

lim inf n → ∞ ∥ 1 2 ( x n − p ) + 1 2 ( x n − 1 − p ) ∥ ≥ d 1 .
(3.11)

On the other hand, we obtain

lim sup n → ∞ ∥ 1 2 ( x n − p ) + 1 2 ( x n − 1 − p ) ∥ ≤ lim sup n → ∞ [ 1 2 ∥ x n − p ∥ + 1 2 ∥ x n − 1 − p ∥ ] = d 1 ,
(3.12)

from (3.11) and (3.12), we have

lim n → ∞ ∥ 1 2 ( x n − p ) + 1 2 ( x n − 1 − p ) ∥ = d 1 .

It follows from Lemma 2.3 that

lim n → ∞ ∥ x n − x n − 1 ∥=0.
(3.13)

Thus, for any i∈I, we have

lim n → ∞ ∥ x n − x n + i ∥=0.
(3.14)

Since 0<α< α n ≤β<1 and from (1.4) and (3.13), we get

lim n → ∞ ∥ x n − T i ( n ) k ( n ) y n ∥ = lim n → ∞ α n 1 − α n ∥ x n − x n − 1 ∥ ≤ 1 1 − β lim n → ∞ ∥ x n − x n − 1 ∥ = 0 .
(3.15)

On the other hand, from (3.13) and (3.15)

lim n → ∞ ∥ x n − 1 − T i ( n ) k ( n ) y n ∥ ≤ lim n → ∞ ∥ x n − 1 − x n ∥+ lim n → ∞ ∥ x n − T i ( n ) k ( n ) y n ∥ =0.
(3.16)

Now,

∥ T i ( n ) k ( n ) x n − x n ∥ ≤ ∥ x n − x n − 1 ∥ + ∥ T i ( n ) k ( n ) y n − x n − 1 ∥ + ∥ T i ( n ) k ( n ) y n − T i ( n ) k ( n ) x n ∥ ≤ ( 1 + L ) ∥ x n − x n − 1 ∥ + ∥ T i ( n ) k ( n ) y n − x n − 1 ∥ + L ∥ y n − x n − 1 ∥ .
(3.17)

Again, by using (1.4), we obtain

∥ y n − x n − 1 ∥ ≤ ∥ r n x n + s n x n − 1 + t n T i ( n ) k ( n ) x n + w n T i ( n ) k ( n ) x n − 1 − x n − 1 ∥ ≤ t n ∥ T i ( n ) k ( n ) x n − x n ∥ + w n ∥ T i ( n ) k ( n ) x n − 1 − x n ∥ + ( r n + t n + w n ) ∥ x n − x n − 1 ∥ ≤ ( t n + w n ) ∥ T i ( n ) k ( n ) x n − x n ∥ + ( r n + t n + w n + w n L ) ∥ x n − x n − 1 ∥ .
(3.18)

Substituting (3.18) into (3.17), we get

∥ T i ( n ) k ( n ) x n − x n ∥ ≤ ( 1 + L ) ∥ x n − x n − 1 ∥ + ∥ T i ( n ) k ( n ) y n − x n − 1 ∥ + L ( t n + w n ) ∥ T i ( n ) k ( n ) x n − x n ∥ + L ( r n + t n + w n + w n L ) ∥ x n − x n − 1 ∥ .

Since t n + w n ≤b/L<1, the above inequality gives

(1−b) ∥ T i ( n ) k ( n ) x n − x n ∥ ≤ [ 1 + L ( 1 + r n + t n + w n + w n L ) ] ∥ x n − x n − 1 ∥+ ∥ T i ( n ) k ( n ) y n − x n − 1 ∥ .

Then from (3.13), (3.16), and the above inequality, we have

lim n → ∞ ∥ T i ( n ) k ( n ) x n − x n ∥ =0.
(3.19)

From (3.13), (3.18), and (3.19), we get

lim n → ∞ ∥ y n − x n − 1 ∥=0.
(3.20)

On the other hand, from (3.13) and (3.20) we have

lim n → ∞ ∥ y n − x n ∥≤ lim n → ∞ ∥ y n − x n − 1 ∥+ lim n → ∞ ∥ x n − 1 − x n ∥=0.
(3.21)

Since for any positive integer n>N, we can write n=(k(n)−1)N+i(n), i(n)∈I.

Let A n =∥ T i ( n ) k ( n ) y n − x n − 1 ∥, then from (3.16), we have A n →0. Also,

∥ x n − 1 − T n x n ∥ ≤ ∥ x n − 1 − T i ( n ) k ( n ) y n ∥ + ∥ T i ( n ) k ( n ) y n − T n x n ∥ = A n + ∥ T i ( n ) k ( n ) y n − T i ( n ) x n ∥ ≤ A n + L ∥ T i ( n ) k ( n ) − 1 y n − x n ∥ ≤ A n + L { ∥ T i ( n ) k ( n ) − 1 y n − T i ( n − N ) k ( n ) − 1 x n − N ∥ + ∥ T i ( n − N ) k ( n ) − 1 x n − N − T i ( n − N ) k ( n ) − 1 y n − N ∥ + ∥ T i ( n − N ) k ( n ) − 1 y n − N − x ( n − N ) − 1 ∥ + ∥ x ( n − N ) − 1 − x n ∥ } .
(3.22)

Since for each n>N, n=(n−N)(modN) and n=(k(n)−1)N+i(n), n−N=((k(n)−1)−1)N+i(n)=(k(n−N)−1)N+i(n−N), i.e.

k(n−N)=k(n)−1andi(n−N)=i(n).

Therefore from (3.22), we have

∥ x n − 1 − T n x n ∥ ≤ A n + L { ∥ T i ( n ) k ( n ) − 1 y n − T i ( n ) k ( n ) − 1 x n − N ∥ + ∥ T i ( n − N ) k ( n − N ) x n − N − T i ( n − N ) k ( n − N ) y n − N ∥ + ∥ T i ( n − N ) k ( n − N ) y n − N − x ( n − N ) − 1 ∥ + ∥ x ( n − N ) − 1 − x n ∥ } ≤ A n + L { L ∥ y n − x n − N ∥ + L ∥ x n − N − y n − N ∥ + A n − N + ∥ x ( n − N ) − 1 − x n ∥ } ≤ A n + L 2 ( ∥ y n − x n ∥ + ∥ x n − x n − N ∥ + ∥ x n − N − y n − N ∥ ) + L ( A n − N + ∥ x ( n − N ) − 1 − x n ∥ ) .
(3.23)

From (3.14), (3.21), and A n →0, we have

lim n → ∞ ∥ x n − 1 − T n x n ∥=0.
(3.24)

It follows from (3.13) and (3.24) that

lim n → ∞ ∥ x n − T n x n ∥≤ lim n → ∞ { ∥ x n − x n − 1 ∥ + ∥ x n − 1 − T n x n ∥ } =0.
(3.25)

Consequently, for any i∈I, from (3.14), (3.25), we obtain

∥ x n − T n + i x n ∥ ≤ ∥ x n − x n + i ∥ + ∥ x n + i − T n + i x n + i ∥ + ∥ T n + i x n + i − T n + i x n ∥ ≤ ( 1 + L ) ∥ x n − x n + i ∥ + ∥ x n + i − T n + i x n + i ∥ → 0 ,

as n→∞. This implies that the sequence

⋃ i = 1 N { ∥ x n − T n + i x n ∥ } n = 1 ∞ →0,as n→∞.

Since for each l=1,2,…,N, {∥ x n − T l x n ∥} is a subsequence of ⋃ i = 1 N {∥ x n − T n + i x n ∥}, therefore, we have

lim n → ∞ ∥ x n − T l x n ∥=0,∀l∈I.
(3.26)

This completes the proof. □

3.1 Strong convergence theorems

First, we prove necessary and sufficient conditions for the strong convergence of the modified general composite implicit iteration process to a common fixed point of a finite family of asymptotically pseudo-contractive mappings.

Theorem 3.1 Let E, K, and T i (i∈I) be as defined above and { α n } be a sequence of real numbers as in Lemma  3.1. Then the sequence { x n } generated by (1.4) converges strongly to a member of ℱ if and only if lim inf n → ∞ d( x n ,F)=0.

Proof The necessity of the condition is obvious. Thus, we will only prove the sufficiency.

Let lim inf n → ∞ d( x n ,F)=0. Then from (ii) in Lemma 3.1, we have lim n → ∞ d( x n ,F)=0.

Next, we show that { x n } is a Cauchy sequence in K. For any given ε>0, since lim n → ∞ d( x n ,F)=0, there exists a natural number n 1 such that d( x n ,F)<ε/4 when n≥ n 1 .

Since lim n → ∞ ∥ x n −p∥ exists for all p∈F, we have ∥ x n −p∥< M ′ , for all n≥1 and some positive number M ′ .

Furthermore ∑ n = 1 ∞ σ n <∞ implies that there exists a positive integer n 2 such that ∑ j = n ∞ σ j <ε/4 M ′ for all n≥ n 2 . Let N ′ =max{ n 1 , n 2 }. It follows from (3.8) that

∥ x n −p∥≤∥ x n − 1 −p∥+ M ′ σ n .

Now, for all n,m≥ N ′ and for all p∈F, we have

∥ x n − x m ∥ ≤ ∥ x n − p ∥ + ∥ x m − p ∥ ≤ ∥ x N ′ − p ∥ + M ′ ∑ j = N ′ + 1 n σ j + ∥ x N ′ − p ∥ + M ′ ∑ j = N ′ + 1 m σ j ≤ 2 ∥ x N ′ − p ∥ + 2 M ′ ∑ j = N ′ ∞ σ j .

Taking the infimum over all p∈F, we obtain

∥ x n − x m ∥≤2d ( x N ′ , F ) +2 M ′ ∑ j = N ′ ∞ σ j <ε.

This implies that { x n } is a Cauchy sequence. Since E is complete, therefore { x n } is convergent.

Suppose lim n → ∞ x n =q.

Since K is closed, we get q∈K, then { x n } converges strongly to q.

It remains to show that q∈F.

Notice that

|d(q,F)−d( x n ,F)|≤∥q− x n ∥,∀n∈N,

since lim n → ∞ x n =q and lim n → ∞ d( x n ,F)=0, we obtain q∈F.

This completes the proof. □

Corollary 3.1 Suppose that the conditions are the same as in Theorem  3.1. Then the sequence { x n } generated by (1.4) converges strongly to u∈F if and only if { x n } has a subsequence { x n j } which converges strongly to u∈F.

A mapping T:K→K with F(T)≠∅ is said to satisfy condition (A) [24] on K if there exists a nondecreasing function f:[0,∞)→[0,∞), with f(0)=0 and f(r)>r, for all r∈(0,∞), such that for all x∈K,

∥x−Tx∥≥f ( d ( x , F ( T ) ) ) .

A family { T i } i = 1 N of N self-mappings of K with F= ⋂ i ∈ I F( T i )≠∅ is said to satisfy

  1. (1)

    condition (B) on K [25] if there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(r)>r for all r∈(0,∞) such that for all x∈K such that

    max 1 ≤ l ≤ N { ∥ x − T l x ∥ } ≥f ( d ( x , F ) ) ;
  2. (2)

    condition ( C ¯ ) on K [26] if there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(r)>r for all r∈(0,∞) such that for all x∈K such that

    { ∥ x − T l x ∥ } ≥f ( d ( x , F ) )

for at least one T l , l=1,2,…,N or, in other words, at least one of the T l ’s satisfies condition (A).

Condition (B) reduces to condition (A) when all but one of the T l ’s are identities. Also condition (B) and condition ( C ¯ ) are equivalent (see [26]).

Senter and Dotson [24] established a relation between condition (A) and demicompactness that the condition (A) is weaker than demicompactness for a nonexpansive mapping T defined on a bounded set. Every compact operator is demicompact. Since every completely continuous mapping T:K→K is continuous and demicompact, it satisfies condition (A).

Therefore in the next result, instead of complete continuity of mappings T 1 , T 2 ,…, T N , we use condition ( C ¯ ).

Theorem 3.2 Let E and K be as defined above, T i (i∈I) be N asymptotically pseudo-contractive mappings as defined above and satisfying condition ( C ¯ ) and { α n } be a sequence of real numbers as in Lemma  3.1. Then the sequence { x n } generated by (1.4) converges strongly to a member of ℱ.

Proof By Lemma 3.1, we see that lim n → ∞ ∥ x n −p∥ and lim n → ∞ d( x n ,F) exist.

Let one of the T i ’s, say T l , l∈I, satisfy condition (A).

By Lemma 3.1, we have lim n → ∞ ∥ x n − T l x n ∥=0. Therefore we have lim n → ∞ f(d( x n ,F))=0. By the nature of f and the fact that lim n → ∞ d( x n ,F) exists, we have lim n → ∞ d( x n ,F)=0. By Theorem 3.1, we find that { x n } converges strongly to a common fixed point in ℱ.

This completes the proof. □

A mapping T:K→K is said to be semicompact, if the sequence { x n } in K such that ∥ x n −T x n ∥→0, as n→∞, has a convergent subsequence.

Theorem 3.3 Let E and K be as defined above, and let T i (i∈I) be N asymptotically pseudo-contractive mappings as defined above such that one of the mappings in { T i } i = 1 N is semicompact, and let { α n } be a sequence of real numbers as in Lemma  3.1. Then the sequence { x n } generated by (1.4) converges strongly to a member of ℱ.

Proof Without loss of generality, we may assume that T s is semicompact for some fixed s∈{1,2,…,N}. Then by Lemma 3.1, we have lim n → ∞ ∥ x n − T s x n ∥=0. So by definition of semicompactness, there exists a subsequence { x n j } of { x n } such that { x n j } converges strongly to x ∗ ∈K. Now again by Lemma 3.1, we have

lim n j → ∞ ∥ x n j − T l x n j ∥=0

for all l∈I. By continuity of T l , we have T l x n j → T l x ∗ for all l∈I.

Thus lim j → ∞ ∥ x n j − T l x n j ∥=∥ x ∗ − T l x ∗ ∥=0 for all l∈I. This implies that x ∗ ∈F. Also, lim inf n → ∞ d( x n ,F)=0. By Theorem 3.1, we find that { x n } converges strongly to a common fixed point in ℱ. □

3.2 Weak convergence theorem

Theorem 3.4 Let E be a uniformly convex and smooth Banach space which admits a weakly sequentially continuous duality mapping, and let K and T i (i∈I) be as defined above and { α n } be a sequence of real numbers as in Lemma  3.1. Then the sequence { x n } generated by (1.4) converges weakly to a member of ℱ.

Proof Since { x n } is a bounded sequence in K, there exists a subsequence { x n k }⊂{ x n } such that { x n k } converges weakly to q∈K. Hence from Lemma 3.1, we have

lim n → ∞ ∥ x n k − T l x n k ∥=0,∀l∈I.

By Lemma 2.4, we find that (I− T l ) is demiclosed at zero, i.e. (I− T l )q=0, so that q∈F( T l ). By the arbitrariness of l∈I, we know that q∈F= â‹‚ l ∈ I F( T l ).

Next we prove that { x n } converges weakly to q.

If { x n } has another subsequence { x n j } which converges weakly to q 1 ≠q, then we must have q 1 ∈F, and since lim n → ∞ ∥ x n − q 1 ∥ exists and since the Banach space E has a weakly sequentially duality mapping, it satisfies Opial’s condition, and it follows from a standard argument that q 1 =q. Thus { x n } converges weakly to q∈F. □

Remark 3.1 Our results improve and generalize the corresponding results of Browder [7], Kirk [8], Goebel and Kirk [1], Schu [4], Xu [9, 10], Liu [11], Zhou and Chang [15], Osilike [27], Osilike and Akuchu [13], Su and Li [16], Su and Qin [17], and many others.

Let K be a nonempty subset of a real Banach space E. Let D be a nonempty bounded subset of K. The set-measure of noncompactness of D, γ(D), is defined as

γ(D)=inf{d>0:D can be covered by a finite number of sets of diameter≤d}.

The ball-measure of compactness of D, χ(D), is defined as

χ ( D ) = inf { r > 0 : D  can be covered by a finite family of balls with centers in  E and radius  r } .

A bounded continuous mapping T:K→E is called

  1. (1)

    k-set-contractive if γ(T(D))≤kγ(D), for each bounded subset D of K and some constant k≥0;

  2. (2)

    k-set-condensing if γ(T(D))<γ(D), for each bounded subset D of K with γ(D)>0;

  3. (3)

    k-ball-contractive if χ(T(D))≤kχ(D), for each bounded subset D of K and some constant k≥0;

  4. (4)

    k-ball-condensing if χ(T(D))<χ(D), for each bounded subset D of K with χ(D)>0.

A mapping T:K→E is called

  1. (5)

    compact if cl(T(A)) is compact whenever A⊂K is bounded;

  2. (6)

    completely continuous if it maps weakly convergence sequences into strongly convergent sequences;

  3. (7)

    a generalized contraction if for each x∈K there exists k(x)<1 such that ∥Tx−Ty∥≤k(x)∥x−y∥ for all y∈K;

  4. (8)

    a mapping T:E→E is called uniformly strictly contractive (relative to E) if for each x∈E there exists k(x)<1 such that ∥Tx−Ty∥≤k(x)∥x−y∥ for all y∈K. Every k-set-contractive mapping with k<1 is set-condensing and also every compact mapping is set-condensing.

Let K be a nonempty closed bounded subset of E and T:K→E a continuous mapping. Then

  1. (a)

    T is strictly semicontractive if there exists a continuous mapping V:E×E→E with T(x)=V(x,x) for x∈E such that for each x∈E, V(⋅,x) is a k-contraction with k<1 and V(x,⋅) is compact;

  2. (b)

    T is of strictly semicontractive type if there exists a continuous mapping V:K×K→E with T(x)=V(x,x), for x∈K such that for each x∈K, V(⋅,x) is a k-contraction with some k<1 independent of x and x↦V(⋅,x) is compact from K into the space of continuous mapping of K into E with the uniform metric;

  3. (c)

    T is of strongly semicontractive type relative to X if there exists a mapping V:E×K→E with T(x)=V(x,x), for x∈K such that x∈K, V(⋅,x) is uniformly strictly contractive on K relative to E and V(x,⋅) is a completely continuous from K to E, uniformly for x∈K.

For details refer to [28–30].

Let K be a nonempty closed convex bounded subset of a uniformly convex Banach space E. Suppose T:K→K. Then T is semicompact if T satisfies any one of the following conditions [[25], Proposition 3.4]:

  1. (i)

    T is either set-condensing or ball-condensing (or compact);

  2. (ii)

    T is a generalized contraction;

  3. (iii)

    T is uniformly strictly contractive;

  4. (iv)

    T is strictly semicontractive;

  5. (v)

    T is of strictly semicontractive type;

  6. (vi)

    T is of strongly semicontractive type.

Remark 3.2 In view of the above, it is possible to replace the semicompactness assumption in Theorem 3.3 with any of the contractive assumptions (i)-(vi).

We now give an example of asymptotically pseudo-contractive mapping with nonempty fixed point set.

Example 3.1 [31]

Let E=R=(−∞,∞) with usual norm and K=[0,1] and define T:K→K by

Tx= { 0 if  x = 0 , 1 9 if  x = 1 , x − 1 3 n + 1 if  1 3 n + 1 ≤ x < 1 3 ( 1 3 n + 1 + 1 3 n ) , 1 3 n − x if  1 3 ( 1 3 n + 1 + 1 3 n ) ≤ x < 1 3 n

for all n≥0. Then F(T)={0} and for any x∈K, there exists j(x−0)∈J(x−0) satisfying

〈 T n x − T n 0 , j ( x − 0 ) 〉 = T n x⋅x≤ 1 3 ∥ x ∥ 2 < ∥ x ∥ 2

for all n≥1. That is, T is an asymptotically pseudo-contractive mapping with sequence { k n }=1.

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Thakur, B.S., Dewangan, R. & Postolache, M. General composite implicit iteration process for a finite family of asymptotically pseudo-contractive mappings. Fixed Point Theory Appl 2014, 90 (2014). https://doi.org/10.1186/1687-1812-2014-90

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