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The convergence theorems of Ishikawa iterative process with errors for Φ-hemi-contractive mappings in uniformly smooth Banach spaces

Abstract

Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and T:DD be a generalized Lipschitz Φ-hemi-contractive mapping with qF(T). Let { a n }, { b n }, { c n }, { d n } be four real sequences in [0,1] and satisfy the conditions (i) a n , b n , d n 0 as n and c n =o( a n ); (ii) n = 0 a n =. For some x 0 D, let { u n }, { v n } be any bounded sequences in D, and { x n } be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the fixed point q of T. A related result deals with the operator equations for a generalized Lipschitz and Φ-quasi-accretive mapping.

MSC:47H10.

1 Introduction and preliminary

Let E be a real Banach space and E be its dual space. The normalized duality mapping J:E 2 E is defined by

J(x)= { f E : x , f = x 2 = f 2 } ,xE,

where , denotes the generalized duality pairing. It is well known that

  1. (i)

    If E is a smooth Banach space, then the mapping J is single-valued;

  2. (ii)

    J(αx)=αJ(x) for all xE and α;

  3. (iii)

    If E is a uniformly smooth Banach space, then the mapping J is uniformly continuous on any bounded subset of E. We denote the single-valued normalized duality mapping by j.

Definition 1.1 ([1])

Let D be a nonempty closed convex subset of E, T:DD be a mapping.

  1. (1)

    T is called strongly pseudocontractive if there is a constant k(0,1) such that for all x,yD,

    T x T y , j ( x y ) k x y 2 ;
  2. (2)

    T is called ϕ-strongly pseudocontractive if for all x,yD, there exist j(xy)J(xy) and a strictly increasing continuous function ϕ:[0,+)[0,+) with ϕ(0)=0 such that

    T x T y , j ( x y ) x y 2 ϕ ( x y ) xy;
  3. (3)

    T is called Φ-pseudocontractive if for all x,yD, there exist j(xy)J(xy) and a strictly increasing continuous function Φ:[0,+)[0,+) with Φ(0)=0 such that

    T x T y , j ( x y ) x y 2 Φ ( x y ) .

It is obvious that Φ-pseudocontractive mappings not only include ϕ-strongly pseudocontractive mappings, but also strongly pseudocontractive mappings.

Definition 1.2 ([1])

Let T:DD be a mapping and F(T)={xD:Tx=x}.

  1. (1)

    T is called ϕ-strongly-hemi-pseudocontractive if for all xD, qF(T), there exist j(xq)J(xq) and a strictly increasing continuous function ϕ:[0,+)[0,+) with ϕ(0)=0 such that

    T x T q , j ( x q ) x q 2 ϕ ( x q ) xq.
  2. (2)

    T is called Φ-hemi-pseudocontractive if for all xD, qF(T), there exist j(xq)J(xq) and the strictly increasing continuous function Φ:[0,+)[0,+) with Φ(0)=0 such that

    T x T q , j ( x q ) x q 2 Φ ( x q ) .

Closely related to the class of pseudocontractive-type mappings are those of accretive type.

Definition 1.3 ([1])

Let N(T)={xE:Tx=0}. The mapping T:EE is called strongly quasi-accretive if for all xE, x N(T), there exist j(x x )J(x x ) and a constant k(0,1) such that TxT x ,j(x x )k x x 2 ; T is called ϕ-strongly quasi-accretive if for all xE, x N(T), there exist j(x x )J(x x ) and a strictly increasing continuous function ϕ:[0,+)[0,+) with ϕ(0)=0 such that TxT x ,j(x x )ϕ(x x )x x ; T is called Φ-quasi-accretive if for all xE, x N(T), there exist j(x x )J(x x ) and a strictly increasing continuous function Φ:[0,+)[0,+) with Φ(0)=0 such that TxT x ,j(x x )Φ(x x ).

Definition 1.4 ([2])

For arbitrary given x 0 D, the Ishikawa iterative process with errors { x n } n = 0 is defined by

{ y n = ( 1 b n d n ) x n + b n T x n + d n v n , n 0 , x n + 1 = ( 1 a n c n ) x n + a n T y n + c n u n , n 0 ,
(1.1)

where { u n }, { v n } are any bounded sequences in D; { a n }, { b n }, { c n }, { d n } are four real sequences in [0,1] and satisfy a n + c n 1, b n + d n 1, for all n0. If b n = d n =0, then the sequence { x n } defined by

x n + 1 =(1 a n c n ) x n + a n T x n + c n u n ,n0
(1.2)

is called the Mann iterative process with errors.

Definition 1.5 ([3, 4])

A mapping T:DD is called generalized Lipschitz if there exists a constant L>0 such that TxTyL(1+xy), x,yD.

The aim of this paper is to prove the convergent results of the above Ishikawa and Mann iterations with errors for generalized Lipschitz Φ-hemi-contractive mappings in uniformly smooth real Banach spaces. For this, we need the following lemmas.

Lemma 1.6 ([5])

Let E be a uniformly smooth real Banach space, and let J:E 2 E be a normalized duality mapping. Then

x + y 2 x 2 +2 y , J ( x + y )
(1.3)

for all x,yE.

Lemma 1.7 ([6])

Let { ρ n } n = 0 be a nonnegative sequence which satisfies the following inequality:

ρ n + 1 (1 λ n ) ρ n + σ n ,n0,
(1.4)

where λ n [0,1] with n = 0 λ n =, σ n =o( λ n ). Then ρ n 0 as n.

2 Main results

Theorem 2.1 Let E be an arbitrary uniformly smooth real Banach space, D be a nonempty closed convex subset of E, and T:DD be a generalized Lipschitz Φ-hemi-contractive mapping with qF(T). Let { a n }, { b n }, { c n }, { d n } be four real sequences in [0,1] and satisfy the conditions (i) a n , b n , d n 0 as n and c n =o( a n ); (ii) n = 0 a n =. For some x 0 D, let { u n }, { v n } be any bounded sequences in D, and { x n } be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the unique fixed point q of T.

Proof Since T:DD is a generalized Lipschitz Φ-hemi-contractive mapping, there exists a strictly increasing continuous function Φ:[0,+)[0,+) with Φ(0)=0 such that

T x T q , J ( x q ) x q 2 Φ ( x q ) ,

i.e.,

x T x , J ( x q ) Φ ( x q ) ,

and

TxTyL ( 1 + x y ) ,

for any x,yD and qF(T).

Step 1. There exists x 0 D and x 0 T x 0 such that r 0 = x 0 T x 0 x 0 qR(Φ) (range of Φ). Indeed, if Φ(r)+ as r+, then r 0 R(Φ); if sup{Φ(r):r[0,+)}= r 1 <+ with r 0 < r 1 , then for qD, there exists a sequence { w n } in D such that w n q as n with w n q. Furthermore, we obtain that { w n T w n } is bounded. Hence, there exists a natural number n 0 such that w n T w n w n q< r 1 2 for n n 0 , then we redefine x 0 = w n 0 and x 0 T x 0 x 0 qR(Φ).

Step 2. For any n0, { x n } is bounded. Set R= Φ 1 ( r 0 ), then from Definition 1.2(2), we obtain that x 0 qR. Denote B 1 ={xD:xqR}, B 2 ={xD:xq2R}. Since T is generalized Lipschitz, so T is bounded. We may define M= sup x B 2 {Txq+1}+ sup n { u n q}+ sup n { v n q}. Next, we want to prove that x n B 1 . If n=0, then x 0 B 1 . Now, assume that it holds for some n, i.e., x n B 1 . We prove that x n + 1 B 1 . Suppose it is not the case, then x n + 1 q>R. Since J is uniformly continuous on a bounded subset of E, then for ϵ 0 = Φ ( R 4 ) 24 L ( 1 + 2 R ) , there exists δ>0 such that JxJy<ϵ when xy<δ, x,y B 2 . Now, denote

τ 0 = min { 1 , R 2 [ L ( 1 + 2 R ) + 2 R + M ] , R 4 [ L ( 1 + R ) + 2 R + M ] , δ 2 [ L ( 1 + 2 R ) + 2 R + M ] , Φ ( R 4 ) 24 R 2 , Φ ( R 4 ) 24 L ( 1 + 2 R ) , Φ ( R 4 ) 48 M R } .

Owing to a n , b n , c n , d n 0 as n, without loss of generality, assume that 0 a n , b n , c n , d n τ 0 for any n0. Since c n =o( a n ), denote c n < a n τ 0 . So, we have

(2.1)
(2.2)
(2.3)
(2.4)

and

(2.5)
(2.6)
(2.7)
(2.8)
(2.9)

Therefore,

J ( x n q ) J ( y n q ) < ϵ 0 ; J ( x n + 1 q ) J ( x n q ) < ϵ 0 .

Using Lemma 1.6 and the above formulas, we obtain

(2.10)

and

(2.11)

Substitute (2.11) into (2.10)

(2.12)

this is a contradiction. Thus, x n + 1 B 1 , i.e., { x n } is a bounded sequence. So, { y n }, {T y n }, {T x n } are all bounded sequences.

Step 3. We want to prove x n q0 as n. Set M 1 =max{ sup n x n q, sup n y n q, sup n T x n q, sup n T y n q, sup n u n q, sup n v n q}.

By (2.10), (2.11), we have

(2.13)

and

(2.14)

where A n =J( x n + 1 q)J( x n q), B n =J( x n q)J( y n q) and A n , B n 0 as n.

Taking (2.14) into (2.13),

(2.15)

where C n = a n 2 M 1 2 + M 1 A n +3 M 1 B n +2 b n M 1 2 +2 d n M 1 2 + c n M 1 2 a n 0 as n.

Set inf n 0 Φ ( y n q ) 1 + x n + 1 q 2 =λ, then λ=0. If it is not the case, we assume that λ>0. Let 0<γ<min{1,λ}, then Φ ( y n q ) 1 + x n + 1 q 2 γ, i.e., Φ( y n q)γ+γ x n + 1 q 2 γ x n + 1 q 2 . Thus, from (2.15) it follows that

(2.16)

This implies that

(2.17)

Let ρ n = x n q 2 , λ n = 2 a n γ 1 + 2 a n γ , σ n = 2 a n C n 1 + 2 a n γ . Then we get that

ρ n + 1 (1 λ n ) ρ n + σ n .

Applying Lemma 1.7, we get that ρ n 0 as n. This is a contradiction and so λ=0. Therefore, there exists an infinite subsequence such that Φ ( y n i q ) 1 + x n i + 1 q 2 0 as i. Since 0 Φ ( y n i q ) 1 + M 1 2 Φ ( y n i q ) 1 + x n i + 1 q 2 , then Φ( y n i q)0 as i. In view of the strictly increasing continuity of Φ, we have y n i q0 as i. Hence, x n i q0 as i. Next, we want to prove x n q0 as n. Let ε(0,1), there exists n i 0 such that x n i q<ϵ, a n , c n < ϵ 8 M 1 , b n , d n < ϵ 16 M 1 , C n < Φ ( ϵ ) 2 , for any n i ,n n i 0 . First, we want to prove x n i + 1 q<ϵ. Suppose it is not the case, then x n i + 1 qϵ. Using (1.1), we may get the following estimates:

(2.18)
(2.19)

Since Φ is strictly increasing, then (2.19) leads to Φ( y n i q)Φ( ϵ 4 ). From (2.15), we have

(2.20)

which is a contradiction. Hence, x n i + 1 q<ϵ. Suppose that x n i + m q<ϵ holds. Repeating the above course, we can easily prove that x n i + m + 1 q<ϵ holds. Therefore, for any m, we obtain that x n i + m q<ϵ, which means x n q0 as n. This completes the proof. □

Theorem 2.2 Let E be an arbitrary uniformly smooth real Banach space, and T:EE be a generalized Lipschitz Φ-quasi-accretive mapping with N(T). Let { a n }, { b n }, { c n }, { d n } be four real sequences in [0,1] and satisfy the conditions (i) a n , b n , d n 0 as n and c n =o( a n ); (ii) n = 0 a n =. For some x 0 D, let { u n }, { v n } be any bounded sequences in E, and { x n } be an Ishikawa iterative sequence with errors defined by

{ y n = ( 1 b n d n ) x n + b n S x n + d n v n , n 0 , x n + 1 = ( 1 a n c n ) x n + a n S y n + c n u n , n 0 ,
(2.21)

where S:EE is defined by Sx=xTx for any xE. Then { x n } converges strongly to the unique solution of the equation Tx=0 (or the unique fixed point of S).

Proof Since T is a generalized Lipschitz and Φ-quasi-accretive mapping, it follows that

TxTyL ( 1 + x y ) ,

i.e.,

S x S y L 1 ( 1 + x y ) , L 1 = 1 + L ; T x T q , J ( x q ) Φ ( x q ) ,

i.e.,

S x S q , J ( x q ) x q 2 Φ ( x q ) ,

for all x,yE, qN(T). The rest of the proof is the same as that of Theorem 2.1. □

Corollary 2.3 Let E be an arbitrary uniformly smooth real Banach space, D be a nonempty closed convex subset of E, and T:DD be a generalized Lipschitz Φ-hemi-contractive mapping with qF(T). Let { a n }, { c n } be two real sequences in [0,1] and satisfy the conditions (i) a n 0 as n and c n =o( a n ); (ii) n = 0 a n =. For some x 0 D, let { u n } be any bounded sequence in D, and { x n } be the Mann iterative sequence with errors defined by (1.2). Then (1.2) converges strongly to the unique fixed point q of T.

Corollary 2.4 Let E be an arbitrary uniformly smooth real Banach space, and T:EE be a generalized Lipschitz Φ-quasi-accretive mapping with N(T). Let { a n }, { d n } be two real sequences in [0,1] and satisfy the conditions (i) a n 0 as n and c n =o( a n ); (ii)  n = 0 a n =. For some x 0 D, let { u n } be any bounded sequence in E, and { x n } be the Mann iterative sequence with errors defined by

x n + 1 =(1 a n c n ) x n + a n S x n + c n u n ,n0,
(2.22)

where S:EE is defined by Sx=xTx for any xE. Then { x n } converges strongly to the unique solution of the equation Tx=0 (or the unique fixed point of S).

Remark 2.5 It is mentioned that in 2006, Chidume and Chidume [1] proved the approximative theorem for zeros of generalized Lipschitz generalized Φ-quasi-accretive operators. This result provided significant improvements of some recent important results. Their result is as follows.

Theorem CC ([[1], Theorem 3.1])

Let E be a uniformly smooth real Banach space and A:EE be a mapping with N(A). Suppose A is a generalized Lipschitz Φ-quasi-accretive mapping. Let { a n }, { b n } and { c n } be real sequences in [0,1] satisfying the following conditions: (i) a n + b n + c n =1; (ii) n = 0 ( b n + c n )=; (iii) n = 0 c n <; (iv) lim n b n =0. Let { x n } be generated iteratively from arbitrary x 0 E by

x n + 1 = a n x n + b n S x n + c n u n ,n0,
(2.23)

where S:EE is defined by Sx:=f+xAx, xE and { u n } is an arbitrary bounded sequence in E. Then, there exists γ 0 such that if b n + c n γ 0 , n0, the sequence { x n } converges strongly to the unique solution of the equation Au=0.

However, there exists a gap in the proof process of above Theorem CC. Here, c n =min{ ϵ 4 β , 1 2 σ Φ( ϵ 2 ) α n } ( α n = b n + c n ) does not hold in line 14 of Claim 2 on page 248, i.e., c n 1 2 σ Φ( ϵ 2 ) α n is a wrong case. For instance, set the iteration parameters: a n =1 b n c n , where { b n }: b 1 = 1 4 , b n = 1 n , n2; { c n }: 1 4 , 1 2 2 , 1 3 2 , 1 4 , 1 5 2 ,, 1 8 2 , 1 9 , 1 10 2 ,, 1 15 2 , 1 16 , 1 17 2 ,, 1 24 2 , 1 25 , 1 26 2 ,, 1 35 2 , 1 36 , 1 37 2 , . Then n = 0 c n <+, but c n o( b n + c n ). Therefore, the proof of above Theorem CC is not reasonable. Up to now, we do not know the validity of Theorem CC. This will be an open question left for the readers!

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Xue, Z., Lv, G. & Rhoades, B. The convergence theorems of Ishikawa iterative process with errors for Φ-hemi-contractive mappings in uniformly smooth Banach spaces. Fixed Point Theory Appl 2012, 206 (2012). https://doi.org/10.1186/1687-1812-2012-206

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Keywords

  • generalized Lipschitz mapping
  • Φ-hemi-contractive mapping
  • Ishikawa iterative sequence with errors
  • uniformly smooth real Banach space