Open Access

A three-step iterative scheme for solving nonlinear ϕ-strongly accretive operator equations in Banach spaces

Fixed Point Theory and Applications20122012:149

https://doi.org/10.1186/1687-1812-2012-149

Received: 30 June 2012

Accepted: 29 August 2012

Published: 12 September 2012

Abstract

In this paper, we study a three-step iterative scheme with error terms for solving nonlinear ϕ-strongly accretive operator equations in arbitrary real Banach spaces.

Keywords

three-step iterative scheme ϕ-strongly accretive operator ϕ-hemicontractive operator

1 Introduction

Let K be a nonempty subset of an arbitrary Banach space X and X be its dual space. The symbols D ( T ) , R ( T ) and F ( T ) stand for the domain, the range and the set of fixed points of T respectively (for a single-valued map T : X X , x X is called a fixed point of T iff T ( x ) = x ). We denote by J the normalized duality mapping from E to 2 E defined by
J ( x ) = { f X : x , f = x 2 = f 2 } .

Let T : D ( T ) X X be an operator. The following definitions can be found in [115] for example.

Definition 1 T is called Lipshitzian if there exists L > 0 such that
T x T y L x y ,

for all x , y K . If L = 1 , then T is called nonexpansive, and if 0 < L < 1 , T is called contraction.

Definition 2
  1. (i)
    T is said to be strongly pseudocontractive if there exists a t > 1 such that for each x , y D ( T ) , there exists j ( x y ) J ( x y ) satisfying
    Re T x T y , j ( x y ) 1 t x y 2 .
     
  2. (ii)
    T is said to be strictly hemicontractive if F ( T ) is nonempty and if there exists a t > 1 such that for each x D ( T ) and q F ( T ) , there exists j ( x y ) J ( x y ) satisfying
    Re T x q , j ( x q ) 1 t x q 2 .
     
  3. (iii)
    T is said to be ϕ-strongly pseudocontractive if there exists a strictly increasing function ϕ : [ 0 , ) [ 0 , ) with ϕ ( 0 ) = 0 such that for each x , y D ( T ) , there exists j ( x y ) J ( x y ) satisfying
    Re T x T y , j ( x y ) x y 2 ϕ ( x y ) x y .
     
  4. (iv)
    T is said to be ϕ-hemicontractive if F ( T ) is nonempty and if there exists a strictly increasing function ϕ : [ 0 , ) [ 0 , ) with ϕ ( 0 ) = 0 such that for each x D ( T ) and q F ( T ) , there exists j ( x y ) J ( x y ) satisfying
    Re T x q , j ( x q ) x q 2 ϕ ( x q ) x q .
     

Clearly, each strictly hemicontractive operator is ϕ-hemicontractive.

Definition 3
  1. (i)
    T is called accretive if the inequality
    x y x y + s ( T x T y )
     
holds for every x , y D ( T ) and for all s > 0 .
  1. (ii)
    T is called strongly accretive if, for all x , y D ( T ) , there exists a constant k > 0 and j ( x y ) J ( x y ) such that
    T x T y , j ( x y ) k x y 2 .
     
  2. (iii)
    T is called ϕ-strongly accretive if there exists j ( x y ) J ( x y ) and a strictly increasing function ϕ : [ 0 , ) [ 0 , ) with ϕ ( 0 ) = 0 such that for each x , y X ,
    T x T y , j ( x y ) ϕ ( x y ) x y .
     

Remark 4 It has been shown in [11, 12] that the class of strongly accretive operators is a proper subclass of the class of ϕ-strongly accretive operators. If I denotes the identity operator, then T is called strongly pseudocontractive (respectively, ϕ-strongly pseudocontractive) if and only if ( I T ) is strongly accretive (respectively, ϕ-strongly accretive).

Chidume [1] showed that the Mann iterative method can be used to approximate fixed points of Lipschitz strongly pseudocontractive operators in L p (or l p ) spaces for p [ 2 , ) . Chidume and Osilike [4] proved that each strongly pseudocontractive operator with a fixed point is strictly hemicontractive, but the converse does not hold in general. They also proved that the class of strongly pseudocontractive operators is a proper subclass of the class of ϕ-strongly pseudocontractive operators and pointed out that the class of ϕ-strongly pseudocontractive operators with a fixed point is a proper subclass of the class of ϕ-hemicontractive operators. These classes of nonlinear operators have been studied by various researchers (see, for example, [725]). Liu et al. [26] proved that, under certain conditions, a three-step iteration scheme with error terms converges strongly to the unique fixed point of ϕ-hemicontractive mappings.

In this paper, we study a three-step iterative scheme with error terms for nonlinear ϕ-strongly accretive operator equations in arbitrary real Banach spaces.

2 Preliminaries

We need the following results.

Lemma 5 [27]

Let { a n } , { b n } and { c n } be three sequences of nonnegative real numbers with n = 1 b n < and n = 1 c n < . If
a n + 1 ( 1 + b n ) a n + c n , n 1 ,

then the limit lim n a n exists.

Lemma 6 [28]

Let x , y X . Then x x + r y for every r > 0 if and only if there is f J ( x ) such that Re y , f 0 .

Lemma 7 [9]

Suppose that X is an arbitrary Banach space and A : E E is a continuous ϕ-strongly accretive operator. Then the equation A x = f has a unique solution for any f E .

3 Strong convergence of a three-step iterative scheme to a solution of the system of nonlinear operator equations

For the rest of this section, L denotes the Lipschitz constant of T 1 , T 2 , T 3 : X X , L = ( 1 + L ) and R ( T 1 ) , R ( T 2 ) and R ( T 3 ) denote the ranges of T 1 , T 2 and T 3 respectively. We now prove our main results.

Theorem 8 Let X be an arbitrary real Banach space and T 1 , T 2 , T 3 : X X Lipschitz ϕ-strongly accretive operators. Let f R ( T 1 ) R ( T 2 ) R ( T 3 ) and generate { x n } from an arbitrary x 0 X by
x n + 1 = a n x n + b n ( f + ( I T 1 ) y n ) + c n v n , y n = a n x n + b n ( f + ( I T 2 ) z n ) + c n u n , z n = a n x n + b n ( f + ( I T 3 ) x n ) + c n w n , n 0 ,
(3.1)

where { v n } n = 0 , { u n } n = 0 and { w n } n = 0 are bounded sequences in X and { a n } , { c n } , { a n } , { b n } , { c n } , { a n } , { b n } , { c n } are sequences in [ 0 , 1 ] and { b n } in ( 0 , 1 ) satisfying the following conditions: (i) a n + b n + c n = 1 = a n + b n + c n = a n + b n + c n , (ii) n = 0 b n = , (iii) n = 0 b n 2 < , n = 0 b n < , (iv) n = 0 c n < , n = 0 c n < and n = 0 c n < . Then the sequence { x n } converges strongly to the solution of the system T i x = f ; i = 1 , 2 , 3 .

Proof By Lemma 7, the system T i x = f ; i = 1 , 2 , 3 has the unique solution x X . Following the techniques of [5, 812, 26, 29], define S i : X X by S i x = f + ( I T i ) x ; i = 1 , 2 , 3 ; then each S i is demicontinuous and x is the unique fixed point of S i ; i = 1 , 2 , 3 , and for all x , y X , we have
( I S i ) x ( I S i ) y , j ( x y ) ϕ i ( x y ) x y ϕ i ( x y ) ( 1 + ϕ i ( x y ) + x y ) x y 2 = θ i ( x , y ) x y 2 ,
where θ i ( x , y ) = ϕ i ( x y ) ( 1 + ϕ i ( x y ) + x y ) [ 0 , 1 ) for all x , y X ; i = 1 , 2 , 3 . Let x i = 1 3 F ( S i ) be the fixed point set of S i , and let θ ( x , y ) = inf min i { θ i ( x , y ) } [ 0 , 1 ] . Thus
( I S i ) x ( I S i ) y , j ( x y ) θ ( x , y ) x y 2 ; i = 1 , 2 , 3 .
(3.2)
It follows from Lemma 6 and inequality (3.2) that
x y x y + λ [ ( I S i ) x θ ( x , y ) x ( ( I S i ) y θ ( x , y ) y ) ] ,
(3.3)

for all x , y X and for all λ > 0 ; i = 1 , 2 , 3 .

Set α n = b n + c n , β n = b n + c n and γ n = b n + c n , then (3.1) becomes
x n + 1 = ( 1 α n ) x n + α n S 1 y n + c n ( v n S 1 y n ) , y n = ( 1 β n ) x n + β n S 2 z n + c n ( u n S 2 z n ) , z n = ( 1 γ n ) x n + γ n S 3 x n + c n ( w n S 3 x n ) , n 0 .
(3.4)
We have
x n = ( 1 + α n ) x n + 1 + α n [ ( I S 1 ) x n + 1 θ ( x n + 1 , x ) x n + 1 ] ( 1 θ ( x n + 1 , x ) ) α n x n + ( 2 θ ( x n + 1 , x ) ) α n 2 ( x n S 1 y n ) + α n ( S 1 x n + 1 S 1 y n ) [ 1 + ( 2 θ ( x n + 1 , x ) ) α n ] c n ( v n S 1 y n ) .
Furthermore,
x = ( 1 + α n ) x + α n [ ( I S 1 ) x θ ( x n + 1 , x ) x ] ( 1 θ ( x n + 1 , x ) ) α n x ,
so that
x n x = ( 1 + α n ) ( x n + 1 x ) + α n [ ( I S 1 ) x n + 1 θ ( x n + 1 , x ) x n + 1 ( ( I S 1 ) x θ ( x n + 1 , x ) x ) ] ( 1 θ ( x n + 1 , x ) ) α n ( x n x ) + ( 2 θ ( x n + 1 , x ) ) α n 2 ( x n S 1 y n ) + α n ( S 1 x n + 1 S 1 y n ) [ 1 + ( 2 θ ( x n + 1 , x ) ) α n ] c n ( v n S 1 y n ) .
Hence,
x n x ( 1 + α n ) x n + 1 x + α n ( 1 + α n ) [ ( I S 1 ) x n + 1 θ ( x n + 1 , x ) x n + 1 ( ( I S 1 ) x θ ( x n + 1 , x ) x ) ] ( 1 θ ( x n + 1 , x ) ) α n x n x ( 2 θ ( x n + 1 , x ) ) α n 2 x n S 1 y n α n S 1 x n + 1 S 1 y n [ 1 + ( 2 θ ( x n + 1 , x ) ) α n ] c n v n S 1 y n ( 1 + α n ) x n + 1 x ( 1 θ ( x n + 1 , x ) ) α n x n x ( 2 θ ( x n + 1 , x ) ) α n 2 x n S 1 y n α n S 1 x n + 1 S 1 y n [ 1 + ( 2 θ ( x n + 1 , x ) ) α n ] c n v n S 1 y n .
Hence,
x n + 1 x [ 1 + ( 1 θ ( x n + 1 , x ) ) α n ] ( 1 + α n ) x n x + 2 α n 2 x n S 1 y n + α n S 1 x n + 1 S 1 y n + [ 1 + ( 2 θ ( x n + 1 , x ) ) α n ] c n v n S 1 y n [ 1 + ( 1 θ ( x n + 1 , x ) ) α n ] [ 1 α n + α n 2 ] x n x + 2 α n 2 x n S 1 y n + α n S 1 x n + 1 S 1 y n + 3 c n v n S 1 y n [ 1 θ ( x n + 1 , x ) α n + α n 2 ] x n x + 2 α n 2 x n S 1 y n + α n S 1 x n + 1 S 1 y n + 3 c n v n S 1 y n .
(3.5)
Furthermore, we have the following estimates:
(3.6)
(3.7)
(3.8)
(3.9)
Using (3.4) and (3.6),
x n y n = β n ( x n S 2 z n ) c n ( u n S 2 z n ) β n x n S 2 z n + c n u n S 2 z n [ [ 1 + L ( 3 L 1 ) ] β n + L ( 3 L 1 ) c n ] x n x + L ( β n + c n ) c n w n x + c n u n x [ [ 1 + L ( 3 L 1 ) ] β n + L ( 3 L 1 ) c n ] x n x + 3 L c n w n x + c n u n x .
(3.10)
Using (3.7),
S 1 y n y n S 1 y n x + y n x ( 1 + L ) y n x ( 1 + L ) [ 3 L ( 3 L 1 ) 1 ] x n x + 3 L ( 1 + L ) c n w n x + ( 1 + L ) c n u n x .
(3.11)
Again, using (3.7),
v n S 1 y n v n x + L y n x L [ 3 L ( 3 L 1 ) 1 ] x n x + v n x + 3 L 2 c n w n x + L c n u n x .
(3.12)
Substituting (3.10)-(3.12) in (3.9), we obtain
S 1 x n + 1 S 1 y n L [ 1 + L ( 3 L 1 ) ] β n + L ( 3 L 1 ) c n + [ 3 L ( 3 L 1 ) 1 ] [ ( 1 + L ) α n + L c n ] x n x + 3 L [ L c n + [ ( 1 + L ) α n + L c n ] c n ] w n x + L [ c n + [ ( 1 + L ) α n + L c n ] c n ] u n x + L c n v n x .
(3.13)
Substituting (3.8), (3.12) and (3.13) in (3.5), we obtain
x n + 1 x [ 1 + [ 3 + L ( 3 + L ) 3 L ( 3 L 1 ) 1 ] ] α n 2 + L [ 3 L ( 3 L 1 ) 1 ] α n β n + L 2 ( 3 L 1 ) α n c n + L [ 3 L ( 3 L 1 ) 1 ] α n c n + 3 L [ 3 L ( 3 L 1 ) 1 ] c n x n x θ ( x n + 1 , x ) α n x n x + [ 3 L ( 1 + 3 L ) α n 2 c n + 3 L 2 α n c n + 3 L 2 α n c n c n + 9 L 2 c n c n ] w n x + [ L ( 3 + L ) α n 2 c n + L α n c n + L 2 α n c n c n + 3 L c n c n ] u n x + ( 2 L + 3 ) c n v n x .
(3.14)
Since { v n } , { u n } and { w n } are bounded, we set
M = sup n 0 v n x + sup n 0 u n x + sup n 0 w n x < .
Then it follows from (3.14) that
x n + 1 x [ 1 + [ 3 + L ( 3 + L ) [ 3 L ( 3 L 1 ) 1 ] ] α n 2 + L [ 3 L ( 3 L 1 ) 1 ] α n β n + L 2 ( 3 L 1 ) α n c n + L [ 3 L ( 3 L 1 ) 1 ] α n c n + 3 L [ 3 L ( 3 L 1 ) 1 ] c n ] x n x θ ( x n + 1 , x ) α n x n x + [ 3 L ( 1 + 3 L ) α n 2 c n + 3 L 2 α n c n + 3 L 2 α n c n c n + 9 L 2 c n c n ] M + [ L ( 3 + L ) α n 2 c n + L α n c n + L 2 α n c n c n + 3 L c n c n ] M + ( 2 L + 3 ) c n M = ( 1 + δ n ) x n x θ ( x n + 1 , x ) α n x n x + σ n ( 1 + δ n ) x n x + σ n ,
(3.15)
where
Since b n ( 0 , 1 ) , the conditions (iii) and (iv) imply that n = 0 δ n < and n = 0 σ n < . It then follows from Lemma 5 that lim n x n x exists. Let lim n x n x = δ 0 . We now prove that δ = 0 . Assume that δ > 0 . Then there exists a positive integer N 0 such that x n x δ 2 for all n N 0 . Since
θ ( x n + 1 , x ) x n x = ϕ ( x n + 1 x ) 1 + ϕ ( x n + 1 x ) + x n + 1 x x n x ϕ ( δ 2 ) δ 2 ( 1 + ϕ ( D ) + D ) ,
for all n N 0 , it follows from (3.15) that
x n + 1 x x n x ϕ ( δ 2 ) δ 2 ( 1 + ϕ ( D ) + D ) α n + λ n for all  n N 0 .
Hence,
ϕ ( δ 2 ) δ 2 ( 1 + ϕ ( D ) + D ) α n x n x x n + 1 x + λ n for all  n N 0 .
This implies that
ϕ ( δ 2 ) δ 2 ( 1 + ϕ ( D ) + D ) j = N 0 n α j x N 0 x + j = N 0 n λ j .
Since b n α n ,
ϕ ( δ 2 ) δ 2 ( 1 + ϕ ( D ) + D ) j = N 0 n b j x N 0 x + j = N 0 n λ j

yields n = 0 b n < , contradicting the fact that n = 0 b n = . Hence, lim n x n x = 0 . □

Corollary 9 Let X be an arbitrary real Banach space and T 1 , T 2 , T 3 : X X be three Lipschitz ϕ-strongly accretive operators, where ϕ is in addition continuous. Suppose lim inf r ϕ ( r ) > 0 or T i x as x ; i = 1 , 2 , 3 . Let { a n } , { b n } , { c n } , { a n } , { b n } , { c n } , { a n } , { b n } , { c n } , { w n } , { u n } , { v n } , { y n } and { x n } be as in Theorem  8. Then, for any given f X , the sequence { x n } converges strongly to the solution of the system T i x = f ; i = 1 , 2 , 3 .

Proof The existence of a unique solution to the system T i x = f ; i = 1 , 2 , 3 follows from [9] and the result follows from Theorem 8. □

Theorem 10 Let X be a real Banach space and K be a nonempty closed convex subset of X. Let T 1 , T 2 , T 3 : K K be three Lipschitz ϕ-strong pseudocontractions with a nonempty fixed point set. Let { a n } , { b n } , { c n } , { a n } , { b n } , { c n } , { a n } , { b n } , { c n } , { w n } , { u n } and { v n } be as in Theorem  8. Let { x n } be the sequence generated iteratively from an arbitrary x 0 K by
x n + 1 = a n x n + b n T 1 y n + c n v n , y n = a n x n + b n T 2 z n + c n u n , z n = a n x n + b n T 3 x n + c n w n , n 0 .

Then { x n } converges strongly to the common fixed point of T 1 , T 2 , T 3 .

Proof As in the proof of Theorem 8, set α n = b n + c n , β n = b n + c n , γ n = b n + c n to obtain
x n + 1 = ( 1 α n ) x n + α n T 1 y n + c n ( v n T 1 y n ) , y n = ( 1 β n ) x n + β n T 2 z n + c n ( u n T 2 z n ) , z n = ( 1 γ n ) x n + γ n T 3 x n + c n ( w n T 3 x n ) , n 0 .
Since each T i ; i = 1 , 2 , 3 is a ϕ-strong pseudocontraction, ( I T i ) is ϕ-strongly accretive so that for all x , y X , there exist j ( x y ) J ( x y ) and a strictly increasing function ϕ : ( 0 , ) ( 0 , ) with ϕ ( 0 ) = 0 such that
( I T i ) x ( I T i ) y , j ( x y ) ϕ ( x y ) x y θ ( x , y ) x y 2 ; i = 1 , 2 , 3 .

The rest of the argument now follows as in the proof of Theorem 8. □

Remark 11 The example in [4] shows that the class of ϕ-strongly pseudocontractive operators with nonempty fixed point sets is a proper subclass of the class of ϕ-hemicontractive operators. It is easy to see that Theorem 8 easily extends to the class of ϕ-hemicontractive operators.

Remark 12
  1. (i)

    If we set b n = 0 = c n for all n 0 in our results, we obtain the corresponding results for the Ishikawa iteration scheme with error terms in the sense of Xu [15].

     
  2. (ii)

    If we set b n = 0 = c n = b n = 0 = c n for all n 0 in our results, we obtain the corresponding results for the Mann iteration scheme with error terms in the sense of Xu [15].

     
Remark 13 Let { α n } and { β n } be real sequences satisfying the following conditions:
  1. (i)

    0 α n , β n 1 , n 0 ,

     
  2. (ii)

    lim n α n = lim n β n = 0 ,

     
  3. (iii)

    n = 0 α n = ,

     
  4. (iv)

    n = 0 β n < , and

     
  5. (v)

    n = 0 α n 2 < .

     

If we set a n = ( 1 β n ) , b n = β n , c n = 0 , a n = ( 1 α n ) , b n = α n , c n = 0 , b n = 0 = c n for all n 0 in Theorems 8 and 10 respectively, we obtain the corresponding convergence theorems for the original Ishikawa [18] and Mann [30] iteration schemes.

Remark 14
  1. (i)

    Gurudwan and Sharma [29] studied a strong convergence of multi-step iterative scheme to a common solution for a finite family of ϕ-strongly accretive operator equations in a reflexive Banach space with weakly continuous duality mapping. Some remarks on their work can be seen in [31].

     
  2. (ii)

    All the above results can be extended to a finite family of ϕ-strongly accretive operators.

     

Declarations

Acknowledgements

The last author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.

Authors’ Affiliations

(1)
Department of Mathematics, Statistics and Physics, Qatar University
(2)
Hajvery University
(3)
Department of Mathematics, King Abdulaziz University

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