Open Access

On solving Lipschitz pseudocontractive operator equations

Journal of Inequalities and Applications20142014:314

https://doi.org/10.1186/1029-242X-2014-314

Received: 16 January 2014

Accepted: 21 March 2014

Published: 21 August 2014

Abstract

We analyze the convergence of the Mann-type double sequence iteration process to the solution of a Lipschitz pseudocontractive operator equation on a bounded closed convex subset of arbitrary real Banach space into itself. Our results extend the result in (Moore in Comp. Math. Appl. 43: 1585-1589, 2002).

MSC:47H10, 54H25.

Keywords

Lipschitz pseudocontractionsMann-type double sequence iterationstrong convergence

1 Introduction

Let E be a real Banach space and E be the dual space of E. Let J be the normalized duality mapping from E to 2 E defined by
J ( x ) = { f E : x , f = x f , f = x }

for all x E where , denotes the generalized duality pairing. A single-valued duality map will be denoted by j.

An operator T : E E is said to be

  • pseudocontractive if there exists j ( x y ) J ( x y ) such that
    T x T y , j ( x y ) x y 2

for any x , y E ;

  • accretive if for any x , y E , there exists j ( x y ) J ( x y ) satisfying
    T x T y , j ( x y ) 0 ;
  • strongly pseudocontractive if there exist j ( x y ) J ( x y ) and a constant λ ( 0 , 1 ) such that
    T x T y , j ( x y ) λ x y 2

for any x , y E ;

  • strongly accretive if for any x , y E , there exist j ( x y ) J ( x y ) and a constant t ( 0 , 1 ) satisfying
    T x T y , j ( x y ) t x y 2

for all x , y E .

As a consequence of a result of Kato [1], the concept of pseudocontractive operators can equivalently be defined as follows:

T is strongly pseudocontractive if there exists λ ( 0 , 1 ) such that the inequality
x y x y + r [ ( I T λ I ) x ( I T λ I ) y ]
(1.1)

holds for all x , y E and r > 0 . If λ = 0 in the inequality (1.1), then T is pseudocontractive.

It is easy to see that T is pseudocontractive if and only if I T is accretive where I denotes the identity mapping on E.

Let C be a compact convex subset of a real Hilbert space and let T : C C be a Lipschitz pseudocontraction. It remains as an open question whether the Mann iteration process always converges to a fixed point of T. In [2] it was proved that the Ishikawa iteration process converges strongly to a fixed point of T. In 2001, Mutangadura and Chidume [3] constructed the following example to demonstrate that the Mann iteration process is not guaranteed to converge to a fixed point of a Lipschitz pseudocontraction mapping a compact convex subset of a real Hilbert space H into itself.

Example [3]

Let H = 2 with the usual Euclidean inner product, and for x = ( a , b ) H define x = ( b , a ) . Now, let C = B 1 ( o ) ; the closed unit ball in H and let C 1 = { x H : x 1 2 } , C 2 = { x H : 1 2 x 1 } . Define the map T : C C by
T x = { x + x , if  x C 1 ; x x x + x , if  x C 2 .
Observe that T is pseudocontractive, Lipschitz continuous (with Lipschitz constant 5) and has the origin ( 0 , 0 ) as its unique fixed point; C is compact and convex. However, for any x C 1 , we have
( 1 λ ) x + λ T x 2 = ( 1 + λ 2 ) x 2 > x 2 , λ ( 0 , 1 ) ,
while for any x C 2 , we have
( 1 λ ) x + λ T x 2 1 2 x 2 , λ ( 0 , 1 ) ,

and therefore no Mann sequence can converge to ( 0 , 0 ) , the unique fixed point of T, unless the initial guess is the fixed point itself.

Moore [4] introduced the concept of a Mann-type double sequence iteration process and proved that it converges strongly to a fixed point of a continuous pseudocontraction which maps a bounded closed convex nonempty subset of a real Hilbert space into itself.

Definition 1.1 [4]

Let N denote the set of all nonnegative integers (the natural numbers) and let E be a normed linear space. By a double sequence in E is meant a function f : N × N E defined by f ( n , m ) = x n , m E . A double sequence { x n , m } is said to converge strongly to x if given any ϵ > 0 , there exist N , M > 0 such that x n , m x < ϵ for all n N , m M . If n , r N , m , t M , we have x n , r x m , t < ϵ , then the double sequence is said to be Cauchy. Furthermore, if for each fixed n, x n , m x n as m and then x n x as n , then x n , m x as n , m .

Theorem 1.1 [4]

Let C be a bounded closed convex nonempty subset of a (real) Hilbert space H, and let T : C C be a continuous pseudocontractive map. Let { α n } n 0 , { a k } k 0 ( 0 , 1 ) be real sequences satisfying the following conditions:
  1. (i)

    lim k a k = 1 ,

     
  2. (ii)

    lim k , r ( a k a r ) / ( 1 a k ) = 0 , 0 < r k ,

     
  3. (iii)

    lim n α n = 0 ,

     
  4. (iv)

    n 0 α n = .

     
For an arbitrary but fixed ω C , and for each k 0 , define T k : C C by T k x = ( 1 a k ) ω + a k T x , x C . Then the double sequence { x k , n } k 0 , n 0 generated from an arbitrary x 0 , 0 C by
x k , n + 1 = ( 1 α n ) x k , n + α n T k x k , n , k , n 0 ,

converges strongly to a fixed point x of T in C.

The following lemma will be useful in the sequel.

Lemma 1.2 [5]

Let { δ n } and { σ n } be two sequences of nonnegative real numbers satisfying the inequality
δ n + 1 γ δ n + σ n , n 0 .

Here γ [ 0 , 1 ) . If lim n σ n = 0 , then lim n δ n = 0 .

It is our purpose in this paper to extend Theorem 1.1 from Hilbert space to an arbitrary real Banach space with no further assumptions on the real sequences { α n } n 0 , { a k } k 0 .

2 Main results

Theorem 2.1 Let C be a bounded closed convex subset of a Banach space E and T : C C be a Lipschitz pseudocontraction with F ( T ) . Let { α n } n 0 , { a k } k 0 ( 0 , 1 ) be real sequences satisfying the following conditions:
  1. (i)

    lim k a k = 1 ,

     
  2. (ii)

    lim n α n = 0 .

     
For an arbitrary but fixed ω C , and for each k 0 , define T k : C C by T k x = ( 1 a k ) ω + a k T x , x C . Then the double sequence { x k , n } k 0 , n 0 generated from an arbitrary x 0 , 0 C by
x k , n + 1 = ( 1 α n ) x k , n + α n T k x k , n , k , n 0
(2.1)

converges strongly to a fixed point x of T in C.

Proof Since T is Lipschitzian, there exists L > 0 such that
T x T y L x y for all  x , y C .
Since T is pseudocontractive, for each k 0 , we have
T k x T k y , j ( x y ) = a k T x T y , j ( x y ) a k x y 2 .

Hence, T k is Lipschitz and strongly pseudocontractive. Also, C is invariant under T k for all k 0 , by convexity. Thus, for each k 0 , T k has a unique fixed point x k , say, in C.

Now, we proceed in the following steps.
  1. (I)

    for each k 0 , x k , n x k C as n .

     
  2. (II)

    x k x C as k .

     
  3. (III)

    x F ( T ) .

     
Proof of (I). In fact, it follows from (2.1) that
x k , n = x k , n + 1 + α n x k , n α n T k x k , n = ( 1 + α n ) x k , n + 1 + α n ( I T k λ I ) x k , n + 1 ( 2 λ ) α n x k , n + 1 + α n x k , n + α n ( T k x k , n + 1 T k x k , n ) = ( 1 + α n ) x k , n + 1 + α n ( I T k λ I ) x k , n + 1 ( 2 λ ) α n [ ( 1 α n ) x k , n + α n T k x k , n ] + α n x k , n + α n ( T k x k , n + 1 T k x k , n ) = ( 1 + α n ) x k , n + 1 + α n ( I T k λ I ) x k , n + 1 ( 1 λ ) α n x k , n + ( 2 λ ) α n 2 ( x k , n T k x k , n ) + α n ( T k x k , n + 1 T k x k , n ) .
Thus, if x k is a fixed point of T k , k 0 , then
x k , n + 1 x k = ( 1 + α n ) ( x k , n + 1 x k ) + α n ( I T k λ I ) ( x k , n + 1 x k ) ( 1 λ ) α n ( x k , n x k ) + ( 2 λ ) α n 2 ( x k , n T k x k , n ) + α n ( T k x k , n + 1 T k x k , n ) .
Using inequality (1.1), it follows that
x k , n + 1 x k ( 1 + α n ) x k , n + 1 x k ( 1 λ ) α n x k , n x k ( 2 λ ) α n 2 x k , n T k x k , n α n T k x k , n + 1 T k x k , n .
(2.2)
On the other hand, by (2.1) we obtain
x k , n + 1 x k , n = α n T k x k , n x k , n α n ( T k x k , n x k + x k , n x k ) = α n ( a k T x k , n x k + x k , n x k ) α n ( a k L x k , n x k + x k , n x k ) α n ( L + 1 ) x k , n x k .
Therefore,
T k x k , n + 1 T k x k , n = a k T x k , n + 1 T x k , n a k L x k , n + 1 x k , n L x k , n + 1 x k , n α n L ( L + 1 ) x k , n x k .
(2.3)
Substituting (2.3) into (2.2), we arrive at
x k , n x k ( 1 + α n ) x k , n + 1 x k ( 1 λ ) α n x k , n x k ( 2 λ ) α n 2 x k , n T k x k , n L ( L + 1 ) α n 2 x k , n x k ,
which implies that
α n x k , n + 1 x k ( 1 λ ) α n x k , n x k + α n 2 [ L ( L + 1 ) x k , n x k + ( 2 λ ) x k , n T k x k , n ] ,
and so
x k , n + 1 x k ( 1 λ ) x k , n x k + α n [ L ( L + 1 ) x k , n x k + ( 2 λ ) x k , n T k x k , n ] .
(2.4)
Since C is bounded, there exists M > 0 such that
M = max { L ( L + 1 ) sup n 0 x k , n x k , ( 2 λ ) sup n 0 x k , n T k x k , n } .
Hence, it follows from (2.4) that
x k , n + 1 x k ( 1 λ ) x k , n x k + α n M .
Since λ ( 0 , 1 ) and lim n α n = 0 , it follows from Lemma 1.2 that
lim n x k , n x k = 0 ,

i.e., x k , n x k as n .

Proof of (II). We prove that { x k } k = 0 = { T k x k } k = 0 converges to some x C . For this purpose, we need only to prove that { x k } 0 is a Cauchy sequence.

In fact, we have
x l x m 2 = x l x m , j ( x l x m ) = T l x l T m x m , j ( x l x m ) = ( 1 a l ) ω + a l T x l ( 1 a m ) ω a m T x m , j ( x l x m ) = ( a m a l ) ω , j ( x l x m ) + a l T x l T x m , j ( x l x m ) + ( a l a m ) T x m , j ( x l x m ) | a l a m | ( ω x l x m + T x m x l x m ) + a l T x l T x m , j ( x l x m ) | a l a m | ( ω + T x m ) x l x m + a l λ x l x m 2 | a l a m | ( ω + T x m ) x l x m + λ x l x m 2 ,
that is,
x l x m [ | a l a m | ( ω + T x m ) + λ x l x m ] ,
hence
x l x m 2 | a l a m | 1 λ d ,
where d = diam C . If follows from condition (i) that
lim l , m x l x m = 0 .

This completes step (II) of the proof.

Proof of (III). In order to accomplish step (III), we first have to prove that { x k } k = 0 is an approximate fixed point sequence for T. In fact, from T k x k = ( 1 a k ) ω + a k T x k , we have
x k T x k = x k 1 a k T k x k + 1 a k a k ω = x k 1 a k x k + 1 a k a k ω = 1 a k a k ( ω x k ) 1 a k a k ( ω + x k ) 1 a k a k 2 d ,

where d = diam C . Hence lim k x k T x k = 0 . Since x k x as k , T is continuous and using continuity of the norm, we get lim k x T x = 0 , i.e., x = T x . This completes the proof. □

Corollary 2.2 Let C be a bounded closed convex subset of a Banach space E and T : C C be a nonexpansive mapping with F ( T ) . Let { α n } n 0 , { a k } k 0 ( 0 , 1 ) be real sequences satisfying conditions (i)-(ii) in Theorem  2.1. For an arbitrary but fixed ω C , and for each k 0 , define T k : C C by T k x = ( 1 a k ) ω + a k T x , x C . Then the double sequence { x k , n } k 0 , n 0 generated from an arbitrary x 0 , 0 C by
x k , n + 1 = ( 1 α n ) x k , n + α n T k x k , n , k , n 0 ,

converges strongly to a fixed point of T in C.

Proof Obvious, observing the fact that every nonexpansive mapping is Lipschitz and pseudocontractive. □

The following corollary follows from Theorem 2.1 on setting ω = 0 C .

Corollary 2.3 Let C, E, T, { α n } n = 0 , { a k } k = 0 be as in Theorem  2.1. For an arbitrary but fixed ω C , and for each k 0 , define T k : C C by T k x = a k T x for all x C . Then the double sequence { x k , n } k 0 , n 0 generated from an arbitrary x 0 , 0 C by
x k , n + 1 = ( 1 α n ) x k , n + α n T k x k , n , k , n 0 ,

converges strongly to a fixed point of T in C.

Remark 2.1 Theorem 2.1 improves and extends Theorem 3.1 of Moore [3] in three respects:
  1. (1)

    It abolishes the condition that lim r , k a k a r 1 a k = 0 .

     
  2. (2)

    It abolishes the condition that n = 1 α n = .

     
  3. (3)

    The ambient space is no longer required to be a Hilbert space and is taken to be the more general Banach space instead.

     
Remark 2.2
  1. (1)

    Whereas the Ishikawa iteration process was proved to converge to a fixed point of a Lipschitz pseudocontractive mapping in compact convex subsets of a Hilbert space, we imposed no compactness conditions to obtain the strong convergence of the double sequence iteration process to a fixed point of a Lipschitz pseudocontraction.

     
  2. (2)

    Our results may easily be extended to the slightly more general classes of Lipschitz hemicontractive and Lipschitz quasi-nonexpansive mappings.

     
  3. (3)
    Prototypes of the sequences { a k } k = 0 and { α n } n = 0 are
    a k = k 1 + k and α n = 1 ( n + 1 ) 2 .
     

Declarations

Authors’ Affiliations

(1)
Mathematics Department, Faculty of Science, Assiut University
(2)
Institute of Applied Mathematics and Department of Mathematics, Hangzhou Normal University

References

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Copyright

© Abdelhakim and Gu; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.