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On solving Lipschitz pseudocontractive operator equations

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.

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 xE where , denotes the generalized duality pairing. A single-valued duality map will be denoted by j.

An operator T:EE is said to be

  • pseudocontractive if there exists j(xy)J(xy) such that

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

for any x,yE;

  • accretive if for any x,yE, there exists j(xy)J(xy) satisfying

    T x T y , j ( x y ) 0;
  • strongly pseudocontractive if there exist j(xy)J(xy) and a constant λ(0,1) such that

    T x T y , j ( x y ) λ x y 2

for any x,yE;

  • strongly accretive if for any x,yE, there exist j(xy)J(xy) and a constant t(0,1) satisfying

    T x T y , j ( x y ) t x y 2

for all x,yE.

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

xy x y + r [ ( I T λ I ) x ( I T λ I ) y ]
(1.1)

holds for all x,yE 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 IT 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:CC 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 ={xH:x 1 2 }, C 2 ={xH: 1 2 x1}. Define the map T:CC by

Tx={ 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×NE 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 nN, mM. If n,rN, m,tM, 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:CC 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<rk,

  3. (iii)

    lim n α n =0,

  4. (iv)

    n 0 α n =.

For an arbitrary but fixed ωC, and for each k0, define T k :CC by T k x=(1 a k )ω+ a k Tx, xC. 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,n0,

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 ,n0.

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:CC 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 ωC, and for each k0, define T k :CC by T k x=(1 a k )ω+ a k Tx, xC. 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,n0
(2.1)

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

Proof Since T is Lipschitzian, there exists L>0 such that

TxTyLxyfor all x,yC.

Since T is pseudocontractive, for each k0, 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 k0, by convexity. Thus, for each k0, T k has a unique fixed point x k , say, in C.

Now, we proceed in the following steps.

  1. (I)

    for each k0, 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 , k0, 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 (1λ) 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=diamC. 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=diamC. 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:CC 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 ωC, and for each k0, define T k :CC by T k x=(1 a k )ω+ a k Tx, xC. 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,n0,

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=0C.

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 ωC, and for each k0, define T k :CC by T k x= a k Tx for all xC. 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,n0,

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 .

References

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Correspondence to Ahmed A Abdelhakim.

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Abdelhakim, A.A., Gu, F. On solving Lipschitz pseudocontractive operator equations. J Inequal Appl 2014, 314 (2014). https://doi.org/10.1186/1029-242X-2014-314

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Keywords

  • Lipschitz pseudocontractions
  • Mann-type double sequence iteration
  • strong convergence