Open Access

Iteration scheme for common fixed points of hemicontractive and nonexpansive operators in Banach spaces

Fixed Point Theory and Applications20132013:247

https://doi.org/10.1186/1687-1812-2013-247

Received: 7 May 2013

Accepted: 5 September 2013

Published: 7 November 2013

Abstract

The purpose of this paper is to characterize the conditions for the convergence of the iterative scheme in the sense of Agarwal et al. (J. Nonlinear Convex. Anal. 8(1): 61-79, 2007), associated with nonexpansive and ϕ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.

Keywords

modified iterative schemenonexpansive mappingsϕ-hemicontractive mappingsBanach spaces

Dedication

Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday

1 Preliminaries

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

Let T : D ( T ) X X be an operator.

Definition 1 T is called L-Lipschitzian if there exists L 0 such that
T x T y L x y

for all x , y D ( T ) . If L = 1 , then T is called non-expansive, and if 0 L < 1 , T is called contraction.

Definition 2 [13]

  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 ) 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 ) 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.

For a nonempty convex subset K of a normed space X , S : K K and T : K K ,
  1. (a)
    the Mann iteration scheme [4] is defined by the following sequence { x n } :

    where { b n } is a sequence in [ 0 , 1 ] ;

     
  2. (b)
    the sequence { x n } defined by

    where { b n } , { b n } are sequences in [ 0 , 1 ] is known as the Ishikawa [2] iteration scheme;

     
  3. (c)
    the sequence { x n } defined by

    where { b n } , { b n } are sequences in [ 0 , 1 ] , is known as the Agarwal-O’Regan-Sahu [5] iteration scheme;

     
  4. (d)
    the sequence { x n } defined by

    where { b n } , { b n } are sequences in [ 0 , 1 ] , is known as the modified Agarwal-O’Regan-Sahu iteration scheme.

     

Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of T in case T is a Lipschitz strongly pseudo-contractive mapping from a bounded closed convex subset of L p (or l p ) into itself. Afterwards, several authors generalized this result of Chidume in various directions [3, 612].

The purpose of this paper is to characterize conditions for the convergence of the iterative scheme in the sense of Agarwal et al. [5] associated with nonexpansive and ϕ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space. Our results improve and generalize most results in recent literature [1, 3, 5, 6, 8, 9, 11, 12].

2 Main result

The following result is now well known.

Lemma 3 [13]

For all x , y X and j ( x + y ) J ( x + y ) ,
x + y 2 x 2 + 2 Re y , j ( x + y ) .

Now, we prove our main result.

Theorem 4 Let K be a nonempty closed and convex subset of an arbitrary Banach space X, let S : K K be nonexpansive, and let T : K K be a uniformly continuous ϕ-hemicontractive mapping such that S and T have the common fixed point. Suppose that { b n } n = 1 and { b n } n = 1 are sequences in [ 0 , 1 ] satisfying conditions
  1. (i)

    lim n ( 1 b n ) = lim n b n = 0 ,

     
  2. (ii)

    n = 1 ( 1 b n ) = .

     
For any x 1 K , define the sequence { x n } n = 1 inductively as follows:
{ y n = b n x n + ( 1 b n ) T x n , x n + 1 = b n S x n + ( 1 b n ) T y n , n 1 .
(2.1)
Then the following conditions are equivalent:
  1. (a)

    { x n } n = 1 converges strongly to the common fixed point q of S and T.

     
  2. (b)

    { S x n } n = 1 , { T x n } n = 1 and { T y n } n = 1 are bounded.

     

Proof First, we prove that (a) implies (b).

Since T is ϕ-hemicontractive, it follows that F ( T ) is a singleton. Let F ( S ) F ( T ) = { q } for some q K .

Suppose that lim n x n = q . Then the continuity of S and T yields that
lim n S x n = q = lim n T x n
and
lim n y n = lim n [ b n x n + ( 1 b n ) T x n ] = q .

Thus, lim n T y n = q . Therefore, { S x n } n = 1 , { T x n } n = 1 and { T y n } n = 1 are bounded.

Second, we need to show that (b) implies (a). Suppose that { S x n } n = 1 , { T x n } n = 1 and { T y n } n = 1 are bounded.

Put
M 1 = x 1 q + sup n 1 S x n q + sup n 1 T x n q + sup n 1 T y n q .

It is clear that x 1 q M 1 . Let x n q M 1 . Next, we will prove that x n + 1 q M 1 .

Note that
x n + 1 q = b n S x n + ( 1 b n ) T y n q = b n ( S x n q ) + ( 1 b n ) ( T y n q ) b n S x n q + ( 1 b n ) T y n q ( b n + ( 1 b n ) ) M 1 = M 1 .
Thus, we can conclude that the sequence { x n q } n 1 is bounded, and hence, there is a constant M > 0 satisfying
M = sup n 1 x n q + sup n 1 S x n q + sup n 1 T x n q + sup n 1 T y n q .
(2.2)
Let w n = T y n T x n + 1 for each n 1 . The uniform continuity of T ensures that
lim n w n = 0 ,
(2.3)
because
y n x n + 1 = b n ( x n T x n ) + ( 1 b n ) ( S x n T y n ) b n x n T x n + ( 1 b n ) S x n T y n 2 M ( b n + ( 1 b n ) ) 0 as  n .
By virtue of Lemma 3 and (2.1), we infer that
x n + 1 q 2 = b n S x n + ( 1 b n ) T y n q 2 = b n ( S x n q ) + ( 1 b n ) ( T y n q ) 2 b n 2 S x n q 2 + 2 ( 1 b n ) Re T y n q , j ( x n + 1 q ) b n 2 x n q 2 + 2 ( 1 b n ) Re T y n T x n + 1 , j ( x n + 1 q ) + 2 ( 1 b n ) Re T x n + 1 q , j ( x n + 1 q ) b n 2 x n q 2 + 2 ( 1 b n ) T y n T x n + 1 x n + 1 q + 2 ( 1 b n ) x n + 1 q 2 2 ( 1 b n ) ϕ ( x n + 1 q ) x n + 1 q b n 2 x n q 2 + 2 M ( 1 b n ) w n + 2 ( 1 b n ) x n + 1 q 2 2 ( 1 b n ) ϕ ( x n + 1 q ) x n + 1 q .
(2.4)
The real function f : [ 0 , ) [ 0 , ) , f ( t ) = t 2 is increasing and convex. For all a [ 0 , 1 ] and t 1 , t 2 > 0 , we have
( ( 1 a ) t 1 + a t 2 ) 2 ( 1 a ) t 1 2 + a t 2 2 .
Hence,
x n + 1 q 2 = b n S x n + ( 1 b n ) T y n q 2 = b n ( S x n q ) + ( 1 b n ) ( T y n q ) 2 b n S x n q 2 + ( 1 b n ) T y n q 2 b n x n q 2 + ( 1 b n ) M 2 ,
(2.5)

where the second inequality holds by the convexity of 2 .

By substituting (2.5) in (2.4), we get
x n + 1 q 2 ( b n 2 + 2 b n ( 1 b n ) ) x n q 2 + 2 M ( 1 b n ) ( w n + M ( 1 b n ) ) 2 ( 1 b n ) ϕ ( x n + 1 q ) x n + 1 q = ( 1 ( 1 b n ) 2 ) x n q 2 + 2 M ( 1 b n ) ( w n + M ( 1 b n ) ) 2 ( 1 b n ) ϕ ( x n + 1 q ) x n + 1 q x n q 2 + 2 M ( 1 b n ) ( w n + M ( 1 b n ) ) 2 ( 1 b n ) ϕ ( x n + 1 q ) x n + 1 q = x n q 2 + ( 1 b n ) l n 2 ( 1 b n ) ϕ ( x n + 1 q ) x n + 1 q ,
(2.6)
where
l n = 2 M ( w n + M ( 1 b n ) ) 0 ,
(2.7)

as n .

Let δ = inf { x n + 1 q : n 0 } . We claim that δ = 0 . Otherwise, δ > 0 . Thus, (2.7) implies that there exists a positive integer N 1 such that l n < ϕ ( δ ) δ for each n N 1 . In view of (2.6), we conclude that
x n + 1 q 2 x n q 2 ϕ ( δ ) δ ( 1 b n ) , n N 1 ,
which implies that
ϕ ( δ ) δ n = N 1 ( 1 b n ) x N 1 q 2 ,
(2.8)
which contradicts (ii). Therefore, δ = 0 . Thus, there exists a subsequence { x n i + 1 } n = 1 of { x n + 1 } n = 1 such that
lim i x n i + 1 = q .
(2.9)
Let ϵ > 0 be a fixed number. By virtue of (2.7) and (2.9), we can select a positive integer i 0 > N 1 such that
x n i 0 + 1 q < ϵ , l n < ϕ ( ϵ ) ϵ , n n i 0 .
(2.10)
Let p = n i 0 . By induction, we show that
x p + m q < ϵ , m 1 .
(2.11)
Observe that (2.6) means that (2.11) is true for m = 1 . Suppose that (2.11) is true for some m 1 . If x p + m + 1 q ϵ , by (2.6) and (2.10), we know that
ϵ 2 x p + m + 1 q 2 x p + m q 2 + ( 1 b p + m ) l p + m 2 ( 1 b p + m ) ϕ ( x p + m + 1 q ) x p + m + 1 q < ϵ 2 + ( 1 b p + m ) ϕ ( ϵ ) ϵ 2 ( 1 b p + m ) ϕ ( ϵ ) ϵ = ϵ 2 ( 1 b p + m ) ϕ ( ϵ ) ϵ < ϵ 2 ,

which is impossible. Hence, x p + m + 1 q < ϵ . That is, (2.11) holds for all m 1 . Thus, (2.11) ensures that lim n x n = q . This completes the proof. □

Taking S = I in Theorem 4, we get the following.

Corollary 5 Let K be a nonempty closed and convex subset of an arbitrary Banach space X, and let T : K K be a uniformly continuous ϕ-hemicontractive mapping. Suppose that { b n } n = 1 and { b n } n = 1 are sequences in [ 0 , 1 ] satisfying conditions (i)-(ii) of Theorem  4. For any x 1 K , define the sequence { x n } n = 1 inductively as follows:
{ y n = b n x n + ( 1 b n ) T x n , x n + 1 = b n x n + ( 1 b n ) T y n , n 1 .
Then the following conditions are equivalent:
  1. (a)

    { x n } n = 1 converges strongly to the unique fixed point q of T.

     
  2. (b)

    { T x n } n = 1 is bounded.

     
Remark 6
  1. 1.

    All the results can also be proved for the same iterative scheme with error terms.

     
  2. 2.

    The known results for strongly pseudocontractive mappings are weakened by the ϕ-hemicontractive mappings.

     
  3. 3.

    Our results hold in arbitrary Banach spaces, where as other known results are restricted for L p (or l p ) spaces and q-uniformly smooth Banach spaces.

     
  4. 4.

    Theorem 4 is more general in comparison to the results of Agarwal et al. [5] in the context of the class of ϕ-hemicontractive mappings. Theorem 4 extends convergence results coercing ϕ-hemicontractive mappings in the literature in the framework of Agarwal-O’Regan-Sahu iteration process (see also [1421]).

     

3 Applications

Theorem 7 Let X be an arbitrary real Banach space, S : X X be nonexpansive, and let T : X X be uniformly continuous ϕ-strongly accretive operators, respectively. Suppose that { b n } n = 1 and { b n } n = 1 are sequences in [ 0 , 1 ] satisfying conditions (i)-(ii) of Theorem  4. For any x 1 X , define the sequence { x n } n = 1 inductively as follows:
{ y n = b n x n + ( 1 b n ) ( f + ( I T ) x n ) , x n + 1 = b n ( f + ( I S ) x n ) + ( 1 b n ) ( f + ( I T ) y n ) , n 1 ,
where f X , and I is the identity operator. Then the following conditions are equivalent:
  1. (a)

    { x n } n = 1 converges strongly to the solution of the system S x = f = T x .

     
  2. (b)

    { ( I S ) x n } n = 1 , { ( I T ) x n } n = 1 and { ( I T ) y n } n = 1 are bounded.

     

Proof Suppose that x is the solution of the system S x = f = T x . Define G , G : X X by G x = f + ( I S ) x and G x = f + ( I T ) x , respectively. Since S and T are nonexpansive and uniformly continuous ϕ-strongly accretive operators, respectively, so are G and G , then x is the common fixed point of G and G . Thus, Theorem 7 follows from Theorem 4. □

Example 8 Let X = R be the reals with the usual norm and K = [ 0 , 1 ] . Define S : K K by
S x = sin x for all  x K
and T : K K by
T x = x tan x for all  x K .
By the mean value theorem, we have
| T ( x ) T ( y ) | sup c ( 0 , 1 ) | T ( c ) | | x y | for all  x , y K .
Noticing that T ( c ) = 1 sec 2 ( c ) and 1 < sup c ( 0 , 1 ) | T ( c ) | = 2.4255 . Hence,
| T ( x ) T ( y ) | L | x y | for all  x , y K ,
where L = 2.4255 . It is easy to verify that T is ϕ-hemicontractive mapping with ϕ : [ 0 , ) [ 0 , ) defined by ϕ ( t ) = tan ( t ) for all t [ 0 , ) . Moreover, 0 is the common fixed point of S and T. Let { b n } n = 1 and { b n } n = 1 be sequences in [ 0 , 1 ] defined by
b n = 1 1 n and b n = 1 n , n 1 .

Then { x n } n = 1 defined by (2.1) in Theorem 4 converges to 0, which is the common fixed point of S and T.

Declarations

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University
(2)
Department of Mathematics, Faculty of Science, Banaras Hindu University
(3)
Department of Mathematics, Lahore Leads University

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© Hussain et al.; licensee Springer. 2013

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