Open Access

Hybrid iterative algorithms for nonexpansive and nonspreading mappings in Hilbert spaces

Fixed Point Theory and Applications20132013:314

https://doi.org/10.1186/1687-1812-2013-314

Received: 31 August 2013

Accepted: 28 October 2013

Published: 25 November 2013

Abstract

Recently, Iemoto and Takahashi considered a weak convergence iterative scheme for a nonspreading mapping and a nonexpansive mapping in Hilbert spaces. In this paper, we suggest two hybrid iterative algorithms by modifying Iemoto and Takahashi’s iterative scheme for a countable family of nonspreading mappings and a nonexpansive mapping in Hilbert spaces.

MSC:47H05, 47H09.

Keywords

nonexpansive mappings nonspreading mappings AKTT-condition hybrid algorithms

1 Introduction and preliminaries

Let H be a Hilbert space and C be a nonempty closed convex subset of H. Let T be a nonlinear mapping of C into itself. We use F ( T ) and P C to denote the set of fixed points of T and the metric projection from H onto C, respectively.

Recall that T is said to be nonexpansive if
T x T y x y
(1.1)

for all x , y C .

For approximating the fixed point of a nonexpansive mapping in a Hilbert space, Mann [1] in 1953 introduced the famous iterative scheme as follows:
x 1 C , x n + 1 = ( 1 α n ) x n + α n T x n , n 1 ,
(1.2)

where T is a nonexpansive mapping of C into itself and { α n } is a sequence in ( 0 , 1 ) . It is well known that { x n } defined in (1.2) converges weakly to a fixed point of T.

Attempts to modify the normal Mann iteration method (1.2) for nonexpansive mappings so that strong convergence is guaranteed have recently been made; see, e.g., [29].

Let T be a mapping from C into itself. Then T is called nonspreading [3] if
2 T x T y 2 T x y 2 + x T y 2

for all x , y C . A mapping T : C C is called quasi-nonexpansive if F ( T ) and T x y x y for all x C and y F ( T ) . If T is a nonspreading mapping from C into itself and F ( T ) is nonempty, then T is quasi-nonexpansive. Further, we know that the set of fixed points of each quasi-nonexpansive mapping is closed and convex; see [10].

In [11], by using Moudafi’s iterative scheme [12], Iemoto and Takahashi considered the following weak convergence theorem.

Theorem IT ([11])

Let H be a Hilbert space, and let C be a nonempty closed convex subset of H. Let S be a nonspreading mapping of C into itself, and let T be a nonexpansive mapping of C into itself such that F ( S ) F ( T ) . Define a sequence { x n } as follows:
{ x 1 C , x n + 1 = ( 1 α n ) x n + α n { β n S x n + ( 1 β n ) T x n }
(1.3)
for all n N , where { α n } , { β n } [ 0 , 1 ] . Then the following hold:
  1. (i)

    If lim inf n α n ( 1 α n ) > 0 and n = 1 ( 1 β n ) < , then { x n } converges weakly to v F ( S ) ;

     
  2. (ii)

    If i = 1 α n ( 1 α n ) = and n = 1 β n < , then { x n } converges weakly to v F ( T ) ;

     
  3. (iii)

    If lim inf n α n ( 1 α n ) > 0 and lim inf n β n ( 1 β n ) > 0 , then { x n } converges weakly to v F ( S ) F ( T ) .

     

In this paper, we modify (1.1) by a hybrid iterative scheme and obtain the strong convergence theorems for a family of nonspreading mappings and a nonexpansive mapping in a Hilbert space.

Let E be a Banach space and K be a nonempty closed convex subset of E. Let { T n } : K K be a family of mappings. Then { T n } is said to satisfy the AKTT-condition [13] if for each bounded subset B of K, one has
n = 1 sup { T n + 1 z T n z : z B } < .

The following is an important result on a family of mappings { T n } n = 1 satisfying the AKTT-condition.

Lemma 1.1 ([13])

Let K be a nonempty and closed subset of a Banach space E, and let { T n } n = 1 be a family of mappings of K into itself which satisfies the AKTT-condition. Then, for each x K , { T n x } converges strongly to a point in K. Moreover, let the mapping T : K K be defined by
T x = lim n T n x , x K .
Then, for each bounded subset B of K,
lim sup n { T z T n z : z B } = 0 .

Obviously, if a family of mappings { T n } n = 1 satisfies the AKTT-condition and T x = lim n T n x for each x K , then it is unnecessary that F ( T ) = i = 1 F ( T i ) . To show this, see the following example.

Example 1.1 Let E = R and K = [ 0 , 2 ] . Define a family of mappings { T n } n = 1 : K K by
T 1 x = 0 , T n = 1 n ( 1 + x ) , n 2 .

Then { T n } n = 1 satisfy the AKTT-condition. It is easy to see that for each x K , lim n T n x = 0 . Define the mapping T : K K by T x = lim n T n x . That is, T x = 0 for all x K . But F ( T ) n = 1 F ( T n ) .

In this paper, we call that { T n , T } satisfy the AKTT-condition if { T n } n = 1 satisfy the AKTT-condition with F ( T ) = n = 1 F ( T n ) .

Lemma 1.2 ([11])

Let C be a nonempty closed subset of a Hilbert space H. Then a mapping T : C C is nonspreading if and only if
T x T y 2 x y 2 + 2 x T x , y T y

for all x , y C .

By using Lemma 1.2, we get the following simple but important result.

Lemma 1.3 Let H be a Hilbert space and C be a nonempty subset of H. Let { T n } be a family of nonspreading mappings of C into itself, and assume that lim n T n x exists for each x C . Define the mapping T : C C by T x = lim n T n x . Then the mapping T is a nonspreading mapping.

Proof In fact, for all x , y C , we have
T x T y 2 = lim n T n x lim n T n y 2 = lim n T n x T n y 2 lim n [ x y 2 + 2 x T n x , y T n y ] = x y 2 + 2 x lim n T n x , y lim n T n y = x y 2 + 2 x T x , y T y .

Lemma 1.2 shows that the mapping T is a nonspreading mapping. □

Lemma 1.4 Let C be a closed convex subset of a real Hilbert space H, and let P C be the metric projection from H onto C (i.e., for x H , P C x is the only point in C such that x P C x = inf { x z : z C } ). Given x H and z C . Then z = P C x if and only if the following relation holds:
x z , y z 0 , y C .

Lemma 1.5 ([14])

Let H be a real Hilbert space. Then the following equation holds:
t x + ( 1 t ) y 2 = t x 2 + ( 1 t ) y 2 t ( 1 t ) x y 2 , x C and t [ 0 , 1 ] .

2 Main results

Theorem 2.1 Let C be a nonempty closed convex subset of a Hilbert space H. Let S : C C be a nonexpansive mapping and { T i } i = 1 : C C be a countable family of nonspreading mappings such that F = F ( S ) [ i = 1 F ( T i ) ] . Let { x n } be a sequence generated in the following manner:
{ x 1 = x C chosen arbitrarily , y n = ( 1 α n ) x n + α n [ β n S x n + i = 1 n ( β i 1 β i ) T i x n ] , C n = { v C : y n v x n v } , D n = j = 1 n C j , x n + 1 = P D n x , n 1 ,
(2.1)
where { α n } , { β n } [ 0 , 1 ] . Assume that { β n } is strictly decreasing and β 0 = 1 . Then the following hold:
  1. (i)

    If lim inf n α n > 0 and lim n β n = 0 , then { x n } strongly converges to q i = 1 F ( T i ) ;

     
  2. (ii)

    If lim inf n α n ( 1 α n ) > 0 and lim inf n β n > 0 , then { x n } converges strongly to q F .

     
Proof Obviously, each C n is closed and convex and hence D n is closed and convex. Next, we show that F D n for all n 1 . To end this, we need to prove that F C n for all n 1 . Indeed, for each p F , we have
y n p ( 1 α n ) x n p + α n [ β n S x n p + i = 1 n ( β i 1 β i ) T i x n p ] ( 1 α n ) x n p + α n [ β n x n p + i = 1 n ( β i 1 β i ) x n p ] = x n p .
(2.2)
This implies that
p C n for all  n 1 .

Therefore, F C n and hence C n is nonempty for all n 1 . On the other hand, from the definition of D n , we see that F D n = i = 1 n C j for all n 1 .

From x n + 1 = P D n x , we have
x n + 1 x v x , v D n , n 1 .
Since P F x F D n , one has
x n + 1 x P F x x , n 1 .
(2.3)

This implies that { x n } is bounded and hence { y n } is bounded.

On the other hand, since D n + 1 D n for all n 1 , we have
x n + 2 = P D n + 1 x D n + 1 D n
for all n 1 . From x n + 1 = P D n x one has
x n + 1 x x n + 2 x
(2.4)

for all n 1 . It follows from (2.3) and (2.4) that the limit of { x n x } exists.

Since D m D n and x m + 1 = P D m x D m D n for all m n and x n + 1 = P D n x , by Lemma 1.4 one has
x n + 1 x , x m + 1 x n + 1 0 .
(2.5)
It follows from (2.5) that
x m + 1 x n + 1 2 = x m + 1 x ( x n + 1 x ) 2 = x m + 1 x 2 + x n + 1 x 2 2 x n + 1 x , x m + 1 x = x m + 1 x 2 + x n + 1 x 2 2 x n + 1 x , x m + 1 x n + 1 + x n + 1 x = x m + 1 x 2 x n + 1 x 2 2 x n + 1 x , x m + 1 x n + 1 x m + 1 x 2 x n + 1 x 2 .
(2.6)
Since the limit of x n x exists, we get
lim m , n x m x n = 0 .
It follows that { x n } is a Cauchy sequence. Since H is a Hilbert space and C is closed and convex, there exists q C such that
x n q , as  n .
(2.7)
By taking m = n + 1 in (2.6), one arrives at
lim n x n + 2 x n + 1 = 0 ,
i.e.,
lim n x n + 1 x n = 0 .
(2.8)
Noticing that x n + 1 = P D n x D n C n , we get
y n x n + 1 x n x n + 1 0 ,
and hence
y n x n y n x n + 1 + x n + 1 x n 0 .
(2.9)
From (2.7) and (2.9) it follows that
lim n y n p = lim n x n p = q p , p F .
(2.10)
Now we prove (i). Note that
y n = ( 1 α n ) x n + α n [ β n S x n + i = 1 n ( β i 1 β i ) ( T i x n x n ) ] + α n ( 1 β n ) x n = ( 1 α n β n ) x n + α n β n S x n + α n i = 1 n ( β i 1 β i ) ( T i x n x n ) .
Hence,
α n i = 1 n ( β i 1 β i ) ( T i x n x n ) = ( 1 α n β n ) ( y n x n ) + α n β n ( y n S x n ) .
(2.11)
On the other hand, for any p F , from Lemma 1.2 we have
x n p 2 = 2 x n T i x n , p T i p + x n p 2 T i x n T i p 2 = T i x n p 2 = T i x n x n + ( x n p ) 2 = T i x n x n 2 + x n p 2 + 2 T i x n x n , x n p ,
and hence
T i x n x n 2 2 x n T i x n , x n p , i N .
(2.12)
Note that { β n } is strictly decreasing. Hence from (2.11) and (2.12) we get
T i x n x n 2 1 2 α n ( β i 1 β i ) [ ( 1 α n β n ) y n x n , x n T i p + α n β n y n S x n , x n p ] , i 1 .
(2.13)
Since lim inf n α n > 0 and lim n β n = 0 , from (2.9) and (2.13) it follows that
lim n T i x n x n = 0 , i N .
(2.14)
Since each T i is a nonspreading mapping, by Lemma 1.2, (2.7) and (2.10), we have
T i q T i x n 2 x n q 2 + 2 q T i q , x n T i x n 0 , i N .
(2.15)
Further, one has
q T i q q x n + x n T i x n + T i x n T i q 0 , i N .
(2.16)

So, we have q i = 1 F ( T i ) .

To prove (ii), first we show that lim n x n S x n = 0 . For any p F , we have
y n p 2 = β n [ ( 1 α n ) x n + α n S x n p ] + i = 1 n ( β i 1 β i ) [ ( 1 α n ) x n + α n T i x n p ] 2 β n ( 1 α n ) x n + α n S x n p 2 + i = 1 n ( β i 1 β i ) ( 1 α n ) x n + α n T i x n p 2 β n ( 1 α n ) x n + α n S x n p 2 + i = 1 n ( β i 1 β i ) [ ( 1 α n ) x n p 2 + α n T i x n p 2 ] β n ( 1 α n ) x n + α n S x n p 2 + i = 1 n ( β i 1 β i ) x n p 2 = β n ( 1 α n ) x n + α n S x n p 2 + ( 1 β n ) x n p 2 x n p 2 ,
and hence by (2.10) we get
0 x n p 2 β n ( 1 α n ) x n + α n S x n p 2 ( 1 β n ) x n p 2 = β n [ x n p 2 ( 1 α n ) x n + α n S x n p 2 ] x n p 2 y n p 2 0 .
(2.17)
Since lim inf n β n > 0 , it follows from (2.17) that
lim n ( x n p 2 ( 1 α n ) x n + α n S x n p 2 ) = 0 .
(2.18)
From (2.18) and
( 1 α n ) x n + α n S x n p 2 = ( 1 α n ) x n p 2 + α n S x n p 2 α n ( 1 α n ) x n S x n 2 ,
we get
α n ( 1 α n ) x n S x n 2 = ( x n p 2 ( 1 α n ) x n + α n S x n p 2 ) α n x n p 2 + α n S x n p 2 ( x n p 2 ( 1 α n ) x n + α n S x n p 2 ) α n x n p 2 + α n x n p 2 = x n p 2 ( 1 α n ) x n + α n S x n p 2 0 .
Since lim inf n α n ( 1 α n ) > 0 , we get
lim n x n S x n = 0 .
(2.19)
Now, using (2.19), (2.7) and
q S q q x n + x n S x n + S x n S q 2 q x n + x n S x n 0 ,

which implies that q F ( S ) .

Note that (2.9) and (2.19) imply that lim n y n S x n = 0 . Then, repeating (2.11) to (2.16), we get q i = 1 F ( T i ) . So, q F . This completes the proof. □

Theorem 2.2 Let C be a nonempty closed convex subset of a Hilbert space H. Let S : C C be a nonexpansive mapping and { T i } i = 1 : C C be a countable family of nonspreading mappings such that F = F ( S ) [ i = 1 F ( T i ) ] . Let { x n } be a sequence generated in the following manner:
{ x 1 = x C chosen arbitrarily , y n = ( 1 α n ) x n + α n [ β n S x n + ( 1 β n ) T n x n ] , C n = { v C : y n v x n v } , D n = j = 1 n C j , x n + 1 = P D n x , n 1 ,
(2.20)
where { α n } , { β n } [ 0 , 1 ] . Assume that { T n , T } satisfies the AKTT-condition. Then the following hold:
  1. (i)

    If lim inf n α n > 0 and lim n β n = 0 , then { x n } strongly converges to v i = 1 F ( T i ) ;

     
  2. (ii)

    If lim inf n α n ( 1 α n ) > 0 and lim inf n β n > 0 , then { x n } converges strongly to z F .

     
Proof By a process similar to the proof of Theorem 2.1, we can conclude that { x n } converges strongly to some q C and
x n y n 0 .
We first prove (i). From (2.20) we have
T n x n x n = 1 α n ( 1 β n ) ( y n x n ) β n 1 β n ( S x n x n ) ,
and hence
T n x n x n 1 α n ( 1 β n ) y n x n + β n 1 β n S x n x n .
Since lim inf n α n > 0 and lim n β n = 0 , we get
lim n T n x n x n = 0 .
(2.21)
Further, by Lemma 1.1 and (2.21), we have
x n T x n x n T n x n + T n x n T x n x n T n x n + sup { T n z T z : z { x n } } 0 .
(2.22)
Since each T n is a nonspreading mapping, Lemma 1.3 shows that T is a nonspreading mapping. Further, by using Lemma 1.2, we have
T q T x n 2 x n q 2 + 2 q T q , x n T x n 0 , i N .
(2.23)
From (2.21) and (2.23) it follows that
q T q q x n + x n T x n + T x n T q 0 .
(2.24)

It follows that q F ( T ) . Since ( { T n } , T ) satisfies the AKTT-condition, one has q i = 1 F ( T i ) = F ( T ) . This completes (i).

Next we show (ii). By a process similar to the proof of Theorem 2.1 and from (2.22) to (2.24), we can get that
lim n x n S x n = 0 , lim n x n T n x n = 0 , lim n T x n T q = 0 and lim n x n T x n = 0 .
Finally, by
q S q q x n + x n S x n + S x n S q 2 x n q + x n S x n 0
and
q T q q x n + x n T x n + T x n T q 0 ,

we get q F ( S ) F ( T ) . Since ( { T n } , T ) satisfies the AKTT-condition, we conclude that q F . This completes (ii). □

Letting T i = T for all i N in Theorem 2.1 and Theorem 2.2, we get the following corollary.

Corollary 2.1 Let C be a nonempty closed convex subset of a Hilbert space H. Let S : C C be a nonexpansive mapping and T : C C be a nonspreading mapping such that F ( S ) F ( T ) . Let { x n } be a sequence generated in the following manner:
{ x 1 = x C chosen arbitrarily , y n = ( 1 α n ) x n + α n [ β n S x n + ( 1 β n ) T x n ] , C n = { v C : y n v x n v } , D n = j = 1 n C j , x n + 1 = P D n x , n 1 ,
where { α n } , { β n } [ 0 , 1 ] . Then the following hold:
  1. (i)

    If lim inf n α n > 0 and lim n β n = 0 , then { x n } strongly converges to x F ( T ) ;

     
  2. (ii)

    If lim inf n α n ( 1 α n ) > 0 and lim inf n β n > 0 , then { x n } converges strongly to q F ( S ) F ( T ) with q = P F x .

     

Letting S = I in Theorems 2.1 and 2.2, we get the following corollary.

Corollary 2.2 Let C be a nonempty closed convex subset of a Hilbert space H. Let { T i } i = 1 : C C be a countable family of nonspreading mappings such that i = 1 F ( T i ) . Let { x n } be a sequence generated in the following manner:
{ x 1 = x C chosen arbitrarily , y n = ( 1 α n ( 1 β n ) ) x n + α n i = 1 n ( β i 1 β i ) T i x n , C n = { v C : y n v x n v } , D n = j = 1 n C j , x n + 1 = P D n x , n 1 ,

where { α n } , { β n } [ 0 , 1 ] . Assume that { β n } is strictly decreasing and β 0 = 1 . Then if lim inf n α n ( 1 α n ) > 0 , then { x n } strongly converges to q i = 1 F ( T i ) .

Corollary 2.3 Let C be a nonempty closed convex subset of a Hilbert space H. Let { T i } i = 1 : C C be a countable family of nonspreading mappings such that i = 1 F ( T i ) . Let { x n } be a sequence generated in the following manner:
{ x 1 = x C chosen arbitrarily , y n = ( 1 α n ( 1 β n ) ) x n + α n ( 1 β n ) T n x n , C n = { v C : y n v x n v } , D n = j = 1 n C j , x n + 1 = P D n x , n 1 ,

where { γ n } [ 0 , 1 ] . Assume that ( { T n , T } ) satisfies the AKTT-condition. Then if lim inf n α n ( 1 α n ) > 0 , then { x n } strongly converges to q i = 1 F ( T i ) .

Declarations

Acknowledgements

This work is supported by the Fundamental Research Funds for the Central Universities (Grant Number: 13MS109).

Authors’ Affiliations

(1)
School of Mathematics and Physics, North China Electric Power University

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© Wang; licensee Springer. 2013

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