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Hybrid iterative algorithms for nonexpansive and nonspreading mappings in Hilbert spaces

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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.

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

TxTyxy
(1.1)

for all x,yC.

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 ,n1,
(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,yC. A mapping T:CC is called quasi-nonexpansive if F(T) and Txyxy for all xC and yF(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 nN, 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 vF(S);

  2. (ii)

    If i = 1 α n (1 α n )= and n = 1 β n <, then { x n } converges weakly to vF(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 vF(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 }:KK 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 xK, { T n x} converges strongly to a point in K. Moreover, let the mapping T:KK be defined by

Tx= lim n T n x,xK.

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 Tx= lim n T n x for each xK, 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 :KK by

T 1 x=0, T n = 1 n (1+x),n2.

Then { T n } n = 1 satisfy the AKTT-condition. It is easy to see that for each xK, lim n T n x=0. Define the mapping T:KK by Tx= lim n T n x. That is, Tx=0 for all xK. 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:CC is nonspreading if and only if

T x T y 2 x y 2 +2xTx,yTy

for all x,yC.

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 xC. Define the mapping T:CC by Tx= lim n T n x. Then the mapping T is a nonspreading mapping.

Proof In fact, for all x,yC, 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 xH, P C x is the only point in C such that x P C x=inf{xz:zC}). Given xH and zC. Then z= P C x if and only if the following relation holds:

xz,yz0,yC.

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 +(1t) y 2 t(1t) x y 2 ,xCandt[0,1].

2 Main results

Theorem 2.1 Let C be a nonempty closed convex subset of a Hilbert space H. Let S:CC be a nonexpansive mapping and { T i } i = 1 :CC 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 qF.

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 n1. To end this, we need to prove that F C n for all n1. Indeed, for each pF, 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 n1.

Therefore, F C n and hence C n is nonempty for all n1. On the other hand, from the definition of D n , we see that F D n = i = 1 n C j for all n1.

From x n + 1 = P D n x, we have

x n + 1 xvx,v D n ,n1.

Since P F xF D n , one has

x n + 1 x P F xx,n1.
(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 n1, we have

x n + 2 = P D n + 1 x D n + 1 D n

for all n1. From x n + 1 = P D n x one has

x n + 1 x x n + 2 x
(2.4)

for all n1. 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 mn 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 qC 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=qp,pF.
(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 pF, 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,iN.
(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,iN.
(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 +2q T i q, x n T i x n 0,iN.
(2.15)

Further, one has

q T i qq x n + x n T i x n + T i x n T i q0,iN.
(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 pF, 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

qSqq x n + x n S x n +S x n Sq2q x n + x n S x n 0,

which implies that qF(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, qF. This completes the proof. □

Theorem 2.2 Let C be a nonempty closed convex subset of a Hilbert space H. Let S:CC be a nonexpansive mapping and { T i } i = 1 :CC 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 zF.

Proof By a process similar to the proof of Theorem 2.1, we can conclude that { x n } converges strongly to some qC 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 +2qTq, x n T x n 0,iN.
(2.23)

From (2.21) and (2.23) it follows that

qTqq x n + x n T x n +T x n Tq0.
(2.24)

It follows that qF(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

qSqq x n + x n S x n +S x n Sq2 x n q+ x n S x n 0

and

qTqq x n + x n T x n +T x n Tq0,

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

Letting T i =T for all iN 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:CC be a nonexpansive mapping and T:CC 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 qF(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 :CC 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 :CC 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 ).

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Acknowledgements

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

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Correspondence to Shenghua Wang.

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Wang, S. Hybrid iterative algorithms for nonexpansive and nonspreading mappings in Hilbert spaces. Fixed Point Theory Appl 2013, 314 (2013) doi:10.1186/1687-1812-2013-314

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Keywords

  • nonexpansive mappings
  • nonspreading mappings
  • AKTT-condition
  • hybrid algorithms