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General alternative regularization methods for nonexpansive mappings in Hilbert spaces

Abstract

Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product , and the norm . Let T:CC be a nonexpansive mapping with a nonempty set of fixed points Fix(T) and let h:CC be a Lipschitzian strong pseudo-contraction. We first point out that the sequence generated by the usual viscosity approximation method x n + 1 = λ n h( x n )+(1 λ n )T x n may not converge to a fixed point of T, even not bounded. Secondly, we prove that if the sequence ( λ n )(0,1) satisfies the conditions: (i) λ n 0, (ii) n = 0 λ n = and (iii)  n = 0 | λ n + 1 λ n |< or lim n λ n + 1 λ n =1, then the sequence ( x n ) generated by a general alternative regularization method: x n + 1 =T( λ n h( x n )+(1 λ n ) x n ) converges strongly to a fixed point of T, which also solves the variational inequality problem: finding an element x such that h( x ) x ,x x 0 for all xFix(T). Furthermore, we prove that if T is replaced with the sequence of average mappings (1 β n )I+ β n T (n0) such that 0< β β n β <1, where β and β are two positive constants, then the same convergence result holds provided conditions (i) and (ii) are satisfied. Finally, an algorithm for finding a common fixed point of a family of finite nonexpansive mappings is also proposed and its strong convergence is proved. Our results in this paper extend and improve the alternative regularization methods proposed by HK Xu.

MSC:47H09, 47H10, 65K10.

1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product , and the norm and let f:CC be a α-contractive mapping, i.e., there exists a constant α[0,1) such that f(x)f(y)αxy holds for all x,yC. Let T:CC be a nonexpansive mapping, i.e., TxTyxy for all x,yC. Throughout this article, the set of fixed points of T, indicated by Fix(T){xCTx=x}, is always assumed to be nonempty.

For every nonempty closed convex subset K of H, the metric (or nearest point) projection indicated by P K from H onto K can be defined, that is, for each xH, P K x is the only point in K such that x P K x=inf{xzzK}. It is well known (e.g., see [1]) that P K is nonexpansive and a characteristic inequality holds: given xH and zK, then z= P K x if and only if

xz,yz0,yK.

Since Fix(T) is a closed convex subset of H, so the metric projection P Fix ( T ) is valid.

Recall that a mapping h:CC is said to be L-Lipschitzian, if there exists a positive constant L such that

h ( x ) h ( y ) Lxy,x,yC,

and a mapping h:CC is said to be α-strongly pseudo-contractive, if there exists a constant α[0,1) such that

h ( x ) h ( y ) , x y α x y 2 ,x,yC.

In this case, we also call h a α-strong pseudo-contraction.

It is very easy to see that a α-contractive mapping is a α-strongly pseudo-contractive and α-Lipschitzian mapping, i.e., the class of contractive mappings is a proper subset of the class of Lipschitzian strong pseudo-contractions. The class of Lipschitzian strong pseudo-contractions will be used repeatedly in the sequel.

Recall that a mapping F:CH is said to be η-strongly monotone, if there exists a positive constant η such that

F ( x ) F ( y ) , x y η x y 2 ,x,yC.

The variational inequality problem [2] can mathematically be formulated as the problem of finding a point x K with the property

F x , x x 0,xK,

where K is a nonempty closed convex subset of H and F:KH is a nonlinear operator. It is well known that [3] if F:KH is a Lipschitzian and strongly monotone operator, then the variational inequality problem has a unique solution.

Many iteration processes are often used to approximate a fixed point of a nonexpansive mapping in a Hilbert space or a Banach space (for example, see [1] and [420]). One of them is now known as Halpern’s iteration process [4] and is defined as follows: take an initial guess x 0 C arbitrarily and define ( x n ) recursively by

x n + 1 = λ n u+(1 λ n )T x n ,n0,
(1.1)

where ( λ n ) is a sequence in the interval [0,1] and u is some given element in C. For Halpern’s iteration process, a classical result in the setting of Hilbert spaces is as follows.

Theorem 1.1 ([4])

If ( λ n ) satisfies the conditions:

  1. (i)

    λ n 0 (n);

  2. (ii)

    n = 0 λ n =;

  3. (iii)

    n = 0 | λ n + 1 λ n |< or lim n λ n + 1 λ n =1;

then the sequence ( x n ) generated by (1.1) converges strongly to a fixed point x of T such that x = P Fix ( T ) u, that is,

u x , x x 0,xFix(T).

Xu [5] proposed an alternative regularization method:

x n + 1 =T ( λ n u + ( 1 λ n ) x n ) ,n0
(1.2)

and studied its strong convergence in the setting of Hilbert spaces and Banach spaces, respectively. Indeed, in the setting of Hilbert spaces, one can proved that for algorithm (1.2) the same convergence result as that of Theorem 1.1 holds under conditions (i)-(iii) above.

As an extension to Halpern’s iteration process, Moudafi proposed [15] the viscosity approximation method: take an initial guess x 0 C arbitrarily and define ( x n ) recursively by

x n + 1 = λ n f( x n )+(1 λ n )T x n ,n0,
(1.3)

where ( λ n ) is a sequence in the interval [0,1]. Moudafi proved the following result in Hilbert spaces.

Theorem 1.2 ([15])

If ( λ n ) satisfies the same conditions (i)-(iii) as above, then the sequence ( x n ) generated by (1.3) converges strongly to a fixed point x of T, which also solves the variational inequality problem: finding an element x Fix(T) such that

f ( x ) x , x x 0,xFix(T).

Xu studied the viscosity approximation method in the setting of Banach spaces and obtained the strong convergence theorems [16].

Similar to algorithm (1.2), we can naturally consider a general alternative regularization method: take an initial guess x 0 C arbitrarily and define ( x n ) recursively by

x n + 1 =T ( λ n f ( x n ) + ( 1 λ n ) x n ) ,n0,
(1.4)

where ( λ n ) is a sequence in the interval [0,1]. In fact, in the setting of Hilbert spaces, it is not difficult to prove by an argument very similar to the proof of Theorem 1.2 that for algorithm (1.4) the same result as that of Theorem 1.2 holds under conditions (i)-(iii) above.

The main purpose of this paper is to consider a very natural question: if algorithms (1.3) and (1.4) can be extended to more general cases, more precisely, can we replace a contractive mapping f with a Lipschitzian strong pseudo-contraction h so that the same convergence result as that of Theorem 1.2 is still guaranteed under conditions (i)-(iii) as above? The answer to this question is negative for algorithm (1.3) unfortunately but is sure for algorithm (1.4) fortunately. In this sense, it seems reasonable to deem that algorithm (1.4) is better that algorithm (1.3).

Now we illustrate a fact via a very simple example that if f in algorithm (1.3) is replaced with a Lipschitzian and strongly pseudo-contractive mapping h, then strong convergence (even boundedness) of the iteration sequence ( x n ) may not be guaranteed, in general. Indeed, take H= R 2 , C= R 2 , and T: R 2 R 2 defined by

( u v )( 0 1 1 0 )( u v )=( v u ),x=( u v ) R 2 .

Noting that T is just a rotation operator over R 2 , we see that T is a nonexpansive mapping. Moreover, noting the fact that

Tx,x=0

holds for all x R 2 , it is easy to see that for any positive constant κ and any α[0,1), hκT is a κ-Lipschitzian and α-strongly pseudo-contractive mapping.

If f in algorithm (1.3) is replaced with h=T (i.e., κ=1), then (1.3) is of the form:

x n + 1 =T x n ,n0.
(1.5)

Since T is a rotation operator over R 2 , the sequence ( x n ) generated by (1.5) does not converge to ( 0 0 ) , the unique fixed point of T, unless we take the initial guess x 0 = ( 0 0 ) . On the other hand, if f in algorithm (1.3) is replaced with h=2T (i.e., κ=2), thus (1.3) is rendered in the form

x n + 1 =(1+ λ n )T x n ,n0.
(1.6)

Consequently, noting that Tx=x, x R 2 , we have

x n + 1 = ( 1 + λ n ) T x n = ( 1 + λ n ) x n = ( 1 + λ n ) ( 1 + λ n 1 ) ( 1 + λ 0 ) x 0 .

Since n = 0 + λ n =, lim n (1+ λ n )(1+ λ n 1 )(1+ λ 0 )= holds and thus this implies that the sequence ( x n ) generated by (1.6) is not bounded provided x 0 ( 0 0 ) .

The rest of this paper is organized as follows. In order to prove our main results, some useful facts and tools are listed as lemmas in the next section. In Section 3, we prove that if a contractive mapping f in algorithm (1.4) is replaced with a Lipschitzian strong pseudo-contraction h, then the same convergence result as that of Theorem 1.2 is still guaranteed under conditions (i)-(iii) as above. Furthermore, we prove that if T is replaced with the sequence of average mappings (1 β n )I+ β n T (n0) such that 0< β β n β <1, where β and β are two positive constants, then the same result still holds provided conditions (i) and (ii) are satisfied. In the last section, an algorithm for finding a common fixed point of a family of finite nonexpansive mappings is also proposed and its strong convergence is proved.

We will use the notations:

  1. 1.

    for weak convergence and → for strong convergence.

  2. 2.

    ω w ( x n )={x( x n k )( x n ) such that  x n k x} denotes the weak ω-limit set of ( x n ).

  3. 3.

    AB means that B is the definition of A.

2 Preliminaries

We need some facts and tools in a real Hilbert space H, which are listed as lemmas below.

Lemma 2.1 The following relation holds in a real Hilbert space H:

x + y 2 x 2 +2y,x+y,x,yH.

Lemma 2.2 ([14, 21])

Assume ( a n ) is a sequence of nonnegative real numbers satisfying the property

a n + 1 (1 γ n ) a n + γ n δ n + σ n ,n=0,1,2.

If ( γ n ) n = 1 (0,1), ( δ n ) n = 1 and ( σ n ) n = 1 satisfy the conditions:

  1. (i)

    n = 1 γ n =,

  2. (ii)

    lim sup n δ n 0,

  3. (iii)

    n = 1 | σ n |<,

then lim n a n =0.

Lemma 2.3 ([22])

Let C be a closed convex subset of a real Hilbert space H and let T:CC be a nonexpansive mapping such that Fix(T). If a sequence ( x n ) in C is such that x n z and x n T x n 0, then z=Tz.

Lemma 2.4 For each λ[0,1], the following identity holds in a real Hilbert space H:

λ u + ( 1 λ ) v 2 =λ u 2 +(1λ) v 2 λ(1λ) u v 2 ,u,vH.

Lemma 2.5 ([23])

Assume ( s n ) is a sequence of nonnegative real numbers such that

s n + 1 (1 γ n ) s n + γ n δ n ,n0,
(2.1)
s n + 1 s n η n + α n ,n0,
(2.2)

where ( γ n ) is a sequence in (0,1), ( η n ) is a sequence of nonnegative real numbers and ( δ n ) and ( α n ) are two sequences in such that

  1. (i)

    n = 0 γ n =,

  2. (ii)

    lim n α n =0,

  3. (iii)

    lim k η n k =0 implies lim sup k δ n k 0 for any subsequence ( n k )(n).

Then lim n s n =0.

3 Algorithms for one mapping

Throughout this section, we will assume that H is a real Hilbert space with the inner product , and the norm , C is a closed convex subset of H and h:CC is a L-Lipschitzian and α-strongly pseudo-contractive mapping, i.e., there exist positive constants L and α[0,1) such that

h ( x ) h ( y ) Lxy,x,yC,
(3.1)
h ( x ) h ( y ) , x y α x y 2 ,x,yC.
(3.2)

It is obvious that if h is a α-strong pseudo-contraction, then Ih is a (1α)-strongly monotone mapping, i.e.,

( I h ) x ( I h ) y , x y (1α) x y 2 ,x,yC,

where I denotes the identity operator.

Our first result is as follows.

Theorem 3.1 Let h:CC be a L-Lipschitzian and α-strongly pseudo-contractive mapping and let T:CC be a nonexpansive mapping. If the sequence ( λ n )(0,1) satisfies the conditions:

  1. (i)

    λ n 0 (n);

  2. (ii)

    n = 0 λ n =;

  3. (iii)

    n = 0 | λ n + 1 λ n |< or lim n λ n + 1 λ n =1;

then the sequence ( x n ) generated by the algorithm

x n + 1 =T ( λ n h ( x n ) + ( 1 λ n ) x n ) ,n0,
(3.3)

where x 0 is selected in C arbitrarily, converges strongly to a fixed point of T, which also solves the variational inequality problem: finding an element x Fix(T) such that

x h ( x ) , x x 0,xFix(T).
(3.4)

Proof Noting that Ih is a (1+L)-Lipschitzian and (1α)-strongly monotone mapping, so the variational inequality problem (3.4) has a unique solution, which is denoted by x . Now we try to prove that x n x .

Firstly, we deduce from (3.1)-(3.3) that

x n + 1 x 2 = T ( λ n h ( x n ) + ( 1 λ n ) x n ) T x 2 λ n ( h ( x n ) x ) + ( 1 λ n ) ( x n x ) 2 = λ n 2 h ( x n ) x 2 + 2 λ n ( 1 λ n ) h ( x n ) x , x n x + ( 1 λ n ) 2 x n x 2 2 λ n 2 [ L 2 x n x 2 + h ( x ) x 2 ] + 2 λ n ( 1 λ n ) ( α x n x 2 + h ( x ) x , x n x ) + ( 1 λ n ) 2 x n x 2 .
(3.5)

Obviously

h ( x ) x , x n x h ( x ) x x n x β x n x 2 + 1 4 β h ( x ) x 2 ,
(3.6)

where β is a positive constant such that α+β<1. Thus, the combination of (3.5) and (3.6) leads to

x n + 1 x 2 [ 1 2 λ n ( 1 λ n ( 1 2 + L 2 ) ( 1 λ n ) ( α + β ) ) ] x n x 2 + 2 λ n 2 h ( x ) x 2 + 2 λ n ( 1 λ n ) 1 4 β h ( x ) x 2 [ 1 2 λ n ( 1 λ n ( 1 2 + L 2 ) ( 1 λ n ) ( α + β ) ) ] x n x 2 + 2 λ n ( 1 + 1 4 β ) h ( x ) x 2 .
(3.7)

Noting the fact that λ n 0 and

lim n [ 1 λ n ( 1 2 + L 2 ) ( 1 λ n ) ( α + β ) ] =1(α+β)>0,

we assert that there exists some integer n 0 such that 2 λ n <1 and

1 λ n ( 1 2 + L 2 ) (1 λ n )(α+β)> 1 2 ( 1 ( α + β ) )
(3.8)

hold for all n n 0 . So we see from (3.7) and (3.8) that for all n n 0 , the following relation holds:

x n + 1 x 2 ( 1 λ n ( 1 ( α + β ) ) ) x n x 2 + λ n ( 1 ( α + β ) ) 2 1 ( α + β ) ( 1 + 1 4 β ) h ( x ) x 2 .
(3.9)

Consequently

x n + 1 x 2 max { x n x 2 , 2 1 ( α + β ) ( 1 + 1 4 β ) h ( x ) x 2 } ,n n 0 ,

and inductively

x n x 2 max { x n 0 x 2 , 2 1 ( α + β ) ( 1 + 1 4 β ) h ( x ) x 2 } ,n n 0 .

This means that ( x n ) is bounded, so is (h( x n )).

From (3.3), we have

x n + 1 T x n = λ n ( h ( x n ) x n ) λ n ( h ( x n ) + x n ) 0 ( n )
(3.10)

due to the boundedness of ( x n ) and (h( x n )). Now we show that x n + 1 x n 0. Setting u n =(1 λ n ) x n (1 λ n 1 ) x n 1 , v n = λ n h( x n ) λ n 1 h( x n 1 ), we derive from Lemma 2.1, (3.1) and (3.2) that

u n 2 = ( 1 λ n ) x n ( 1 λ n 1 ) x n 1 2 = ( 1 λ n ) ( x n x n 1 ) ( λ n λ n 1 ) x n 1 2 ( 1 λ n ) 2 x n x n 1 2 2 ( λ n λ n 1 ) x n 1 , ( 1 λ n ) x n ( 1 λ n 1 ) x n 1 ( 1 λ n ) 2 x n x n 1 2 + 2 | λ n λ n 1 | ( x n 1 2 + x n 1 x n ) ,
(3.11)
v n 2 = λ n h ( x n ) λ n 1 h ( x n 1 ) 2 = λ n ( h ( x n ) h ( x n 1 ) ) + ( λ n λ n 1 ) h ( x n 1 ) 2 λ n 2 h ( x n ) h ( x n 1 ) 2 + 2 ( λ n λ n 1 ) h ( x n 1 ) , λ n h ( x n ) λ n 1 h ( x n 1 ) λ n 2 L 2 x n x n 1 2 + 2 | λ n λ n 1 | ( h ( x n 1 ) 2 + h ( x n 1 ) h ( x n ) ) ,
(3.12)

and

u n , v n = ( 1 λ n ) ( x n x n 1 ) ( λ n λ n 1 ) x n 1 , λ n ( h ( x n ) h ( x n 1 ) ) + ( λ n λ n 1 ) h ( x n 1 ) λ n ( 1 λ n ) α x n x n 1 2 + | λ n λ n 1 | x n 1 ( h ( x n ) + h ( x n 1 ) ) + | λ n λ n 1 | ( x n h ( x n 1 ) + x n 1 h ( x n 1 ) ) + | λ n λ n 1 | 2 x n 1 h ( x n 1 ) .
(3.13)

Noting the boundedness of ( x n ) and (h( x n )), it follows from (3.11)-(3.13) that

x n + 1 x n 2 u n + v n 2 u n 2 + v n 2 + 2 u n , v n [ 1 γ n ] x n x n 1 2 + | λ n λ n 1 | M ,

where γ n = λ n (2 λ n (1+ L 2 )2(1 λ n )α) and M is a positive constant independent on n. Observing that

lim n [ 2 λ n ( 1 + 2 L 2 ) 2 ( 1 λ n ) α ] =2(1α)>0,

we get from conditions (i) and (ii) that γ n 0 and n = 0 γ n = hold. Hence, this together with condition (iii) allows us to assert x n + 1 x n 0 by using Lemma 2.2. From this together with (3.10) one concludes that x n T x n 0 and hence we obtain ω( x n )Fix(T) by using Lemma 2.3.

Finally, we prove x n x (n). Again using (3.5), we also have

x n + 1 x 2 [ 1 λ n ( 2 λ n ( 1 + 2 L 2 ) 2 ( 1 λ n ) α ) ] x n x 2 + 2 λ n 2 h ( x ) x 2 + 2 λ n ( 1 λ n ) h ( x ) x , x n x ( 1 γ n ) x n x 2 + γ n δ n ,
(3.14)

where γ n = λ n (2 λ n (1+ L 2 )2(1 λ n )α) and

δ n = 2 λ n h ( x ) x 2 + 2 ( 1 λ n ) h ( x ) x , x n x 2 λ n ( 1 + L 2 ) 2 ( 1 λ n ) α .

Take a subsequence ( x n k ) such that

lim sup n h ( x ) x , x n x = lim k h ( x ) x , x n k x .

Without loss of the generality, we assume that there exists some x ¯ Fix(T) (noting that ω( x n )Fix(T) holds) such that x n k x ¯ (otherwise, we may select some subsequence of ( x n k ) with this property). Hence, we have

lim sup n h ( x ) x , x n x = lim k h ( x ) x , x n k x = h ( x ) x , x ¯ x 0
(3.15)

noting that x is the unique solution of the variational inequality (3.4). Consequently, it is easy to verify that lim sup n δ n 0 by using (3.15). This together with γ n 0 and n = 0 γ n = allows us to use Lemma 2.2 to (3.14) to obtain

x n x 0(n).

 □

Theorem 3.2 Let h:CC be a L-Lipschitzian and α-strongly pseudo-contractive mapping and let T:CC be a nonexpansive mapping. Let T n =(1 β n )I+ β n T, where ( β n )(0,1). Take x 0 C arbitrarily and define a sequence ( x n ) by the process

x n + 1 = T n ( λ n h ( x n ) + ( 1 λ n ) x n ) ,n0,
(3.16)

where ( λ n )(0,1). If ( λ n ) and ( β n ) satisfy the conditions:

  1. (i)

    λ n 0 (n);

  2. (ii)

    n = 0 λ n =;

  3. (iii)

    there exist two constants β and β such that 0< β β n β <1 for all n0,

then the sequence ( x n ) converges strongly to a fixed point of T, which also solves the variational inequality problem: finding a point x Fix(T) such that

x h ( x ) , x x 0,xFix(T).

Proof Firstly, noting that

x n + 1 x ( 1 β n ) ( λ n h ( x n ) + ( 1 λ n ) x n ) x + β n T ( λ n h ( x n ) + ( 1 λ n ) x n ) T x λ n h ( x n ) + ( 1 λ n ) x n x ,
(3.17)

we assert that ( x n ) is bounded by an argument similar to the proof of Theorem 3.1. Moreover, in a way similar to getting (3.5), we have from (3.17)

x n + 1 x 2 λ n ( h ( x n ) x ) + ( 1 λ n ) ( x n x ) 2 [ 1 λ n ( 2 λ n ( 1 + 2 L 2 ) 2 ( 1 λ n ) α ) ] x n x 2 + 2 λ n 2 h ( x ) x 2 + 2 λ n ( 1 λ n ) h ( x ) x , x n x .
(3.18)

Secondly, setting z n = λ n h( x n )+(1 λ n ) x n and using Lemma 2.4, we have from (3.16) and (3.18)

x n + 1 x 2 = ( 1 β n ) ( z n x ) + β n ( T z n x ) 2 = ( 1 β n ) z n x 2 + β n T z n x 2 β n ( 1 β n ) z n T z n 2 z n x 2 β n ( 1 β n ) z n T z n 2 x n x 2 β n ( 1 β n ) z n T z n 2 + 2 λ n 2 h ( x ) x 2 + 2 λ n ( 1 λ n ) h ( x ) x x n x .
(3.19)

Setting

s n = x n x 2 , γ n = λ n ( 2 λ n ( 1 + 2 L 2 ) 2 ( 1 λ n ) α ) , δ n = 1 2 λ n ( 1 + 2 L 2 ) 2 ( 1 λ n ) α [ 2 λ n h ( x ) x 2 + 2 ( 1 λ n ) h ( x ) x , x n x ] , η n = β n ( 1 β n ) z n T z n 2 , α n = 2 λ n 2 h ( x ) x 2 + 2 λ n ( 1 λ n ) h ( x ) x x n x ,

thus (3.18) and (3.19) can be rewritten as the form

s n + 1 (1 γ n ) s n + γ n s n ,
(3.20)
s n + 1 s n η n + α n .
(3.21)

Since γ n 0, n = 0 γ n = ( lim n (2 λ n (1+2 L 2 )2(1 λ n )α)=2(1α)>0) and α n 0 hold obviously, so in order to complete the proof by using Lemma 2.5, it suffices to verify that η n k 0 (k) implies that

lim ¯ k δ n k 0

for any subsequence ( n k )(n).

Indeed, η n k 0 (k) implies that z n k T z n k 0 due to condition (iii). Furthermore, the relation

x n k T x n k x n k z n k + z n k T z n k + T z n k T x n k 2 x n k z n k + z n k T z n k
(3.22)

together with the fact that

x n z n λ n [ h ( x n ) + x n ] 0

allows us to get x n k T x n k 0 (k). Using Lemma 2.3, we get ω( x n k )Fix(T). Thus we have

lim ¯ k h ( x ) x , x n k x 0

and hence lim ¯ k δ n k 0 holds. □

4 Algorithm for several mappings

In this section, we turn to considering an algorithm for finding a common fixed point of a family of finite nonexpansive mappings.

Let H be a real Hilbert space and let C be a closed convex subset of H. Let N be an integer such that N2 and let T i :CC (i=1,2,,N) be a family of finite nonexpansive mappings. Set

T i n = ( 1 β i n ) I+ β i n T i ,n0,i=1,2,,N,

where ( β i n )(0,1) for all n0 and i=1,2,,N.

Our main result in this section is as follows.

Theorem 4.1 Let h:CC be a L-Lipschitzian and α-strongly pseudo-contractive mapping. Take a initial guess x 0 C arbitrarily and define a sequence ( x n ) by

x n + 1 = T N n T N 1 n T 1 n ( λ n h ( x n ) + ( 1 λ n ) x n ) ,n0,
(4.1)

where ( λ n )(0,1) and T i n is given as above. If ( λ n ) and ( β i n ) satisfy the conditions:

  1. (i)

    λ n 0 (n);

  2. (ii)

    n = 0 λ n =;

  3. (iii)

    there exist two constants β and β such that 0< β β i n β <1 for all n0 and i=1,2,,N,

then the sequence ( x n ) generated by (4.1) converges strongly to a common fixed point of T 1 , T 2 ,, T N , which also solves the variational inequality problem: finding an element x i = 1 N Fix( T i ) such that

x h ( x ) , x x 0,x i = 1 N Fix( T i ).
(4.2)

Proof Without loss of the generality, we only give the proof for the case where N=2.

It is clear that the variational inequality (4.2) has a unique solution, which is denoted by x in the sequel. An argument very similar to the proof of Theorem 3.2 allows us to verify that ( x n ) is bounded (so is (h( x n ))) and the following relation holds:

x n + 1 x 2 [ 1 λ n ( 2 λ n ( 1 + 2 L 2 ) 2 ( 1 λ n ) α ) ] x n x 2 + 2 λ n 2 h ( x ) x 2 + 2 λ n ( 1 λ n ) h ( x ) x , x n x .
(4.3)

On the other hand, setting z n = λ n h( x n )+(1 λ n ) x n , we have, by using Lemma 2.4,

x n + 1 x 2 = ( 1 β 2 n ) ( T 1 n z n x ) + β 2 n ( T 2 T 1 n z n x ) 2 = ( 1 β 2 n ) T 1 n z n x 2 + β 2 n T 2 T 1 n z n x 2 β 2 n ( 1 β 2 n ) T 2 T 1 n z n T 1 n z n 2 T 1 n z n x 2 β 2 n ( 1 β 2 n ) T 2 T 1 n z n T 1 n z n 2 = ( 1 β 1 n ) z n x 2 + β 1 n T 1 z n x 2 β 1 n ( 1 β 1 n ) z n T 1 z n 2 β 2 n ( 1 β 2 n ) T 1 n z n T 2 T 1 n z n 2 z n x 2 β 1 n ( 1 β 1 n ) z n T 1 z n 2 β 2 n ( 1 β 2 n ) T 1 n z n T 2 T 1 n z n 2 .
(4.4)

Similar to (3.18), it is easy to verify that

z n x 2 x n x 2 + 2 λ n 2 h ( x ) x 2 + 2 λ n ( 1 λ n ) h ( x ) x x n x .
(4.5)

Combining (4.4) and (4.5), we derive that

x n + 1 x 2 x n x 2 β 1 n ( 1 β 1 n ) z n T 1 z n 2 β 2 n ( 1 β 2 n ) T 1 n z n T 2 T 1 n z n 2 + 2 λ n 2 h ( x ) x 2 + 2 λ n ( 1 λ n ) h ( x ) x x n x .
(4.6)

Setting

s n = x n x , γ n = λ n [ 2 λ n ( 2 λ n ( 1 + 2 L 2 ) 2 ( 1 λ n ) α ) ] , δ n = 1 1 λ n ( 2 λ n ( 1 + 2 L 2 ) 2 ( 1 λ n ) α ) [ 2 λ n h ( x ) x 2 + 2 ( 1 λ n ) h ( x ) x , x n x ] , η n = β 1 n ( 1 β 1 n ) z n T 1 z n 2 + β 2 n ( 1 β 2 n ) T 1 n z n T 2 T 1 n z n 2 , α n = 2 λ n 2 h ( x ) x 2 + 2 λ n ( 1 λ n ) h ( x ) x x n x ,

thus (4.3) and (4.6) can be rewritten in the following forms, respectively:

s n + 1 (1 γ n ) s n + γ n s n ,
(4.7)
s n + 1 s n η n + α n .
(4.8)

By using Lemma 2.5, in order to complete the proof, it suffices to show that η n k 0 (k) implies that

lim ¯ k δ n k 0.

In fact, noting condition (iii), we get from η n k 0, z n k T 1 z n k 0 and T 1 n k z n k T 2 T 1 n k z n k 0 all hold. Consequently, x n k T 1 x n k 0 follows from the inequality

x n k T 1 x n k x n k z n k + z n k T 1 z n k + T 1 z n k T 1 x n k 2 x n k z n k + z n k T 1 z n k

and the fact that x n z n 0 (n) and hence ω( x n k )Fix( T 1 ) holds. On the other hand, using z n k T 1 z n k 0, T 1 n k z n k T 2 T 1 n k z n k 0 and the relation

x n k T 2 x n k x n k z n k + z n k T 1 n k z n k + T 1 n k z n k T 2 T 1 n k z n k + T 2 T 1 n k z n k T 2 z n k + T 2 z n k T 2 x n k 2 x n k z n k + 2 z n k T 1 z n k + T 1 n k z n k T 2 T 1 n k z n k ,

we conclude that x n k T 2 x n k 0 and this means that ω( x n k )Fix( T 2 ) also holds. Thus we have proved that

ω( x n k )Fix( T 1 )Fix( T 2 ).
(4.9)

Moreover, we have

lim ¯ k h ( x ) x , x n x 0,

and hence

lim ¯ k δ n k 0.

 □

References

  1. Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.

    Google Scholar 

  2. Stampacchia G: Formes bilineaires coercivites sur les ensembles convexes. C. R. Acad. Sci. Paris 1964, 258: 4413–4416.

    MathSciNet  Google Scholar 

  3. Lions JL, Stampacchia G: Variational inequalities. Commun. Pure Appl. Math. 1967, 20: 493–512. 10.1002/cpa.3160200302

    Article  MathSciNet  Google Scholar 

  4. Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0

    Article  Google Scholar 

  5. Xu HK: An alternative regularization method for nonexpansive mappings with applications. Contemp. Math. 2010, 513: 239–263.

    Article  Google Scholar 

  6. Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3

    Article  Google Scholar 

  7. Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 2003, 279: 372–379. 10.1016/S0022-247X(02)00458-4

    Article  MathSciNet  Google Scholar 

  8. Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 1979, 67: 274–276. 10.1016/0022-247X(79)90024-6

    Article  MathSciNet  Google Scholar 

  9. Reich S: Strong convergence theorem for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 1980, 75: 287–292. 10.1016/0022-247X(80)90323-6

    Article  MathSciNet  Google Scholar 

  10. Reich S, Shemen L: Two algorithms for nonexpansive mappings. Fixed Point Theory 2011, 12: 443–448.

    MathSciNet  Google Scholar 

  11. Shioji N, Takahashi W: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 1997, 125: 3641–3645. 10.1090/S0002-9939-97-04033-1

    Article  MathSciNet  Google Scholar 

  12. Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178(2):301–308. 10.1006/jmaa.1993.1309

    Article  MathSciNet  Google Scholar 

  13. Wittmann R: Approximation of fixed points of nonexpansive mappings. Arch. Math. 1992, 58: 486–491. 10.1007/BF01190119

    Article  MathSciNet  Google Scholar 

  14. Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332

    Article  Google Scholar 

  15. Moudafi A: Viscosity approximation methods for fixed points problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615

    Article  MathSciNet  Google Scholar 

  16. Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059

    Article  MathSciNet  Google Scholar 

  17. Wang F, Xu HK: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl. 2010., 2010: Article ID 102085 10.1155/2010/102085

    Google Scholar 

  18. Diestel J Lecture Notes in Mathematics 485. In Geometry of Banach Spaces-Selected Topics. Springer, Berlin; 1975.

    Google Scholar 

  19. Beauzamy B: Introduction to Banach Spaces and Their Geometry. North-Holland, Amsterdam; 1982.

    Google Scholar 

  20. Reich S: Approximating fixed points of nonexpansive mappings. Panam. Math. J. 1994, 4(2):23–28.

    Google Scholar 

  21. Liu LS: Iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289

    Article  MathSciNet  Google Scholar 

  22. Goebel K, Kirk WA Cambridge Studies in Advanced Mathematics 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.

    Chapter  Google Scholar 

  23. He SN, Yang CP: Solving the variational inequality problem defined on intersection of finite level sets. Abstr. Appl. Anal. 2013., 2013: Article ID 942315

    Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant no. 11201476) and in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing.

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Yang, C., He, S. General alternative regularization methods for nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl 2014, 203 (2014). https://doi.org/10.1186/1687-1812-2014-203

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Keywords

  • fixed point
  • nonexpansive mapping
  • strong pseudo-contraction
  • viscosity approximation method
  • general alternative regularization method