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Boundary point algorithms for minimum norm fixed points of nonexpansive mappings

Abstract

Let H be a real Hilbert space and C be a closed convex subset of H. Let T:CC be a nonexpansive mapping with a nonempty set of fixed points Fix(T). If 0C, then Halpern’s iteration process x n + 1 =(1 t n )T x n cannot be used for finding a minimum norm fixed point of T since x n may not belong to C. To overcome this weakness, Wang and Xu introduced the iteration process x n + 1 = P C (1 t n )T x n for finding the minimum norm fixed point of T, where the sequence { t n }(0,1), x 0 C arbitrarily and P C is the metric projection from H onto C. However, it is difficult to implement this iteration process in actual computing programs because the specific expression of P C cannot be obtained, in general. In this paper, three new algorithms (called boundary point algorithms due to using certain boundary points of C at each iterative step) for finding the minimum norm fixed point of T are proposed and strong convergence theorems are proved under some assumptions. Since the algorithms in this paper do not involve P C , they are easy to implement in actual computing programs.

MSC:47H09, 47H10, 65K10.

1 Introduction and preliminaries

Let H be a real Hilbert space with the inner product , and the norm , and let C be a nonempty closed convex subset of H. Recall that a mapping T:CC is nonexpansive if TxTyxy for all x,yC. We use Fix(T) to denote a set of fixed points of T, i.e., Fix(T){xCTx=x}. Throughout this article, Fix(T) 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.

Lemma 1.1 Let K be a closed convex subset of a real Hilbert space H. Given xH and zK. Then z= P K x if and only if there holds the relation

xz,yz0,yK.

Since Fix(T) is a closed convex subset of H, so the metric projection P Fix ( T ) is valid and thus there exists a unique element, denoted by x , in Fix(T) such that x = inf x Fix ( T ) x, that is, x = P Fix ( T ) 0. x is called a minimum norm fixed point of T. Because the minimum norm fixed point of a nonexpansive mapping is closely related to convex optimization problems, it is favored by people.

An extensive literature on iteration methods for fixed point problems of nonexpansive mappings has been published (for example, see [117]). Many iteration processes are often used to approximate a fixed point of a nonexpansive mapping in a Hilbert space or a Banach space. One of them is now known as Halpern’s iteration process [2] and is defined as follows: take an initial guess x 0 C arbitrarily and define { x n } recursively by

x n + 1 = t n u+(1 t n )T x n ,n=0,1,2,,
(1.1)

where { t 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 is as follows.

Theorem 1.2 ([13, 14])

If { t n } satisfies the conditions:

  1. (i)

    t n 0 (n);

  2. (ii)

    n = 1 t n =;

  3. (iii)

    lim n t n + 1 t n =1 or n = 1 | t n + 1 t n |<;

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 = inf x Fix ( T ) ux.

Now we consider how to get the minimum norm fixed point of T. In the case where 0C, taking u=0 in (1.1), we assert by using Theorem 1.2 that { x n } generated by (1.1) converges strongly to x under conditions (i)-(iii) above. But, in the case where 0C, the iteration process x n + 1 =(1 t n )T x n becomes invalid because x n may not belong to C. In order to overcome this weakness, Wang and Xu [15] introduced the iteration process

x n + 1 = P C (1 t n )T x n ,n=1,2,.
(1.2)

They proved that if { t n } satisfies the same conditions in Theorem 1.2, then the sequence { x n } generated by (1.2) converges strongly to x .

However, it is difficult to implement the iteration process (1.2) in actual computing programs because the specific expression of P C cannot be obtained, in general.

The purpose of this paper is to propose three new algorithms for finding the minimum norm fixed point of T. The strong convergence theorems are proved under some assumptions. The main advantage of the algorithms in this paper is that they have nothing to do with the metric projection P C and thus they are easy to implement in actual computing programs. Because the key of our algorithms is replacing a fixed element u in (1.1) by a certain sequence { u n } in the boundary of C, they are called boundary point algorithms.

We will use the following 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.

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

Lemma 1.3 ([18])

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 1.4 There holds the identity in a real Hilbert space H:

u v 2 = u 2 v 2 2v,uv,u,vH.

Lemma 1.5 ([12, 19])

Assume that { 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.

2 Main results

In this section, C is always assumed to be a nonempty closed convex subset of H such that 0C. We use ∂C to denote the boundary of C. In order to give our main results, we first introduce a function h:C(0,1] by the definition

h(x)=inf { λ ( 0 , 1 ] λ x C } ,xC.

It is easy to see that h(x)xC and h(x)>0 hold for each xC due to the assumption 0C.

Since our iteration processes will involve the function h(x), it is necessary to explain how to calculate h(x) for any given xC in actual computing programs. In order to get the value h(x) for a given xC, we often need to deal with an algebraic equation. But dealing with an algebraic equation is easier than calculating the metric projection P C , in general. To illustrate this viewpoint, let us consider the following simple example.

Example 1 Let H be a real Hilbert space. Define a convex function φ:H R 1 by

φ(x)= x x 0 2 +x,u,xH,

where x 0 and u are two given points in H such that x 0 ,u<0. Setting C={xHφ(x)0}, then it is easy to show that C is a nonempty convex closed subset of H such that 0C (note that φ( x 0 )= x 0 ,u<0 and φ(0)= x 0 2 >0). For a given xC, we have φ(x)0. In order to get h(x), let φ(λx)=0, where λ(0,1] is an unknown number. Thus we obtain an algebraic equation

x 2 λ 2 + ( x , u 2 x , x 0 ) λ+ x 0 2 =0.

Consequently, we get

λ= 2 x , x 0 x , u ± ( x , u 2 x , x 0 ) 2 4 x 2 x 0 2 2 x 2 .

By the definition of h, we have

h(x)= 2 x , x 0 x , u ( x , u 2 x , x 0 ) 2 4 x 2 x 0 2 2 x 2 .

Next we give our first iteration process for finding the minimum norm fixed point of T: take u 0 C arbitrarily and define { x n } recursively by

{ x n = P Fix ( T ) u n , u n = λ n x n 1 ,
(2.1)

where λ n =h( x n 1 ) (n1).

Remark 1 How to implement the iteration process (2.1)? In actual computing programs, we can use the standard Halpern’s iteration process to get x n from u n for each n0. Indeed, taking x n ( 0 ) = u n and { x n ( m ) } is generated inductively by

x n ( m + 1 ) = t m u n +(1 t m )T x n ( m ) ,m0,

then, using Theorem 1.2, x n ( m ) x n P Fix ( T ) u n as m. Thus we can take x n = x n ( M n ) approximately for a sufficiently large integer M n in actual computing programs.

Geometric intuition seems to encourage us to guess x n P Fix ( T ) 0 as n under some certain assumptions. As a matter of fact, it is true.

Theorem 2.1 If { λ n } satisfies n = 1 (1 λ n )=, then { x n } generated by (2.1) converges strongly to x = P Fix ( T ) 0.

Proof Noticing the fact that x = P Fix ( T ) 0= P Fix ( T ) λ x holds for all λ[0,1], we have from (2.1) that

x n x = P Fix ( T ) u n x = P Fix ( T ) λ n x n 1 P Fix ( T ) λ n x λ n x n 1 x ,

consequently,

x n x λ n λ n 1 λ 2 λ 1 x 0 x .
(2.2)

Thus this together with the condition n = 1 (1 λ n )= leads to the conclusion. □

Remark 2 Is the condition n = 1 (1 λ n )= reasonable? In other words, can we find an example which satisfies this condition? The answer is yes. The following result implies that this condition is not harsh.

Corollary 1 If d(Fix(T),C)inf{xyxFix(T),yC}>0, then { x n } generated by (2.1) converges strongly to x = P Fix ( T ) 0.

Proof Obviously, it suffices to verify that if d(Fix(T),C)>0, then n = 1 (1 λ n )=. In fact, setting dd(Fix(T),C)>0, we have from (2.1) and (2.2) that

λ n = u n x n 1 = x n 1 x n 1 u n x n 1 1 d x 0 + x 0 x ,

hence

1 λ n d x 0 + x 0 x .

This implies that n = 1 (1 λ n )= holds. □

Our second iteration process for finding the minimum norm fixed point of T is defined by

x n = t n λ n x n 1 +(1 t n )T x n ,n1,
(2.3)

where { t n }(0,1), λ n =h( x n 1 ) (n1) and x 0 is taken in C arbitrarily.

Remark 3 Equation (2.3) is an implicit iteration process. A natural question is how to get x n from x n 1 . Indeed, suppose that we have got x n 1 , define the mapping T n :CC by T n :x t n λ n x n 1 +(1 t n )Tx (xC), then T n is (1 t n )-contractive and x n is just its unique fixed point. So we can use Picard’s iteration process

x n ( m + 1 ) = t n λ n x n 1 +(1 t n )T x n ( m ) ,m0,

to calculate x n approximately since x n ( m ) x n as m, where x n ( 0 ) can be taken in C arbitrarily, for example, x n ( 0 ) = x n 1 .

Theorem 2.2 Assume that n = 1 (1 λ n )= and n = 1 t n <, then { x n } generated by (2.3) converges strongly to x = P Fix ( T ) 0.

Proof We first show that { x n } is bounded. Indeed, take a pFix(T) to derive that

x n p = t n λ n ( x n 1 p ) + ( 1 t n ) ( T x n p ) t n ( 1 λ n ) p t n λ n x n 1 p + ( 1 t n ) x n p + t n ( 1 λ n ) p .

It follows that

x n p λ n x n 1 p+(1 λ n )p.

By induction,

x n pmax { x 1 p , p }
(2.4)

and { x n } is bounded, so are {T x n }. This together with (2.3) implies that x n T x n 0 (n). Thus it follows from Lemma 1.3 that ω w ( x n )Fix(T).

Next we show that

lim n sup x , x n x 0.
(2.5)

Indeed, take a subsequence { x n k } of { x n } such that

lim n sup x , x n x = lim k x , x n k x ,

without loss of generality, we may assume that x n k x ¯ . Noticing x = P Fix ( T ) 0, we obtain from x ¯ Fix(T) and Lemma 1.1 that

lim n sup x , x n x = x , x ¯ x 0.

Finally, we show that x n x 0 (n). As a matter of fact, we have by using Lemma 1.4 that

x n x 2 = t n λ n ( x n 1 x ) + ( 1 t n ) ( T x n x ) t n ( 1 λ n ) x 2 t n λ n ( x n 1 x ) + ( 1 t n ) ( T x n x ) 2 + 2 t n ( 1 λ n ) x , x n x t n 2 λ n 2 x n 1 x 2 + ( 1 t n ) 2 x n x 2 + 2 t n λ n ( 1 t n ) x n 1 x x n x + 2 t n ( 1 λ n ) x , x n x .

Hence,

( 2 t n ) x n x 2 t n λ n 2 x n 1 x 2 + 2 λ n ( 1 t n ) x n 1 x x n x + 2 ( 1 λ n ) x , x n x t n λ n 2 x n 1 x 2 + λ n 2 x n 1 x 2 + ( 1 t n ) 2 x n x 2 + 2 ( 1 λ n ) x , x n x .

Consequently,

x n x 2 [ 1 ( 1 λ n ) ] x n 1 x 2 + 2 ( 1 λ n ) x , x n x + t n x n 1 x 2 .

Using Lemma 1.5, we conclude from (2.5) and conditions n = 1 (1 λ n )= and n = 1 t n < that x n x . □

By a similar argument as above, we easily get the following result.

Corollary 2 If d(R(T),C)inf{xyxFix(T),yC}>0 and 1 t n <, then { x n } generated by (2.3) converges strongly to x = P Fix ( T ) 0, where R(T) is the range of T.

Proof It suffices to verify that d(R(T),C)>0 implies n = 1 (1 λ n )=. Indeed,

λ n = u n x n 1 = x n 1 x n 1 u n x n 1 =1 x n 1 T x n 1 + T x n 1 u n x n 1 .

Setting dd(R(T),C)>0, we have from (2.4) that

1 λ n T x n 1 u n x n 1 x n 1 T x n 1 x n 1 d x 1 x + 2 x x n 1 T x n 1 d ( 0 , C ) .

Note that x n 1 T x n 1 0, it follows that n = 1 (1 λ n )=. □

Finally, we propose an explicit iteration process for finding the minimum norm fixed point of T which is defined by

x n + 1 = t n λ n x n +(1 t n )T x n ,n0,
(2.6)

where { t n }(0,1), λ n =h( x n ) (n0) and x 0 is taken in C arbitrarily.

Theorem 2.3 Assume that { t n } and { λ n } satisfy the following conditions:

  1. (i)

    t n 0 and n = 0 t n =;

  2. (ii)

    lim sup n λ n λ ¯ <1;

  3. (iii)

    n = 1 | t n t n 1 |< or lim n t n t n 1 =1;

  4. (iv)

    n = 1 t n | λ n λ n 1 |< or lim n λ n λ n 1 =1.

Then { x n } generated by (2.6) converges strongly to x = P Fix ( T ) 0.

Proof We first show that { x n } is bounded. Indeed, we have by taking pFix(T) arbitrarily that

x n + 1 p t n λ n x n p + ( 1 t n ) T x n p t n [ λ n x n p + ( 1 λ n ) p ] + ( 1 t n ) x n p t n max { x n p , p } + ( 1 t n ) x n p max { x n p , p } .

Inductively,

x n pmax { x 0 p , p } ,n0.

This means that { x n } is bounded, so are {T x n }.

We next show that x n + 1 x n 0. Using (2.6), it follows from a direct calculation that

x n + 1 x n = [ t n λ n x n + ( 1 t n ) T x n ] [ t n 1 λ n 1 x n 1 + ( 1 t n 1 ) T x n 1 ] = ( 1 t n ) ( T x n T x n 1 ) ( t n t n 1 ) T x n 1 + t n λ n ( x n x n 1 ) + ( t n λ n t n 1 λ n 1 ) x n 1 [ 1 t n ( 1 λ n ) ] x n x n 1 + | t n t n 1 | ( T x n 1 + λ n 1 x n 1 ) + t n | λ n λ n 1 | x n 1 .

Using Lemma 1.5, we conclude from conditions (i)-(iv) that x n + 1 x n 0. Noticing the boundedness of { x n } and {T x n } and condition (i), we have from (2.6) that x n + 1 T x n 0. Consequently, x n T x n 0. Using Lemma 1.3, we derive that ω w ( x n )Fix(T).

Then we show that

lim n sup x , x n 1 x 0.
(2.7)

As a matter of fact, this is derived by the same argument as in the proof of Theorem 2.3.

Finally, we show that x n x 0. Using Lemma 1.4 and (2.6), it is easy to verify that

x n + 1 x 2 = t n ( λ n x n x ) + ( 1 t n ) ( T x n x ) 2 ( 1 t n ) 2 T x n x 2 + 2 t n λ n x n x , x n + 1 x ( 1 t n ) 2 x n x 2 + 2 t n λ n x n x , x n + 1 x + 2 t n ( 1 λ n ) x , x n + 1 x ( 1 t n ) 2 x n x 2 + 2 t n λ n x n x x n + 1 x + 2 t n ( 1 λ n ) x , x n + 1 x .

Hence,

x n + 1 x 2 (1 γ n ) x n x 2 + γ n σ n ,

where

γ n = t n 2 ( 1 λ n ) t n 1 t n λ n , σ n = 2 ( 1 λ n ) 2 ( 1 λ n ) t n x , x n + 1 x .

It is easily seen that γ n 0, n = 0 γ n = by conditions (i) and (ii), and lim n sup σ n 0 by (2.7). By Lemma 1.5, we conclude that x n x . □

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Acknowledgements

This work was supported in part by the Fundamental Research Funds for the Central Universities (3122013k004) and in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing.

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Correspondence to Songnian He.

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He, S., Yang, C. Boundary point algorithms for minimum norm fixed points of nonexpansive mappings. Fixed Point Theory Appl 2014, 56 (2014). https://doi.org/10.1186/1687-1812-2014-56

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Keywords

  • minimum norm fixed point
  • nonexpansive mapping
  • metric projection
  • boundary point algorithm
  • Hilbert space