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A damped algorithm for the split feasibility and fixed point problems

Abstract

The purpose of this paper is to study the split feasibility problem and the fixed point problem. We suggest a damped algorithm. Convergence theorem is proven.

MSC:47J25, 47H09, 65J15, 90C25.

1 Introduction

Let C and Q be two closed convex subsets of two Hilbert spaces H 1 and H 2 , respectively, and let A: H 1 H 2 be a bounded linear operator. Finding a point x satisfies

x CandA x Q.
(1.1)

This problem, referred to as the split problem, has been studied by some authors. See, e.g., [18] and [9]. Some algorithms for solving (1.1) have been presented. One is Byrne’s CQ algorithm [1]

x n + 1 = P C ( x n τ A ( I P Q ) A x n ) ,nN,

where τ(0, 2 L ) with L being the largest eigenvalue of the matrix A A, I is the unit matrix or operator, and P C and P Q denote the orthogonal projections onto C and Q, respectively. Motivated by Byrne’s CQ algorithm, Xu [6] suggested a single step regularized method

x n + 1 = P C ( ( 1 α n γ n ) x n γ n A ( I P Q ) A x n ) ,nN.
(1.2)

Very recently, Dang and Gao [5] introduced the following damped projection algorithm

x n + 1 =(1 β n ) x n + β n P C ( ( 1 α n ) ( x n τ A ( I P Q ) A x n ) ) ,nN.

If every closed convex subset of a Hilbert space is the fixed point set of its associating projection, then the split feasibility problem becomes a special case of the split common fixed point problem of finding a point x with the property

x Fix(U)andA x Fix(T).

This problem was first introduced by Censor and Segal [10], who invented an algorithm, which generates a sequence { x n } according to the iterative procedure

x n + 1 =U ( x n γ A ( I T ) A x n ) ,nN.

Recently, Cui, Su and Wang [11] extended the damped projection algorithm to the split common fixed point problems. For some related work, please refer to [12] and [13, 14].

Motivated by these results, the purpose of this paper is to study the following split feasibility problem and fixed point problem

Find  x CFix(T) such that A x QFix(S),
(1.3)

where S:QQ and T:CC are two nonexpansive mappings. We suggest a damped algorithm for solving (1.3). Convergence theorem is proven.

2 Preliminaries

Let H be a real Hilbert space with the inner product , and the norm , respectively. Let C be a nonempty closed convex subset of H.

Definition 2.1 A mapping T:CC is called nonexpansive if

TxTyxy

for all x,yC.

We will use Fix(T) to denote the set of fixed points of T, that is, Fix(T)={xC:x=Tx}.

Definition 2.2 We call P C :HC the metric projection if for each xH

x P C ( x ) =inf { x y : y C } .

It is well known that the metric projection P C :HC is characterized by

x P C ( x ) , y P C ( x ) 0

for all xH, yC. From this, we can deduce that P C is firmly-nonexpansive, that is,

P C ( x ) P C ( y ) 2 x y , P C ( x ) P C ( y )
(2.1)

for all x,yH. Hence P C is also nonexpansive.

It is well known that in a real Hilbert space H, the following two equalities hold

t x + ( 1 t ) y 2 =t x 2 +(1t) y 2 t(1t) x y 2
(2.2)

for all x,yH and t[0,1], and

x + y 2 = x 2 +2x,y+ y 2
(2.3)

for all x,yH. It follows that

x + y 2 x 2 +2y,x+y
(2.4)

for all x,yH.

Lemma 2.3 [15]

Let C be a closed convex subset of a real Hilbert space H, and let S:CC be a nonexpansive mapping. Then, the mapping IS is demiclosed. That is, if { x n } is a sequence in C such that x n x weakly and (IS) x n y strongly, then (IS) x =y.

Lemma 2.4 [16]

Assume that { a n } is a sequence of nonnegative real numbers such that

a n + 1 (1 γ n ) a n + δ n ,nN,

where { γ n } is a sequence in (0,1), and { δ n } is a sequence such that

  1. (1)

    n = 1 γ n =;

  2. (2)

    lim sup n δ n γ n 0 or n = 1 | δ n |<.

Then lim n a n =0.

3 Main results

Let C and Q be two nonempty closed convex subsets of real Hilbert spaces H 1 and H 2 , respectively. Let A: H 1 H 2 be a bounded linear operator with its adjoint A . Let S:QQ and T:CC be two nonexpansive mappings. We use Γ to denote the set of solutions of (1.3), that is, Γ={ x | x CFix(T),A x QFix(S)}. Now, we present our algorithm.

Algorithm 3.1 For x 0 H 1 arbitrarily, let { x n } be a sequence defined by

x n + 1 =T P C ( ( 1 α n ) ( x n δ A ( I S P Q ) A x n ) ) for all nN,
(3.1)

where { α n } n N and { β n } n N are two real number sequences in (0,1) and δ(0, 1 A 2 ).

Theorem 3.2 Suppose Γ. Assume the sequence { α n } n N satisfies three conditions

(C1) lim n α n =0;

(C2) n = 1 α n =;

(C3) lim n α n + 1 α n =1.

Then the sequence { x n }, generated by algorithm (3.1), converges strongly to x = P Γ (0).

Proof For the convenience, we write z n = P Q A x n , y n =(1 α n )( x n δ A (IS P Q )A x n ) and u n = P C ((1 α n )( x n δ A (IS P Q )A x n )) for all nN. Thus u n = P C y n for all nN.

Let x = P Γ (0). Hence, x CFix(T) and A x QFix(S). By the firmly-nonexpansivity of P C and P Q , we can deduce the following conclusions

z n A x = P Q A x n P Q A x A x n A x ,
(3.2)
u n x = P C y n P C x y n x ,
(3.3)
S z n A x 2 z n A x 2 A x n A x 2 z n A x n 2 ,
(3.4)
u n + 1 u n = P C y n + 1 P C y n y n + 1 y n
(3.5)

and

z n + 1 z n = P Q A x n + 1 P Q A x n A x n + 1 A x n .
(3.6)

From (3.1) and (3.3), we have

x n + 1 x = T u n x u n x y n x .
(3.7)

Using (2.3), we get

y n x 2 = ( 1 α n ) ( x n x + δ A ( S z n A x n ) ) α n x 2 ( 1 α n ) ( x n x + δ A ( S z n A x n ) 2 + α n x 2 = ( 1 α n ) [ x n x + δ 2 A ( S z n A x n ) 2 + 2 δ x n x , A ( S z n A x n ) ] + α n x 2 .
(3.8)

Since A is a linear operator with its adjoint A , we have

x n x , A ( S z n A x n ) = A ( x n x ) , S z n A x n = A x n A x + S z n A x n ( S z n A x n ) , S z n A x n = S z n A x , S z n A x n S z n A x n 2 .
(3.9)

Again using (2.3), we obtain

S z n A x , S z n A x n = 1 2 ( S z n A x 2 + S z n A x n 2 A x n A x 2 ) .
(3.10)

By (3.4), (3.9) and (3.10), we get

x n x , A ( S z n A x n ) = 1 2 ( S z n A x 2 + S z n A x n 2 A x n A x 2 ) S z n A x n 2 1 2 ( A x n A x 2 z n A x n 2 + S z n A x n 2 A x n A x 2 ) S z n A x n 2 = 1 2 z n A x n 2 1 2 S z n A x n 2 .
(3.11)

Substituting (3.11) into (3.8), we deduce

y n x 2 ( 1 α n ) [ x n x 2 + δ 2 A 2 S z n A x n 2 + 2 δ ( 1 2 z n A x n 2 1 2 S z n A x n 2 ) ] + α n x 2 = ( 1 α n ) [ x n x 2 + ( δ 2 A 2 δ ) S z n A x n 2 δ z n A x n 2 ] + α n x 2 ( 1 α n ) x n x 2 + α n x 2 .
(3.12)

It follows from (3.7) that

x n + 1 x 2 y n x 2 ( 1 α n ) x n x 2 + α n x 2 max { x n x 2 , x 2 } .

The boundedness of the sequence { x n } yields.

Next, we estimate x n + 1 x n . Set v n = x n δ A (IS P Q )A x n . According to (2.3) and (3.5), we have

v n + 1 v n 2 = x n + 1 x n + δ [ A ( S P Q I ) A x n + 1 A ( S P Q I ) A x n ] 2 = x n + 1 x n 2 + δ 2 A [ ( S P Q I ) A x n + 1 ( S P Q I ) A x n ] 2 + 2 δ x n + 1 x n , A [ ( S P Q I ) A x n + 1 ( S P Q I ) A x n ] x n + 1 x n 2 + δ 2 A 2 S z n + 1 S z n ( A x n + 1 A x n ) 2 + 2 δ A x n + 1 A x n , S z n + 1 S z n ( A x n + 1 A x n ) = x n + 1 x n 2 + δ 2 A 2 S z n + 1 S z n ( A x n + 1 A x n ) 2 + 2 δ S z n + 1 S z n , S z n + 1 S z n ( A x n + 1 A x n ) 2 δ S z n + 1 S z n ( A x n + 1 A x n ) 2 = x n + 1 x n 2 + δ 2 A 2 S z n + 1 S z n ( A x n + 1 A x n ) 2 + δ ( S z n + 1 S z n 2 + S z n + 1 S z n ( A x n + 1 A x n ) 2 A x n + 1 A x n 2 ) 2 δ S z n + 1 S z n ( A x n + 1 A x n ) 2 = x n + 1 x n 2 + ( δ 2 A 2 δ ) S z n + 1 S z n ( A x n + 1 A x n ) 2 + δ ( S z n + 1 S z n 2 A x n + 1 A x n 2 ) x n + 1 x n 2 + ( δ 2 A 2 δ ) S z n + 1 S z n ( A x n + 1 A x n ) 2 + δ ( z n + 1 z n 2 A x n + 1 A x n 2 ) .
(3.13)

Since δ(0, 1 A 2 ), we derive by virtue of (3.6) and (3.13) that

v n + 1 v n x n + 1 x n .
(3.14)

From (3.5) and (3.14), we have

x n + 1 x n y n + 1 y n = ( 1 α n + 1 ) v n + 1 ( 1 α n ) v n = ( 1 α n + 1 ) ( v n + 1 v n ) + ( α n α n + 1 ) v n ( 1 α n + 1 ) v n + 1 v n + | α n + 1 α n | v n ( 1 α n + 1 ) x n + 1 x n + | α n + 1 α n | v n .

It follows that

x n + 1 x n | α n + 1 α n | α n + 1 v n .

This, together with condition (C3), implies that

lim n x n + 1 x n =0.
(3.15)

That is,

lim n x n T u n =0.
(3.16)

Using the firmly-nonexpansiveness of P C , we have

u n x 2 = P C y n x 2 y n x 2 P C y n y n 2 = y n x 2 u n y n 2 .
(3.17)

Thus,

x n + 1 x 2 u n x 2 y n x 2 u n y n 2 ( 1 α n ) x n x 2 + α n x 2 u n y n 2 .
(3.18)

It follows that

u n y n 2 x n x 2 x n + 1 x 2 + α n x 2 ( x n x + x n + 1 x ) x n + 1 x n + α n x 2 .

This, together with (3.15) and (C1), implies that

lim n u n y n =0.
(3.19)

Returning to (3.18) and using (3.12), we have

x n + 1 x 2 y n x 2 ( 1 α n ) x n x 2 + ( 1 α n ) ( δ 2 A 2 δ ) S z n A x n 2 ( 1 α n ) δ z n A x n 2 + α n x 2 .

Hence,

( 1 α n ) ( δ δ 2 A 2 ) S z n A x n 2 + ( 1 α n ) δ z n A x n 2 x n x 2 x n + 1 x 2 + α n x 2 ( x n x + x n + 1 x ) x n + 1 x n + α n x 2 ,

which implies that

lim n S z n A x n = lim n z n A x n =0.
(3.20)

So,

lim n S z n z n =0.
(3.21)

Note that

y n x n = δ A ( S P Q I ) A x n + α n v n δ A S z n A x n + α n v n .

It follows from (3.20) that

lim n x n y n =0.
(3.22)

From (3.16), (3.19) and (3.22), we get

lim n x n T x n =0.
(3.23)

Now, we show that

lim sup n x , y n x 0.

Choose a subsequence { y n i } of { y n } such that

lim sup n x , y n x = lim i x , y n i x .
(3.24)

Since the sequence { y n i } is bounded, we can choose a subsequence { y n i j } of { y n i } such that y n i j z. For the sake of convenience, we assume (without loss of generality) that y n i z. Consequently, we derive from the above conclusions that

x n i z, u n i z,A x n i Azand z n i Az.
(3.25)

By the demiclosed principle of the nonexpansive mappings S and T (see Lemma 2.3), we deduce that zFix(T) and AzFix(S) (according to (3.23) and (3.21), respectively). Note that u n i = P C y n i C and z n i = P Q A x n i Q. From (3.25), we deduce zC and AzQ. To this end, we deduce that zCFix(T) and AzQFix(S). That is to say, zΓ. Therefore,

lim sup n x , y n x = lim i x , y n i x = lim i x , z x 0 .
(3.26)

Finally, we prove that x n x . From (3.1), we have

x n + 1 x 2 y n x 2 = ( 1 α n ) ( v n x ) α n x 2 ( 1 α n ) v n x 2 2 α n x , y n x ( 1 α n ) x n x 2 2 α n x , y n x .
(3.27)

Applying Lemma 2.4 and (3.26) to (3.27), we deduce that x n x . The proof is completed. □

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Acknowledgements

Cun-lin Li was supported in part by NSFC 71161001-G0105. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3. Yonghong Yao was supported in part by NSFC 11071279, NSFC 71161001-G0105 and LQ13A010007.

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Li, Cl., Liou, YC. & Yao, Y. A damped algorithm for the split feasibility and fixed point problems. J Inequal Appl 2013, 379 (2013). https://doi.org/10.1186/1029-242X-2013-379

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Keywords

  • split feasibility problem
  • fixed point problem
  • nonexpansive mapping
  • damped algorithm