Strong convergence of an algorithm about quasi nonexpansive mapping for the split common fixed point...

Strong Convergence of an Algorithm about Quasi-

Non Expansive Mappings for The Split Common

Fixed-Point Problem in Hilbert Space

Abstract: Based on the very recent work by Censor-Segal [4] and inspired by Xu [6], Zhao-

Yang [10] and Bauschke-Combettes [2], in this paper, we study the algorithm proposed by

Moudafi [11] for the class of quasi – non expansive operators to solve the split common fixed-

point problem (SCFP) in the framework of Hilbert space. Furthermore we proved the strong

convergence for the (SCFP), which extend and improve the result of Moudafi [11] from a weak

convergence to a strong convergence.

Keywords: Feasibility problem, Fejer Monotonicity, Iterative Algorithm, Strong Convergence,

Quasi-Nonresponsive.

Lawan Bulama Mohammed

Department Of Mathematics, P.M.B 65,

Bauchi State University, Nigeria.

ISSN 2319-9725

August, 2013 www.ijirs.com Vol 2 Issue 8

International Journal of Innovative Research and Studies 98

1. Introduction:

Throughout, this paper is a real Hilbert space, ⟨ ⟩ denote the inner product space and ‖ ‖

stand for the corresponding norm. The split common fixed–point problem (SCFP) is a

generalization of the split feasibility problem (SFP) and the convex feasibility problem

(CFP). In this paper, we study the strong convergence algorithm for solving the SCFP for the

class of quasi-non expansive operators such that is closed at origin. This general

class, which properly include the class of directed operator is more desirable, for example, in

fixed-point methods, in image recovery where, in many cases, it is possible to map the set of

image possessing a certain property to the fixed point set of non linear quasi-non expansive

operator . Also for the hybrid steepest decent method, which is an algorithmic solution to

the variational inequality problem over the fixed-point set of certain quasi-non expansive

mappings and is applicable to a broad range of convex constrained nonlinear inverse

problems in real Hilbert spaces. See also Mainge, which proved the convergence of a

viscosity algorithm forsuch a class of operator.

It was in 2011, that Moudafi give an extension of the unified framework developed in to a

general class of quasi-nonexpansive operators by proposing a weak convergence result and

the work was done in content of general Hilbert space. In this paper we study the algorithm

proposed by Moudafi and we give the strong convergence result of the Moudafi’s algorithm

for general class of quasi-non expansive operator, our work improved the result of Moudafi

from a weak convergence to a strong convergence and also this result improved some recent

result announced.

To begin with, let us recall that the split feasibility problem (SFP) is to find a point

x (1,1)

where C and Q are nonempty closed convex subset of Hilbert spaces , and

is bounded and linear operators. Assuming that the SFP is consistence (i.e. (1.1)

has a solution), it is not hard to see that solves (1.1) if and only if it solves the fixed-

point equation

( ( ) ) (1.2)

where orthogonal projection onto C and Q, respectively, is any positive

constant and denotes adjoint of A.

To solve (1.2), Byne proposed his CQ algorithm, which generate a sequence { } by

( ( ) ) (1.3)


International Journal of Innovative Research and Studies Page 299

where (

) with being the spectral radius of the operator .

The CQ algorithm (1.3) is a special case of the K-M algorithm. Due to the fixed –point

formulation (1.2) of the SFP, we can apply the K-M algorithm to the operator (

( ) ) to obtain a sequence given by

( ) ( ( ) )

( )

where (

) and again is the spectral radius of the operator . Then as long as ( )

satisfies the condition∑ ( ) we have weak convergence of the sequence

generated by (1.4). It should be noticed that when the (SFP) has no solution, the CQ

algorithm converges to a minimizer of ‖ ( ) ‖over , whenever such a

minimizer exists. For a complete and exhaustive study on algorithms for solving convex

feasibility problem, including comments about their applications and an excellent

bibliography see, for example, H. H. Bauschke and J. M. Borwein . Now let us first recall the

definition of quasi-nonexpansive operators which appear naturally when using sub-gradient

projection operator techniques in solving some feasibility problems and also some definitions

of classes of operators often used in fixed-point theory and which are commonly encountered

in the literature.

An operator:

T belongs in the general class of (possibly discontinuous) quasi-nonexpansive mappings if

( ) ‖ ‖

‖ ‖ ( )

T belongs to the set of nonexpansive mappings if

( ) ‖ ‖

‖ ‖ ( )

T belongs to the set of firmly nonexpansive mappings if

( ) ‖ ‖

‖ ‖ ‖( ) ( )‖ ( )

T belongs to the set of _firmly quasi-nonexpansive mappings if



( ) ( ) ‖ ‖

‖ ‖ ‖ ‖ ( )

It is easily observed that and that . Furthermore, is well-

known to include re-solvents and projection operators, while contains subgradient

projection operators. Recently Bauschke and Combettes have considered a class of mappings

satisfying the condition

⟨ ⟩ ( ) ( )

It can be seen easily that the class of mappings satisfying the latter condition coincideswith

that of firmly quasi-nonexpansive mappings. Usually, the convergence of fixed-point

algorithms requires some additional smoothness properties of the mapping like

demiclosedness.

Definition 1.1:

A mapping T is said to be demiclosed, if for any sequence ( ) which weakly converges to y,

and if the sequence { ( )+ strongly converges to z, then ( )

In what follows, only the particular case of demiclosedness at zero will be used, which is the

particular case when z = 0. This notion is frequently used in the studying of Mann-type

iteration.

Surprisingly, even the computation of fixed points of operators in the wide class of quasi-

nonexpansive mappings can be easily reduced, by a simple relaxation process on the

operator, to computing fixed points of a type of firmly quasi-nonexpansive operators.

In what follows we summarize some key properties of the relaxed operator

( ) which will be needed in the proof of our results.

Lemma 1.1: Let be a quasi-nonexpansive mapping and set ( ) , ( .

Then, the following properties are reached ( ) ( )

1. ⟨ ⟩

‖ ‖ and ⟨ ⟩

‖ ‖

2. ‖ ‖ ‖ ‖ ( )‖ ‖ ;

3. ⟨ ⟩

‖ ‖



2. Main Results:

In the sequel, we will focus our attention on the following general two operators split

common fixed point problem find such that (2.1)

where is a bounded linear operator, and are two

quasi-nonexpansive operators with nonempty fixed-point sets ( ) ( )

and denote the solution set of the two-operators (SCFP) by:

* + (2.2)

To solve (2.1), Censor and Segal proposed and proved, in finite dimensional spaces, the

convergence of the following algorithm

( ( ) ), (2.3)

Where ( ) with being the largest eigenvalue of the matrix (t stands formatrix

transposition).

Inspired by their work, Moudafi introduce the following relaxed algorithm

( ( ) ), (2.4)

Where ( ), ( ) and

Using definition of relaxed operators, observe that the latter takes the following form

Algorithm:

Initialization: Let be arbitrary.

Iterative step: for

{ ( )

( ) ( ) (2.5)

Where ( ), ( )and (

), with being the spectral radius of the operator

.

To prove its convergence we will need the two following lemmas.

Lemma 2.6: Given a bounded linear operator , two Hilbert spaces and

let be two quasi-nonexpansive operators with

nonempty ( ) ( ) . Any sequence ( ) generated by the algorithm

(2.12) is Fejer-monotone with respect to , namely: for every ;

‖ ‖ ‖ ‖

Provided that ( ) (

)



Lemma 2.7: Let H be a Hilbert space and let * + be a sequence in H such that there exist a

nonempty set S H satisfying the following:

1. For every sequence ‖ ‖ exist;

2. Any weak-cluster point of the sequence * + belongs in S. Then there exist x

in S such that * + weakly converges to x.

Now, we are in a position to prove our convergence result.

Theorem 2.8: Given a bounded linear operator , let

be two quasi-nonexpansive operators with nonempty ( ) ( )

. Assume that ( ) and ( ) are both demiclosed at zero, and let be a metric

projection from H onto satisfying⟨ ⟩ . If then

anysequence* + generated by the algorithm (2.5) converges strongly to a split common

fixed-point , provided that ( ), and (

), for a small enough

Proof. Take i.e. ( ) ( ) and using statement 2 in lemma 1.1, we

obtain

‖ ‖ ‖ ‖ ( )‖ ( ) ‖ (2.8.1)

On the other hand, we have

‖ ‖ ‖ ( ) ‖

= ‖ ‖ + ‖ ( ) ‖ + 2 ⟨ ( ) ⟩

=‖ ‖ ⟨( ) ( ) ⟩ + 2 ⟨ ⟩ ( )

From definition of follows

⟨( ) ( ) ⟩ ⟨( ) ( ) ⟩

= ‖( ) ‖ (2.8.2)

Now, by setting ⟨ ( ) ⟩and by using property 1 of lemma 1.1,

we obtain

⟨ ( ) ( ) ( ) ( ) ⟩



= ⟨ ( ) ( ) ( ) ⟩+ ⟨ ( ) ( ) ⟩

= ⟨ ( ) ( ) ⟩ ‖( ) ‖

(

‖( ) ‖ ‖( ) ‖

)

= ‖( ) ‖ (2.8.3)

The key inequalities above combined with (2.13) yield

‖ ‖ ‖ ‖ ( )‖ ( ) ‖

‖ ‖ ⟨( ) ( ) ⟩ + 2 ⟨ ⟩ (

)

‖ ‖ ‖( ) ‖ ‖( ) ‖

(

)‖ ( ) ‖

= ‖ ‖ ( )‖( ) ‖ ( )‖ ( ) ‖

‖ ‖ ( )‖( ) ‖ ‖ ( ) ‖ (2.8.4)

then from equation (2.8.4), we deduce that * + is a fejer monotone and moreover *‖

‖+ is monotonically decreasing sequence, hence converges. Therefore we have

‖( ) ‖ ( )

From the fejer monotonicity of * + , it follows that the sequence is bounded. Denoting by

a weak cluster point of * +, let j = 0,1,2,... be the sequence of indices, such that

( )

Then, from (2.8.5) and the demi closeness of ( )at zero, we obtain

( ) ( )

From which it follows from (2.5), by considering ( ) , it

follows that

( )



Again from (2.8.4) and the convergence of the sequence*‖ ‖+ , we also have

‖( ) ‖ ( )

Which, combined with the demi closeness of ( ) at zero and weak convergence

of{ } , yields

( )

Hence , and therefore . Since there is no morethan one weak-cluster point, the weak

convergence of the whole sequence follows by applying Lemma (1.2) with . i.e.

( )

Next, we show that

‖ ‖ ‖ ‖as k

To show this,

it suffices to show that

‖ ‖ ‖ ‖

Now, from (3.8) we deduce that

|‖ ‖ ‖ ‖| ‖ ‖ ‖ ‖

Therefore, we have

|‖ ‖ ‖ ‖| ‖ ‖

|‖ ‖ ‖ ‖| ‖ ‖ =‖ ‖

‖ ‖+ ‖ ‖ ( )

Claim ‖ ‖ ‖ ‖

Proof of claim

‖ ‖ ‖ ‖



‖ ‖ ⟨ ⟩ +‖ ‖

‖ ‖ ⟨ ⟩ + ⟨ ⟩ +‖ ‖

‖ ‖ + ⟨ ⟩ +‖ ‖

‖ ‖

‖ ‖

‖ ‖ ( )

Now, put ( ) in ( ) , it follows that

|‖ ‖ ‖ ‖| ‖ ‖

0 |‖ ‖ ‖ ‖| ‖ ‖

|‖ ‖ ‖ ‖| .

Hence

‖ ‖ ‖ ‖ (2.8.15)

By (2.8.15) and (2.8.12), we have that

as

n



References:

1. H.H. Bauschke and J.M. Borwein: On projection algorithms for solving convex

feasibility problems, SIAM Review 38(3), pp. 367-426, 1996.

2. H. H. Bauschke, P. L. Combettes: A Weak-to-Strong Convergence Principle for

Fejer-Monotone Methods in Hilbert Spaces. Math.Oper.Res. 26 (2), pp. 248-264

(2001).

3. C. Byrne, Iterative oblique projection onto convex sets and the split feasibility

problem. Inverse Problems 18 (2002), pp. 441-453.

4. Y. Censor and A. Segal, The split common fixed point problem for directed

operators, Journal of Convex Analysis, 16 (2009), pp. 587-600.

5. P.-E. Mainge, The viscosity approximation process for quasi-nonexpansive mappings

in Hilbert spaces Computers & Mathematics with Applications, In Press, Available

online.

6. S. Maruster and C. Popirlan, On the Mann-type iteration and convex feasibility

problem, J. Comput. Appl. Math. 212 (2008), pp. 390-396.

7. H. K. Xu, A variable Krasnosel'skiiMann algorithm and the multiple-set split

feasibility problem. Inverse Problems 22 (2006), pp. 2021-2034.

8. I. Yamada, Hybrid steepest descent method for variational inequality problem over

the fixed point set of certain quasi-nonexpansive mappings, Numer. Funct Anal.

Optim. 25 (2004), pp. 619-655.

9. I. Yamada, Signal processing applications of a pair of simple fixed point algorithms,

Approximation and optimization in image restoration and reconstruction, Poquerolles,

France 2009.

10. J. Zhao and Q. Yang, Several solution methods for the split feasibility

problem,Inverse Problems 21 (2005) No 5, pp. 1791-1799.

11. A. Moudafi, “A note on the split common fixed-point problem for the quasi

nonexpansive operators”, Nonlinear Analysis, Volume 74, no. 12, pp. 4083-4087,

2011.

Strong convergence of an algorithm about quasi nonexpansive mapping for the split common fixed point...

Documents

Transcript of Strong convergence of an algorithm about quasi nonexpansive mapping for the split common fixed point...