Y.SHEHUDepartment of Mathematics，University of Nigeria，Nsukka，NigeriaE-mail:yekini.shehu@unn.edu.ngO.T.MEWOMO　F.U.OGBUISISchool of Mathematics，Statistics and Computer Science，University of Kwazulu-Natal，Durban，South AfricaE-mail:mewomoo@ukzn.ac.za;fudochukwu@yahoo.com

FURTHER INVESTIGATION INTO APPROXIMATION OF A COMMON SOLUTION OF FIXED POINT PROBLEMS AND SPLIT FEASIBILITY PROBLEMS?

Y.SHEHU
Department of Mathematics，University of Nigeria，Nsukka，Nigeria
E-mail:yekini.shehu@unn.edu.ng
O.T.MEWOMOF.U.OGBUISI
School of Mathematics，Statistics and Computer Science，University of Kwazulu-Natal，Durban，South Africa
E-mail:mewomoo@ukzn.ac.za;fudochukwu@yahoo.com

The purpose of this paper is to study and analyze an iterative method for finding a common element of the solution set Ω of the split feasibility problem and the set F（T）of fixed points of a right Bregman strongly nonexpansive mapping T in the setting of puniformly convex Banach spaces which are also uniformly smooth.By combining Mann’s iterative method and the Halpern’s approximation method，we propose an iterative algorithm for finding an element of the set F（T）∩Ω;moreover，we derive the strong convergence of the proposed algorithm under appropriate conditions and give numerical results to verify the efficiency and implementation of our method.Our results extend and complement many known related results in the literature.

strong convergence;split feasibility problem;uniformly convex;uniformly smooth;fixed point problem;right Bregman strongly nonexpansive mappings

2010 MR Subject Classification49J53;65K10;49M37;90C25

1　Introduction

Let E1and E2be Banach spaces.Let C and Q be nonempty closed and convex subsets of E1and E2respectively.Let A:E1→E2be a bounded linear operator.The split feasibility problem（SFP）is to find an element x∈E1satisfying

We will denote the solution set of（1.1）byIt is worth mentioning that SFP in finite-dimensional spaces was first introduced by Censor and Elfving［5］for modeling inverse problems which arise from phase retrievals and in medicalimage reconstruction［2］.Recently，it was found that the SFP can also be used in various disciplines such as image restoration，computer tomograph，and radiation therapy treatment planning［3，4，7］.For some existing results on SFP，see［8，14-16，29］.In［2］，Byrne used the CQ algorithm generated iteratively for any initial x0by

where ΠCdenotes the Bregman projection and J the duality mapping.Using algorithm（1.2），Sch¨opfer et.al.［20］obtained a weak convergence result in a p-uniformly convex and uniformly smooth Banach space，with the condition that the duality mapping of E is sequentially weakto-weak continuous.To obtain strong convergence，Wang［25］based on an idea in Nakajo-Takahashi［17］，introduced the following algorithm:for any initial guess x0，definerecursively by

where Tnis defined for each n∈N by

i:N→I is the cyclic control mapping i（n）=nmod（r+s）+1，and tnsatisfies

Very recently，Zegeye and Shahzad［31］proved a strong convergence theorem for a common fixed point of a finite family of right Bregman strongly nonexpansive mappings in the framework of real reflexive Banach spaces.Furthermore，they applied their method to approximate a common zero of a finite family of maximal monotone mappings and a solution of a finite family of convex feasibility problems in reflexive real Banach spaces.In particular，they proved this theorem.

Theorem 1.1（Zegeye and Shahzad，［31］）Let f:E→R be a cofinite function which is bounded，uniformly Fr′echet differentiable and totally convex on bounded subsets of E.Let C be a nonempty，closed，and convex subset of int（dom f）and let Ti:C→C，for i=1，2，···，N， be a finite family of right Bregman strongly nonexpansive mappings such thatfor eachAssume thatis nonempty.Fora sequence generated by

The purpose of this paper is to study and analyze an iterative method for finding a common element of the solution set Ω of the split feasibility problem and the set F（T）of fixed points of a right Bregman strongly nonexpansive mapping T in the setting of p-uniformly convex Banach spaces which are also uniformly smooth.By combining Mann’s iterative method and the Halpern’s approximation method，we propose an iterative algorithm for finding an element of the set F（T）∩Ω;moreover，we derive the strong convergence of the proposed algorithm under appropriate conditions.It is worth emphasizing that our results are new and novel in p-uniformly convex Banach spaces which are also uniformly smooth.Our results represent the supplement，improvement and extension of many known related results in the literature. Furthermore，in all our results，we dispense with the assumption of the weak-to-weak continuity of the duality mapping imposed in Sch¨opfer et.al.［20］.

2　Preliminaries

Let E1and E2be real Banach spaces and let A:E1→E2be a bounded linear operator. The dual（adjoint）operator of A，denoted by A?，is a bounded linear operator defined by

E is uniformly smooth if and only if

q-uniformly smooth if there exists asuch that

Lemma 2.1Let x，y∈E.If E is q-uniformly smooth，then there exists a Cq＞0 such that

Let dim E≥2.The modulus of convexity of E is the function δE:（0，2］→［0，1］defined by

E is uniformly convex if and only if δE（∈）＞0 for all∈∈（0，2］and p-uniformly convex if there is afor any∈∈（0，2］.

It is known that E is p-uniformly convex and uniformly smooth if and only if its dual E?is q-uniformly smooth and uniformly convex.It is also a common knowledge that the dualitymapping Jpis one-to-one，single valued and satisfiesis the duality mapping of

Definition 2.2The duality mappingis defined by

holds true for any y∈E.We note here that lp（p＞1）spaces has such a property，but the Lp（p＞2）does not share this property.The domain of a convex function f:E→R is defined to be

Definition 2.3Given a G?ateaux differentiable convex function f:E→R，the Bregman distance with respect to f is defined as

The duality mapping Jpis actually the derivative of the functionGiven that，then the Bregman distance with respect to fpnow becomes

The Bregman distance is not symmetric therefore is not a metric but it posses the following important properties

For the p-uniformly convex space，the metric and Bregman distance has the following relation（see［20］）:

where τ＞0 is some fixed number.Let C be a nonempty，closed and convex subset of E.The metric projection

is the unique minimizer of the norm distance，which can be characterized by a variational inequality

Similarly，to the metric projection，the Bregman projection is defined asthe unique minimizer of the Bregman distance（see［19］）.The Bregman projection can also be characterized by a variational inequality

from which one has

Let C be a convex subset of int domfp，whereand let T be a self-mapping of C.A point p∈C is said to be an asymptotic fixed point of T if C contains a sequencewhich converges weakly to p andThe set of asymptotic fixed points of T is denoted by

Definition 2.4Recalling that the Bregman distance is not symmetric，we define the following operators.

A mapping T with a nonempty asymptotic fixed point set is said to be:

（i）right Bregman strongly nonexpansive（see［12］）with respect to a nonempty

According to Martin-Marquez et.al.［12］，a right Bregman strongly nonexpansive mapping T with respect to a nonemptyis called strictly right Bregman strongly nonexpansive mapping.

（ii）An operator T:C→ intdomf is said to be:right Bregman firmly nonexpansive（R-BFNE）if

for any x，y∈C，or equivalently，

See［12］for more information and examples of R-BFNE operators.From［12］，we know that every right Bregman firmly nonexpansive mapping is right Bregman strongly nonexpansive with respect to

Following［1］and［6］，we make use of the functionwhich is defined by

Then Vpis nonnegative andMoreover，by the subdifferential inequality，

So，we have

And from（2.8），we obtain

where

3　Main Results

By combining Mann’s iterative method and the Halpern’s approximation method，we propose an iterative method and obtain strong convergence result for finding a common element of the set of solutions of the SFP and the set of fixed points of a right Bregman strongly nonexpansive mapping in the setting of p-uniformly convex Banach spaces which are also uniformly smooth.

Theorem 3.1Let E1and E2be two p-uniformly convex real Banach spaces which are also uniformly smooth.Let C and Q be nonempty，closed and convex subsets of E1and E2respectively，be a bounded linear operator andbe the adjoint of A.Suppose that SFP（1.1）has a nonempty solution set Ω.Let T be a right Bregman strongly nonexpansive mapping of C into C such thatSupposeare sequences in（0，1）such thatFor a fixed，let sequencesbe iteratively generated

with the conditions

From（3.2）and Lemma 2.1 we obtain that

Then from condition（iii）we have that

We now consider two cases to prove the strong convergence.

Case 1Suppose that there existsis monotonically nonincreasing.Then obviouslyconverges and

Therefore，since T is right Bregman strongly nonexpansive，we have that

which implies that

Also from（3.3）we obtain

Hence we obtain from the definition of ynthat

It follows from（3.13）and（3.15）that

Now from（2.5），we have

Then from（2.6），we have

Therefore，by Lemma 2.5，we conclude thatthen we have that

Case 2Suppose there exists a subsequencesuch thatfor all i∈N.Then，by Lemma 2.6 there exists a nondecreasing sequencesuch that

By same arguments as in Case 1，we obtain that

Again from（3.5）we have

which implies that

for all k∈N，we conclude that

Corollary 3.2Let E1and E2be two p-uniformly convex real Banach spaces which are also uniformly smooth.Let C and Q be nonempty，closed and convex subsets of E1and E2respectively，A:E1→E2be a bounded linear operator andbe the adjoint of A. Suppose that SFP（1.1）has a nonempty solution set Ω.Supposeis a sequence in（0，1） such that αn≤b＜1.For a fixed u∈C，let sequencesbe iteratively generated by u0∈E1，

with the conditions

Next，using the idea in［13］，we consider the mappingare right Bregman strongly nonexpansive mappings on E.Using Theorem 3.1，we have the following corollary.

Corollary 3.3Let E1and E2be two p-uniformly convex real Banach spaces which are also uniformly smooth.Let C and Q be nonempty，closed and convex subsets of E1and E2respectively，A:E1→ E2be a bounded linear operator andbe the adjoint of A.Suppose that SFP（1.1）has a nonempty solution set Ω.Letwhereis a finite family of right Bregman strongly nonexpansive mappings such thatare sequences in（0，1）such thatFor a fixed u∈C，let sequencesbe iteratively generated by

with the conditions

Corollary 3.4Let H1and H2be two real Hilbert spaces.Let C and Q be nonempty，closed and convex subsets of H1and H2respectively，A:H1→H2be a bounded linear operator and A?:H2→H1be the adjoint of A.Suppose that SFP（1.1）has a nonempty solution set Ω.Let T be a strongly quasi-nonexpansive operator of C into C such thatdemi-closed at zero andSupposeare sequences in（0，1） such thatFor a fixedletsequencesbe iteratively generated by

with the conditions

Remark 3.5We now highlight our contributions in this paper.

1.In［2］，a weak convergence result was obtained for split feasibility problems in real Hilbert spaces，while in our result a strong convergence result is obtained for split feasibility problems and fixed point problems for right Bregman strongly nonexpansive mappings in p-uniformly convex real Banach spaces which are also uniformly smooth.Hence，our result improve and extend the results of Byrne［2］and more applicable than the results of Sch¨opfer et.al.［20］and Wang［25］.

2.In［31］，Zegeye and Shahzad proved strong convergence result for a common fixed point of a finite family of right Bregman strongly nonexpansive mappings in the framework of real reflexive Banach spaces while in this our result，we obtained strong convergence result for approximation of a common solution of split feasibility problem and fixed point problem for a finite family of right Bregman strongly nonexpansive mappings p-uniformly convex real Banach spaces which are also uniformly smooth（see Corollary 3.3）.Therefore，our result complements the result of Zegeye and Shahzad［31］in p-uniformly convex real Banach spaces which are also uniformly smooth.

Remark 3.6A prototype for sequencesin Theorem 3.1 are

4　Numerical Example

In this section，we present some preliminary numerical results.All codes were written in Matlab 2012b and run on Hp i-5 Dual-Core laptop.

Also，let

Let us assume that

Then A is a bounded linear operator withSuppose that we take operator T in Theorem 3.1 as T:=PC，the metric projection on C.Then the problem considered in Theorem 3.1 reduces to:

We observe that if Ω denotes the set of solutions of（4.1），thenFurthermore，our iterative scheme（3.1）becomes

We make different choices ofwith a choice ofas our stopping criterion.

Table 1　Example 4，Case I

Fig.1　Click here to download high resolution image

Case 2tn=0.0002，u1=t.We have the numerical analysis tabulated in Table 2 and shown in Fig.2.

Table 2　Example 4，Case II

Fig.2　Click here to download high resolution image

Case 3tn=0.0000002，u1=t.We have the numerical analysis tabulated in Table 3 and shown in Fig.3.

Table 3　Example 4，Case II

Fig.3　Click here to download high resolution image

Remark 4.2We remark from this numerical example that different choices of u1and tn，within the specified spaces and range，have no effect on the number of iterations required for convergence and with very insignificant effect on the cpu run time as can be seen from the Tables and the Figures presented.Please observe in the figures presented，there is representation for xn+1-xn（blue line）since the values of xn+1-xnare the same as the values of un+1-un. Thus，the representation of xn+1-xn（blue line）is the same as the representation of un+1-un（red line）in the figures presented.This is because the columns from the table have same values or values very very close to each other，so the curves（blue and red lines）overlapped each other; the red is on the blue and that is why the colour of the blue cannot be seen.

We can see from the tables and graphs that the closer the values of tnto zero，the less the number of iterations required for the convergence.

References

［1］Alber Y I.Metric and generalized projection operator in Banach spaces:properties and applications//Theory and Applications of Nonlinear Operators of Accretive and Monotone Type.Vol 178 of Lecture Notes in Pure and Applied Mathematics.New York:Dekker，1996:15-50

［2］Byrne C.Iterative oblique projection onto convex sets and the split feasibility problem.Inverse Problems，2002，18（2）:441-453

［3］Censor Y，Bortfeld T，Martin B，Trofimov A.A unified approach for inversion problem in intensitymodulated radiation therapy.Phys Med Biol，2006，51:2353-2365

［4］Censor Y，Elfving T，Kopf N，Bortfeld T.The multiple-sets split feasibility problem and its applications for inverse problems.Inverse Problems，2005，21（6）:2071-2084

［5］Censor Y，Elfving T.A multiprojection algorithm using Bregman projections in a product space.Numerical Algorithms，1994，8（2-4）:221-239

［6］Censor Y，Lent A.An iterative row-action method for interval convex programming.J Optim Theory Appl，1981，34:321-353

［7］Censor Y，Motova A，Segal A.Perturbed projections and subgradient projections for the multiple-sets split feasibility problem.J Math Anal Appl，2007，327（2）:1244-1256

［8］Censor Y，Segal A.The split common fixed point problem for directed operators.J Convex Anal，2009，16（2）:587-600

［9］Cioranescu I.Geometry of Banach Spaces，Duality Mappings and Nonlinear Problems.Dordrecht:Kluwer，1990

［10］Dunford N，Schwartz J T.Linear Operators I.New York:Wiley Interscience，1958

［11］Maing′e P E.Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization.Set-Valued Anal，2008，16:899-912

［12］Mart′?n-M′arquez V，Reich S，Sabach S.Right Bregman nonexpansive operators in Banach spaces.Nonlinear Analy，2012，75:5448-5465

［13］Mart′?n-M′arquez V，Reich S，Sabach S.Bregman strongly nonexpansive operators in reflexive Banach spaces. J Math Anal Appl，2013，400:597-614

［14］Masad E，Reich S.A note on the multiple-set split convex feasibility problem in Hilbert space.J Nonlinear Convex Anal，2007，8（3）:367-371

［15］Moudafi A.The split common fixed-point problem for demicontractive mappings.Inverse Problems，2010，26（5）:Article ID 055007

［16］Moudafi A.A note on the split common fixed-point problem for quasi-nonexpansive operators.Nonlinear Anal，2011，74（12）:4083-4087

［17］Nakajo K，Takahashi W.Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups.J Math Anal Appl，2003，279:372-379

［18］Phelps R P.Convex Functions，Monotone Operators，and Differentiability.2nd ed.Berlin:Springer-Verlag，1993

［19］Sch¨opfer F.Iterative Regularisation Method for the Solution of the Split Feasibility Problem in Banach Spaces［D］.Saabr¨ucken，2007

［20］Sch¨opfer F，Schuster T，Louis A K.An iterative regularization method for the solution of the split feasibility problem in Banach spaces.Inverse Problems，2008，24（5）:055008

［21］Shehu Y，Iyiola O S，Enyi C D.Iterative algorithm for split feasibility problems and fixed point problems in Banach spaces.Numer Algor，DOI:10.1007/s11075-015-0069-4

［22］Shehu Y，Ogbuisi F U，Iyiola O S.Convergence Analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces.Optimization，DOI:10.1080/02331934.2015.1039533

［23］Takahashi W.Nonlinear Functional Analysis-Fixed Point Theory and Applications.Yokohama:Yokohama Publishers Inc，2000（in Japanese）

［24］Takahashi W.Nonlinear Functional Analysis.Yokohama:Yokohama Publishers，2000

［25］Wang F.A new algorithm for solving the multiple sets split feasibility problem in Banach spaces.Numerical Functional Anal Opt，2014，35（1）:99-110

［26］Wang F，Xu H K.Cyclic algorithms for split feasibility problems in Hilbert spaces.Nonlinear Anal，2011，74（12）:4105-4111

［27］Xu H K.Iterative algorithms for nonlinear operators.J London Math Soc，2002，66（2）:240-256

［28］Xu H K.A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem.Inverse Problems，2006，22（6）:2021-2034

［29］Xu H K.Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces.t Inverse Problems，2010，26（10）:Article ID 105018

［30］Yang Q.The relaxed CQ algorithm solving the split feasibility problem.Inverse Problems，2004，20（4）: 1261-1266

［31］Zegeye H，Shahzad N.Convergence Theorems for Right Bregman Strongly Nonexpansive Mappings in Reflexive Banach Spaces.Abstr Appl Anal，2014，2014:Article ID 584395

［32］Zhao J，Yang Q.Several solution methods for the split feasibility problem.Inverse Problems，2005，21（5）: 1791-1799

?January 27，2015;revised July 13，2015.