Strong convergence of hybrid Bregman projection algorithm for split feasibility and fixed point

Research Article

Strong convergence of hybrid Bregman projection algorithm for split feasibility and fixed point

problems in Banach spaces

Jin-Zuo Chen, Lu-Chuan Ceng^∗

Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China.

Abstract

In this paper, we consider and study split feasibility and fixed point problems involved in Bregman quasi- strictly pseudo-contractive mapping in Banach spaces. It is proven that the sequences generated by the proposed iterative algorithm converge strongly to the common solution of split feasibility and fixed point problems.

Keywords: Split feasibility problem, fixed point problem, Bregman quasi-strictly pseudo-contractive mapping, Bregman projection, strong convergence.

2010 MSC: 47J25, 47H10, 65J15, 90C25.

1. Introduction

Throughout this paper, we assume thatC and Qbe nonempty closed convex sets inp-uniformly convex and uniformly smooth real Banach spacesE1 andE2, respectively. Let A be a bounded linear operator from E₁ toE₂ with its adjointA^∗. LetT be a nonlinear mapping from C to itself. We use F ix(T) to denote the set of all fixed points of the mapping T, that is, F ix(T) ={x∈C :T x=x}.

This paper is concerned on studying the following split feasibility and fixed point problems:

Find x^∗ ∈C∩F ix(T) such that Ax^∗ ∈Q. (1.1) Let Γ ={x^∗ : x^∗ ∈C∩F ix(T) such that Ax^∗ ∈ Q} be the set of all solutions of (1.1). In the sequel, we assume Γ6=∅.A special case of (1.1) is the following split feasibility problem (in short, SFP):

Find x^∗∈C such thatAx^∗ ∈Q. (1.2)

∗Corresponding author

Email addresses: [email protected](Jin-Zuo Chen),[email protected](Lu-Chuan Ceng) Received 0000-0-0

Let Γ₀ ={x^∗ :x^∗ ∈ C such that Ax^∗ ∈ Q} be the set of all solutions of (1.2). Then, we have that Γ₀ is a closed and convex subset ofE1.

The theory of the SFP was first introduced and learned by Censor [7] in finite-dimensional space for modeling inverse problems which arise from phase retrievals and in image reconstruction has been discussed in the last two decades and much intensively in the last ten years. A large number of algorithms related to the SFP have been studied; see, for example, [5, 8, 4] and the references therein. Recently, it has been found that the SFP can be used in intensity modulated radiation therapy, please see [6, 8, 9] and the references therein. The algorithm suggested by Censor in [7] involves the computation of the inverseA⁻¹, so, it can not be widely used. A seemingly more popular algorithm is theCQalgorithm [5]:

xn+1=PC(I−γA^∗(I −PQ)A)xn, n≥0, (1.3) where x₀ ∈ H₁ (a Hilbert space) and γ ∈ (0,²_λ), with λ being the largest eigenvalue of the matrix A^∗A.

Recently, the SFP was studied in a more general framework, for example, Banach spaces. More specifically, Sch¨opfer et al. [13] proposed the following algorithm in p-uniformly convex and uniformly smooth real Banach spaces:

xn+1 = ΠCJ^∗(J xn−γA^∗J(I−PQ)Axn), (1.4) where ΠC denotes the Bregman projection andJ the duality mapping, they established weak convergence of algorithm (1.4) under some mild conditions. Obviously the above algorithm (1.4) convers theCQalgorithm (1.3) as a special case.

It is worth pointing out that only weak convergence result is established in [13]. However, the strong convergence is more acceptable than the weak convergence in some practical applications. Wang [16] con- sidered the following iterative algorithm for multiple-sets split feasibility problem inp-uniformly convex and uniformly smooth real Banach spaces:















y_n=T_nx_n,

Dn={v ∈E1 : ∆p(yn, v)≤∆p(xn, v)}, En={v∈E1:hx_n−v, Jpx−Jpxni ≥0}, x_n+1= Π_Dn^T_Enx.

(1.5)

Using the idea in the work of Nakajo [11], Wang proved the strong convergence of iterative algorithm (1.5).

Subsequently, Takahashi [14] proposed the following hybrid projection algorithm for the SFP in uniformly convex and uniformly smooth real Banach spaces:















z_n=x_n−rJ_E⁻¹

1A^∗J_E2(Ax_n−P_QAx_n), Cn={v ∈E1 :hz_n−v, JE1(xn−zn)i ≥0}, Q_n={v∈E₁ :hx_n−v, J_E1(x₁−x_n)i ≥0}, x_n+1 =P_Cn^T_Qnx₁.

(1.6)

Basing mainly on the hybrid method, he proved the strong convergence of iterative algorithm (1.6).

On the other hand, in 1967, Bregman [3] used the so-called Bregman distance function to design and analyze feasibility and optimization algorithms. After that, many authors found that the so-called Bregman distance function could be applied in different ways in order to construct iterative algorithms for solving not only feasibility and optimization problems, but also variational inequality problems, equilibria problems, fixed points problems and so on (see,e.g, [2, 10, 12, 18, 19, 20] and the references therein). The fixed point theory with respect to Bregman distance has been studied in the last decade and a lot of good results were published intensively in the last five years. Many authors concentrated their energies on constructing the fixed point of Bregman nonlinear operators by utilizing the Bregman distance and the Bregman projection, see [15, 17] and the references therein. In 2015, Wang [17] studied a new hybrid Bregman projection iterative

algorithm for Bregman quasi-strictly pseudo-contractive mapping and proved strong convergence result in reflexive Banach spaces. In particularly, he proposed the following iterative method:



















x₀ ∈C chosen arbitrarily, C1 =C,

x₁ =P_C1x₀,

Cn+1 ={v∈Cn:Df(xn, T xn)≤ _1−κ¹ h∇f(xn)− ∇f(T xn), xn−vi}, xn+1=PCn+1x0,

(1.7)

where κ ∈ [0,1). Then the sequence {x_n} converges strongly to p = P_{F ix(T)}x0, P_{F ix(T)} is the Bregman projection ofE ontoF ix(T).

In this paper, motivated and inspired by the above research work going on in this field, we propose a new hybrid projection method for solving split feasibility and fixed point problems (1.1) involved in Bregman quasi-strictly pseudo-contractive mapping inp-uniformly convex and uniformly smooth real Banach spaces.

Our modification is mainly based on the schemes (1.5), (1.6) and (1.7). Furthermore, we will prove the strong convergence theorem for the proposed algorithm.

2. Preliminaries

Let 1< q≤2≤p with 1/p+ 1/q = 1.LetE be a real Banach space.

The modulus of convexity of E is the function δE : (0,2]→[0,1] defined by δ_E() = inf

1− kx+yk

2 :x, y∈S(E),kx−yk ≥

,

for anyx, y on the unit sphereS(E) ={x∈E :kxk= 1}. E is called uniformly convex ifδ_E()>0 for any ∈(0,2]; p-uniformly convex if there exists cp >0 such thatδE()≥cp^p for any ∈(0,2].

The modulus of smoothness ofE is the functionρ_E : [0,∞)→[0,∞) defined by ρ_E(t) = sup

1

2(kx+yk − kx−yk)−1 :x∈S(E),kyk=t

.

E is called uniformly smooth if limt→0ρE(t)/t = 0. By setting 1< q ≤2 ≤p, a Banach space E is called q-uniformly smooth if there exists C_q > 0 such that ρ_E(t) ≤ C_qt^q for all t > 0. We assume that E is p-uniformly convex and uniformly smooth, which implies that its dual space,E^∗, isq-uniformly smooth and uniformly convex. In this situation, it is known that the duality mapping J_E^p is one-to-one, single-valued and satisfiesJ_E^p = J_E^q∗

−1

,where J_E^q∗ is the duality mapping ofE^∗. The q-uniformly smooth spaces have the following conclusion.

Lemma 2.1. [16] If E is a q-uniformly smooth space, then there is a constant Cq >0 such that kx−yk^q ≤ kxk^q−qhy, J_E^q(x)i+C_qkyk^q,

for all x, y∈E, where Cq >0 is the q-uniformly smoothness constant of E and J_E^q is the duality mapping fromE into 2^E∗ defined by

J_E^q(x) ={x^∗∈E^∗ :hx, x^∗i=kxk^q, kx^∗k=kxk^q−1}, ∀x, y∈E.

Given a Gˆateaux differentiable convex function f :E → R, the Bregman distance with respect to f is defined by

∆_f(x, y) =f(y)−f(x)− hf⁰(x), y−xi, ∀x, y∈E.

It is note worthy that the duality mappingJ_pis the derivative of the functionf_p = ¹_pkxk^p.Then the Bregman distance with respect tofp is given by

∆p(x, y) = 1

qkxk^p− hJ_E^px, yi+1 pkyk^p

= 1

p(kyk^p− kxk^p) +hJ_E^px, x−yi

= 1

q(kxk^p− kyk^p)− hJ_E^px−J_E^py, xi.

From the definition of ∆p(·,·), we get

∆_p(x, y) = ∆_p(x, z) + ∆_p(z, y) +

z−y, J_E^px−J_E^pz

, (2.1)

and

∆p(x, y) + ∆p(y, x) =

x−y, J_E^px−J_E^py

, (2.2)

for any x, y, z ∈E. All in all, the Bregman distance is not a metric because of the lack of symmetry. For thep-uniformly convex space, the metric and Bregman distance has the following relation

τkx−yk^p ≤∆p(x, y)≤

x−y, J_E^px−J_E^py

, (2.3)

whereτ >0. Obviously, if{x_n}and{y_n}are both bounded sequences of ap-uniformly convex and uniformly smooth spaceE, then xn−yn→0 as n→ ∞ implies that ∆p(xn, yn)→0 asn→ ∞.

Projections are an important tool for our work in this paper. We can define metric projection P_C as follows

PCx= argmin_y∈Ckx−yk, x∈E,

metric projection P_C can be characterized by the following variational inequality J_E^p(x−PCx), z−PCx

≤0, z∈C. (2.4)

Likewise, one can define the Bregman projection

Π_Cx= argmin_y∈C∆_p(x, y), x∈E,

is the unique minimizer of the Bregman distance, The Bregman projection can also be characterized by the following variational inequality

J_E^px−J_E^pΠ_Cx, z−Π_Cx

≤0, z∈C, (2.5)

from which one has

∆_p(Π_Cx, z)≤∆_p(x, z)−∆_p(x,Π_Cx), z∈C. (2.6) The metric projection and the Bregman projection with respect tof2 are coincident in a Hilbert space, but in a more general framework, they are totally different. What is important is that the metric projection can not share property (2.6) as the Bregman projection in Banach spaces.

Following [1], we study the functionVp :E^∗×E→[0,∞) associated withfp, which is defined by V_p(x, x) = 1

qkxk^q− hx, xi+ 1

pkxk^p, x∈E, x∈E^∗. Then Vp is nonnegative and

V_p(x, x) = ∆_p(J_E^q∗x, x), x∈E, x∈E^∗.

Moreover, by the subdifferential inequality, we have V_p(x, x) +

y, J_E^q∗x−x

≤V_p(x+y, x), x∈E, x, y∈E^∗. In addition,V_p is convex in the first variable. Thus, for all z∈E,

∆_p J_E^q∗

N

X

i=1

t_iJ_E^p(x_i)

! , z

!

≤

N

X

i=1

t_i∆_p(x_i, z), (2.7)

where{x_i}^N_i=1 ⊂E and {t_i}^N_i=1⊂(0,1) withPN

i=1t_i = 1. For more details aboutV_p, please see [1].

Very recently, Ugwunnadiet al.[15] introduced the concept of Bregman quasi-strictly pseudo-contractive mapping and proved the strong convergence by using hybrid Bregman projection iterative algorithm for a Bregman quasi-strictly pseudo-contractive mapping.

Definition 2.2. A mapping T :C→C is said to be Bregman quasi-strictly pseudo-contractive mapping if there exists a constantκ∈[0,1) and F ix(T)6=∅ such that

∆p(T x, x^∗)≤∆p(x, x^∗) +κ∆p(T x, x), ∀x∈C, x^∗ ∈F ix(T).

Definition 2.3. A mappingT :C →C is said to be Bregman quasi-nonexpansive mapping if F ix(T)6=∅ such that

∆_p(T x, x^∗)≤∆_p(x, x^∗), ∀x∈C, x^∗ ∈F ix(T).

Definition 2.4. A mapping T :C →C is said to be closed if for any sequence{x_n} ⊂C withx_n→x∈C and T x_n→y∈C asn→ ∞, then T x=y.

We shall adopt the notation: xn → x means that {x_n} converges to x strongly. Now, we give some examples of a Bregman quasi-strictly pseudo-contractive mapping.

Example 2.5. [17] Let E be a smooth space, and define f(x) = kxk² for all x ∈ E. Letx0 6= 0 be any element ofE,T :E →E be defined as follows:

T(x) =

((1/2 + 1/2ⁿ⁺¹)x₀, x= (1/2 + 1/2ⁿ)x₀,

−x, x6= (1/2 + 1/2ⁿ)x0

for all n≥1.Then T is a Bregman quasi-strictly pseudo-contractive mapping.

Example 2.6. [15] Let E=R and defineT, f : [−1,0]→Rby f(x) =x and T(x) = 2x for all x∈[−1,0].

Then T is a Bregman quasi-strictly pseudo-contractive mapping.

3. Main results

In this section, we will introduce the following algorithm and prove strong convergence theorem for finding the common solution of split feasibility and fixed point problems.

Theorem 3.1. Let C and Q be nonempty closed convex sets in p-uniformly convex and uniformly smooth real Banach spaces E1 and E2, respectively. Let A: E1 → E2 be a bounded linear operator with its adjoint A^∗: E₂^∗ → E₁^∗. Let T be a closed Bregman quasi-strictly pseudo-contractive mapping from C to itself. Let the sequence{x_n} be iteratively generated by x₁=x₀ ∈C, D₁ =C₁ =C,































x1∈C, yn= ΠCJ_E^q∗

1

J_E^p

1xn−λnA^∗J_E^p

2(Axn−PQAxn)

, z_n=J_E^q∗

1

α_nJ_E^p

1y_n+ (1−α_n)J_E^p

1T x_n , Dn+1={w∈Dn: ∆p(yn, w)≤∆p(xn, w)}, C_n+1=n

w∈C_n: ∆_p(z_n, x_n)≤ _1−κ^κ D

w−x_n, J_E^p

1T x_n−J_E^p

1x_nE +D

w−x_n, J_E^p

1z_n−J_E^p

1x_nEo , x_n+1= Π_Dn+1^T_Cn+1x₀, n≥1,

(3.1)

where κ ∈ [0,1). Assume that {α_n} ⊂ [c, d] ⊂ (0,1) and {λ_n} ⊂ [a, b] ⊂

0, q

CqkAk^q

_q−1¹

. Then the sequence{x_n} defined by (3.1)converges strongly to ΠΓx0.

Proof. Taking x^∗ ∈Γ,x⁰∈C, by the definition ofT, we have

∆p(T x⁰, x^∗)≤∆p(x⁰, x^∗) +κ∆p(T x⁰, x⁰).

From (2.1), we get

∆p(T x⁰, x^∗) = ∆p(T x⁰, x⁰) + ∆p(x⁰, x^∗) + D

x⁰−x^∗, J_E^p

1T x⁰−J_E^p

1x⁰ E

, which implies that

∆_p(T x⁰, x⁰)≤ 1 1−κ

D

x^∗−x⁰, J_E^p

1T x⁰−J_E^p

1x⁰E

. (3.2)

Let{x_n} be a sequence in F ix(T) such thatxn→z asn→ ∞. From (3.2), we obtain

∆p(T z, z)≤ 1 1−κ

D

xn−z, J_E^p₁T z−J_E^p₁z E

,

settingn→ ∞in the above inequality, we have ∆_p(T z, z)≤0,it follows from (2.3) thatT z =z. Therefore, F ix(T) is closed.

Next, let z₁, z₂ ∈ F ix(T), for given t ∈ (0,1), putting z = tz₁ + (1 −t)z₂. From (3.2), we obtain, respectively,

∆p(T z, z)≤ 1 1−κ

D

z1−z, J_E^p

1T z−J_E^p

1z E

, (3.3)

and

∆p(T z, z)≤ 1 1−κ

D

z2−z, J_E^p

1T z−J_E^p

1z E

. (3.4)

Multiplying (3.3) by tand (3.4) by 1−t, we have

∆_p(T z, z)≤ 1 1−κ

D

z−z, J_E^p

1T z−J_E^p

1zE ,

settingn→ ∞in the above inequality, we have ∆p(T z, z)≤0,it follows from (2.3) thatT z =z. Therefore, F ix(T) is convex. Since Γ₀ is a closed convex subset ofE₁, we obtain that Γ is closed convex.

Now, from (3.1), we knowDn is closed for eachn≥1. Note that ∆p(yn, w)≤∆p(xn, w) is equivalent to D

J_E^p

1x_n−J_E^p

1y_n, wE

≤ 1

q(kx_nk^p− ky_nk^p), so thatDn is a halfspace, therefore, we get Dn is convex immediately.

Forn= 1,C₁=Cis closed convex essentially. Assume thatC_nis closed convex forn >1. Forw∈C_n+1, we obtain

∆_p(z_n, x_n)≤ κ 1−κ

D

w−x_n, J_E^p

1T x_n−J_E^p

1x_nE +D

w−x_n, J_E^p

1z_n−J_E^p

1x_nE , since

D

·, J_E^p

1T xn−J_E^p

1xn

E and

D

·, J_E^p

1zn−J_E^p

1xn

E

are continuous and linear in E1, we get Cn is closed convex.

Let x^∗∈Γ and letvn=Axn−PQAxn. It follows from (2.4) that D

J_E^p

2v_n, Ax_n−Ax^∗E

=kAx_n−P_QAx_nk^p+D J_E^p

2v_n, P_QAx_n−Ax^∗E

≥ kAx_n−P_QAx_nk^p,

applying Lemma 2.1, we have

∆_p(y_n, x^∗)≤∆_p J_E^q∗

1

J_E^p

1x_n−λ_nA^∗J_E^p

2v_n , x^∗

= 1 q

J_E^p₁xn−λnA^∗J_E^p₂vn

q−D

J_E^p₁xn, x^∗ E

+λn

D

J_E^p₂vn, Ax^∗ E

+1 pkx^∗k^p

≤ 1 q

J_E^p

1xn

q−λn

D

Axn, J_E^p

2vn

E

+Cq(λnkAk)^q q

J_E^p

2vn

q

−D J_E^p

1xn, x^∗ E

+λn

D J_E^p

2vn, Ax^∗ E

+1 pkx^∗k^p

= 1

qkx_nk^p−D J_E^p

1xn, x^∗ E

+1

pkx^∗k^p+λn

D J_E^p

2vn, Ax^∗−Axn

E

+Cq(λnkAk)^q q

J_E^p

2v_n

q

= ∆_p(x_n, x^∗) +λ_nD J_E^p

2v_n, Ax^∗−Ax_nE

+C_q(λ_nkAk)^q q

J_E^p

2v_n

q

≤∆_p(x_n, x^∗)−

λ_n−C_q(λ_nkAk)^q q

kv_nk^p.

(3.5)

By the assumption of{λ_n}, we have

∆p(yn, x^∗)≤∆p(xn, x^∗), (3.6) so that Γ⊂ D_n+1 for all n ≥ 1. Next, we show Γ⊂ C_n+1. Note that Γ ⊂ C₁ =C. Suppose Γ ⊂ C_n for n≥1, then for all x^∗ ∈Γ⊂Cn, from (2.7), (3.1), (3.2) and (3.6), we obtain

∆p(zn, x^∗) = ∆p

J_E^q∗

1

αnJ_E^p

1yn+ (1−αn)J_E^p

1T xn

, x^∗

≤α_n∆_p(y_n, x^∗) + (1−α_n)∆_p(T x_n, x^∗)

≤α_n∆_p(y_n, x^∗) + (1−α_n) (∆_p(x_n, x^∗) +κ∆_p(T x_n, x_n))

≤∆_p(x_n, x^∗) +κ∆_p(T x_n, x_n)

≤∆p(xn, x^∗) + κ 1−κ

D

x^∗−xn, J_E^p

1T xn−J_E^p

1xn

E .

(3.7)

From (2.1), we get

∆p(zn, x^∗) = ∆p(zn, xn) + ∆p(xn, x^∗) + D

xn−x^∗, J_E^p

1zn−J_E^p

1xn

E

. (3.8)

By (3.7) and (3.8), we obtain

∆p(zn, xn)≤ κ 1−κ

D

x^∗−xn, J_E^p

1T xn−J_E^p

1xn

E +

D

x^∗−xn, J_E^p

1zn−J_E^p

1xn

E . This shows that x^∗ ∈ Cn+1, which implies Γ ⊂ Cn+1 for all n ≥1. Thus, Dn+1T

Cn+1 is nonempty. So, {x_n}is well defined.

From (3.1) and (2.5), we have D

J_E^p

1x₀−J_E^p

1x_n, z−x_nE

≤0, z∈C_n. Since Γ⊂C_n, we have

D J_E^p

1x0−J_E^p

1xn, x^∗−xn

E

≤0, x^∗∈Γ. (3.9)

By (2.6) and for allx^∗ ∈Γ,we have

∆_p(x_n, x₀)≤∆_p(x^∗, x₀)−∆_p(x^∗, x_n)

≤∆_p(x^∗, x₀),

this shows that {∆_p(x_n, x₀)} is bounded. Hence, {x_n} is bounded. By the construction of C_n, we get xm ∈Cm⊂Cn and xn= ΠCnx0 for any m≥n. From (2.6), we obtain

∆p(xm, xn) = ∆p(xm,ΠCnx0)≤∆p(xm, x0)−∆p(ΠCnx0, x0)

= ∆p(xm, x0)−∆p(xn, x0). (3.10) Since xn = Π_Cnx0 and xm = Π_Cmx0 ∈ Cm ⊂ Cn, we have ∆p(xn, x0) ≤∆p(xm, x0) for all m ≥ n. This implies that {∆_p(x_n, x₀)} is nondecreasing and hence the limit limn→∞∆_p(x_n, x₀) exists. From (3.10), we obtain ∆p(xn, xm) →0 as m, n→ ∞. From (2.3), we have kx_n−xmk →0 as m, n → ∞. Hence, {x_n}is a Cauchy sequence inC ⊂E1, so there existsx∈E1 such that xn→x asn→ ∞.

By using (2.1) and (2.5), we have

∆_p(x₀,Π_Γx₀)≥∆_p(x₀, x_n+1)

= ∆p(x0, xn) + ∆p(xn, xn+1) + D

xn−xn+1, J_E^p

1x0−J_E^p

1xn

E

≥∆p(x0, xn) + ∆p(xn, xn+1)

≥∆p(x0, xn−1) + ∆p(xn−1, xn) + ∆p(xn, xn+1) ...

≥

n

X

i=0

∆p(xi, xi+1).

Consequently,P∞

i=0∆_p(x_i, x_i+1)<∞, which from (2.3) yieldsP∞

i=0kx_n−x_n+1k^p <∞. This implies that

n→∞lim kx_n−x_n+1k= 0. (3.11)

Since xn+1= Π_Cn+1x0∈Cn+1, we have,

∆_p(z_n, x_n)≤ κ 1−κ

D

x_n+1−x_n, J_E^p

1T x_n−J_E^p

1x_nE +D

x_n+1−x_n, J_E^p

1z_n−J_E^p

1x_nE , from (3.11) and (2.3), we obtain

n→∞lim kx_n−znk= 0. (3.12)

Since x_n+1= Π_Dn+1x₀∈D_n+1, we get,

∆_p(y_n, x_n+1)≤∆_p(x_n, x_n+1), from (3.11), we have

n→∞lim ky_n−xn+1k= 0, so,

n→∞lim kx_n−y_nk= 0. (3.13)

Since J_E^p

1 is norm-to-norm uniformly continuous, from (3.1), we get kJ_E^p

1zn−J_E^p

1xnk=kα_n(J_E^p

1yn−J_E^p

1xn) + (1−αn)(J_E^p

1T xn−J_E^p

1xn)k

≥(1−αn)kJ_E^p

1T xn−J_E^p

1xnk −αnkJ_E^p

1yn−J_E^p

1xnk, this implies that

(1−α_n)kJ_E^p

1T x_n−J_E^p

1x_nk ≤α_nkJ_E^p

1y_n−J_E^p

1x_nk+kJ_E^p

1z_n−J_E^p

1x_nk, (3.14)

sinceJ_E^q∗

1 is norm-to-norm uniformly continuous, setting n→ ∞in (3.14), from (3.12), (3.13) and {α_n} ⊂ [c, d]⊂(0,1), we have

n→∞lim kT x_n−x_nk= 0, by the closedness ofT, fromxn→x, we obtain

T x=x. (3.15)

From (2.3) and (2.5), we have

∆p(x,ΠCx)≤D

x−ΠC, J_E^p

1x−J_E^p

1ΠCx E

=D

x−x_n, J_E^p

1x−J_E^p

1Π_CxE +D

x_n−y_n, J_E^p

1x−J_E^p

1Π_CxE +

D

yn−ΠCx, J_E^p₁x−J_E^p₁ΠCx E

≤D

x−xn, J_E^p

1x−J_E^p

1Π_CxE +D

xn−yn, J_E^p

1x−J_E^p

1Π_CxE . Settingn→ ∞yields ∆p(x,Π_Cx) = 0, we get x∈C.

From (3.5), we have

λ_n−C_q(λ_nkAk)^q q

kv_nk^p≤∆_p(x_n, x^∗)−∆_p(y_n, x^∗), this together with v_n=Ax_n−P_QAx_n and (3.13) implies that

n→∞lim kAx_n−P_QAx_nk= 0. (3.16)

By (2.4), we have

kAx−P_QAxk^p =D J_E^p

2(Ax−P_QAx), Ax−P_QAxE

= D

J_E^p₂(Ax−PQAx), Ax−Axn

E +

D

J_E^p₂(Ax−PQAx), Axn−PQAxn

E

+D J_E^p

2(Ax−P_QAx), P_QAxn−P_QAxE

≤D J_E^p

2(Ax−P_QAx), Ax−Ax_nE +D

J_E^p

2(Ax−P_QAx), Ax_n−P_QAx_nE .

From (3.16) andAx_n→Ax asn→ ∞, setting n→ ∞yieldskAx−P_QAxk^p = 0, we have Ax∈Q. Thus, we conclude that xn→x∈Γ.

Setting n→ ∞ in (3.9), we obtain D

J_E^p

1x₀−J_E^p

1x, x^∗−xE

≤0, x^∗∈Γ.

By (2.5), we havex= Π_Γx₀.

Remark 3.2. Compared with the known results in the literature, our result is very different from those in the following aspects.

• The corresponding iterative algorithms in [14, Theorem 3.2], [16, Theorem 3.1], [17, Theorem 3.1] are extended for developing our algorithm which couples modified CQ method with Nakajo’s iteration involved in Bregman quasi-strictly pseudo-contractive mapping in Theorem 3.1. Our iterative scheme in Theorem 3.1 can be viewed as a merger between corresponding iterative algorithms in [14, Theorem 3.2], [16, Theorem 3.1], [17, Theorem 3.1].

• The construction of sets

(

^{such as C}n+1 =n

w∈C_n: ∆_p(z_n, x_n)≤ _1−κ^κ D

w−x_n, J_E^p

1T x_n−J_E^p

1x_nE + D

w−x_n, J_E^p

1z_n−J_E^p

1x_nEo

)

in our iterative scheme is very different from the iterative algorithm in [14, Theorem 3.2] because our construction is mainly based on the definition of Bregman quasi-strictly pseudo-contractive mapping. Moreover, we attain strong convergence result in a broader framework, thep-uniformly convex and uniformly smooth Banach spaces.

• The technique of proving strong convergence in Theorem 3.1 is different from those in [14, Theorem 3.2], [17, Theorem 3.1] because our technique depends on Lemma 2.1 in Banach spaces.

• The problem of finding a common element of the set of solutions of split feasibility problem and the set of fixed points of a Bregman quasi-strictly pseudo-contractive mapping in our Theorem 3.1 is more general than the problem of finding a solution of split feasibility problem in [14, Theorem 3.2] and the problem of finding an element of the set of fixed points of a Bregman quasi-strictly pseudo-contractive mapping in [17, Theorem 3.1].

Since the class of Bregman quasi-nonexpansive mappings is Bregman quasi-strict pseudo-contractive, the following corollary is obtained by using Theorem 3.1.

Corollary 3.3. Let C and Q be nonempty closed convex sets inp-uniformly convex and uniformly smooth real Banach spaces E₁ and E₂, respectively. Let A: E₁ → E₂ be a bounded linear operator with its adjoint A^∗: E₂^∗→E₁^∗. Let T be a closed Bregman quasi-nonexpansive mapping fromC to itself. Let sequence{x_n} be iteratively generated by x1 =x0 ∈C, D1=C1 =C,



























x₁ ∈C, y_n= Π_CJ_E^q∗

1

J_E^p

1x_n−λ_nA^∗J_E^p

2(Ax_n−P_QAx_n) , zn=J_E^q∗

1

αnJ_E^p

1yn+ (1−αn)J_E^p

1T xn

, D_n+1 ={w∈D_n: ∆_p(y_n, w)≤∆_p(x_n, w)}, C_n+1 =n

w∈C_n: ∆_p(z_n, x_n)≤D

w−x_n, J_E^p

1z_n−J_E^p

1x_nEo , x_n+1 = Π_Dn+1^T_Cn+1x₀, n≥1,

(3.17)

Assume that {α_n} ⊂[c, d]⊂(0,1)and {λ_n} ⊂[a, b]⊂

0,

q CqkAk^q

_q−1¹

. Then the sequence {x_n} defined by (3.17) converges strongly to Π_Γx₀.

Typical examples of both uniformly convex and uniformly smooth Banach spaces are L^p, where p >1.

Then we have the following corollary.

Corollary 3.4. Let E1 and E2 be two L^p spaces with 2≤ p < ∞, C ⊂E1 and Q⊂ E2 be two nonempty closed convex sets. Let A: E1→E2 be a bounded linear operator with its adjoint A^∗: E₂^∗ →E₁^∗. LetT be a closed Bregman quasi-strictly pseudo-contractive mapping fromC to itself. Let sequence {x_n} be iteratively generated by x₁ =x₀ ∈C, D₁=C₁ =C,































x1∈C, yn= ΠCJ_E^q∗

1

J_E^p

1xn−λnA^∗J_E^p

2(Axn−PQAxn)

, z_n=J_E^q∗

1

α_nJ_E^p

1y_n+ (1−α_n)J_E^p

1T x_n , Dn+1={w∈Dn: ∆p(yn, w)≤∆p(xn, w)}, C_n+1=n

w∈C_n: ∆_p(z_n, x_n)≤ _1−κ^κ D

w−x_n, J_E^p

1T x_n−J_E^p

1x_nE +D

w−x_n, J_E^p

1z_n−J_E^p

1x_nEo , x_n+1= Π_Dn+1^T_Cn+1x₀, n≥1,

(3.18)

where κ ∈ [0,1). Assume that {α_n} ⊂ [c, d] ⊂ (0,1) and {λ_n} ⊂ [a, b] ⊂

0, q

CqkAk^q

_q−1¹

. Then the sequence{x_n} defined by (3.18)converges strongly to ΠΓx0.

Acknowledgements

This work was supported by the Innovation Program of Shanghai Municipal Education Commission (15ZZ068), Ph.D. Program Foundation of Ministry of Education of China (20123127110002) and Program for Shanghai Outstanding Academic Leaders in Shanghai City (15XD1503100).

References

[1] Y.I. Alber,Metric and generalized projection operators in Banach spaces: properties and Applications, Kartsatos A G(ed). Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. New York: Marcel Dekker.178(1996), 15–50. 2, 2

[2] H.H. Bauschke, J.M. Borwein and P.L. Combettes,Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math.3:4(2001), 615–647. 1

[3] L.M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys.7:3(1967), 200–217. 1 [4] C. Byrne,A unified treatment of some iterative algorithm in signal processing and image reconstruction, Inverse

Probl.20:1(2004), 103–120. 1

[5] C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Probl.18:2 (2002), 441–453. 1

[6] Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol.51:10(2006), 2353–2365. 1

[7] Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer.

Algorithm.8:2(1994), 221–239. 1

[8] Y. Censor, T. Elfving, N. Kopf and T. Bortfeld,The mulitiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl. 21:6(2005), 2071–2084. 1

[9] Y. Censor, A. Motova and A. Segal,Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl.327:2(2007), 1244–1256. 1

[10] J.W. Chen, Y.J. Cho and R.P. Agarwal,Strong convergence theorems for equilibrium problems and weak Bregman relatively nonexpansive mappings in Banach spaces, J. Inequal. Appl.2013, 2013:119. 1

[11] K. Nakajo and W. Takahashi,Strong convergence theorems for nonexpansive mappings and nonexpansive semi- groups, J. Math. Anal. Appl.279:2(2003), 372–379. 1

[12] X.L. Qin, S.Y. Cho and L. Wang,Algorithms for treating equilibrium and fixed point problems, Fixed Point Theory Appl.2013, 2013:308. 1

[13] F. Sch¨opfer, T. Schuster and A.K. Louis,An iterative regularization method for the solution of the split feasibility problem in Banach space, Inverse Probl.24:5(2008), 055008. 1, 1

[14] W. Takahashi,The split feasibility problem in Banach spaces, Journal of Nonlinear and Convex Analysis. 15:6 (2014), 1349–1355. 1

[15] G.C. Ugwunnadi, A. Bashir, I. Ibrahim and S.M. Maaruf,Strong convergence theorem for quasi-Bregman strict- ly pseudocontractive mappings and equilibrium problems in Banach spaces, Fixed Point Theory Appl. 2014, 2014:231. 1, 2, 2.6

[16] F.H. Wang,A New Algorithm for Solving The Multiple-Sets Split Feasibility Problem in Banach Spaces, Numer.

Func. Anal. Opt.35:1(2014), 99–110. 1, 2.1

[17] Z.M. Wang,Strong convergence theorems for Bregman quasi-strict pseudo-contractions in reflexive Banach spaces with applications, Fixed Point Theory Appl.2015, 2015:91. 1, 2.5

[18] Y. Yao, G. Marino, L. Muglia,A modified Korpelevich’s method convergent to the minimum-norm solution of a variational inequality, Optim.63:4(2014), 559–569. 1

[19] Y. Yao, G. Marino, H.K. Xu, Y.C. Liou, Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces, J. Inequal. Appl.2014, 2014:206. 1

[20] Y. Yao, M. Postolache, S.M. Kang,Strong convergence of approximated iterations for asymptotically pseudocon- tractive mappings, Fixed Point Theory Appl.2014, 2014:100. 1