and Abdul Rahim Khan

https://doi.org/10.1051/ro/2020085 www.rairo-ro.org

SPLIT VARIATIONAL INCLUSIONS FOR BREGMAN MULTIVALUED MAXIMAL MONOTONE OPERATORS

Mujahid Abbas

, Faik G¨ ursoy

, Yusuf Ibrahim

and Abdul Rahim Khan

Abstract.We introduce a new algorithm to approximate a solution of split variational inclusion prob- lems of multivalued maximal monotone operators in uniformly convex and uniformly smooth Banach spaces under the Bregman distance. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. As application, we solve a split minimization problem and provide a numerical example to support better findings of our result.

Mathematics Subject Classification. 47J25.

Received January 27, 2020. Accepted July 29, 2020.

1. Introduction

Censor [8] imposed the well known split feasibility problem (SFP), which is formulated as finding a point x^∗∈Csuch thatAx^∗∈Q, whereCandQare nonempty closed and convex subsets ofRⁿ andR^m, respectively, whereAis anm×nmatrix. Byrne [3], defined CQ-algorithm as follows:

xn+1=PC(xn+γA^T(PQ−I)Axn), n≥0,

where x0 ∈ Rⁿ is an initial value, γ ∈ (0,_kAk²2) and PC and PQ denote the metric projections onto C and Q, respectively. The split feasibility problem has been considered by many authors and in many aspects [1–3,5,8,9,13,16,25,26,30]. In practice, SFP serves as a model in the intensity-modulation radiation ther- apy (IMRT) treatment planning [2,5]. Censor et al. [10] introduced a concept of Split Variational Inequality Problem (SVIP), which is a problem of finding a pointx^∗∈H1solves

hf(x^∗), x−x^∗i ≥0 for allx∈C, and the pointy^∗=Ax^∗∈H2 such that

hg(y^∗), y−y^∗i ≥0 for ally∈Q,

Keywords.Split variational inclusion problem, maximal monotone operators, Bregman distance, strong convergence, uniformly convex and uniformly smooth Banach space.

1 Department of Mathematics, Government College University, Katchery Road, Lahore 54000, Pakistan.

2 Department of Mathematics, Adiyaman University, Adiyaman 02040, Turkey.

3 Department of Mathematics, Sa’adatu Rimi College of Education, Kumbotso Kano, P.M.B. 3218 Kano, Nigeria.

4 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.

∗Corresponding author:faikgursoy02@hotmail.com

Article published by EDP Sciences c EDP Sciences, ROADEF, SMAI 2021

where C and Q are closed and convex subsets of Hilbert spaces H₁ and H₂, respectively,A : H₁ → H₂ is a bounded linear operator and A^∗ : H2 → H1 is adjoint of A, f : H1 → H1 and g : H2 → H2 are two given operators. Furthermore, they proposed the following algorithm. Let λ >0 and x1 ∈ H1 be arbitrary chosen.

Define the sequence{xn} by

xn+1=P_C^f,λ(xn+γA^∗(P_Q^g,λ−I)Axn)),∀n≥0, (1.1) where γ∈(0,_kAk¹2), and denoted byP_C^f,λ and P_Q^g,λ the expressionsPC(I−λf) and PQ(I−λg), respectively.

By some assumptions imposed on the operatorsf andg, they proved weak convergence result for the sequence {xn} to a solution point of split variational inequality problem.

Let E be a real normed space with dual E^∗ and J(x) ={x^∗ ∈E^∗;hx, x^∗i=kxkkx^∗k,kx^∗k =kxk} be the normalized duality. A mapB :E →E^∗ is called monotone if for each x, y∈E, the following inequality holds:

hη−ν, x−yi ≥ 0∀η ∈ Bx, ν ∈ By. It is called maximal monotone if, in addition, the graph of B is not properly contained in the graph of any other monotone operator. Also, B is maximal monotone if and only if it is monotone and for all t > 0,R(J +tB) = E^∗, where R(J +tB) is the range of (J +tB); see [4]. By using maximal monotone mappings, Moudafi [15] introduced the following Split Monotone Variational Inclusion (SMVI).

(findx^∗∈H₁: 0∈f(x^∗) +B₁(x^∗), and

y^∗=Ax^∗∈H₂: 0∈g(y^∗) +B₂(y^∗), (1.2) where B1 : H1 → 2^H1 andB2 : H2 →2^H2 are multi-valued maximal monotone mappings on Hilbert spaces, H₁ andH₂, respectively, and A:H₁ →H₂ is a bounded linear operator, f :H₁→H₁ and g :H₂ →H₂ are two given single-valued operators. Whenf andgare zero functions in (1.2), we have the usual Split Variational Inclusion Problem (SVIP). The algorithm introduced by Sch¨opferet al. [20] involves computations in terms of Bregman distance in the setting of p-uniformly convex and uniformly smooth real Banach spaces. Their iterative algorithm given below, converges weakly under some suitable conditions.

xn+1= ΠCJ⁻¹(J xn+γA^∗J(PQ−I)Axn), n≥0, (1.3) where Π_C denotes the Bregman Projection andA^∗the adjoint operator ofA. It is obvious that, strong conver- gence is more useful than the weak convergence in some applications. Recently, strong convergence theorems for SFP have been studied in the setting of p-uniformly convex and uniformly smooth real Banach spaces; see for example [11,17,22,23].

In this paper, inspired by the above cited works, we use a modified version of (1.1) and (1.3) to approximate a solution of the problem proposed here. Both the iterative methods and the underlying space used here are improvements of those employed in [6,7,10,11,13,17,20,22,23,28] and the references therein.

Definition 1.1. For eachp >1, letg:R⁺−→R⁺ given byg(t) =t^p−1be a gauge function such thatg(0) = 0 and lim_t→∞g(t) =∞.We define the generalized duality mapJ^p:E−→2^E∗ given by

J_g(t)=J^p(x) ={x^∗∈E^∗;hx, x^∗i=kxkkx^∗k,kx^∗k=g(kxk) =kxk^p−1}.

Definition 1.2. Let E be a smooth Banach space, the Bregman distance 4p of xto y, with respect to the convex continuous functionf :E→Rgiven byf(x) =¹_pkxk^p, is defined as

4p(x, y) = 1

qkxk^p− hJ^p(x), yi+1 pkyk^p, for allx, y∈E andp, q∈(1,∞) such that ¹_p+¹_q = 1.

Definition 1.3. Let E be a smooth Banach space and E^∗ its dual, the bifunctional V_p with respect to the convex continuous functionf :E→Rgiven byf(x) =¹_pkxk^p, is defined by

Vp(x^∗, x) = 1

qkx^∗k^q− hx^∗, xi+1 pkxk^p, for allx∈E,x^∗∈E^∗ andp, q∈(1,∞) such that ¹_p+¹_q = 1.

Definition 1.4. A Banach space E is said to be uniformly convex, if for x, y ∈ E, 0 < δE() ≤ 1, where δE() = inf{1− k¹₂(x+y)k;kxk=kyk= 1,kx−yk ≥,where 0≤≤2}.

Definition 1.5 ([19]). A Banach space E is said to be uniformly smooth, if for x, y ∈ E and r > 0, lim_r→0(^ρE_r^(r)) = 0 whereρE(r) =¹₂sup{kx+yk+kx−yk −2 :kxk= 1,kyk ≤r}. Moreover,

(1) ρE is continuous, convex and nondecreasing withρE(0) = 0 andρE(r)≤r.

(2) The functionr7→ ^ρE_r^(r) is nondecreasing and fulfills ^ρE_r^(r) >0 for allr >0.

Lemma 1.6([19]). Let {xn} be a sequence in a smooth Banach spaceE. Consider the following assertions;

(1) limn→∞kxn−xk= 0

(2) lim_n→∞kxnk=kxk andlim_n→∞hJ^p(xn), xi=hJ^p(x), xi (3) lim_n→∞4p(xn, x) = 0.

The assertions (1)=⇒(2)=⇒(3) are valid. If E is also uniformly convex, then the assertions are equivalent.

Lemma 1.7. Let E be a reflexive and smooth Banach space and E^∗ its dual. Let 4p andV_p be the mappings defined as above andJ_E^p the generalized duality map onE. Then4p(x, y) =Vp(J_E^px, y)for all x, y∈E.

Proof. Forp, q∈(1,∞) let J_E^q∗:E^∗→E andJ_E^p :E →E^∗ be duality mappings, whereJ_E^q∗J_E^p =I. It follows from ¹_p+¹_q = 1 thatp(q−1) =q. So, we have that

4p(x, y) =1

qkxk^p− hJ_E^px, yi+1 pkyk^p

=1

qkJ_E^q∗J_E^pxk^p− hJ_E^px, yi+1 pkyk^p

=1

qkJ_E^pxk^p(q−1)− hJ_E^px, yi+1 pkyk^p

=1

qkJ_E^pxk^q− hJ_E^px, yi+1 pkyk^p

=Vp(J_E^px, y).

Lemma 1.8([19]). LetE be a reflexive, strictly convex and smooth Banach space andJ^pbe the duality mapping of E. Then

(i) for every closed and convex subset C ⊂E andx∈E, there exists a unique element Π^p_C(x)∈C such that 4_p(x,Π^p_C(x)) = min_y∈C4_p(x, y); Π^p_C(x) is called the Bregman projection of xonto C, with respect to the function f(x) =¹_pkxk^p. Moreover, x₀∈C is the Bregman projection of xontoC if

hJ^p(x0−x), y−x0i ≥0 or equivalently

4p(x0, y)≤ 4p(x, y)− 4p(x, x0) for everyy∈C.

(ii) the Bregman projection and the metric projection are related via P_C(x)−x= Π^p_C−x(0),∀x∈E. Especially, we have PC(0) = Π^p_C(0)and thuskΠ^p_C(0)k= miny∈Ckyk.

The uniform convexity of E implies thatE is reflexive andE^∗ is uniformly smooth. Therefore, Theorem 2 in [27], forx, y∈E andx^∗, y^∗∈E^∗ andkx+yk^p replaced bykx^∗−y^∗k^q gives the following technical result.

Lemma 1.9. For the uniformly smooth space E^∗, with the duality mapJ^q,∀x^∗, y^∗∈E^∗, we have kx^∗−y^∗k^q ≤ kx^∗k^q−qhJ^q(x^∗), y^∗i+ ¯σq(x^∗, y^∗) where

¯

σq(x^∗, y^∗) =qGq

Z 1 0

(kx^∗−ty^∗k ∨ kx^∗k)^q

t ρE^∗

tky^∗k

2(kx^∗−ty^∗k ∨ kx^∗k)

dt (1.4)

andGq = 8∨64cK_q⁻¹ with c, Kq>0.

Lemma 1.10 ([19]). Let E be a reflexive, strictly convex and smooth Banach space. We write 4^∗_q(x, y) = ¹_pkx^∗k^q − hJ_E^q∗x^∗, y^∗i+ ¹_qky^∗k^q for x^∗ = J_E^p(x), y^∗ = J_E^p(y) for the Bregman distance on the dual spaceE^∗ with respect to the functionf_q^∗(x^∗) = ¹_qkx^∗k^q. Then we have4p(x, y) =4^∗_q(x^∗, y^∗).

Lemma 1.11. Let E be a reflexive, smooth and strictly convex Banach space. Then for all x, y, z ∈ E and x^∗=J_E^px,z^∗=J_E^pz, the following hold:

(1) 4p(x, y)≥0 and4p(x, y) = 0 ifx=y;

(2) 4p(x, y) =4p(x, z) +4p(z, y) +hx^∗−z^∗, z−yi.

Proof. The property (1) is proved in [19]. For (2) we have that 4_p(x, z) +4_p(z, y) = 1

qkxk^p− hx^∗, zi+1

pkzk^p+1

qkzk^p− hz^∗, yi+1 pkyk^p

= 1

qkxk^p− hx^∗, zi+kzk^p− hz^∗, yi+1

pkyk^p+hx^∗, yi − hx^∗, yi

= 1

qkxk^p− hx^∗, yi+1 pkyk^p

+hz^∗, zi − hz^∗, yi+hx^∗, yi − hx^∗, zi

⇔ 4p(x, y) =4p(x, z) +4p(z, y) +hx^∗−z^∗, z−yi.

IfE is smooth andf(x) = ¹_pkxk^p, then the following result holds (cf.Prop. 5 in [18]).

Lemma 1.12. Let E be a smooth Banach space and f : E → R be a continuous convex function given by f(x) = ¹_pkxk^p. Ifx0∈E and the sequence{4p(xn, x0)}^∞_n=1 is bounded, then the sequence{xn}is also bounded.

2. Main results

Let E1 and E2 be uniformly convex and uniformly smooth Banach spaces and E₁^∗ and E₂^∗ be their duals, respectively. LetU :E1→2^E1∗andT :E2→2^E∗2 be multi-valued maximal monotone operators. ForK ⊂ E1, closed and convex, δ > 0 andp, q ∈(1,∞), let A : E₁ → E₂ be a bounded and linear operator, A^∗ denotes the adjoint of A and AK be closed and convex. Suppose that Π^p_AK : E2 → AK is the Bregman projection onto a closed and convex subset AK. Let B^U_δ : E1 → E1 be the generalized resolvent operator defined by B_δ^U = (J_E^p

1 +δU)⁻¹J_E^p

1 and B^T_δ : E2 → E2 be another generalized resolvent operator defined by B_δ^T = (J_E^p

2 +δT)⁻¹J_E^p

2. Let us denote the solutions of variational inclusion problem with respect to U and T by SOLV IP(U) and SOLV IP(T), respectively. Let the set of solutions of split variational inclusion problem be

given by Ω ={x^∗∈SOLV IP(U);Ax^∗∈SOLV IP(T)} 6=∅. Letx₁∈E₁be chosen arbitrarily and the sequence {xn} ⊂E1be defined as follows;







un=B_δ^U

n

J_E^q∗

1 J_E^p

1xn−λnA^∗J_E^p

2(I−Π^p_AKB_δ^T

n)Axn

, Kn+1={v∈Kn:4p(un, v)≤ 4p(xn, v)},

xn+1= Π^p_K

n+1(x1), n≥1,

(2.1)

whereδ_n∈(0,∞). It is remarked that we have replaced the gradient algorithm in (1.1) [the projection maps in (1.3), respectively] with the resolvent operators and used the generalized duality map in our algorithm.

We shall strictly employ the above terminology in the sequel.

Lemma 2.1. Suppose that σ¯_q is the function in (1.4) for the characteristic inequality of the uniformly smooth space E₁^∗. For the sequence {xn} ⊂ E1 defined by (2.1), let 0 6= xn ∈ E1, 0 6= A and 06=J_E^p

2(I−Π^p_AKB_δ^T

n)Axn ∈ E₂^∗. Let λn >0 andµn >0 be defined, respectively, by λn= 1

kAk

1 kJ_E^p

2(I−Π^p_AKB^T_δ

n)Ax_nk and µn= 1

kxnk^p−1· (2.2)

Then 1 qσ¯_q J_E^p

1x_n, λ_nA^∗J_E^p

2(I−Π^p_AKB^T_δ

n)Ax_n

≤

(2^qGqkJ_E^p

1xnk^qρE₁^∗(µn) if µn ∈(0,1],

2^qGqρE^∗₁(µn) if µn ∈(1,∞), (2.3) whereGq is the constant defined in Lemma1.9andρE^∗₁ is the modulus of smoothness ofE₁^∗.

Proof. By Lemma1.9, we have 1

qσ¯q J_E^p

1xn, λnA^∗J_E^p

2(I−Π^p_AKB_δ^T_n)Axn

=Gq

Z 1 0

J_E^p

1xn−λnA^∗J_E^p

2(I−Π^p_AKB^T_δ

n)Axn

∨ kJ_E^p

1xnk^q t

×ρ_E^∗ tkλnA^∗J_E^p

2(I−Π^p_AKB^T_δ

n)Axnk J_E^p

1x_n−λ_nA^∗J_E^p

2(I−Π^p_AKB_δ^T

n)Ax_n

∨ kJ_E^p₁x_nk

! dt, (2.4) for every t∈[0,1].

We note that J_E^p

1xn−λn,iA^∗J_E^p

2(I−Π^p_AKB_δ^T_n)Axn

≤ kxnk^p−1+

λnA^∗J_E^p

2(I−Π^p_AKB_δ^T_n)Axn

. By (2.2), withxn6= 0

λn= µn

kAk

kxnk^p−1 J_E^p

2(I−Π^p_AKB_δ^T

n)Axn

(2.5) and so we have that

J_E^p

1xn−λnA^∗J_E^p

2(I−Π^p_AKB^T_δn)Axn

≤(1 +µn)kxnk^p−1 and

(kxnk^p−1≤ J_E^p

1x_n−λ_nA^∗J_E^p

2 I−Π^p_AKB_δ^T

n

Ax_n

∨ kJ_E^p₁x_nk ≤2kxnk^p−1 if µ_n∈(0,1]

kxnk^p−1≤ J_E^p

1xn−λnA^∗J_E^p

2 I−Π^p_AKB_δ^T_n Axn

∨ kJ_E^p

1xnk ≤2 if µn∈(1,∞). (2.6)

By (2.6), (2.5) and Definition 1.5(2), we get ρE₁^∗

t

λ_nA^∗J_E^p

2(I−Π^p_AKB_δ^T

n)Ax_n J_E^p

1xn−tλnA^∗J_E^p

2(I−Π^p_AKB_δ^T

n)Axn

∨ kJ_E^p

1xnk

!

≤ρE^∗₁

t

λ_nA^∗J_E^p

2(I−Π^p_AKB_δ^T

n)Ax_n kx_nk^p−1

!

=ρ_E^∗

1(tµ_n). (2.7)

Substituting (2.7) and (2.6) into (2.4), and using nondecreasingness of ρE₁^∗, we get (2.3) as required.

Lemma 2.2. For the sequence {xn} ⊂E1 defined by (2.1), let 06=xn, 06=J_E^p

2(I−Π^p_AKB_δ^T

n)Axn∈E₂^∗, and λn>0 andµn>0 be defined by (2.2) andλn andµn are chosen such that

ρE₁^∗(µn) =











ι 2^qGqkAk×

D J_E^p

2(I−Π^p_AKB_δn^T )Ax_n,(I−Π^p_AKB^T_δn)Ax_n E

kJ_E^p

1x_nk^qkJ_E^p

2(I−Π^p_AKB^T

δn)Ax_nk , if µ_n ∈(0,1],

ι 2^qGqkAk×

D J_E^p

2(I−Π^p_AKB_δn^T )Ax_n,(I−Π^p_AKB^T_δn)Ax_n E

kJ_E^p

2(I−Π^p_AKB^T_δn)Axnk , if µn ∈(1,∞),

(2.8)

whereι∈(0,1). Then, for all v∈Ω, we get 4p(un, v)≤ 4p(xn, v)−[1−ι]

J_E^p

2(I−Π^p_AKB_δ^T

n)Axn,(I−Π^p_AKB_δ^T

n)Axn kAkkJ_E^p

2(I−Π^p_AKB_δ^T

n)Ax_nk · (2.9)

Proof. Forv=B^U_γv andAv=B^T_γAv, by Lemma1.7, we have that 4p(un, v) =4p

B^U_δ

n

J_E^q∗

1 J_E^p

1xn−λnA^∗J_E^p

2 I−Π^p_AKB_δ^T

n

Axn , v

=4p

B^U_δn

J_E^q∗ 1 J_E^p

1xn−λnA^∗J_E^p

2 I−Π^p_AKB_δ^T_n Axn

, B_δ^U_nv

≤Vp J_E^p

1xn−λnA^∗J_E^p

2 I−Π^p_AKB_δ^T_n

Axn, v

=1 q

J_E^p

1xn−λnA^∗J_E^p

2 I−Π^p_AKB_δ^T_n Axn

q+1 pkvk^p

− hJ_E^p

1xn, vi+

λnA^∗J_E^p

2 I−Π^p_AKB^T_δn

Axn, v

, (2.10)

where, λnA^∗J_E^p

2 I−Π^p_AKB_δ^T_n

Axn, v

= λnJ_E^p

2 I−Π^p_AKB_δ^T_n

Axn, Av−Axn+Axn−Π^p_AKB_δ^T_nAxn +

λnJ_E^p

2 I−Π^p_AKB^T_δn

Axn,Π^p_AKB_δ^T_nAxn−Axn+Axn

=− λ_nJ_E^p

2 Π^p_AKB^T_δ

n−I

Ax_n,(Av−Ax_n)− Π^p_AKB_δ^T

n−I Ax_n

− λ_nJ_E^p

2 I−Π^p_AKB^T_δ

n

Ax_n, I−Π^p_AKB_δ^T

n

Ax_n +

λnJ_E^p

2 I−Π^p_AKB^T_δn

Axn, Axn

.

As AK is closed and convex so by Lemma1.8(i) and the variational inequality for the Bregman projection of zero ontoAK−Ax_n, as in Lemma 1.8(ii), we arrive at

λnJ_E^p

2(Π^p_AKB_δ^T_n−I)Axn,(Av−Axn)−(Π^p_AKB^T_δn−I)Axn

≥0 and therefore, we obtain

λ_nA^∗J_E^p

2(I−Π^p_AKB^T_δ

n)Ax_n, v

≤ − λ_nJ_E^p

2(I−Π^p_AKB_δ^T

n)Ax_n,(I−Π^p_AKB^T_δ

n)Ax_n +

λnJ_E^p

2(I−Π^p_ΓB_δ^T_n)Axn, Axn

. (2.11)

In addition, by Lemma1.9, we have that 1

q J_E^p

1xn−λnA^∗J_E^p

2(I−Π^p_AKB_δ^T_n)Axn

q ≤ 1 qkJ_E^p

1xnk^q−λn

Axn, J_E^p

2(I−Π^p_AKB^T_δn)Axn +1

qσ¯q J_E^p

1xn, λnA^∗J_E^p

2(I−Π^p_AKB_δ^T_n)Axn

(2.12) By Lemma2.1and (2.12), we have that

1 q

J_E^p

1xn−λnA^∗J_E^p

2(I−Π^p_AKB_δ^T_n)Axn

q ≤ 1 qkJ_E^p

1xnk^q−λn

Axn, J_E^p

2(I−Π^p_AKB^T_δn)Axn

+ 2^qGqkJ_E^p

1xnk^qρE₁^∗(µn). (2.13) Substituting (2.13) and (2.11) into (2.10), we have that

4p(un, v)≤ 1 qkJ_E^p

1xnk^q+1

pkvk^p− hJ_E^p

1xn, vi+ 2^qGqkJ_E^p

1xnk^qρE^∗₁(µn)

− λ_nJ_E^p

2(I−Π^p_AKB^T_δ

n)Ax_n,(I−Π^p_AKB_δ^T

n)Ax_n

= 4p(xn, v) + 2^qGqkJ_E^p

1xnk^qρE^∗₁(µn)

− λ_nJ_E^p

2(I−Π^p_AKB^T_δ

n)Ax_n,(I−Π^p_AKB_δ^T

n)Ax_n

. (2.14)

Substituting (2.2) and (2.8) into (2.14), we have that 4p(un, v)≤ 4p(xn, v) +ι

(J_E^p

2(I−Π^p_AKB_δ^T

n)Ax_n,(I−Π^p_AKB^T_δ

n)Ax_n kAkkJ_E^p

2(I−Π^p_AKB^T_δ

n)Aunk

− (J_E^p

2(I−Π^p_AKB^T_δ

n)Ax_n,(I−Π^p_AKB_δ^T

n)Ax_n kAkkJ_E^p

2(I−Π^p_AKB_δ^T

n)Axnk

= 4_p(x_n, v)−[1−ι]

J_E^p

2(I−Π^p_AKB^T_δ

n)Ax_n,(I−Π^p_AKB_δ^T

n)Ax_n kAkkJ_E^p

2(I−Π^p_AKB_δ^T

n)Axnk ·

Thus, (2.9) holds.

We now prove our main result.

Theorem 2.3. For δ > 0 and p, q ∈ (1,∞), let (I−Π^p_AKB_δ^T) be demiclosed at zero. Let x₁ ∈E₁ be chosen arbitrarily and the sequence{xn}be defined by (2.1), where

λn =







1 kAk

1 kJ_E^p

2(I−Π^p_AKB^T

δn)Axnk, xn 6= 0

1 kAk^p

D J_E^p

2(I−Π^p_AKB_δn^T )Ax_n,(I−Π^p_AKB_δn^T )Ax_n E_p−1

kJ_E^p

2(I−Π^p_AKB^T

δn)Ax_nk^p , x_n = 0

and µn= 1

kxnk^p−1 (2.15) are chosen such that equation (2.8) holds. If Ω = {x^∗ ∈SOLV IP(U);Ax^∗ ∈ SOLV IP(T)} 6= ∅, then {xn} converges strongly to x^∗∈Ω, whereΠ^p_AKB_δ^T

n(Ax^∗) =B_δ^T

n(Ax^∗).

Proof. We will divide the proof into two steps.

Step one. We show that{xn}is a bounded sequence.

Assume thatkJ_E^p

2(I−Π^p_AKB_δ^T

n)Axnk= 0. Then fromv=B_γ^Uv, Lemma1.7and v∈Ω, we get 4p(un, v) =4p

B_δ^U_n(J_E^q∗ 1(J_E^p

1xn)), B^U_δnv

≤Vp(J_E^p

1xn, v) =4p(xn, v). (2.16)

Next assume thatkJ_E^p₂(I−Π^p_AKB^T_δ

n)Ax_nk 6= 0 and x_n 6= 0. Then forv∈Ω, by Lemma2.2, we get 4p(un, v)≤ 4p(xn, v)−[1−ι]

J_E^p

2(I−Π^p_AKB_δ^T

n)Ax_n,(I−Π^p_AKB_δ^T

n)Ax_n kAkkJ_E^p

2(I−Π^p_AKB_δ^T

n)Axnk (2.17)

≤ 4p(x_n, v). (2.18)

Forxn = 0, we have

4p(xn, v) =1

pkvk^p (2.19)

and so by (2.19), we have that

4p(un, v) =1 q

λnA^∗J_E^p

2(I−Π^p_AKB^T_δn)Axn

q

+4p(xn, v) +λn

J_E^p

2(I−Π^p_AKB_δ^T_n)Axn, Av

. (2.20)

Substituting (2.11) in (2.20), we have that 4p(u_n, v)≤1

q

λ_nA^∗J_E^p

2(I−Π^p_AKB_δ^T

n)Ax_n

q

+4p(xn, v) +λn J_E^p

2(I−Π^p_AKB_δ^T_n)Axn, Axn

−λn J_E^p

2(I−Π^p_AKB^T_δ

n)Axn,(I−Π^p_AKB_δ^T

n)Axn

. (2.21)

By (2.15), we have that 1

q

λnA^∗J_E^p

2(I−Π^p_AKB^T_δn)Axn

q =1 q

1 kAk^p

J_E^p

2(I−Π^p_AKB^T_δ

n)Axn,(I−Π^p_AKB_δ^T

n)Axn^p J_E^p

2(I−Π^p_AKB_δ^T

n)Ax_n

p · (2.22)

Substituting (2.22) into (2.21), we have that 4_p(u_n, v)≤1

q 1 kAk^p

J_E^p

2(I−Π^p_AKB^T_δ

n)Axn,(I−Π^p_AKB_δ^T

n)Axn

p

J_E^p

2(I−Π^p_AKB_δ^T

n)Ax_n

p

+4p(x_n, v) +λ_nhJ_E^p

2(I−Π^p_AKB_δ^T

n)Ax_n, Ax_ni

−λnhJ_E^p

2(I−Π^p_AKB_δ^T_n)Axn,(I−Π^p_AKB^T_δn)Axni

≤

1−1 p

1 kAk^p

hJ_E^p

2(I−Π^p_AKB_δ^T

n)Axn,(I−Π^p_AKB_δ^T

n)Axni^p kJ_E^p

2(I−Π^p_AKB_δ^T

n)Ax_nk^p +4p(xn, v) +λnkJ_E^p

2(I−Π^p_AKB_δ^T_n)AxnkkAxnk

− 1 kAk^p

hJ_E^p

2(I−Π^p_AKB^T_δ

n)Ax_n,(I−Π^p_AKB_δ^T

n)Ax_ni^p kJ_E^p

2(I−Π^p_AKB_δ^T

n)Axnk^p

= 4p(xn, v)− 1 pkAk^p

hJ_E^p

2(I−Π^p_AKB^T_δ

n)Axn,(I−Π^p_AKB_δ^T

n)Axni^p kJ_E^p

2(I−Π^p_AKB_δ^T

n)Ax_nk^p · (2.23)

This implies that

4_p(u_n, v)≤ 4_p(x_n, v). (2.24) By (2.1), (2.16), (2.18) and (2.24),v∈K_n so that Ω⊂K_n.

We know from (2.1),x_n= Π^p_K

nx₁. Then, by Lemma 1.8, we have

4p(xn, x1) =4p(Π^p_Kx1, x1)≤ 4p(v, x1)− 4p(v, xn)⇒ 4p(xn, x1)≤ 4p(v, x1)∀v∈Ω⊂Kn. (2.25) By (2.25), the sequence{4p(xn, x1)} is bounded and therefore by Lemma1.12,{xn} is bounded. Hence,{un} is also bounded. Consequently, there exists a subsequence xn_j such that xn_j * x^∗ as j → ∞ (* stands for weak convergence).

Step two. We show thatx_n→x^∗∈Ω.

Sincex_n+1= Π^p_K

n+1x₁⊂K_n+1⊂K_n andJ^pis weakly sequentially continuous, we have by Lemma1.11 4p(un, xn) = 4p(un, xn+1) +4p(xn+1, xn) +

un−xn+1, J_E^p

1xn+1−J_E^p

1xn

≤ 4_p(x_n, x_n+1) +4_p(x_n+1, x_n) +

u_n−x_n+1, J_E^p

1x_n+1−J_E^p

1x_n

= 4p(xn, xn) +

un−xn, J_E^p

1xn+1−J_E^p

1xn

−→0 asn→ ∞. (2.26)

It follows from (2.1) that (J_E^p

1x_n−J_E^p

1u_n)−λ_nA^∗J_E^p

2(I−Π^p_AKB^T_δ

n)Ax_n

δ_n ∈U(un). (2.27)

By (2.17), we have that

4p(un, v)≤ 4p(xn, v)−[1−ι]

J_E^p

2(I−Π^p_AKB^T_δ

n)Axn,(I−Π^p_AKB_δ^T

n)Axn kAkkJ_E^p

2(I−Π^p_AKB_δ^T

n)Ax_nk , and

k(I−Π^p_AKB_δ^T

n)Ax_nk ≤

4p(xn, v)− 4p(un, v) kAk⁻¹[1−ι]

−→0 asn→ ∞. (2.28)

By (2.23), we have that

4p(u_n, v)≤ 4p(x_n, v)− 1 pkAk^p

hJ_E^p

2(I−Π^p_AKB_δ^T

n)Axn,(I−Π^p_AKB_δ^T

n)Axni^p kJ_E^p

2(I−Π^p_AKB_δ^T

n)Axnk^p and therefore

k(I−Π^p_AKB_δ^T_n)Axnk ≤

4p(xn, v)− 4p(un, v) (pkAk)⁻¹

¹_p

−→0 asn→ ∞. (2.29)

By (2.26) to (2.29) and weak sequential continuity property ofJ^p, we have that 0∈ U(x^∗). This means that x^∗∈SOLV IP(U).But, since4p(·, x) is lower semi continuous and convex and thus weakly lower semi contin- uous on int(domf) then from the fact thatx_nj * x^∗ asj → ∞, wee see that

4p(x^∗, x₁)≤lim inf

j→∞ 4p(x_nj, x₁)≤ 4p(v, x₁).

From the definition ofv, that isv=B_δ^U(v), we can conclude thatx^∗=vand the sequencex_n* x^∗. In addition, it is clear thatAxn * Ax^∗. So by using (2.28), (2.29) and applying the demicloseness of (I−Π^p_AKB_δ^T

n) at zero, we have that 0∈T(Ax^∗) as Π^p_AKB_δ^T(Ax^∗) =B^T_δ(Ax^∗). ThereforeAx^∗∈SOLV IP(T).Hence,x^∗∈Ω.